Conference Proceedings of the Society for Experimental Mechanics Series
Series EditorTom ProulxSociety for Experimental Mechanics, Inc.,Bethel, CT, USA
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D. Adams • G. Kerschen • A. Carrella
Editors
Topics in Nonlinear Dynamics,Volume 3
Proceedings of the 30th IMAC, A Conference on StructuralDynamics, 2012
EditorsD. AdamsPurdue UniversityWest Lafayette, IN, USA
G. KerschenUniversity of LiegeBelgium
A. CarrellaLMS InternationalLeuven, Belgium
ISSN 2191-5644 e-ISSN 2191-5652ISBN 978-1-4614-2415-4 e-ISBN 978-1-4614-2416-1DOI 10.1007/978-1-4614-2416-1Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2012936657
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Preface
Topics in Nonlinear Dynamics represents one of six volumes of technical papers presented at the 30th IMAC, A Conference
and Exposition on Structural Dynamics, 2012 organized by the Society for Experimental Mechanics, and held in
Jacksonville, Florida, January 30–February 2, 2012. The full proceedings also include volumes on Dynamics of Civil
Structures; Substructuring and Wind Turbine Dynamics; Model Validation and Uncertainty Quantification; and Modal
Analysis, I & II.
Each collection presents early findings from experimental and computational investigations on an important area within
Structural Dynamics. Nonlinearity is one of these areas. The vast majority of real engineering structures behave nonlinearly.
Therefore, in order to go From the Laboratory to the Real World it is necessary to include nonlinear effects in all the steps ofthe engineering design: in the experimental analysis tools (so that the nonlinear parameters can be correctly identified) and inthe mathematical and numerical models of the structure (in order to run accurate simulations). In so doing, it will be possibleto create a model representative of the reality which (once validated) can be used for better predictions. This volume
addresses theoretical and numerical aspects of nonlinear dynamics (covering rigorous theoretical formulations and robust
computational algorithms) as well as experimental techniques and analysis methods.
The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in
this track.
West Lafayette, IN, USA D. Adams
Belgium G. Kerschen
Leuven, Belgium A. Carrella
Contents
1 Application of the Restoring Force Surface Method to a Real-life Spacecraft Structure . . . . . . . . . . . . . . . . . . . . . . . 1J.P. Noel, G. Kerschen, and A. Newerla
2 Nonlinear Dynamic Model and Simulation of Morphing Wing Profile
Actuated by Shape Memory Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Cassio T. Faria, Carlos De Marqui Jr., Daniel J. Inman, and Vicente Lopes Jr.
3 Environmental Testing and Data Analysis for Non-linear Spacecraft Structures:
Impact on Virtual Shaker Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Simone Manzato, Bart Peeters, Raphael Van der Vorst, and Jan Debille
4 Using Impact Modulation to Detect Loose Bolts in a Satellite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Janette Jaques and Douglas E. Adams
5 Nonlinear Modal Analysis of the Smallsat Spacecraft. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45L. Renson, G. Kerschen, and A. Newerla
6 Filter Response to High Frequency Shock Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Jason R. Foley, Jacob C. Dodson, and Alain L. Beliveau
7 Simplified Nonlinear Modeling Approach for a Bolted Interface Test Fixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Charles Butner, Douglas Adams, and Jason R. Foley
8 Transmission of Guided Waves Across Prestressed Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Jacob C. Dodson, Janet Wolfson, Jason R. Foley, and Daniel J. Inman
9 Equivalent Reduced Model Technique Development for Nonlinear
System Dynamic Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Louis Thibault, Peter Avitabile, Jason R. Foley, and Janet Wolfson
10 Efficient Computational Nonlinear Dynamic Analysis Using Modal
Modification Response Technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Tim Marinone, Peter Avitabile, Jason R. Foley, and Janet Wolfson
11 Spectral Domain Force Identification of Impulsive Loading in Beam Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157Pooya Ghaderi, Andrew J. Dick, Jason R. Foley, and Gregory Falbo
12 Free-Pendulum Vibration Absorber Experiment Using Digital Image Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167Richard Landis, Atila Ertas, Emrah Gumus, and Faruk Gungor
13 Suppression of Regenerative Instabilities by Means of Targeted Energy Transfers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181A. Nankali, Y.S. Lee, and T. Kalmar-Nagy
14 Force Displacement Curves of a Snapping Bistable Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191Alexander D. Shaw and Alessandro Carrella
15 Characterization of a Strongly Nonlinear Vibration Absorber for Aerospace Applications. . . . . . . . . . . . . . . . . . . . 199Sean A. Hubbard, Timothy J. Copeland, D. Michael McFarland,Lawrence A. Bergman, and Alexander F. Vakakis
vii
16 Identifying and Computing Nonlinear Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209A. Cammarano, A. Carrella, L. Renson, and G. Kerschen
17 Nonlinear Identification Using a Frequency Response Function With the Jump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217A. Carrella
18 Nonlinear Structural Modification and Nonlinear Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225Taner KalaycIoglu and H. Nevzat Ozg€uven
19 Nonlinear Dynamic Response of Two Bodies Across an Intermittent Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239Christopher Watson and Douglas Adams
20 Application of Continuation Methods to Nonlinear Post-buckled Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245T.C. Lyman, L.N. Virgin, and R.B. Davis
21 Comparing Measured and Computed Nonlinear Frequency Responses
to Calibrate Nonlinear System Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255Michael W. Sracic, Shifei Yang, and Matthew S. Allen
22 Identifying the Modal Properties of Nonlinear Structures Using Measured
Free Response Time Histories from a Scanning Laser Doppler Vibrometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269Michael W. Sracic, Matthew S. Allen, and Hartono Sumali
23 Nonlinear System Identification of the Dynamics of a Vibro-Impact Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287H. Chen, M. Kurt, Y.S. Lee, D.M. McFarland, L.A. Bergman, and A.F. Vakakis
24 Modeling of Subsurface Damage in Sandwich Composites Using
Measured Localized Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301Sara S. Underwood and Douglas E. Adams
25 Parametric Identification of Nonlinearity from Incomplete FRF Data
Using Describing Function Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311Murat Aykan and H. Nevzat Ozg€uven
26 Finding Local Non-linearities Using Error Localization from Model Updating Theory . . . . . . . . . . . . . . . . . . . . . . . . 323Andreas Linderholt and Thomas Abrahamsson
viii Contents
Chapter 1
Application of the Restoring Force Surface Method
to a Real-life Spacecraft Structure
J.P. Noel, G. Kerschen, and A. Newerla
Abstract Many nonlinear system identification methods have been introduced in the technical literature during the last 30
years. However, few of these methods were applied to real-life structures. In this context, the objective of the present paper is
to demonstrate that the Restoring Force Surface (RFS) method can provide a reliable identification of a nonlinear spacecraft
structure. The nonlinear component comprises an inertia wheel mounted on a support, the motion of which is constrained by
eight elastomer plots and mechanical stops. Several adaptations to the RFS method are proposed, which include the
elimination of kinematic constraints and the regularization of ill-conditioned inverse problems. The proposed methodology
is demonstrated using numerical data.
Keywords Nonlinear system identification • Space structure • Restoring force surface method
1.1 Introduction
Nonlinear structural dynamics has been studied for a relatively long time, but the first contributions to the identification of
nonlinear structural models date back to the 1970s. Since then, numerous methods have been developed because of the
highly individualistic nature of nonlinear systems [1]. A large number of these methods were targeted to Single-Degree-Of-
Freedom (SDOF) systems, but significant progress in the identification of Multi-Degree-Of-Freedom (MDOF) lumped
parameter systems has been realized during the last 10 or 20 years. To date, simple continuous structures with localized
nonlinearities are within reach. Among the well-established methods, there exist
• Time-domain methods such as the Restoring Force Surface (RFS) and Nonlinear Auto-Regressive Moving Average with
eXogeneous input (NARMAX) methods [2, 3];
• Frequency-domain methods such as the Conditioned Reverse Path (CRP) [4] and Nonlinear Identification through
Feedback of the Output (NIFO) methods [5];
• Time-frequency analysis methods such as the Wavelet Transform (WT) [6].
The RFS method, introduced in 1979 by Masri and Caughey [7], constitutes the first attempt to identify nonlinear
structures. Since then, many improvements of the RFS method were introduced in the technical literature. Without being
comprehensive, we mention the replacement of Chebyschev expansions in favour of more intuitive ordinary polynomials
[8], the design of optimized excitation signals [9] or the direct use of the state space representation of the restoring force as
nonparametric estimate [10].
In theory [11], the RFS method can handle MDOF systems. However, a number of practical considerations diminish this
capability and its scope is, in fact, bound to systems with a few degrees of freedom only. For example, Al-Hadid and Wright
J.P. Noel (*) • G. Kerschen
Space Structures and Systems Laboratory (S3L), Structural Dynamics Research Group Department of Aerospace
and Mechanical Engineering, University of Liege, Belgium
e-mail: [email protected]; [email protected]
A. Newerla
European Space Agency (ESTEC), Noordwijk, The Netherlands
e-mail: [email protected]
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_1, # The Society for Experimental Mechanics, Inc. 2012
1
[12] studied a T-beam structure with well-separated bending and torsion modes. Another extensively studied system of this
kind is the automotive damper [13, 14].
The objective of the present paper is to demonstrate the usefulness of the RFS method in the particular case of a real-life
nonlinear spacecraft structure: the SmallSat spacecraft from EADS-Astrium. Starting from a review of the required
ingredients for a RFS-based identification, we will propose solutions to the inherent limits of the method. First, we will
simplify the kinematics of the nonlinear device of the spacecraft, termed WEMS, in order to explicitely formulate its
dynamic equations. This formulation will be based on the necessary use of the coordinates of its center of gravity.
Eventually, we will discuss why our estimation of coefficients is ill-conditioned and how to circumvent this final issue.
The whole procedure will then be demonstrated using numerical data.
1.2 The SmallSat Spacecraft and Its Finite Element Modelling
The SmallSat structure has been conceived as a low cost structure for small low-earth orbit satellite [15]. It is a monocoque
tube structure which is 1.2 m long and 1 m large. It incorporates eight flat faces for equipment mounting purposes, creating
an octagon shape, as shown in Fig. 1.1a. The octagon is manufactured using carbon fibre reinforced plastic by means of a
filament winding process. The structure thickness is 4.0 mm with an additional 0.25 mm thick skin of Kevlar applied to both
the inside and outside surfaces to provide protection against debris. The interface between the spacecraft and launch vehicle
is achieved through four aluminium brackets located around cut-outs at the base of the structure. The total mass including the
interface brackets is around 64 kg.
The SmallSat structure supports a telescope dummy composed of two stages of base-plates and struts supporting various
concentrated masses; its mass is around 140 kg. The telescope dummy plate is connected to the SmallSat top floor via three
shock attenuators, termed SASSA (Shock Attenuation System for Spacecraft and Adaptator) [16], the behaviour of which is
considered as linear in the present study. The top floor is a 1 m2 sandwich aluminium panel, with 25 mm core and 1 mm
skins. Finally, as shown in Fig. 1.1c a support bracket connects to one of the eight walls the so-called Wheel Elastomer
Mounting System (WEMS) device which is loaded with an 8 kg reaction wheel dummy. The purpose of this device is to
isolate the spacecraft structure from disturbances coming from reaction wheels through the presence of a soft interface
between the fixed and mobile parts. In addition, mechanical stops limit the axial and lateral motion of the WEMSmobile part
during launch, which gives rise to nonlinear dynamic phenomena. Figure 1.1d depicts the WEMS overall geometry, but
details are not disclosed for confidentiality reasons.
The Finite Element (FE) model in Fig. 1.1b was created in Samcef software and is used in the present study to conduct
numerical experiments. The comparison with experimental measurements revealed the good predictive capability of this
model. The WEMS mobile part (the inertia wheel and its cross-shaped support) was modeled as a flexible body, which is
connected to the WEMS fixed part (the bracket and, by extension, the spacecraft itself) through four nonlinear connections,
labeled NC 1–4 in Fig. 1.1d. Black squares signal such connections. Each nonlinear connection possesses
• A nonlinear spring (elastomer in traction plus 2 stops) in the axial direction,
• A nonlinear spring (elastomer in shear plus 1 stop) in the radial direction,
• A linear spring (elastomer in shear) in the third direction.
The spring characteristics (piecewise linear) are listed in Table 1.1 and are displayed in Fig. 1.1e. We stress the presence
of two stops at each end of the cross in the axial direction. This explains the corresponding symmetric bilinear stiffness
curve. In the radial direction, a single stop is enough to limit the motion of the wheel. For example, its +x motion is
constrained by the lateral stop number 2 while the connection 1�x limits the opposite �x motion. The corresponding
stiffness curves are consequently asymmetric. Dissipation is introduced in the FEM through proportional damping and local
dampers to model the elastomer plots.
Sine-sweep excitation was applied locally at the bracket in different directions. The frequency band of interest spans the
range from 5 to 50 Hz and the sweeping rate is chosen equal to four octaves per minute. This frequency range encompasses
the local modes of the WEMS device and some elastic modes of the structure. More precisely, around 11 Hz, the WEMS
vibrates according to two symmetric bending modes (around x and y axis). Around 30 Hz, two other symmetric modes
appear combining bending (around x and y axis) and translation (along x and y axis). A mode involving the WEMS and the
bracket is also present around 30 Hz. The first lateral bending modes and the first axial mode of the structure finally appear
between 30 and 50 Hz.
2 J.P. Noel et al.
Axial nonlinearity
In-plane nonlinearities
NC 1 (-x)
NC 2 (+x)
NC 3 (-y)
NC 4 (+y)
Inertiawheel
SmallSat
Inertia wheel
Bracket
Metalliccross
Filteringelastomer plot
Metallicstop
a b
c d
e
Fig. 1.1 SmallSat structure. (a) Real structure without the WEMSmodule; (b) finite element model; (c) WEMSmodule mounted on a bracket and
supporting a dummy inertia wheel; (d) close-up of the WEMS mobile part (NC stands for nonlinear connection) and (e) graphical display of the
nonlinear restoring forces
1 Application of the Restoring Force Surface Method to a Real-life Spacecraft Structure 3
1.3 Methodology of Identification
In this paper, we address the identification of nonlinear mechanical systems whose nonlinearities are supposed to be
localized and for which there exists an underlying linear regime of vibration. The amplitude, the direction and the frequency
content of the excitation determine in which regime the structure vibrates. For example, we exclude from our scope
distributed or essential nonlinearities.
Such nonlinear systems are modelled through the equations
M€qðtÞ þ C _qðtÞ þKqðtÞ þXn
j¼1WjgjðtÞ ¼ pðtÞ (1.1)
where M, C and K are the mass, damping and stiffness matrices, respectively, q is the generalized displacement vector and
p is the external force vector. The n nonlinear forces acting on the structure are described through some weighted basis
functions gj. Our interest lies in the estimation of the weights, or the coefficients, introduced in these nonlinear expressions
and denoted Wj.
We now give a practical introduction to the RFS method. Other ways to get onto this method can be found in [11, 17, 18].
First, (1.1) is recast into
M€qðtÞ þ fnlðtÞ ¼ pðtÞ (1.2)
where fnl(t) contains all the restoring forces of the system. This offers a simple way to assess the coefficients Wj. Indeed, if
we know the modelling gj(t) of the nonlinear forces hidden in fnl(t), the excitation force p(t), an estimate of the mass matrix
M and the kinematic signals q(t), _qðtÞ and €qðtÞ, the matrixWj (along with K and C) can be estimated, for instance, in a least
squares sense.
For simplicity, we restrict ourselves to the underlying conservative system: neither the damping C of the underlying
linear structure nor the localized dissipation of the elastomer plots of the WEMS will be identified. Concerning the
characterizing functions gj, the nature of the nonlinearity can guide us to its functional form. This is the case of the
WEMS where a bilinear model is obvious.
The access to the excitation signal p(t) can appear to be trivial. However, space structures are universally tested under a
base excitation for which the actual force produced by the shaker is often unknown. In our case, since the exciting force is
applied locally and not directly onto the nonlinear connections, it will not complicate our identification procedure.
The practical knowledge of the mass matrix and of the kinematic signals is more questionable and will entail the whole
methodology developed in this paper. Note that, in practice, only acceleration signals only are recorded. Integration and/or
differentiation are then used to compute displacement q(t) and velocity _qðtÞ [19].
1.3.1 An Illustrative Example
It is interesting to examine the access to these two pieces of information (M and €qðtÞ) in the case of a simple continuous
structure comprising one lumped nonlinearity. The structure of interest is here a linear clamped-free beam with a cubic
nonlinear spring at its free end. This numerical set-up models the geometrical nonlinearity induced by a thin beam part
positioned at the main beam free end as in [20]. Figure 1.2 displays the Finite Element Model (FEM) of the structure where
ten 2D beam elements are considered. Each element possesses a translational (vertical) and a rotational Degree Of Freedom
(DOF) denoted yi and yi, respectively. They are both numbered from 1 to 10, the “nonlinear” DOF being y10.
Table 1.1 Nonlinear spring
characteristics (adimensional
values for confidentiality)Spring Clearance
Stiffness of the
elastomer plot
Stiffness of the
mechanical stop
Axial caxial ¼ 1 1 13.2
Lateral cradial ¼ 1.27 0.26 5.24
4 J.P. Noel et al.
The nonlinear vibrations of that DOF are governed by the equation (assuming that there is no excitation applied at beam tip)
M½y10;y9� €y9ðtÞ þ M½y10;y9�€y9ðtÞ þ M½y10;y10� €y10ðtÞ þ M½y10;y10�
€y10ðtÞ þ . . .
C½y10;y9� _y9ðtÞ þ C½y10;y9�_y9ðtÞ þ C½y10;y10� _y10ðtÞ þ C½y10;y10�
_y10ðtÞ þ . . .
K½y10;y9� y9ðtÞ þ K½y10;y9�y9ðtÞ þ K½y10;y10� y10ðtÞ þ K½y10;y10�y10ðtÞ þ knl y10ð Þ3 ¼ 0: (1.3)
We group the restoring forces together to finally obtain
M½y10;y9� €y9ðtÞ þ M½y10;y9�€y9ðtÞ þ M½y10;y10� €y10ðtÞ þ M½y10;y10�
€y10ðtÞ þ fnlðtÞ ¼ 0: (1.4)
This equation shows that the computation of the restoring forces fnl(t) requires the knowledge of the mass matrix M and
of the accelerations measured at the translational DOF’s y9 and y10 and at the rotational DOF’s y9 and y10. Without either
resorting to a FEM or complicating the experimental procedure, the access to a reliable estimate ofM is a first serious issue.
In addition, in practice, the measurement of rotational DOF’s, such as y9 and y10, is not usually carried out.
This example immediately reveals why there exists almost no application of the RFS method to large-scale structures in
the literature. Most often [21, 22], (1.3) is truncated and adopted under the form
M½y10;y10� €y10ðtÞ þ C½y10;y10� _y10ðtÞ þ K½y10;y10� y10ðtÞ þ knl y10ð Þ3 ¼ 0: (1.5)
The scope of the method is then reduced to qualitative information, i.e. nonlinearity characterization, where it proves to
be a useful tool. It is, however, no longer capable of assessing parameters. In the next subsection, we show how it is possible
to perform RFS-based system identification of the SmallSat spacecraft under an assumption concerning the WEMS
kinematics.
1.3.2 Assumption of a Rigid WEMS Device
One should observe that rotational DOF’s, such as yi in the previous subsection, are central to describe the kinematics of a
flexible body. For instance, the bending of the beam elements in Fig. 1.2 is linked, by essence, to the rotation of their ends.
On the contrary, rotations can be avoided in the description of the motion of a rigid bar element. More generaly, it is possible
to completely define the kinematics of a rigid body through the measure of six translations only, without entailing rotations.
It thus appears that a rigid body assumption is a way to prevent the use of unmeasurable rotational DOF’s.
However, such an assumption is not applicable in the case of the clamped-free beam with cubic nonlinearity. Indeed, this
latter is caused by large deflections (or deformations) of the beam and thus needs flexibility to be activated. On the other
hand, several types of nonlinearity do not resort to such a flexibility in their dynamics. We can cite the geometrical
nonlinearities due to large displacements which always arise in fully rigid multibody systems. This is also the case for the
nonlinearities that are lumped in essence or, in other words, that are caused by localized mechanisms (e.g., friction in a
bolted connection or a damper in an automotive suspension). Alternatively stated, such nonlinearities are not denatured
whether the masses they connect are taken to be rigid. That is not to say that the physics of the structure itself is not modified
(see next subsection).
The WEMS case belongs to this latter class. Its bilinear behaviour in stiffness is indeed localized since it originates from
the combination of lumped elements that are the elastomer plots and the stops. We consequently assume the rigidity of the
inertia wheel and of its cross-shaped support. We simplify further our model by reducing the inertia wheel to a point mass
whose inertia properties are allocated to the center of the cross.
1 2 3 4 5 6 7 8 9 10
knl × y3 [N/m3]
x
y
Fig. 1.2 FEM of the
nonlinear beam
1 Application of the Restoring Force Surface Method to a Real-life Spacecraft Structure 5
To assess this rigidity assumption, we propose in Table 1.2 the first six elastic frequencies of the WEMS mobile part
alone. Their magnitude gives sense to our approach. We will come back later on the verification of this assumption. Indeed,
in the case of large vibrations, the shocks between the cross and the stops can give rise to flexible effects that have to be
monitored.
The fourth point of the required information listing established above is fulfiled: the kinematic signals q(t), _qðtÞ and €qðtÞare within reach. Furthermore, rigidity gives an easy access to an analytical computation of the mass matrixM. The different
elements of the WEMS as we model it were already displayed in Fig. 1.1d. The rigid metallic cross and the point mass
inertia wheel can thereof be seen.
1.3.3 Kinematic Constraints
The three x–y–z displacements of each end of the cross (twelve in total) naturally describe the kinematic of the WEMS.
Since this description requires the knowledge of six coordinates only, six of them turn out to be redundant. If the vector q
collects this set as
qT ¼ x1 y1 z1 x2 y2 z2 x3 y3 z3 x4 y4 z4ð Þ; (1.6)
it can thus be split into two subsets qI and qD, the former containing six user-chosen independent coordinates.
As for the kinematic constraints, they express the invariance of the distance between any two points of the WEMS
module. In general, this leads to a set of
r!
2! r � 2ð Þ! (1.7)
relationships. In the case of the WEMS, six distance invariants result from the last formula in agreement with the sizes of the
sets q, qI and qD.
Considering the scheme in Fig. 1.3, they can be formulated as
r1 � r2k k2 ¼ d2
r3 � r4k k2 ¼ d2
r2 � r3k k2 ¼ c2
r3 � r1k k2 ¼ c2
r1 � r4k k2 ¼ c2
r4 � r2k k2 ¼ c2
8>>>>>>>>>><>>>>>>>>>>:
(1.8)
where d and c are the length of an arm of the cross and the distance between two of its adjacent ends, respectively. These six
relations indicate that the four ends of the cross are, in fact, the vertices of a square. They intrinsically express the
invariability of the lengths of the two arms, their perpendicularity and their common Center of Gravity (CoG).
It is actually possible, but herein skipped, to demonstrate that the set of constraints (1.8) can be transformed into the more
intuitive relationships of length, orthogonality and coincidence.
Table 1.2 Six first elastic
frequencies of the WEMS
mobile part in free–free
conditions
Natural frequencies (Hz)
1 3,109
2 3,163
3 3,175
4 3,176
5 6,624
6 7,036
6 J.P. Noel et al.
r1 � r2k k2 ¼ d2
r3 � r4k k2 ¼ d2
ðr1 � r2Þ � ðr3 � r4Þ¼ 0
r1 þ r2ð Þ=2 ¼ r3 þ r4ð Þ=2:
8>>>><>>>>:
(1.9)
Already at this stage, we want to underline the mathematical complexity of these relations. Their impossibility to
explicitely define qD, no matter its definition, will play a major role further in this paper.
In the previous subsection, we gave in a first argument in favour of the rigid body assumption (in terms of elastic
frequencies). We can herein make use of the geometrical conditions of rigidity to control on-line the quality of our
assumption. This is central as our approach would loose its suitability in case of flexibility effects. Indeed, significant
terms in (1.1) would then be erroneously neglected.
To that purpose, we formulate the aforementioned geometrical constraints in terms of relative errors and seek potential
deviations from rigidity during the increase of the excitation frequency:
r1 � r2k k2=d2 � 1 ¼ 0
r3 � r4k k2=d2 � 1 ¼ 0
cos�1 r1 � r2ð Þ � r3 þ r4ð Þr1 � r2k k2 r3 þ r4k k2
!2
p� 1 ¼ 0
r1 þ r2ð Þ= r3 þ r4ð Þ � 1 ¼ 0:
8>>>>>>><>>>>>>>:
(1.10)
In Fig. 1.4, we propose a first example of this verification means (z excitation on the bracket at 300 N). At this excitation
level, the system is nonlinear. As intuitively expected, the perpendicularity is almost exactly verified. The 4�z stop is
actually reached and this is visible in the deviations of the second and sixth constraints (explained by impacts on the stops).
The influence of the resonances of the structure are also clearly detectable on these six plots.
We can inspect a second set of constraints under x excitation at 300 N (for which the system is now linear) in Fig. 1.5.
Their verification is improved mainly on the sixth constraint. This highlights the role of the impacts in the relevancy of the
rigid body assumption.
In conclusion, we see that our geometrical verification approach provides a qualitative measure of the confidence in our
identification strategy, and therefore in the subsequently estimated coefficients.
1.3.4 Explicit Formulation of the WEMS Dynamics
We already explained in Sect. 1.3.1 that a rigorous and thorough writing of Newton’s law of motion is crucial to the RFS
method. It is worth pointing out that an unconstrained form of these equations is also obviously sought. Whereas the writing
of such a form is direct for classical vibrating structures, the situation gets more complicated in the presence of kinematic
constraints. This problem is addressed in the present Subsection.
1
2
3 4
d
c
y
xr3
S/C
Fig. 1.3 Top view of the
square-shaped WEMS
mobile part
1 Application of the Restoring Force Surface Method to a Real-life Spacecraft Structure 7
0 20 40−0.03
0.03
Swept frequency (Hz)
Len
gth
r 1−
r 2
0 20 40−1
1
Swept frequency (Hz)
Len
gth
r 3−
r 4
0 20 40−1
1x 10
−6
Swept frequency (Hz)
Per
pend
icul
arity
0 20 40−1.5
1.5x 10
−5
Swept frequency (Hz)
CoG
x
0 20 40−1
1x 10
−4
Swept frequency (Hz)
CoG
y
0 20 40−0.02
0.02
Swept frequency (Hz)
CoG
zFig. 1.4 On-line verification (in percents) of the geometrical conditions of rigidity at 300 N (z excitation)
0 20 40−0.2
0.2
Swept frequency (Hz)
Len
gth
r 1−
r 2
0 20 40−0.4
0.4
Swept frequency (Hz)
Len
gth
r 3−
r 4
0 20 40−1.5
1.5x 10
−7
Swept frequency (Hz)
Per
pend
icul
arity
0 20 40−1
1x 10
−4
Swept frequency (Hz)
CoG
x
0 20 40−3
3x 10
−5
Swept frequency (Hz)
CoG
y
0 20 40−5
5x 10
−6
Swept frequency (Hz)
CoG
z
Fig. 1.5 On-line verification (in percents) of the geometrical conditions of rigidity at 300 N (x excitation)
8 J.P. Noel et al.
First, we explicitely introduce the elastic restoring forces in the system through the trilinear form (for which we assume
the asymmetry of the WEMS)
fnlðqiÞ ¼ðk0i þ k1i Þqri þ k1i d
1i if qri<� d1i
k0i qri if � d1i � qri � d2i
ðk0i þ k2i Þqri � k2i d2i if qri>d2i
8>><>>: (1.11)
where
• qi is the ith component of q,
• ki0, ki
1 and ki2 are the stiffnesses of the elastomer and of the stops, respectively,
• di1 and di
2 are the associated clearances.
Figure 1.6 displays this stiffness curve. Note that we use the notation qir rather than qi to remind that the force in the
springs is linked to the relative motion of their ends. The relative displacement qir thus designates the difference between two
opposite displacements of the fixed and mobile parts of the WEMS. We also draw attention to the connections noted 1�y,2�y, 3�x and 4�x that are linear and for which we simply write
fnlðqiÞ ¼ k0i qri : (1.12)
In general, the expression of the elastic forces in the WEMS can be shortened following
fnlðqiÞ ¼ kiqri þ k�
i (1.13)
where ki and ki∗ are piecewise constant.
The requirement for reaching unconstrained equations of motion is the possibility to free them from qD by substitution.
In other words, the requirement is an explicit knowledge of the relation
qD ¼ F qI� �
: (1.14)
Such a relation is out of reach because of the complexity of the set of constraints (1.9). This can be clarified by
considering the writing of the potential energy in the system. This energy, stored in the twelve linear and nonlinear
stiffnesses, has the form
V ¼ VðqÞ ¼X12
i¼1
1
2ki q
ri
� �2 þ k�i q
ri : (1.15)
qri
fnl(qi)
d2i
d1i
k0i
k2i
k1i
Fig. 1.6 Bilinear (or here
trilinear since asymmetry
is supposed) force to
be identified
1 Application of the Restoring Force Surface Method to a Real-life Spacecraft Structure 9
After elimination of qD, we should reach its unconstrained expression, depending on qI only,
V� ¼ V� qI� �
: (1.16)
More particularly, we are interested in the elastic forces which require its gradient, in the Lagrange’s formalism,
@V�
@qI¼ @V
@q
@q
@qI: (1.17)
We can detail this latter as
@V�
@qIj¼X
i:qi2qI@V
@qidij þ
Xi:qi2qD
@V
@qi
@qi@qIj
(1.18)
where it becomes obvious that the last derivate cannot be computed. Indeed, it needs the explicit relationship (1.14) while we
only possess its implicit definition (1.9) of the form
C qI; qD� � ¼ 0: (1.19)
We are consequently compelled to choose a new describing set of independant coordinates and we naturally turn to the
CoG of the WEMS mobile part. We formulate the new definition
qI ¼ xCoG yCoG zCoG a b gð ÞT (1.20)
where xCoG, yCoG and zCoG measure the translation of the CoG in the x–y–z frame and a, b and g parametrize its rotation. We
also redefine qD as
qD ¼ x1 y1 z1 x2 y2 z2 x3 y3 z3 x4 y4 z4ð ÞT (1.21)
and q as qIS
qD.
The immediate advantage of this choice is the possibility to write the transformation qD ¼ F(qI) as
xi
yi
zi
0BB@
1CCA ¼
xCoG
yCoG
zCoG
0BB@
1CCAþ Rða; b; gÞ
Xi
Yi
Zi
0BB@
1CCA; i ¼ 1; . . . ; 4 (1.22)
where Xi Yi ZiT is the initial (undeformed) position of one end of the cross and R is a rotation matrix.
In the parametrization of the WEMS rotation, we opt for Bryant angles because of their intuitive interpretation. The roll
and pitch angles directly quantify rotations around the x and y arms of the cross. As for the yaw angle, it corresponds to a
linear torsion of the WEMS around z axis.
One can go back to the computation of the elastic forces and write, since the potential energy V (see (1.15)) is independant
of the CoG coordinates,
@V
@qIj¼X
i:qi2qD@V
@qi
@qi@qIj
: (1.23)
It thus appears that the restoring force ∂V / ∂qjI can be computed from the product between the gradient ∂V / ∂qi of V
and the jth column of the Jacobian J associated with the set of (1.22).
In addition, the kinetic energy in the system takes the simple form
T ¼ 1
2m _x2CoG þ m _y2CoG þ m _z2CoG þ Ix _a2 þ Iy _b
2 þ Iz _g2� �
(1.24)
and leads to the diagonal mass matrix
10 J.P. Noel et al.
M ¼
m
m 0
m
Ix
0 Iy
Iz
0BBBBBBBB@
1CCCCCCCCA: (1.25)
Under the rigid body assumption, the following unconstrained equation describes the motion of the WEMS rigourously
and is suited to a RFS-based parameter estimation:
Mj€qIj þ ~rV � Jj ¼ 0 (1.26)
where the coefficients to be assessed are hidden in ~rV and whereMj and Jj designate the jth diagonal term ofM and column
of J, respectively. It can be shown that the first three scalar (1.26) give the access to the estimation of the twelve stiffnesses of
the system since they write
m€xCoG þ fnlðx1Þ þ fnlðx2Þ þ fnlðx3Þ þ fnlðx4Þ ¼ 0
m€yCoG þ fnlðy1Þ þ fnlðy2Þ þ fnlðy3Þ þ fnlðy4Þ ¼ 0
m€zCoG þ fnlðz1Þ þ fnlðz2Þ þ fnlðz3Þ þ fnlðz4Þ ¼ 0:
8>><>>: (1.27)
We will consequently restrict our results to the use of these three equations. Rigidity rises a last issue, discussed in the
following subsection, and linked to the rank deficiency of the matrix ~rV � J.
1.3.5 Identification of a Rigid Body: An Ill-Conditioned Problem
Let us consider the identification of the WEMS stiffnesses in the x direction and in linear regime. The first equation of (1.27)
then becomes
m€xCoG þ k0x1 xr1 þ k0x2 xr2 þ k0x3 x
r3 þ k0x4 x
r4 ¼ 0: (1.28)
For the purpose of the identification, this equation is written as a least squares problem:
xr1 xr2 xr3 xr4� �|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}
A
k0x1 k0x2
k0x3 k0x4
� �T|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
x
¼ �m€xCoG|fflfflfflffl{zfflfflfflffl}b
:(1.29)
In practice, matrixA turns out to be badly conditioned preventing an accurate estimation of the coefficients k0xi . More precisely,
thismatrix appears to be not of full rank. The explanation is twofold. On the one hand, and according the fourth and last equation
of constraint (1.9), there exists a linear relationship between the displacements of theWEMSmobile part considered direction per
direction. On the other hand, since theWEMS fixed part (actually the bracket) is almost at rest in our frequency band of interest
(5–50 Hz), this linear dependance is not altered when moving to relative displacements as in matrix A.
As noted in the literature [23], without perturbations and rounding errors, the solution to the rank-deficient system of
equations A x ¼ b is straightforward. Indeed, if we introduce the Singular Value Decomposition (SVD) of A
A ¼Xn
i¼1ui si vTi (1.30)
this solution writes
xideal ¼XrankðAÞ
i¼1
uTi b
sivi: (1.31)
1 Application of the Restoring Force Surface Method to a Real-life Spacecraft Structure 11
However, practically speaking, A is never exactly rank deficient since it has one or several small but non-zero singular
values [23]. It is said to be numerically rank deficient and causes the aforementioned least squares problem to be ill-
conditioned, i.e. its solution is dominated by the errors. A simple regularization strategy consists in truncating the singular
value spectrum of A and thus replacing its smallest elements with exact zeros. In other words, A is seen as a noisy
representation of the mathematically rank deficient matrix Ak defined as
Ak ¼Xk<n
i¼1ui si vTi : (1.32)
The stiffness coefficients of the WEMS are then computed in a numerically stable way through
x ¼ Ayk b ¼
Xk
i¼1
uTi b
sivi (1.33)
where { denotes the inverse of a rectangular matrix in a least squares sense. This approach is known as the Truncated
Singular Value Decomposition (TSVD). Its difficulty lies in the choice of k, i.e. the number of sufficiently large singular
values, that is to say in the definition of what a small singular value is.
1.4 Identification Results
We first consider an axial (z) excitation applied to the WEMS bracket at 200 N. At this excitation level, no mechanical stop
is reached and the dynamics thus remains linear. Our interest lies in the estimation of the axial stiffnesses since the relative
displacements in the lateral directions are negligible. Figure 1.7 presents our results, summarized in Table 1.3. The
estimation is of high quality as proved by a simple visual inspection of the force-displacement curves.
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
Relative displacement
Res
tori
ng for
ce
Nonparametric RFS estimate
Fitted stiffness curve
Exact stiffness curve
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1x 10−3
x 10−3x 10−3
Res
tori
ng for
ce
Relative displacement
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1
Relative displacement
Res
tori
ng for
ce
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1
Res
tori
ng for
ce
Relative displacement
Fig. 1.7 Parametric estimation of the stiffnesses 1�z (top left), 2�z (top right), 3�z (bottom left) and 4�z (bottom right)
12 J.P. Noel et al.
Figure 1.8 shows a qualitative application of the RFS method. As explained in Sect. 1.3.1, this simplified approach
constitutes a powerful characterization tool. However, it cannot be used for parameter estimation purposes since it is based
on truncated equations of motion, as obvious in this figure.
As a last result in the linear case, we prove the relevancy of a TSVD regularization in Table 1.4. The four singular values of
thematrixA to be inverted in our least squares resolution present a large gap between the second and the third ones. It reveals a
numerical rank equal to 2. The consequent truncation of this singular value spectrum led us to the results in Fig. 1.7.
Table 1.3 Summary of the identification results in the axial direction. Each estimate is given with a relative deviation from its exact value
Identification case 1 � z 2 � z 3 � z 4 � z
200 N Linear est. 1.00 1.06 1.03 1.02
(linear estim.) 0.26% 5.50% 2.55% 2.12%
300 N Nonlinear est. – – – 12.93
(only nonlinear estim.) 2.08%
300 N Linear est. 1.66 0.84 0.82 0.80
(full estim.) 66.29% 16.41% 17.79% 20.03%
Nonlinear est. – – – 12.90
2.32%
400 N Nonlinear est. – – 12.26 12.29
(only nonlinear estim.) 7.14% 6.92%
500 N Nonlinear est. – – 13.92 13.10
(only nonlinear estim.) 5.41% 0.78%
Table 1.4 The third and fourth singular values should be equal to zero. They differ
because of rounding errors
200 N axial case Singular value (%)
1 55.6
2 43.3
3 0.6
4 0.5
−1 −0.5 0 0.5 1−4
−2
0
2
4x 10−3
Relative displacement
Res
tori
ng for
ce
Qualitative RFS
Exact RFS
Fig. 1.8 Qualitative application of the RFS method to the 4�z connection. The different WEMSmodes in the frequency band of excitation appear
with various and inexact slopes
1 Application of the Restoring Force Surface Method to a Real-life Spacecraft Structure 13
Prior to addressing a nonlinear case, we anticipate a discussion about displacement coupling tackled later in this paper.
The confidence factor [17] shown in Fig. 1.9 shows that there is no contribution of the lateral motion in the fitting of the
axial restoring forces.
We now move to a 300 N excitation level. The axial nonlinear connection numbered 4 � z is activated when sweeping
the modes of vibration around 30 Hz. This activation is clear on the corresponding time series (Fig. 1.10) where a jump
phenomenon is highlighted.
We chose to assess the 4 � z nonlinear stiffness using the four linear estimates obtained in the previous step (Fig. 1.7).
This approach turns out to be the most accurate. Our result is depicted in Fig. 1.11 and also listed in Table 1.3. Here again,
we note an excellent agreement with the exact result (relative error of 2%). We can however try to simultaneously fit both the
0
5
10
15
20
25
30
Con
fiden
cefa
ctor
z1 z2 z3 z4 x1 x2 x3 x4 y1 y2 y3 y4
Fig. 1.9 The recourse to lateral contributions appears to be useless in axial identification
27 29 31 33 35 37 39Frequency (Hz)
FR
Fm
agni
tude
Linear
Nonlinear Jump
0 10 20 30 40 50−2
0
2
Swept frequency(Hz)
Rel
ativ
e 4–
z di
spla
cem
ent
Fig. 1.10 Jump phenomenon caused by the distortion of the corresponding FRF
14 J.P. Noel et al.
four linear and the nonlinear forces (next case in Table 1.3). The nonlinear estimate remains of high quality whereas the
linear parameters get worse. This gives sense to our decoupled estimation.
We finally address 400 N and 500 N excitations and focus on the first modes of the WEMS around 11 Hz. Indeed,
considering their shapes, the modes of vibration around 30 Hz cannot lead to the activation of several nonlinearities. The
reach of the 4 � z nonlinear regime actually prevents other stops from being impacted. This explains our interest in the first
modes which are activated at higher energy but involve two nonlinear restoring forces. Here again, we exploit our
aforementioned linear estimates in order to focus on a nonlinear estimation problem. The estimates take place in Table 1.3
and a graphical display at 500 N is given in Fig. 1.12. As one may expect, the higher the level of activation of the
nonlinearities, the more accurate their estimation.
The estimation of the lateral stiffnesses (under a lateral excitation) turns out to be much more challenging and will clearly
underline the limitations of our RFS-based identification strategy. Their explanation lies in the existence of kinematic
couplings.
To detail this issue, we now excite the structure along an oblique direction with respect to the x and y reference axis (at the
linear 200 N regime) and try to estimate the y-stiffness coefficients. The result provided in Table 1.5 is of unsatisfactory
quality. This quality can be enhanced by considering the existence of kinematic couplings. To that purpose, we include in
our fitting basis all the available x, y and z restoring forces as potential candidates. The estimation is depicted in Fig. 1.13
and summarized in Table 1.5. Its quality appears to be acceptable and the analysis of the stiffness curves confirms that
judgement. The need for an extended fitting panel is obvious in Fig. 1.14. In particular, the 3 � z and 4 � z contributions aremore relevant than the y stiffnesses themselves. Finally, we stress that a much improved quality would require additional
fitting terms (mainly modelling dissipation) as demonstrated by the remaining disturbances on the force-displacement curves
(Fig. 1.13).
Another proof of interdependence between the directions of motion of the WEMS is given in Fig. 1.15 (x excitation).
This restoring force curve reveals a trilinear behaviour. Such a form is actually caused by the simultaneous reach of an axial
and a lateral stop. Their appearance on a single force-displacement curve is indeed an additional proof of coupling.
An experimental campaign was also performed on the SmallSat spacecraft, and the results are currently analysed.
A preliminary result in Fig. 1.16 clearly highlights the piecewise linear nature of the WEMS device.
−1.5 −1 −0.5 0 0.5 1 1.5−7.5
−5
−2.5
0
2.5
5
7.5x 10−3
Relative displacement
Res
tori
ng for
ce
Nonparametric RFS estimate
Fitted stiffness curveExact stiffness curve
Fig. 1.11 Excellent estimation of the 4 � z nonlinear force
1 Application of the Restoring Force Surface Method to a Real-life Spacecraft Structure 15
−1.5 −1 −0.5 0 0.5 1 1.5−7.5
−5
−2.5
0
2.5
5
7.5x 10−3
Relative displacement
Res
tori
ng for
ce
Nonparametric RFS estimate
Fitted stiffness curve
Exact stiffness curve
−1.5 −1 −0.5 0 0.5 1 1.5−7.5
−5
−2.5
0
2.5
5
7.5x 10−3
Relative displacement
Res
tori
ng for
ce
Fig. 1.12 Simultaneous identification of the 3�z (left) and 4�z (right) restoring forces
−0.25 0 0.25 −1
0
Relative displacement
Res
tori
ng for
ce
−0.25 0 0.25−1
0
1x 10−4
Relative displacement
Res
tori
ng for
ce
−0.25 0 0.25−1
0
1x 10−4
Relative displacement
Res
tori
ng for
ce
−0.25 0 0.25−1
0
1x 10−4
Relative displacement
Res
tori
ng for
ce
Nonparametric RFS estimate
Fitted stiffness curve
Exact stiffness curve
Fig. 1.13 Parametric estimation of the stiffnesses 1�y (top left), 2�y (top right), 3�y (bottom left) and 4�y (bottom right)
Table 1.5 Summary of the identification results in the y lateral direction. Each estimate is given with a relative deviation from its exact value
Identification case 1 � y 2 � y 3 � y 4 � y
200 N Linear est. � 0.73 0.96 0.12 0.12
(linear estim.) 379.42% 268.95% 55.18% 55.37%
200 N Linear est. 0.32 0.24 0.28 0.28
(extended basis) 22.83% 6.75% 7.00% 8.00%
0
2
4
6
8
10
12
14
16
18
Con
fiden
cefa
ctor
y1 y2 y3 y4 x1 x2 x3 x4 z1 z2 z3 z4
Fig. 1.14 The whole fitting basis is relevant in the case of the y identification
−1.5 −1 −0.5 0 0.5 1 1.5−5
−2.5
0
2.5
5x 10−4
Relative displacement
1–z
rest
orin
gfo
rce
Fig. 1.15 The reach of two perpendicular stops induces a coupled trilinear behaviour
1 Application of the Restoring Force Surface Method to a Real-life Spacecraft Structure 17
1.5 Conclusions
This paper aimed at applying the Restoring Force Surface method to a real-life space structure: the SmallSat spacecraft. The
elimination of kinematic constraints and the regularization of ill-conditioned inverse problems are the two main
contributions of the study. To date, the identification results are promising. A particular attention should be devoted to
damping contributions in a further work. A possible approach would be a non-physical modelling through Fourier series. In
addition, the SmallSat identification from experimental data is also in progress.
Acknowledgements This paper has been prepared in the framework of the ESA Technology Research Programme study “Advancement of
Mechanical Verification Methods for Non-linear Spacecraft Structures (NOLISS)” (ESA contractNo.21359/08/NL/SFe).
The author J.P. Noel would like to acknowledge the Belgian National Fund for Scientific Research (FRIA fellowship) for its financial support.
The authors finally thank Astrium SAS for sharing information about the SmallSat spacecraft.
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22. Peeters M, Kerschen G, Golinval JC, Stephan C, Lubrina P (2011) Nonlinear norma modes of a full-scale aircraft. In: International modal
analysis conference (IMAC) XXIX, Jacksonville, FL, 2011
23. Hansen PC (1996) Rank-deficient and discrete ill-posed problems. PhD thesis, Technical University of Denmark
1 Application of the Restoring Force Surface Method to a Real-life Spacecraft Structure 19
Chapter 2
Nonlinear Dynamic Model and Simulation of Morphing
Wing Profile Actuated by Shape Memory Alloys
Cassio T. Faria, Carlos De Marqui Jr., Daniel J. Inman, and Vicente Lopes Jr.
Abstract Morphing aircraft have the ability to actively adapt and change their shape to achieve different missions
efficiently. The development of morphing structures is deeply related with the ability to model precisely different designs
in order to evaluate its characteristics. This paper addresses the dynamic modeling of a sectioned wing profile (morphing
airfoil) connected by rotational joints (hinges). In this proposal, a pair of shape memory alloy (SMA) wires are connected to
subsequent sections providing torque by reducing its length (changing airfoil camber). The dynamic model of the structure is
presented for one pair of sections considering the system with one degree of freedom. The motion equations are solved using
numerical techniques due the nonlinearities of the model. The numerical results are compared with experimental data and
a discussion of how good this approach captures the physical phenomena associated with this problem.
Keywords Morphing wing • Shape memory alloy • Nonlinear dynamics
2.1 Introduction
Since the beginning of self propelled flight in 1903 the design of aircraft have been based on recreate nature’s ability of
flight. However, the recent design trend lines shifted towards a more rigid structure powered by engines, a different approach
compared to the one developed in nature during the course of evolution. The advances achieved by this different approach
were magnificent, creating machines that could flight faster and higher then any other bird or flying animal in nature.
However, these advances, came with improved efficiencies limited to specific flight conditions for a single vehicle.
Inspired by birds that adapt their wing aerodynamics (change in shape) to improve their performance when executing a
specific task [1], a new concept in wing design has been developed, called “morphing wing”. Several authors have been
developing these concepts, such as in [2] where small and continuous adjustments in the wing shape were made, or like in [3]
where small changes in the flow field around the wing improved the aircraft control. A more complete description of the
evolution of this concept is presented in [4].
This paper addresses the modeling of a novel morphing wing configuration that uses shape memory alloys (SMA) as
actuators to provide the desired change in shape [5]. By sectioning a wing profile into two or more pieces and connecting
them back with rotational joints (hinges), the main structure of the proposed morphing wing is built, such as illustrated in
Fig. 2.1a. One pair of SMA wires is positioned over each joint to allow the structure to be actuated in both directions.
Figure 2.1b shows the motion created when the top wire is activated and Fig. 2.1c illustrates the actuation of the bottom wire.
The actuation mechanism consists of one pair of deformed shape memory alloy wires positioned over each hinge. The
shape memory effect [6] is activated by heating one wire, that reduce its length or, if restricted, apply a force. This reduction
in length apply a torque in the structure. To provide a good aerodynamic response an elastic skin has to be designed in order
C.T. Faria (*) • D.J. Inman
Virginia Polytechnic Institute and State University, 310 Durham Hall, 24061 Blacksburg, VA, USA
e-mail: [email protected]
C. De Marqui Jr.
Engineering School of Sao Carlos, Uni of Sao Paulo, Av. Trabalhador Sancarlense 400, 13566-590 Sao Carlos, SP, Brazil
V. Lopes Jr.
Uni Estadual Paulista – UNESP, Av. Brasil Nº56, 15385-000 Ilha Solteira, SP, Brazil
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_2, # The Society for Experimental Mechanics, Inc. 2012
21
to eliminate the discontinuities. The skin has to be compliant to reduce the load on the actuator, but has to be stiff enough to
support aerodynamic loads, as presented by [7].
In practice such structure needs to be controlled in order to maintain or achieve a desired shape, to do so an appropriate
dynamic model has to be derived. During this process two sources of nonlinearities were identified, one is related with
geometrical problem, since the force is always applied in the direction of the wire, and another one is related with the
hysteretic nature of the shape memory alloy transformations. Both explored and treated further in this paper.
2.2 Modeling of the Proposed Morphing Wing
Consider a wing profile sectioned and connected by a rotational joint (hinge) at midchord position. The leading edge section
is attached to an inertial reference frame (I) XY, presented in Fig 2.2a, while the other rotating half is linked to a second
mobile frame (B1) X1Y1. Other important points are illustrated in Fig. 2.2b, such as the center of mass (G), the rotational
point (O), the connection points of the upper actuator (A and B) and the connection points of the lower actuator (C and D). It
is important to notice that the dimension presented for points A and C are related to the inertial frame I, for points B, D and G
they are described in terms of the mobile frame B1.The transformation matrix that relates a vector from the inertial frame (I) to the mobile frame (B1) can be simply derived
considering a rotation angle y.
TB1 ¼cos y sin y 0
� sin y cos y 0
0 0 1
264
375 (2.1)
One can calculate the acceleration of the center of mass (G) of the second body based on the position vector ( IrOG),
considering that, for this case, there is no relative velocity ( IVrel) or acceleration ( Iarel) between the two reference frames.
IaG ¼ IaO þ Iw� Iw� IrOG þ I _w� IrOG þ 2 Iw� IVrel þ Iarel ¼�€y L1 siny� _y
2L1 cos y
€y L1 cosy� _y2L1 sin y
0
264
375 (2.2)
The angular velocity vector ( Iw) is defined as the time derivative of the angular position (y) with orientation defined by
the right hand rule, being positive in the z direction when y moves counter clockwise. The over-dot denotes differentiation
with respect to time.
The distance vector between A-B and C-D can be defined in the inertial frame as a linear operation between the position
vectors ( IrOA, B1rOB, IrOC and B1rOD), such that:
IrAB ¼ � IrOA þ TTB1B1rOB ¼
b1 cos y� b2 sin yþ a1b1 sin yþ b2 cos y� a2
0
24
35 (2.3)
aP1 P2 P3
q
q
P4
b
c
Fig. 2.1 (a) Proposed model
of morphing wing with four
sections with a pair of
actuators between P2 and P3.
(b) Actuation of the top wirein the model. (c) Actuation of
the bottom wire
22 C.T. Faria et al.
IrCD ¼ � IrOC þ TTB1B1rOD ¼
d1 cos yþ d2 sin yþ c1d1 sin y� d2 cos y� c2
0
24
35 (2.4)
It is important to notice that the subscript located to the left of the vector symbol indicates the frame where it is defined,
and for position vectors the subscript index to the right indicates the initial and final point were it is defined, respectively.
By activating the shape memory effect on the top wire/actuator (SMA1), the force with direction A–B is applied in both
bodies ( IFAB and IFBA) but with different orientation, a similar effect can be described for the bottom actuator (SMA2) thatwill apply a force in the C-D direction in both bodies ( IFCD and IFDC).
IFAB ¼ FSMA1IrAB
IrABk k ¼ � IFBA (2.5)
IFCD ¼ FSMA2IrCD
IrCDk k ¼ � IFDC (2.6)
Both parameters FSMA1 and FSMA2 are functions of the angle y and the electrical current in each wire. These forces are
defined in the next section based on the constitutive relation for shape memory alloys. Euller’s principle applied in the
second (moving) body at the origin point (O) is:
XMO ¼ I2 I _wþ Iw� I2 Iwþ m2 IrOG � IaG ¼
0
0
IZZ€yþ m2L21€y
24
35 (2.7)
where I2 is the inertial matrix (for this problem it is a diagonal matrix), m2 is the total mass of the body and IrOG is the
distance vector between point O and G. The external moments applied at point O in the second body are calculated based on
the free body diagram presented by Fig. 2.3, where aerodynamic forces (FAER), weight forces (FP), reaction forces (RX and
RY) and actuator forces (FBA and FDC) are acting on the body.
External moments are calculated by the cross product between the position vector of the application point and the force
vector, for the moments applied by the actuators one can get:
IMOB ¼ IFBA � IrOB ¼0
0
�FSMA1K1 sin yþK2 cos yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K3þK1 cos y�K2 sin yp
24
35 (2.8)
Y1 Y
a1
a
b
c1
c2
A B
DG
C
a2 b2
d2
d1
A
CO
OB
DG
L1b1
X
X1
θ
Fig. 2.2 (a) Frame of references illustration and positive angle measurement. (b) Structures points of interest
2 Nonlinear Dynamic Model and Simulation of Morphing Wing Profile Actuated by Shape Memory Alloys 23
IMOD ¼ IFDC � IrOD ¼0
0
�FSMA2H1 sin y�H2 cos yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
H3þH2 cos yþH1 sin yp
24
35 (2.9)
where
K1 ¼ a1b1 � a2b2
K2 ¼ a1b2 þ a2b1
K3 ¼ a21 þ a22 þ b21 þ b22
(2.10)
H1 ¼ c1d1 � c2d2
H2 ¼ c1d2 þ c2d1
H3 ¼ c21 þ c22 þ d21 þ d22
(2.11)
By setting the weight and aerodynamic forces to zero one can sum equations (2.8) and (2.9), and insert them into equation
(2.7), resulting in the following expression in the Z direction.
IZZ þ m2L21
� �€yþ FSMA1
K1 sin yþ K2 cos yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK3 þ K1 cos y� K2 sin y
p þ FSMA2H1 sin y� H2 cos yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
H3 þ H2 cos yþ H1 sin yp ¼ 0 (2.12)
Equation (2.12) is the nonlinear dynamic equation of themorphingwingwith a single degree of freedom.Nonlinear terms can
be isolated by looking at the expression that multiply the force applied by each actuator. These expressions can be interpreted as
the moment arm for each actuator force and they are functions of the angular position. A second source of non-linearity arises
from the hysteretic behavior of the shape memory alloys that is defined for each actuator force (FSMA1 and FSMA2).
FSMA1 ¼ ASMA1sSMA1 (2.13)
where ASMA1 represent the cross section area of the first actuator and sSMA1 the tension associated with it. A similar expression
can be written for the second actuator. The tension in each actuator is a function of the deformation of the wire and the
temperature, as described in detail by [6]. The deformation in each wire can be related with the angular position, such that:
eSMA1 ¼ IrABk k � L0SMA1
L0SMA1
(2.14)
where L0SMA1 is the initial (undeformed) length of the top actuator. Similar expression can be written for the second actuator.
The temperature in the actuator is the main mechanism to produce actuation by the SMA wire. The change in temperature
will induce a phase transformation in the material microstructure, as illustrated in Fig. 2.4a. Another important effect is the
microstructural transformation induced by stress, illustrated in Fig. 2.4b. More details about these transformation process
and modeling of these phenomena can be found in [6].
To relate the input electric current with a wire temperature [8] presents the following relation:
T ¼ T1 þ R
hCAC1� e�t=th
� �I2 þ ðT0 � T1Þ e�t=th (2.15)
Fig. 2.3 Free body diagram
of the second (moving) body
24 C.T. Faria et al.
where R is electrical resistance of the wire/actuator, T0 the temperature in the beginning of the process, I the input current, hCthe convection heat transfer coefficient, AC is the heat transfer area, T1 the environment temperature and th is the time
constant defined by:
th ¼ rASMAcPhCAC
(2.16)
where cP is the specific heat of the wire and r the SMA density.
2.3 Simulations and Experimental Verification
The solution of the proposed model that includes nonlinearities, such as the geometrical relation and hysteresis in the phase
transformation, was obtained through numerical approach [6]. Equation (2.12) is rewritten using the state space format.
_y€y
� �¼ 0 1
0 0
� �y_y
� ��
0FSMA1
IZZ þ m2L21� � K1 sin yþ K2 cos yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K3 þ K1 cos y� K2 sin yp þ FSMA2
IZZ þ m2L21� � H1 sin y� H2 cos yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
H3 þ H2 cos yþ H1 sin yp
24
35 (2.17)
A fifth order Runge–Kutta-Fehlberg method was used to solve the numerical method [9]. The method was implemented
using a Matlab® code with variable time step. A new time step (smaller) is defined if the local error, i.e. the difference
between the solutions provided by a higher order solution, is bigger then a threshold value. This technique was implemented
to increase the code speed and keep the solution accurate in regions with fast dynamics.
Table 2.1 presents all the parameters used in the simulations. It is important to notice that both actuators are considered to
have the same dimensions and properties. One may notice the large number of variables that have to be defined in the
proposed model once the system dynamics arises from the coupling of different physical phenomena. This fact decreases
model accuracy once that all uncertainties in parameter estimation comes into play.
Table 2.2 presents the initial conditions applied to the model. These conditions correspond to the ones observed during
the experimental tests described in detail further in this paper.
Input electrical current applied to the problem was a step current with amplitude of 1.98A, as illustrated by Fig. 2.5.
The experimental setup, Fig. 2.6, consists of the prototype presented by Fig. 2.1, where section P3 and P4 were rigidly
connected with each other, and section P1 and P2 connected to an inertial table. The relative rotation motion could only be
possible (by this configuration) between section P2 and P3, where a pair of SMA actuators were positioned. Angular
information was collected by a linear potentiometer (TRIMER 3386cw), and the resistance measured was related to a
specific angle by previous calibration. This information was filtered by an analog second order Butterworth filter with cutting
frequency of 1.5 Hz.
A data acquisition system from DSpace® (DS1103) outputted a control signal to the current source (WORNDER BOX
from LORD®) which applied the desired current in one of the actuators as shown by Fig. 2.5. This DAQ was also responsible
for collecting the filtered data from the potentiometer with a sampling frequency of 10 Hz.
It is important to notice that the system had to be actuated a number of times before achieving a constant replicable cycle.
Figure 2.7 presents the experimental results as well as the simulation results, both being actuated as described by Fig. 2.5,
and with the same initial conditions.
Fig. 2.4 (a) Hysteretic
transformation of martensite
into austenite for high
temperatures, b represents the
total martensite fraction in
the material microstructure.
(b) Creation of martensite
induced by stress
2 Nonlinear Dynamic Model and Simulation of Morphing Wing Profile Actuated by Shape Memory Alloys 25
2.4 Discussions and Conclusions
The proposed model has two different types of nonlinearities, one is associated with the geometric feature that is functional
of the angle and the other one is associated with the hysteretic nature of SMA transformations. The first one can be linearized
for small angles around the origin and even though it is not possible to obtain a linear expression of the moment arm as
a function of the angle (y). Based on the geometrical parameters presented in Table 2.1, one can easily plot these moment
arm functions for different angles. The maximum difference between these functions and a linear expression that interpolate
Table 2.2 Initial conditions
applied in the numerical
simulation
y sSMA1 TSMA1 bS SMA1 bT SMA1 ISMA1
�0.82 rad 0 Pa 301 K 100% 0% 0 A
o sSMA2 TSMA2 bS SMA2 bT SMA2 ISMA2
0 rad/s 0 Pa 301 K 0% 0% 0A
Fig. 2.5 Current input pattern
for experimental and
numerical simulations
Fig. 2.6 Experimental
prototype
Table 2.1 Model parameters applied in the numerical simulation
a1 b2 d1 ASMA CP MF AS CM L1 IZZ54 mm 13 mm 50 mm 0.203 mm2 857 J/kg K 313 K 328 K 8 MPa/K 130 mm 18.636 kg mm2
a2 c1 d2 AC L SMA1 MS AF CA m2 hC
17 mm 54 mm 13 mm 1595 mm2 103 mm 323 K 343 K 13 MPa/K 0.021 kg 30 W/K m2
b1 c2 s SCRIT s F
CRIT EA EM T1 R eL r50 mm 17 mm 100 MPa 170 MPa 67 GPa 26 GPa 301 K 3.75 O 2.4% 6450 kg/m3
26 C.T. Faria et al.
the initial and final moment arm value were less then 5% of the maximum moment arm. In practice this nonlinearity will not
be important and can be easily substituted by a linear expression. This nonlinearity can be used to increase model precision,
if necessary.
Hysteretic nonlinearity will play an important role in this problem. This phenomenon will be responsible for the change
in the system equilibrium points, resulting in a new camber without additional input (electrical current) after the desired form
is achieved. In this case, if the applied external forces and temperatures change does not exceed the transformation limits of
the actuators, both will maintain their new shape, shifting the equilibrium point to a new angular position. Dynamics
associated with the transformation process is also an important system feature once they will dictate the transition between
equilibrium points. This process is function of the stress state in the actuators and temperature (or indirectly it is function of
the electrical current).
Another important issue with the presented model is the large number of parameters, for the simulated problem 30
physical properties were used, after considering symmetry between actuators. With such a large number of parameters the
errors associated with the estimation of each one can distort the simulation result. Estimation problems are mainly
concentrated in the actuator properties, once they require a series of specialized equipment. Overcome of our limitation
was possible by executing another experiment, were a SMA wire was connected to a linear spring, in such conditions the
alloy properties could be determined by parameters adjustments to fit the SMA model to the experimental data. Further
investigation is necessary in other to identify the effects of perturbations in the model parameters.
Figure 2.7 is the comparison between experimental data and numerical simulation. In such problem a series of energy
conversions from different domains occur, i.e. electrical to thermal and thermal to mechanical, and also hysteretic behavior
are present. Modeling the interaction of these phenomena is not an easy task. The presented model was capable to capture all
these effects, getting the simulated model to a new equilibrium point. It is important to notice that both curves have a very
similar slope during the transformation process between 3 and 7 seconds, indicating a good agreement between model and
experiment in that case.
The mismatch between both curves can be explained by problems in parameter estimation. For example, if the
transformation temperature (AS) were smaller and the maximum recoverable strain (eL) was bigger both curves would
match during the transformation period (between 2 and 5 s) and the steady value. Numerical simulation also presents another
issue, a reverse transformation, such effect occurred by the cooling of the actuator and after the 22nd second a small recovery
can be noticed. Such phenomenon does not occur in practice, and this discrepancy can be explained by problem in the
parameters used during the simulation. Some noise can also be noticed in the experimental data despite the filtering process
applied.
It is worth to mention that the presented model can capture the most important phenomena associated with the dynamics
of the proposed morphing wing. Such design has an exciting potential to be applied in aerodynamic structures that demand a
Fig. 2.7 Experimental and
simulation results of the
morphing wing being actuated
in one direction
2 Nonlinear Dynamic Model and Simulation of Morphing Wing Profile Actuated by Shape Memory Alloys 27
low frequency change in shape to adapt itself for different stages during the mission. The main advantage is the fact that a
new equilibrium point is achieved by activating the shape memory effect, demanding none (or small amount) energy to keep
the aerodynamic shape.
Acknowledgments The authors are thankful to CNPq and FAPEMIG for partially funding the present research work through the INCT-EIE.
References
1. Akl W, Poh S, Baz A (2007) Wireless and distributed sensing of the shape of morphing structures. Sensor Actuat A Phys Amst 140(1):94–102
2. Hall JM (1989) Executive summary AFTI/F-111 mission adaptive wing. Ohio, Wright Research and Development Center Technical Report. ID.
WRDC-TR-89-2083
3. Natarajan A, Kapania RK, Inman DJ (2004) Aeroelastic optimization of adaptive bumps for yaw control. J Aircraft NY 41(1):175–185
4. Seigler TM, Neal DA, Bae JS, Inman DJ (2007) Modeling and flight control of large-scale morphing aircraft. J Aircraft NY 44(4):1077–1087
5. Faria CT (2010) Controle da variacao do arqueamento de um aerofolio utilizando atuadores de memoria de forma. Dissertation Uni Estadual
Paulista – UNESP. Print
6. Brinson LC (1993) One dimensional constitutive behavior of shape memory alloys: themomechanical derivation with non constant material
functions and redefined martensite internal variable. J Intell Mater Syst Struct Va 4(2):229–242
7. Gandhi F, Anusonti-Inthra P (2008) Skin design studies for variable camber morphing airfoils. Smart Mater Struct NY 17(1):15–25
8. Leo DJ (2007) Engineering analysis of smart material systems. Wiley, New York, p 331
9. Mathews JH, Fink KK (2004) Numerical methods using matlab, 4th edn. Prentice-Hall, New Jersey, pp 497–499
28 C.T. Faria et al.
Chapter 3
Environmental Testing and Data Analysis for Non-linear
Spacecraft Structures: Impact on Virtual Shaker Testing
Simone Manzato, Bart Peeters, Raphael Van der Vorst, and Jan Debille
Abstract This paper reports on the results of the environmental testing and data analysis that was performed on a satellite
structure that incorporates some typical structural non-linearities present in actual flight hardware. The project coordinator
EADS Astrium provided the bread-board satellite model and LMS International was responsible for the execution of the
environmental tests including sine and random tests at various vibration levels and in multiple directions. Next to a
presentation of the test results with an emphasis on the non-linear behaviour, advanced experimental modal estimation
technique were applied on the data.
Keywords Non-linear • Environmental testing • Vibration control • Modal analysis
3.1 Introduction
The European Space Agency (ESA) launched a research project on the advancement of mechanical verification methods for
non-linear spacecraft [1]. For the purpose of spacecraft structure design development and verification by analysis, the
structures are generally assumed to behave linear. However, experience has shown that various non-linearities might exist in
spacecraft structures and the consequences of their dynamic effects can significantly affect the design verification
procedures, in particular (1) the evaluation of flights loads with linearized models employed in launcher/satellite coupled
dynamic loads analyses (CLA); (2) the performance of dynamic verification tests; and (3) the demonstration that non-linear
dynamic effects have been well covered by the satellite verification tests.
In most cases the encountered structural non-linearities are associated with the presence of non-linear damping and/or
stiffness characteristics. Several sources of such non-linearities are well known: backlash, joint gapping, slippage when
friction forces are overcome, rattling, non-linear material characteristics, and discontinuities in force–deflection curves when
pre-loads are exceeded. ESA [1] shows a summary of typical spacecraft structural non-linearities associated with damping
and stiffness characteristics and their consequences on qualification testing (Table 3.1).
In the frame of this research project, EADS Astrium provided a bread-board satellite structure, incorporating some typical
structural non-linearities. LMS International was responsible for the execution of the environmental tests including sine and
random tests at various vibration levels and in multiple directions. This paper reports on these tests, the experimental study
of the non-linear behaviour, and the application of advanced experimental modal parameter estimation technique.
3.2 Structure and Tests
In this section, the representative structure will be introduced as well as the vibration test programme. For cost-saving
reasons, it was a requirement specified in the statement of work of the research project, to use existing hardware to the largest
extent possible. Therefore, Astrium proposed the SMALLSAT structure (Fig. 3.1 left), originally developed in 2000, to
S. Manzato (*) • B. Peeters • R. Van der Vorst • J. Debille
LMS International, Interleuvenlaan 68, B-3001 Leuven, Belgium
e-mail: [email protected]
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_3, # The Society for Experimental Mechanics, Inc. 2012
29
which non-linear elements representative of typical space hardware were added specifically for the present research project
[2]. The following non-linearities have been included:
• Dummy Instrument on Top of SMALLSAT: i.e. a non-linearity involving a large mass and mainly revealing itself as input-
level dependent damping for the first lateral modes.
• Instrument Isolation: The main non-linear effect is introduced by the isolation system located at the interface where this
element is connected to the rest of the structure, i.e. the so-called SASSA (Shock Attenuation System for Spacecraft and
Adaptor [3]) modules; see Fig. 3.1 right.• Suspended Actuator: A local non-linearity was added using an actuator dummy suspended by elastomeric elements,
called WEMS (Wheel Elastomer Mounting System). This element acts by varying the frequency at resonance for low-
levels input, while at higher levels it works as a mechanical stop.
The purpose of this test is not to qualify either the SMALLSAT structure or the non-linear devices which have already
been tested in previous sub-system test campaigns, statically and dynamically. The test will consist of two axis, vertical (Z)
Table 3.1 Overview of typical structural non-linearities in spacecraft structures and their consequences for the test [1]
Physical source of non-linearity Effect on dynamic responses Characterisation criteria
Consequences on
qualification process
Weak non-linearity (sliding at
interface)
Variation of damping
factor
Transmissibility variations Potential need for intermediate
run(s)
Small or no effect on
eigenfrequencies
Estimation of qualification levels
possible
Small gap or material nonlinearity Eigenfrequency shift Eigenfrequency variations Intermediate run is required
Large variations of amplitude Close monitoring of qualification
levels is required
Large gap or nonlinear stiffness Large perturbations of dynamic
test runs
Differences between “global”
and “fundamental” levels
Difficult control of excitation
input
High frequency content (“shocks”) Successful qualification is
jeopardized
Fig. 3.1 Smallsat installed on the shaker table at theAstrium testing site at Stevenage,UK (left) and SASSAmodule introducing non-linearities (right)
30 S. Manzato et al.
and lateral (X) direction, sine and random tests with the objectives to identify the effects of non-linear behaviour and to
identify at what level the non-linear effects impact the spacecraft behaviour. Following tests took place:
• Z-direction increasing and decreasing sine sweep 5–100 Hz: low level (0.1 g), intermediate level (0.4 g), intermediate
level (0.6 g), qualification level (1 g). After each level, in addition a low level (0.1 g) test took place allowing comparing
results with initial low-level test and checking integrity of structure. All tests took place at sweep rate of 2 Oct/min;
except for qualification level in which sweep rate of 4 Oct/min was used.
• Z-direction narrowband sine sweep 15–25 Hz at 0.6 g and sweep rate of 0.5 Oct/min for detailed non-linearity assessment.
• Z-direction random 0–2,000 Hz, 0.001 g2/Hz.
• X-direction increasing and decreasing sine sweep 5–100 Hz: low level (0.1 g), intermediate level (0.2 g), intermediate
level (0.4 g), qualification level (0.6 g). After each level, in addition a low level (0.1 g) test took place allowing comparing
results with initial low-level test and checking integrity of structure. All tests took place at sweep rate of 2 Oct/min.
Multiple entities took part in the tests: Astrium Ltd (test facility, mechanical engineering, quality inspection), Astrium
SAS Satellites (mechanical engineering, quality inspection), LMS (driving test facility, data post-processing), University of
Liege (data post-processing for non-linearity assessment), ESA (project sponsor). In [4], simulation data from the same
structure is used to verify non-linearity assessment methods.
3.3 Modal Parameter Estimation
The purpose of the processing is to apply experimental modal analysis estimation techniques to the test data, both for sine
and random excitation and for different level of excitation, with the following objectives:
• Extract the modal parameters using different processing techniques.
• Identify from the processed data at what level the non-linear effects impact the spacecraft.
There is quite some literature available on the topic of using qualification test data for modal parameter estimation. For
instance, F€ullekrug and Sinapius of the German Aerospace Center (DLR) carried out thorough investigations on the problem
of modal parameter identification from base-driven vibration data [5]. If only the outputs are measured, i.e. accelerations at
the interface between the shaker and the structure and accelerations of the structure, the eigenfrequencies, damping ratios
and mode shapes of the fixed interface structure can be obtained. If a Force Measurement Device (FMD) is used to measure
also the six DOF force input between shaker and structure, both the free and fixed interface modes can be obtained from
(multi-axial) base excitation test (Table 3.2).
In [6], the possibilities to integrate both the modal survey and the vibration qualification test are investigated. Among
other things the use of Frequency Response Functions (FRFs), transmissibilities or spectra as primary data in the modal
parameter estimation process is discussed (Table 3.3). In [7], an approach to non-linear Experimental Modal Analysis was
proposed that starts from base-excitation data. Some recent evolutions in the field of modal parameter estimation are
described in [8, 9]. In this section, this new technology will be applied to the data from the bread-board model tests. The
additional challenge, next to the fact that modal parameters need to be extracted from a test which was not designed to be a
modal test, is that the tested structure exhibits non-linearities.
Table 3.2 Relation between measured quantities and interpretation (boundary conditions) of the identified modal parameters
Measured quantities Fixed interface structure Free structure
Interface and structure accelerations (output-only) Eigenfrequencies, damping ratios, mode shapes –
+ Interface forces (+ inputs) + Modal participation factors All modal parameters
Table 3.3 Interpretation of resonances of FRFs, transmissibilities and cross-spectra in terms of structural modes
Type of base excitation
Measurement functions
FRFs Transmissibilities Cross spectra
White noise (flat force spectrum) Free modes Fixed interface modes Free modes
Shaped noise such that the control acceleration has a flat spectrum Free modes Fixed interface modes Fixed interface modes
3 Environmental Testing and Data Analysis for Non-linear Spacecraft Structures: Impact on Virtual Shaker Testing 31
Usually, modal methods require force measurements that are combined with structural response measurements to obtain
Frequency Response Functions. The issue of not having any information about the actual load conditions during the carried
out environmental tests can be solved by properly processing the acquired data. The results, in terms of transmissibilities
(ratio between output and input accelerations), can then be used in standard modal analysis tools to obtain the modal
parameters of the structure. A first method to obtain the transmissibilities is to use the directly available data during the sine
sweep test. When defining the measurement process, four different amplitude estimators can be selected: peak, RMS,
average and harmonic. In order to obtain also phase information, the latter method is here applied. This filter method, which
works completely in the time domain, offers the best estimate of the amplitude of the fundamental frequency and provides
an excellent harmonic rejection. It is also to be used when noise or harmonic acceleration levels should be filtered out as
much as possible.
During the measurement, also the time histories were acquired with an oversampling factor of 5. For most of the
measurement runs, this led to a sampling frequency of 1,600 Hz. The transmissibilities are computed from the time data
using the so-called H1-estimator, which boils down in this case to the ratio of the cross spectrum between response and input
accelerations SyuðoÞ and the power spectrum of the input acceleration SuuðoÞ:
H1ðoÞ ¼ SyuðoÞSuuðoÞ ; SyuðoÞ ¼ 1
Nb
XNb
b¼1
YðbÞðoÞ UðbÞðoÞ� ��
(3.1)
where Nb is the number of blocks and superindex b denotes the block index. The so-called weighted averaged periodogram
(also known as modified Welch’s periodogram) method allows obtaining the cross- and power spectra from time histories. It
is traditionally used in combination with random data, but may also be used in case of sine sweep data. Welch’s method
consists of cutting the time data in possibly overlapping blocks, applying a weighting (e.g. a Hanning window) to reduce
leakage, and then average the product of the Fourier transform of each block UðbÞðoÞ; YðbÞðoÞ with complex conjugated
Fourier transforms; see (3.1). Figure 3.2 compares online and post-processed transmissibilities from the sine control test.
The quality of the transmissibilities obtained by applying the Welch’s method to sine sweep measurement is typically much
higher than those estimated online.
Also in the case of a random test a distinction can be made between online data and post-processed data. Although in both
cases, the transmissibility estimation is the same (H1 estimator), there are some differences: (1) in the online method, the
cross-spectra estimations are based on a combination of linear and exponential averages, whereas typically for the post-
processing of the time data, just linear averages are used and (2) there is a difference in frequency resolution of the obtained
transmissibilities. Indeed, to have a faster control update, a coarse spectral resolution is typically defined in the online
process: faster control update means lower acquisition time means coarser frequency resolution. On the other hand, for
further (modal) processing, a finer frequency resolution may be desired, and therefore, the transmissibilities acquired online
are not optimal. Figure 3.3 compares online and post-processed transmissibilities from the random control test. The post-
processed results have a much finer resolution.
FRF (Harmonic) 211Z: + Z/013Z:+Z Online - upFRF (Harmonic) 211Z: + Z/013Z:+Z Online - downFRF 211Z:+Z/013Z:+Z H1 estimator
−180.00
180.00
0.00
2.20
Pha
seI
Am
plitu
de
80.005.00 Hz
Fig. 3.2 Sine control—
comparison between
transmissibility estimates:
online during run-up, online
during run-down, H1
estimator from post-processed
time data
32 S. Manzato et al.
The PolyMAX modal parameter estimation method is applied to all the available data. Depending on the direction of
application of the load, the load level, the sweep direction and transmissibilities calculation method, modal parameter
estimates may differ. A detailed discussion of all the results can be found in [10]. In this paper, a synthesis of the results is
presented.
An example of a PolyMAX stabilization diagram can be found in Fig. 3.4. Figure 3.5 shows some typical mode shapes
retrieved from the sine control data and Fig. 3.6 shows the agreement between the measured transmissibilities and the
synthesized ones (i.e. obtained from curve-fitting the data with a modal model). In general, the modal parameter estimation
results are of high quality: a clear stabilization diagram is obtained, the mode shape animations look physically sound,
frequency and damping ratio estimates are within (FE) expectations, and the data is well fitted by the modal model.
5.00−70.00
20.00
l dB
500.00Hz
FRF 123Y:-Y/013Z:+Z H1 OnlineFRF 123Y:-Y/013Z:+Z H1 Post-processing
Fig. 3.3 Random control—comparison between transmissibility estimate online versus post-processing of the time data
1.46
232e-35.00 Linear
Hz94.5
43424140393837363534333231302928272625242322212019181716151413121110987
/A
mpl
itude
Fig. 3.4 PolyMAX stabilization diagram for the X-direction 0.4 g sweep data
3 Environmental Testing and Data Analysis for Non-linear Spacecraft Structures: Impact on Virtual Shaker Testing 33
However, some of the modes that are clearly present at a certain excitation level are not present at another level. This is
probably caused by the following reasons: (1) in some cases the level of excitation is not high enough to properly excite the
mode and (2) due to the presence of non-linearities, as will be explained more in detail in the following, the parameter
estimation method increases the number of identified modes around such a non-linearity in an attempt to better fit the
measured transmissibilities. In Fig. 3.6 (bottom), it can be observed that the data between 20 and 30 Hz is not fitted very well.In this frequency region and for this excitation level (0.6 g in Z-dir), the transmissibility estimates are heavily distorted due
to non-linearity in the data.
Following concluding remarks can be made on the modal parameter estimation:
• The results from the online estimation are of sufficient quality to perform a quick check of the modal properties
immediately after performing a vibration control test on the structure.
• As a general rule, however, it can be observed that the results obtained from post-processing the time histories (i.e. using
the transmissibility H1 estimates) are better than those obtained using the online processing (i.e. transmissibilities
estimated as ratio of harmonic spectra).
• Some differences between the sweep up and down can be immediately observed, but they will be discussed more in detail
in the following section as these differences can be seen as indicators of non linear behavior.
Fig. 3.5 Some typical mode shapes obtained using sine sweep data from both X and Z direction: (left) WEMS mode; (middle) top instrument
mode; (right) global torsion mode
Fig. 3.6 Quality assessment of modal curve fitting: comparison of sum of measured and synthesized transmissibilities from online run-up data:
(top) Z-dir 0.4 g; (bottom) Z-dir 0.6 g. Frequency range from 5 till 100 Hz is shown
34 S. Manzato et al.
3.4 Non-linearity Assessment
After the modal parameters have been extracted, some further processing techniques can be applied to the data in order to
highlight the presence of non-linear behaviour. It can also be useful to understand at which excitation the non-linearities
become apparent. The following processing will be applied to check for non-linearities in the acquired sine sweep data:
• Visual analysis of time histories and spectral maps.
• Comparison between transmissibilities measured at the same point for positive and negative sweep.
• Comparison between transmissibilities at same point for different excitation level.
• Tracking eigenfrequencies identified at different excitation levels.
One of the easiest methods to assess the presence of non-linearities is to analyze the time histories to check for example,
the presence of peaks in the signal when the excitation level increases. Another possibility is to perform a time-frequency
analysis and present the results as a spectrogram: the activation of non-linearities can be recognized from the presence of
higher order harmonics of the excitation frequency or from time intervals in which the whole frequency band is excited. The
example given in the following concerns the second WEMS mode around 20–22 Hz. Non-linear behavior is expected due to
the presence of the elastomeric support and the mechanical stop. In Fig. 3.7, the spectrograms of the control channel at the
shaker–structure interface and of a sensor at the satellite side of the WEMS are represented. Both an intermediate (0.4 g) and
the qualification level are shown. The range of the dB scale of the colormaps is adapted to the excitation as to better highlight
the relative importance of higher-order harmonics.
As can be seen in Fig. 3.7 (top), the non-linearities do not have a major impact on the control channel. However,
occurrence of non-linearity is clear from WEMS sensor (Fig. 3.7, bottom). At the time instant when the excitation reaches
roughly 22 Hz, an horizontal line appears, meaning that at that instant all frequency are excited. In particular, this
phenomenon became clearer as the excitation level increases and is related to the elastomers in the WEMS, which are no
more able to absorb enough energy and impacts are transmitted to the structure. Moreover, in the frequency region
Fig. 3.7 Spectrogram with time-frequency analysis of Z sine sweep data: (top) control accelerometer at shaker table; (bottom) response
accelerometer at WEMS; (left) intermediate level 0.4 g; (right) qualification level 1 g
3 Environmental Testing and Data Analysis for Non-linear Spacecraft Structures: Impact on Virtual Shaker Testing 35
between 8 and 10 Hz, the higher order harmonics of the excitation frequency are clearly excited (Fig. 3.7, bottom-right).The behavior of the structure excited at 22 Hz can be further investigated by analyzing directly the time response at the
different excitation levels (Fig. 3.8). By looking at the plots, it is immediately clear how, as the excitation level increases,
the response is distorted and, at 0.6 g, sharp peaks appear in the response. Moreover, it can be observed how, starting from
the 0.4 g level, the acceleration are distorted and no more centered around zero. According to [11], such a response can be
related to a bilinear stiffness characteristic of the element with some initial preloading (in this case the weight of the
suspended actuator module).
Another indicator of non-linearity in the system is to check for a “jump” in the spectral response of some degrees of
freedom when sweeping through some resonances with different sweep directions. This effect was for instance detected
when observing the WEMS degrees of freedom during the sine control test in X direction. In Fig. 3.9, the run-up and run-
down transmissibilities for different excitation level are presented, to be able to identify which excitation level activates this
effect. At low excitation the transmissibilities from both directions coincide, whereas with increasing levels, the discrepancy
grows.
Linear behaviour can also be observed when comparing transmissibilities at different excitation level. Indeed, for
linear mechanical structures, scaling the magnitude of the excitation force will result in the same scaling of the responses
and hence, “linear” transmissibilities are independent from the excitation level. However, by observing the response of
sensors located on the WEMS or SASSA module, it is clear that, as the level increases, there is both a shift in the
frequency and the amplitude of peaks corresponding to natural frequencies (Figs. 3.10, 3.11), as the level increases, the
peaks are shifted to lower values of frequencies and amplitudes. Non-linearities are apparent for the modes around 10, 36,
55 and 89 Hz. It can be verified that these frequencies correspond to identified modes related to the top instrument and
SASSA module.
Figure 3.11 represents a sensor location at the WEMS: for most of the modes, increasing the input level corresponds to a
decrease in the frequency and the amplitude. An exception to this rule is the peak at 8 Hz, where, as the level increases, the
peak appear to be shifted at higher frequency: this can be explained by assuming that the elastomeric element supporting the
component has a hardening characteristic and, as the excitation level increases, acts as a mechanical stop.
2.10
67.58 68.65s
69:213Y:+Y 0.1 g69:213Y:+Y 0.4 g69:213Y:+Y 0.6 g
Rea
lg
−2.10
Fig. 3.8 Time histories from WEMS channel around the resonance at 22 Hz for different Z-dir excitation levels
15.00 15.00
5.00 5.00 5.00Hz Hz30.00 30.00
time run-uptime run-down
time run-uptime run-down
time run-uptime run-down
Hz 30.00
−35.00 −35.00
/ dB
/ dB
15.00
−35.00
/ dB
Fig. 3.9 WEMS transmissibilities of X-dir sine sweep at different excitation levels: (left) 0.2 g; (middle) 0.4 g; (right) 0.6 g. Sweep up and sweepdown directions are compared with each other
36 S. Manzato et al.
Finally, the effect of frequency shifting due to non-linearities can also be visualized by comparing the estimated
eigenfrequencies at different excitation levels. Figure 3.12 shows the frequency values for the same modes at different
excitation levels. It is clear how, for the analyzed modes, the frequency is decreasing as the level of excitation increases,
indicating a non-linear behavior of the local stiffness characteristics.
15.00
11.07
11.07
5.00
36.32 56.45
56.45
FRF 321Y:-Y/013Z:+Z time run- 01ZFRF 321Y:+Y/013Z:+Z time run- 02ZFRF 321Y:+Y/013Z:+Z time run- 04ZFRF 321Y:+Y/013Z:+Z time run- 06Z
88.56
88.56
100.00Hz
36.32
180.00
−180.00
−40.00
/ dBP
hase
Fig. 3.10 Transmissibilities for SASSA channel at different excitation levels in Z direction
70
60
50
40
30
20
10
00.1g
57.988 55.373 55.26554.115 32.895
32.623
31.308
30.428
32.59433.9935.07837.062
0.4g 0.6g
Excitation level
Evolution of natural frequency with load level Evolution of natural frequency with load level
Fre
qu
ency
[H
z]
Fre
qu
ency
[H
z]
Excitation level
1g 0.1g 0.4g 0.6g 1g
33.5
33
32.5
32
31.5
31
30.5
30
29.5
29
Fig. 3.12 Evolution of natural frequency with excitation level. (left) Two modes identified from processing of the Z-dir sine sweep transmissi-
bilities; (right) mode from X-dir sine sweep
20.00
5.00
8.67 20.50 33.92 45.10
FRF 221Y:-Y/013X:-X time run-up 01XFRF 221Y:-Y/013X:-X time run-up 02XFRF 221Y:-Y/013X:-X time run-up 04XFRF 221Y:-Y/013X:-X time run-up 06X
45.1033.9220.508.67
Hz 100.00
180.00
−180.00
Pha
se
−70.00
/ dB
Fig. 3.11 Transmissibilities for WEMS channel at different excitation level in X direction
3 Environmental Testing and Data Analysis for Non-linear Spacecraft Structures: Impact on Virtual Shaker Testing 37
3.5 Conclusions
In this paper, advancedmodal parameter identification techniques have been applied to the acquired data from the bread-board
model tests performed at the Astrium facility in Stevenage, UK. Usually, modal parameter estimation methods require
measurements of all applied forces (the inputs to the system) in order to compute FrequencyResponse Functions. In the present
case, no interface forces could be measured, and transmissibilities between satellite responses and shaker interface
accelerations have been used as primary data in the identification. For both sine sweep and random excitation, two methods
to compute the transmissibilities are compared: one that is performed online during the control test and the otherwhich is based
on a post-processing of the time histories that have been acquired in parallel with the same measurement system during the
control test. It is observed that the post-processed transmissibilities provide clearer stabilization diagrams and amore accurate
estimation of the parameters. It should be noted however, that the online estimated data are accurate enough to provide a very
fast verification of the behavior of the system immediately after the test is performed. Finally, an overview is given of
methodologies to assess non-linear behavior. By combining the results of the proposed methods, it is possible to clearly
identify a non-linearity, understand some of its properties and identify the level of excitation at which it is activated. In general,
for this processing, sine sweep data are used.
Interesting to mention is that the observed satellite non-linearities presented in this paper can also be predicted using a so-
called “virtual shaker testing” approach, consisting of a coupled electro-mechanical shaker model, a vibration controller
software model, and a structural model with linear sub-components that may be interconnected with non-linear elements.
Betts et al. [12] and Ricci et al. [13] discuss this virtual shaker testing approach and demonstrate the high degree of realism
that can be obtained from such coupled shaker-controller-structure simulations.
References
1. ESA European Space Agency directorate of technical and quality management (2007) Advancement of mechanical verification methods for
non-linear spacecraft structures. Statement of work, reference TEC-MCS/2007/1558/ln/AN, June 20072. Russell AG (2000) Thick skin, faceted, CFRP, monocoque tube structure for smallsats. In: Proceedings of the European conference on
spacecraft structures, materials and mechanical Testing, ESTEC, Noordwijk, 2000
3. Camarasa P, Kirylenko S (2009) Shock attenuation system for spacecraft adaptors (SASSA) final verification. In: Proceedings of
theWworkshop on spacecraft shock environments and verification, ESTEC, Noordwijk, 2009
4. Kerschen G, Soula B, Vergiaud JB, Newerla A (2011) Assessment of non-linear system identification method using the SmallSat spacecraft
structure. In: Proceedings of IMAC 29, Jacksonville, 2011
5. Sinapsius JM (1996) Identification of free and fixed interface normal modes by base excitation. In: Proceedings of IMAC 14, Dearborn, 1996
6. Peeters B, Van der Auweraer H, Guillaume P (2003) Modal survey testing and vibration qualification testing: the integrated approach. J IEST
46:110–118
7. Link M, B€oswald M, Laborde S, Weiland M, Calvi A (2010) An approach to non-linear experimental modal analysis. In: Proceedings of IMAC
28, Jacksonville,2010
8. Peeters B, van der Auweraer H, Guillaume P, Leuridan J (2004) The polyMAX frequency-domain method: a new standard for modal parameter
estimation? Shock Vib 11:395–409
9. LMS International (2011) LMS test lab modal analysis. Leuven, www.lmsintl.com, 2011
10. Manzato S, Peeters B (2010) WP4240: application of advanced experimental modal estimation techniques to test data. Technical report issued
under ESA(2007), LMS International, Leuven, 2010
11. Boeswald M, Link M, Meyer S, Weiland M (2002) Investigation on the non-linear behaviour of a cylindrical bolted casing joint using high
level base excitation test. In: Proceedings of the ISMA 2002, Leuven, 2002
12. Betts JF, Vansant K, Paulson C, Debille J (2008) Smart testing using virtual vibration testing. In: Proceedings of the 24th aerospace testing
seminar (ATS), Manhattan Beach, 2008
13. Ricci S, Peeters B, Fetter R, Boland D, Debille J (2009) Virtual shaker testing for predicting and improving vibration test performance.
In: Proceedings of IMAC 2009, Orlando, 2009
38 S. Manzato et al.
Chapter 4
Using Impact Modulation to Detect Loose Bolts in a Satellite
Janette Jaques and Douglas E. Adams
Abstract Quickly-assembled, on-demand satellites are being developed to meet the needs of responsive space initiatives.
The short testing times and rapid assembly procedures associated with these satellites create the need for an efficient method
to verify the satellite’s structural integrity. In particular, the ability to identify loose bolts within the satellite structure is of
interest. In this work, Impact Modulation is explored as a possible means of detecting loose bolts. A Torque Index metric is
developed which was able to identify the presence of loose bolts within a satellite panel without the use of historical data by
using a dot product analysis to quantify the difference in response amplitudes at the natural frequencies and those at the
sideband frequencies across an array of impact locations.
4.1 Introduction
Today’s satellites are often designed to perform specific tasks and can take months, even years, to develop, assemble, test,
and launch [1]. New efforts are underway to develop modular satellites that can accomplish a wide range of tasks and can be
ready for launch within days of when their need is established [2]. These quickly-built satellites have unique issues due to
time constraints and the variability of their geometry which are not typically associated with traditional satellites. One such
issue is the ability to quickly assure the structural integrity of the satellite after rapid assembly in order to make certain that it
will survive the launch environment.
The goal of this work is to develop a method to diagnose the condition of the bolted joints of the satellite. One requirement
for this method is that the method be insensitive to changes in the geometry of the satellite, because the satellites configuration
of each satellite depends on the requirements of the mission. Methods that require extensive baseline readings are not
applicable because changes in the geometry of the satellite would require the time consuming task of collecting new baseline
data. One method that meets this criteria is Impact Modulation. Impact Modulation (IM) is a nonlinear, vibrations-based
method which uses a combination of low and high frequency excitations to interrogate structures for damage. This work seeks
to use IM testing to identify the presence of loose bolts in a satellite structure without the use of baseline, or historical, data.
Other researchers have also addressed the issue of loose bolt detection using methods that range from laser vibrometry [3]
to wave propagation [4, 5], although the majority of these works depend on detailed analytical models or access to extensive
baseline data sets. A small number of works in the literature develop methods which do not rely on baseline data or an
analytical model to assess the level of torque on the bolts. Milanese et al. [6] developed a method which looked for frequency
content in the measured strain response of a test specimen above the maximum excitation frequency to indicate that the bolt
was loose. A damage index was developed based on probabilistic analysis that was proven effective in identifying loose bolts.
In the method presented by Nichols et al. [7], surrogate baseline data was generated from the response of a structure by using
the iterative amplitude adjusted Fourier transform method (IAAFT). IAAFT operates under the assumption that a healthy
structure is linear and a damaged structure is nonlinear. The linear part of the response is extracted from the full data and
compared to the full data itself. The level of nonlinearity present in the response is used as an indicator of the presence of loose
bolts within the system. Finally, in [8], Amerini and Meo use a technique similar to IM called Vibro-Acoustic Modulation
(VM). For a two-plate, one-bolt structure, they were able to detect torque loss by measuring the difference in the amplitude of
J. Jaques (*) • D.E. Adams
Purdue Center for Systems Integrity, Purdue University, 1500 Kepner Road, IN 47905, Lafayette, USA
e-mail: [email protected]
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_4, # The Society for Experimental Mechanics, Inc. 2012
39
response at the actuator frequency and the average amplitude of the first sidebands. To quantify the torque loss, they fit a
hyperbolic tangent curve to the data. There are numerous examples in the literature of using IM to detect cracks in various
materials and structures, but up to this point, IM has not been applied to loose bolt detection. The work presented in this paper
demonstrates the applicability of IM to detect loose bolts in a satellite structure.
4.2 Experimental Setup and Test Procedure
Testing was performed on a prototype Plug and Play satellite panel provided by the Air Force Research Laboratory (AFRL)
in Albuquerque, New Mexico. The aluminum panel, shown in Fig 4.1a, is a 31 34inch square and has threaded bolt holes
evenly spaced at 2 in. intervals in both directions. While the top side of the panel is smooth and plate-like, the underside is
quite complex. Figure 4.1b shows that the underside has been designed with channels for cable management and with space
for interior components.
The goal of the experiments that were performed on the satellite panel was to determine if IM could identify loose bolts in
the connection of an external component without reference data. To simulate an external component, a 4� 4� 38in.
aluminum plate with four through holes drilled to match the bolt hole pattern on the panel was machined. This plate was
bolted on to the face of the satellite panel, as shown in Fig. 4.1a. Four 8–32 bolts were used to secure the plate to the panel.
Throughout the experiments presented here, the torques on all four bolts were kept at the same level. Three torque levels
were used: 24, 2, and 1 in �lb. According to data provided by the AFRL, 24 in �lbs is considered proper bolt torque for
external component connections.
The panel was equipped with ten PCB356A32 100 mV/g triaxial accelerometers and a PI P-010.10P piezo stack actuator
whose locations are shown in Fig. 4.2. An 8 �8 grid of impact locations was marked with 4 in. spacing between points.
A PCB 086C01 impact hammer was used to impact the satellite and an Agilent E8408A VXI data acquisition system used to
drive an actuator and collect sensor data. In addition a power amplifier was added to the setup to increase the amplitude range
of the actuator. Four bolts that were screwed into bolt holes at each corner on the underside of the panel provided support to
the panel during testing, which created pin-like boundary conditions. IM was performed by impacting the panel at each
impact location while simultaneously exciting the panel with a high frequency (7,500 Hz) signal produced by the actuator.
Acceleration time histories for each of the 64 impacts were collected from each of the ten sensors. The time data was
windowed using a Tukey window with a ratio of tapered section to constant section of 0.5. The windowed data was then
transformed into the frequency domain for analysis via the Discrete Fourier Transform algorithm in MATLAB. In the
frequency domain, the results of the interaction between the modal response, or the response due to the impact, and the high
frequency response, or the response due to the actuator input, are response peaks at frequencies which are linear
combinations of the actuator frequency and the natural frequencies. These peaks are called sidebands. The analysis
procedure which is presented below is based on the amplitudes of the sidebands in the response spectra.
The first step in the analysis was to pick out the mode of vibration to analyze. After analyzing the response specrta (not
shown here) from initial IM tests at several impact locations, it was determined that the 72 Hz mode would be used for the
analysis because of its strong response at both the low frequency and the corresponding sideband frequencies. The sidebands
Fig. 4.1 Satellite panel. (a) Top view of the satellite panel. (b) Underside of the satellite panel
40 J. Jaques and D.E. Adams
that correspond to the 72 Hz mode occur at 7,428 Hz (7,500–72 Hz) and at 7,572 Hz (7,500 + 72 Hz). Next, the amplitudes
of the response at 72, 7,428, and 7,572 Hz were recorded for each of the 64 IM tests (one test per impact location). Figure 4.3
shows the shapes that result when these amplitudes are plotted as a function of impact location for the case when the bolts are
tightened to 24 in �lbs. Qualitatively, the shapes appear to correlate very well. During initial testing, it was noted that when
the bolts were loosened, the shape of the response at the sideband frequencies changed dramatically while the shape of the
response at the natural frequency showed little change. Based on this observation, a metric was developed to quantify the
correlation between the shape of the response at the natural frequency and the shapes at the sideband frequencies. Strong
correlation between the shapes indicates that the bolts are tight. Weak correlation between the shapes indicates that loose
bolts are present. The metric, called the Torque Index, (TI) is calculated by averaging the sum of the dot products of the
derivatives (slopes) of each row and column of the response matrices as follows:
TI ¼ TIL þ TIR2
where
TIL ¼XNrow
ii¼1
ðMrowiiÞslope � ðSBLrowii
ÞslopekðMrowii
ÞslopekkðSBLrowiiÞslopek
þXNcol
jj¼1
ðMcoljjÞslope � ðSBLcoljjÞslopekðMcoljjÞslopekkðSBLcoljjÞslopek
" #=ðNrow þ NcolÞ (4.1)
where M is the amplitude of the response at the natural frequency, SBL is the amplitude of the response at the left sideband
frequency, and Nrow and Ncol are the number of rows and columns in the response matrices. TIR is calculated using SBR, theamplitude of response at the right sideband frequency, in place of SBL. As shown in [9], the dot product of the slope of two
x
xxxxxxxx
xxxxxxxxxxxxxxxx
xxxxxxxx
xxxxxxxx
xxxxxxx xxxxx
xxxxxxxx
xxxxxxxx1 2 3 4 5 6 7 8
9 10 12 13 14 15 16
17
25
33
41
49
57
= Impact Location
= Sensor Location
= Actuator Location
= Component Location
##
11
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26 28 29 30 31 3227
34 36 37 38 39 4035
42 44 45 46 47 4843
50 52 53 54 55 5651
58 60 61 62 63 6459
2019
Impact Amplitude: 600-675 N Actuator Frequency: 7,500 HzActuator Amplitude: 15 V Boundary Conditions: PinnedBolt Torque: Case All Bolts
24 24 in·lbs2 2 in·lbs1 1 in·lb
Fig. 4.2 Setup and parameters for the satellite panel IM testing
Response atNatural Frequency
Response atMode Sideband Frequencies
1
Fig. 4.3 Scaled response amplitudes versus impact location for IM tests for Case 24
4 Using Impact Modulation to Detect Loose Bolts in a Satellite 41
mode shapes better accentuates differences in the shapes as compared to using the shapes themselves. Note that a normalized
dot product is a measure of the orthogonality of two vectors and can take on a value between 0 and 1. A value of 1 indicates
that the vectors are very well correlated. A value of 0 indicates that the vectors have no correlation. Therefore, a TI valuenear 1 indicates thatMslope and SBLslope and SBRslope are nearly identical, indicating that all the bolts in the structure are tight.
A lower value would indicate that one or more of the bolts is loose.
After performing an IM test at each of the 64 impact locations, TIwas calculated. This procedure was repeated for each ofthe three torque cases.
4.2.1 Results
Figure 4.4 shows TI for the three torque cases. The values for Case 1 (0.764), Case 2 (0.826), and Case 24 (0.924) show the
expected trend of a decrease in the TI value with a decrease in torque. In addition, the value of TI for Case 24 can be
considered close to 1, the maximum possible TI value. The other TI values can be considered much less than 1. These
distinctions indicate the possibility of using IM to detect loose bolts in a structure without the use of historical data, because
the upper threshold for TI is independent of the structure being evaluated.
4.3 Conclusions
In this work, the effectiveness of using IM to detect loose bolts on a satellite structure was demonstrated. A metric called the
Torque Index, TI, was defined to quantify the difference between the shape of the response amplitudes across impact
locations at a natural frequency and the shapes at the corresponding sideband frequencies. An important characteristic of TIis that its value is limited to the range between zero and one, eliminating the need for a historical reference data. It was shown
that the value of TI was relatively close to one when all the bolts within the satellite structure were tight. The TI valuedropped over 10% when four connection bolts were loosened.
Case
Tor
que
Inde
x(T
I)
24 in·lb 2 in·lb 1 in·lb0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 4.4 Torque index for each of the three torque cases
42 J. Jaques and D.E. Adams
References
1. Arritt BJ, Buckley SJ, Ganley JM, Welsh JS, Henderson BK, Lyall ME, Williams AD, Prebble JC, DiPalma J, Mehle G, Roopnarine R (2008)
Development of a satellite structural architecture for operationally responsive space. In: Proceedings of the international society for optical
engineering (SPIE), vol 6930, 2008
2. Bhopale A, Finley C (2009) How ORS will answer the 7-day tier-2 challenge. In: 7th responsive space conference, Los Angeles, 27–30
Apr 2009
3. Driesch P, Mann JA III, Gangala H (1996) Identification of loose bolts using wavenumber filtering of low frequency vibration data.
In: Proceedings of the national conference on noise control engineering, vol 2, 1996, pp 769–774
4. Lovell PA, Pines DJ (1998) Damage assessment in a bolted lap joint. In: 5th annual SPIE smart materials and structures symposium: smart
buildings, bridges, and highways, vol 3325, 1998, pp 112–126
5. Reynolds WD, Doyle D, Arritt B (2010) Active loose bolt detection in a complex satellite structure. In: Health monitoring of structural and
biological systems 2010, 7650(1):76500E, San Diego, 2010
6. Milanese A, Marzocca P, Nichols JM, Seaver M, Trickey ST (2008) Modeling and detection of joint loosening using output-only broad-band
vibration data. Struct Health Monit 7(4):309–328
7. Nichols JM, Trickey ST, Seaver M, Motley SR, Eisner ED (2007) Using ambient vibrations to detect loosening of a composite-to-metal bolted
joint in the presence of strong temperature fluctuations. J Vib Acoust 129(6):710–717
8. Amerini F, Meo M (2011) Structural health monitoring of bolted joints using linear and nonlinear acoustic/ultrasound methods. Struct Health
Monit. doi:10.1177/1475921710395810
9. Pandey AK, Biswas M, Samman MM (1991) Damage detection from changes in curvature mode shapes. J Sound Vib 145(2):321–332
4 Using Impact Modulation to Detect Loose Bolts in a Satellite 43
Chapter 5
Nonlinear Modal Analysis of the Smallsat Spacecraft
L. Renson, G. Kerschen, and A. Newerla
Abstract Non-linear elements are present in practically all spacecraft structures. The assumption of a (quasi-)linear
structure is nevertheless adequate for structural analyses and design verification purposes in those cases where these
structural non-linearities are relatively weak or not substantially activated by the mechanical environments encountered
during the launch or during ground testing. However, when significant non-linear effects in spacecraft structures are no
longer negligible then linear modal analysis will not be able to handle non-linear dynamical phenomena in an adequate
manner: the development of a non-linear analogue of linear modal analysis becomes an urgent and important issue. The
objective of this paper is to show that nonlinear normal modes (NNMs) represent a useful and practical tool in this context.
A full-scale spacecraft structure is considered and is modeled using the finite element method. Its NNMs are computed using
advanced numerical algorithms, and the resulting dynamics is then carefully analyzed. Nonlinear phenomena with no linear
counterpart including nonlinear modal interactions are also highlighted.
Keywords Nonlinear dynamics • Modal analysis • Nonlinear normal modes • Space structure
5.1 Introduction
Spacecraft structures are subjected to severe dynamic environments during the launch phase. In order to ensure the structural
integrity of the spacecraft (SC) and the payload (PL) items minimum frequency requirements are usually defined for the SC
in order to avoid dynamic coupling between the main frequency ranges of the launch vehicle (LV) excitations and the SC
fundamental linear normal modes (LNMs), i.e., those modes where the effective modal masses are important. The launcher
authority might however accept a non-compliance with the requirements in those cases where the SC eigenmode represents a
“localized” dynamic effect with only a small effective mass involved.
From a linear point of view, these requirements avoid potentially disastrous coupling and energy exchanges between the
LV and the SC. However, for nonlinear structures, this article will show on a representative SC structure developed by
EADS Astrium, the SmallSat, that these requirements might not be sufficient. In particular, the excitation of global SC
structure modes and PL modes involving local nonlinear effects is presented.
The paper is organized as follows. In Sect. 5.2, a brief review of nonlinear normal modes (NNMs) is achieved. In Sect. 5.3
the SC structure and its finite element model are described. Themodeling of the nonlinearities is also discussed. In Sect. 5.4, a
linear modal analysis of the SC is performed and employed as an introduction to the nonlinear modal analysis of Sect. 5.6.
Section 5.5 shortly presents the algorithm used for the computation of nonlinear normal modes. Finally, Sect. 5.6 presents and
discuss different NNMs of the structure.
L. Renson • G. Kerschen
Space Structures and Systems Laboratory (S3L), Structural Dynamics Research Group, Department of Aerospace
and Mechanical Engineering, University of Liege, Liege, Belgium
e-mail: [email protected]
A. Newerla
European Space Agency (ESTEC), Noordwijk, The Netherlands
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_5, # The Society for Experimental Mechanics, Inc. 2012
45
5.2 Review of Normal Modes for Nonlinear Systems
A detailed description of NNMs and of their fundamental properties (e.g., frequency-energy dependence, bifurcations, and
stability) is given in [1, 2] and is beyond the scope of this paper. For completeness, the main definition of a conservative
NNM is briefly reviewed in this section.
The free response of discrete mechanical systemswithN degrees of freedom (DOFs) is considered, assuming that continuous
systems (e.g., beams, shells, or plates) have been spatially discretized using the FE method. The equations of motion are
M €xðtÞ þKxðtÞ þ fnl xðtÞf g ¼ 0 (5.1)
where M and K are the mass and stiffness matrices, respectively; x and €x are the displacement and acceleration vectors,
respectively; fnl is the nonlinear restoring force vector.
Targeting a straightforward nonlinear extension of the concept of LNMs, Rosenberg defined an NNM motion as a
synchronous periodic oscillation. This definition requires that all material points of the system reach their extreme values
and pass through zero simultaneously and allows all displacements to be expressed in terms of a single reference
displacement. At first glance, Rosenberg’s definition may appear restrictive in two cases:
1. In the presence of internal resonances, an NNM motion is no longer synchronous, but it is still periodic. This is why an
extended definition was considered in [2, 3]; an NNM motion was defined as a (non-necessarily synchronous) periodicmotion of the undamped mechanical system.
2. The definition cannot be easily extended to nonconservative systems. However, as shown in [2], the damped dynamics
can be interpreted based on the topological structure of the NNMs of the underlying conservative system, provided that
damping has a purely parasitic effect.
For illustration, the system depicted in Fig. 5.1 and governed by the equations
€x1 þ ð2x1 � x2Þ þ 0:5 x31 ¼ 0
€x2 þ ð2x2 � x1Þ ¼ 0 (5.2)
is considered. The NNMs corresponding to in-phase and out-of-phase motions are represented in the frequency-energy plot
(FEP) of Fig. 5.2. An NNM is represented by a point in the FEP, which is drawn at a frequency corresponding to the minimal
period of the periodic motion and at an energy equal to the conserved total energy during the motion. A branch, represented
by a solid line, is a family of NNM motions possessing the same qualitative features (e.g., in-phase NNM motion).
5.3 The SmallSat Spacecraft and Its Finite Element Modelling
The SmallSat structure has been conceived as a low cost structure for small low-earth orbit satellite [4]. It is a monocoque
tube structure which is 1.2 m long and 1 m large. It incorporates eight flat faces for equipment mounting purposes, creating
an octagon shape, as shown in Fig. 5.3a. The octagon is manufactured using carbon fibre reinforced plastic by means of a
filament winding process. The structure thickness is 4.0 mm with an additional 0.25 mm thick skin of Kevlar applied to both
the inside and outside surfaces to provide protection against debris. The interface between the spacecraft and launch vehicle
is achieved through four aluminium brackets located around cut-outs at the base of the structure. The total mass including the
interface brackets is around 64 kg.
The SmallSat structure supports a telescope dummy composed of two stages of base-plates and struts supporting various
concentrated masses; its mass is around 140 kg. The telescope dummy plate is connected to the SmallSat top floor via three
shock attenuators, termed SASSA (Shock Attenuation System for Spacecraft and Adaptator) [5], the behaviour of which is
1 1
1 1 1
0.5 x1 x2
Fig. 5.1 Schematic
representation of the 2DOF
system example
46 L. Renson et al.
considered as linear in the present study. The top floor is a 1 square meter sandwich aluminium panel, with 25 mm core and
1 mm skins. Finally, as shown in Fig. 5.3c, a support bracket connects to one of the eight walls the so-called Wheel
Elastomer Mounting System (WEMS) device which is loaded with an 8 kg reaction wheel dummy. The purpose of this
device is to isolate the spacecraft structure from disturbances coming from reaction wheels through the presence of a soft
interface between the fixed and mobile parts. In addition, mechanical stops limit the axial and lateral motion of the WEMS
mobile part during launch, which gives rise to nonlinear dynamic phenomena. Figure 5.3d depicts the WEMS overall
geometry, but details are not disclosed for confidentiality reasons.
The Finite Element (FE) model in Fig. 5.3b was created in Samcef software and is used in the present study to conduct
numerical experiments. The comparison with experimental measurements revealed the good predictive capability of this
model. The WEMS mobile part (the inertia wheel and its cross-shaped support) was modeled as a flexible body, which is
connected to the WEMS fixed part (the bracket and, by extension, the spacecraft itself) through four nonlinear connections,
labeled NC 1–4 in Fig. 5.3d. Black squares signal such connections. Each nonlinear connection possesses:
• A nonlinear spring (elastomer in traction plus 2 stops) in the axial direction,
• A nonlinear spring (elastomer in shear plus 1 stop) in the radial direction,
• A linear spring (elastomer in shear) in the third direction.
The spring characteristics (piecewise linear) are listed in Table 5.1 and are displayed in Fig. 5.3e. We stress the presence
of two stops at each end of the cross in the axial direction. This explains the corresponding symmetric bilinear stiffness
curve. In the radial direction, a single stop is enough to limit the motion of the wheel. For example, its +x motion is
constrained by the lateral stop number 2 while the connection 1 limits the opposite -x motion. The corresponding stiffness
curves are consequently asymmetric.
5.3.1 Nonlinearities Modeling
From a computational standpoint, the use of piecewise linear stiffnesses requires special numerical treatments, which
increase the computational burden and complicate convergence processes. Therefore, piecewise behaviors are usually
regularized.
One can avoid the introduction of piecewise-linear stiffnesses in replacing themby polynomials (e.g., using a single nonlinear
term as in (5.3)). Despite its simplicity, this approach has the disadvantage to consider a nonlinear behavior from the origin.
Fr ¼ klinxþ knlxn (5.3)
10−5 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Energy (J, log scale)
Fre
quen
cy(H
z)
Fig. 5.2 Frequency-energy
plot of system (5.2).
NNM motions depicted
in the configuration space
are inset
5 Nonlinear Modal Analysis of the Smallsat Spacecraft 47
Axial nonlinearity
In-plane nonlinearities
NC 1 (-x)
NC 2 (+x)
NC 3 (-y)
NC 4 (+y)
Inertiawheel
SmallSat
Inertia wheel
Bracket
Metalliccross
Filteringelastomer plot
Metallicstop
a b
c d
e
Fig. 5.3 SmallSat structure. (a) real structure without the WEMS module; (b) finite element model; (c) WEMS module mounted on a bracket and
supporting a dummy inertia wheel; (d) close-up of the WEMS mobile part (NC stands for nonlinear connection) and (e) graphical display of the
nonlinear restoring forces
48 L. Renson et al.
In order to avoid such approximation, the regularization technique developed in this paper uses Hermite polynomials to
smooth the transition between both linear stiffnesses. A regularization area a� D; aþ D½ � is considered where a is the
transition point between linear regimes and where 2D is the size of the regularization area (Fig. 5.4). This approach has the
advantage to keep the restoring force behavior purely linear out of the regularization area (Fig. 5.4).
The nonlinear force is now given by (5.4) and displayed in Fig. 5.4.
f nlðxÞ ¼signðxÞðk1aþ k2ð xj j � aÞÞ xj j � aþ D
p�ðtðxÞÞ aþ D> xj j>a� D
k1x a� D � xj j � 0
8>>><>>>:
(5.4)
where t(x) is defined by (5.5) and is used in the definition of the Hermite interpolation polynomials p � (t(x)) (5.6).
tðxÞ ¼ x� xkxkþ1 � xk
(5.5)
p�ðtÞ ¼ h00ðtÞpk þ h10ðtÞðxkþ1 � xkÞmk þ h01ðtÞpkþ1 þ h11ðtÞðxkþ1 � xkÞmkþ1 (5.6)
where pk and pk + 1 are the values of the restoring force at the points xk and xk + 1. xk ¼ signðxÞða� DÞ and
xkþ1 ¼ signðxÞðaþ DÞ. mk and mk + 1 are the values of the derivatives at the same points. For the piecewise-linear stiffness,
mk ¼ k1 and mkþ1 ¼ k2. The hij(t) are given by (5.7)–(5.10).
h00ðtÞ ¼ 2t3 � 3t2 þ 1 (5.7)
h10ðtÞ ¼ t3 � 2t2 þ t (5.8)
h01ðtÞ ¼ �2t3 þ 3t2 (5.9)
h11ðtÞ ¼ t3 � t2 (5.10)
Table 5.1 Nonlinear spring characteristics (adimensional values for confidentiality)
Spring Clearance Stiffness of the elastomer plot
Stiffness of the
mechanical stop
Axial caxial ¼ 1 1 13.2
Lateral cradial ¼ 1. 27 0.26 5.24
Displacement Displacement
Res
toring
for
ce
Res
toring
for
ce
2Δ
a b
2Δ
Fig. 5.4 Piecewise-linear stiffness (�) and regularized stiffness (��). (a) Restoring force overview. (b) Zoom on regularization areas
5 Nonlinear Modal Analysis of the Smallsat Spacecraft 49
5.4 Linear Modal Analysis
Before the computation of NNMs (Sect. 5.6), a linear modal analysis is performed. The frequency range of interest for this
study is 0–100 Hz. Table 5.2 presents the resonance frequencies of the LNMs of the structure included in this frequency
range. Among these LNMs, four are considered of particular interest for the nonlinear modal analysis of this paper. The first
one is the first mode (Fig. 5.5a) which presents a local WEMS motion with a concave trajectory about the axis A (Fig. 5.6).
Table 5.2 Summary of the
linear normal mode frequencies– Frequency (Hz)
Mode 1 10.66
Mode 2 11.00
Mode 3 28.12
Mode 4 28.38
Mode 5 30.18
Mode 6 30.50
Mode 7 31.60
Mode 8 32.65
Mode 9 37.42
Mode 10 38.26
Mode 11 43.30
Mode 12 52.59
Mode 13 71.36
Mode 14 75.73
Mode 15 80.44
Mode 16 84.35
Mode 17 90.52
Mode 18 95.41
Mode 19 101.56
Mode 20 101.93
Fig. 5.5 First (a), third (b), seventh (c), and ninth (d) LNM modal shapes of the SmallSat
50 L. Renson et al.
The second mode considered is the third LNM (Fig. 5.5b) which again presents a local WEMS motion with convexe
trajectory about A. The seventh LNM involving a SASSAmode with alternate compression of springs 1, 2, and 3 (Fig. 5.7) is
considered. This mode also includes a vertical motion of the WEMS. Finally, the ninth mode which presents a local SASSA
mode without WEMS motion is considered.
5.4.1 Reduced-Order Model
As presented in Sect. 5.3, the finite element model contains more than 65,000 dofs. It appears that the computation of the
NNMs for such a large number of dofs is not currently feasible in a reasonable amount of time. Therefore, a reduced-order
model (ROM) was created using the Craig-Bampton technique [6].
This method consists in describing the system in terms of some retained DOFs and internal vibration modes. By
partitioning the complete system in terms of nR remaining xR and nC ¼ n� nR condensed xC DOFs, the n governing
equations of motion of the global finite element model are written as
A
B
Fig. 5.6 WEMS local axis
SASSA 2
SASSA 1
SASSA 3
D
C
Fig. 5.7 SASSA local axis
5 Nonlinear Modal Analysis of the Smallsat Spacecraft 51
MRR MRC
MCR MCC
� �€xR€xC
� �þ KRR KRC
KCR KCC
� �xRxC
� �¼ gR
0
� �(5.11)
The Craig-Bampton method expresses the complete set of initial DOFs in terms of: (1) the remaining DOFs through the static
modes (resulting from unit displacements on the remaining DOFs) and (2) a certain number m < nC of internal vibration
modes (relating to the primary structure fixed on the remaining nodes). Mathematically, the reduction is described by relation
xRxC
� �¼ I 0
�K�1CCKCR Fm
� �xRy
� �¼ R
xRy
� �(5.12)
which defines the n �(nR + m) reduction matrix R. y are the modal coordinates of the m internal linear normal modes
collected in the nC �m matrix Fm ¼ [f(1). . .f(m)]. These modes are solutions of the linear eigenvalue problem
corresponding to the system fixed on the remaining nodes
KCC � o2ðjÞMCC
� �fðjÞ ¼ 0 (5.13)
The reduced model is thus defined by the ðnR þ mÞ � ðnR þ mÞ reduced stiffness and mass matrices given by
M ¼ R�MR
K ¼ R�KR (5.14)
where star denotes the transpose operation. After reduction, the system configuration is expressed in terms of the reduced
coordinates (i.e., the remaining DOFs and the modal coordinates). The initial DOFs of the complete model are then
determined by means of the reduction matrix using relation (5.12).
Table 5.3 summarizes the features of the different ROMs investigated. The eight nodes involved in the superelement
definition are the minimum ones required to define the different nonlinearities of the WEMS.
Before proceeding to nonlinear analysis, the accuracy of the reduced-order linear model is assessed. To this end, the linear
normal modes of the initial complete finite element model are compared to those predicted by the reduced model. The
deviation between the mode shapes of the original model x(o) and of the reduced model x(r) is determined using the Modal
Assurance Criterion (MAC)
MAC ¼x�ðoÞxðrÞ��� ���2
x�ðoÞxðoÞ��� ��� x�ðrÞxðrÞ��� ��� (5.15)
MAC values range from 0 in case of no correlation to 1 for a complete coincidence. The minimum correlation criteria are a
maximum relative error on frequencies of 1% andMAC values above 0. 9 in the frequency range 0–200 Hz. Both criteria are
displayed for each ROM in Fig. 5.8 while Table 5.4 presents the frequency range accuratly covered by the different models.
ROMs with 100 and 500 internal modes both satisfy accuracy requirements. However the selection of the appropriate
ROM for Sect. 5.6 is not trivial. A ROM with numerous internal modes provides the best chances to observe the modal
interactions but this larger model increases the computation time. In addition, the number of modal interactions observed
tends to become prohibitive and dramatically increases the computation time too. Therefore, in the following parts of this
study, the reference ROM used for the computation of NNMs is “ROM85”.
Table 5.3 Features of the
different reduced-order
models created
Model Nodes Internal modes
ROM84 8 50
ROM85 8 100
ROM86 8 500
52 L. Renson et al.
5.5 Numerical Computation of NNMs
The numerical method proposed here for the NNM computation relies on two main techniques, namely a shooting technique
and the pseudo-arclength continuation method. A detailed description of the algorithm is given in [7].
5.5.1 Shooting Method
The equations of motion of system (5.1) can be recast into state space form
_z ¼ gðzÞ (5.16)
where z ¼ x� _x�½ �� is the 2n-dimensional state vector, and star denotes the transpose operation, and
gðzÞ ¼ _x
�M�1 Kxþ fnlðx; _xÞ½ �
� �(5.17)
is the vector field. The solution of this dynamical system for initial conditions zð0Þ ¼ z0 ¼ x�0 _x�0 �
is written as z(t) ¼z(t, z0) in order to exhibit the dependence on the initial conditions, z(0, z0) ¼ z0. A solution zp(t, zp0) is a periodic solutionof the autonomous system (5.16) if zpðt; zp0Þ ¼ zpðtþ T; zp0Þ, where T is the minimal period.
Table 5.4 Summary of the
different ROMs performancesModel Valid modes Frequency range covered
ROM84 1–18 0–95.4 Hz
ROM85 1–50 0–248.2 Hz
ROM86 1–246 0–1020.8 Hz
0 50 100 150 200 250 300 350 400 450 500 5500
1
2
3
4
5
Mode number [−]
Rel
.er
ror
onfr
eq.
[%]
0 50 100 150 200 250 300 350 400 450 500 5500
0.5
1
Mode number [−]
MA
C
Fig. 5.8 Relative error on frequencies and MAC for the different ROMs investigated
5 Nonlinear Modal Analysis of the Smallsat Spacecraft 53
The NNM computation is carried out by finding the periodic solutions of the governing nonlinear equations of motion
(5.16). In this context, the shooting method is probably the most popular numerical technique. It solves numerically the two-
point boundary-value problem defined by the periodicity condition
Hðzp0; TÞ � zpðT; zp0Þ � zp0 ¼ 0 (5.18)
Hðz0; TÞ ¼ zðT; z0Þ � z0 is called the shooting function and represents the difference between the initial conditions and the
system response at time T. Unlike forced motion, the period T of the free response is not known a priori.
The shooting method consists in finding, in an iterative way, the initial conditions zp0 and the period T that realize a
periodic motion. To this end, the method relies on direct numerical time integration and on the Newton-Raphson algorithm.
Starting from some assumed initial conditions zp0(0), the motion zp
(0)(t, zp0(0)) at the assumed period T (0) can be obtained
by numerical time integration methods (e.g., Runge-Kutta or Newmark schemes). In general, the initial guess (zp0(0), T (0))
does not satisfy the periodicity condition (5.18). A Newton-Raphson iteration scheme is therefore to be used to correct an
initial guess and to converge to the actual solution. The corrections Dzp0(k) and DT (k) at iteration k are found by expanding the
nonlinear function
H zðkÞp0 þ DzðkÞp0 ; T
ðkÞ þ DTðkÞ� �
¼ 0 (5.19)
in Taylor series and neglecting higher-order terms (H.O.T.).
The phase of the periodic solutions is not fixed. If z(t) is a solution of the autonomous system (5.16), then z(t + Dt) isgeometrically the same solution in state space for any Dt . Hence, an additional condition, termed the phase condition, has tobe specified in order to remove the arbitrariness of the initial conditions. This is discussed in detail in [7].
In summary, an isolated NNM is computed by solving the augmented two-point boundary-value problem defined by
Fðzp0; TÞ �Hðzp0; TÞ ¼ 0
hðzp0Þ ¼ 0
((5.20)
where h(zp0) ¼ 0 is the phase condition.
5.5.2 Continuation of Periodic Solutions
Due to the frequency-energy dependence, the modal parameters of an NNM vary with the total energy. An NNM family,
governed by (5.20), therefore traces a curve, termed an NNM branch, in the (2n + 1)-dimensional space of initial conditions
and period (zp0, T). Starting from the corresponding LNM at low energy, the computation is carried out by finding
successive points (zp0, T) of the NNM branch using methods for the numerical continuation of periodic motions (also
called path-following methods) [8, 9]. The space (zp0, T) is termed the continuation space.
Different methods for numerical continuation have been proposed in the literature. The so-called pseudo-arclength
continuation method is used herein.
Starting from a known solution (zp0, (j), T(j)), the next periodic solution ðzp0;ðjþ1Þ; Tðjþ1ÞÞ on the branch is computed using a
predictor step and a corrector step.
5.5.2.1 Predictor Step
At step j, a prediction ð~zp0;ðjþ1Þ; ~Tðjþ1ÞÞ of the next solution ðzp0;ðjþ1Þ; Tðjþ1ÞÞ is generated along the tangent vector to the
branch at the current point zp0, (j)
~zp0;ðjþ1Þ~Tðjþ1Þ
� �¼ zp0;ðjÞ
TðjÞ
� �þ sðjÞ
pz;ðjÞpT;ðjÞ
� �(5.21)
54 L. Renson et al.
where s(j) is the predictor stepsize. The tangent vector p(j) ¼[pz, (j)∗ pT, (j)]
∗ to the branch defined by (5.20) is solution of the
system
@H
@zp0
����ðzp0;ðjÞ;TðjÞÞ
@H
@T
����ðzp0;ðjÞ;TðjÞÞ
@h
@zp0
�����ðzp0;ðjÞÞ
0
266664
377775
pz;ðjÞpT;ðjÞ
" #¼ 0
0
" #(5.22)
with the condition pðjÞ��� ��� ¼ 1. The star denotes the transpose operator. This normalization can be taken into account by
fixing one component of the tangent vector and solving the resulting overdetermined system using the Moore-Penrose matrix
inverse; the tangent vector is then normalized to 1.
5.5.2.2 Corrector Step
The prediction is corrected by a shooting procedure in order to solve (5.20) in which the variations of the initial conditions
and the period are forced to be orthogonal to the predictor step. At iteration k, the corrections
zðkþ1Þp0;ðjþ1Þ ¼ z
ðkÞp0;ðjþ1Þ þ DzðkÞp0;ðjþ1Þ
Tðkþ1Þðjþ1Þ ¼ T
ðkÞðjþ1Þ þ DTðkÞ
ðjþ1Þ (5.23)
are computed by solving the overdetermined linear system using the Moore-Penrose matrix inverse
@H@zp0
���ðzðkÞ
p0;ðjþ1Þ;TðkÞðjþ1ÞÞ
@H@T
��ðzðkÞ
p0;ðjþ1Þ;TðkÞðjþ1ÞÞ
@h@zp0
����ðzðkÞ
p0;ðjþ1ÞÞ0
p�z;ðjÞ pT;ðjÞ
2666664
3777775
DzðkÞp0;ðjþ1Þ
DTðkÞðjþ1Þ
264
375 ¼
�HðzðkÞp0;ðjþ1Þ;TðkÞðjþ1ÞÞ
�hðzðkÞp0;ðjþ1ÞÞ
0
266664
377775 (5.24)
where the prediction is used as initial guess, i.e., zð0Þp0;ðjþ1Þ ¼ ~zp0;ðjþ1Þ and T
ð0Þðjþ1Þ ¼ ~Tðjþ1Þ. The last equation in (5.24)
corresponds to the orthogonality condition for the corrector step.
This iterative process is carried out until convergence is achieved. The convergence test is based on the relative error of
the periodicity condition:
Hðzp0; TÞ�� ��
zp0�� �� ¼ zpðT; zp0Þ � zp0
�� ��zp0
�� �� <E (5.25)
where e is the prescribed relative precision.
5.5.3 Sensitivity Analysis
Each shooting iteration involves the time integration of the equations of motion to evaluate the current shooting residue
H zðkÞp0 ; T
ðkÞ� �
¼ zðkÞp ðTðkÞ; zðkÞp0 Þ � z
ðkÞp0 . As evidenced by (5.24), the method also requires the evaluation of the 2n �2n
Jacobian matrix
@H
@z0z0;Tð Þ ¼ @zðt; z0Þ
@z0
����t¼T
� I (5.26)
5 Nonlinear Modal Analysis of the Smallsat Spacecraft 55
where I is the 2n �2n identity matrix.
The classical finite-difference approach requires to perturb successively each of the 2n initial conditions and integrate thenonlinear governing equations of motion. This approximate method therefore relies on extensive numerical simulations and
may be computationally intensive for large-scale finite element models.
Targeting a reduction of the computational cost, a significant improvement is to use sensitivity analysis for determining
∂z(t, z0) / ∂z0 instead of a numerical finite-difference procedure. The sensitivity analysis consists in differentiating the
equations of motion (5.16) with respect to the initial conditions z0 which leads to
d
dt
@z t; z0ð Þ@z0
� �¼ @gðzÞ
@z
����zðt;z0Þ
@zðt; z0Þ@z0
� �(5.27)
with
@zð0; z0Þ@z0
¼ I (5.28)
since z(0, z0) ¼ z0. Hence, the matrix ∂z(t, z0) / ∂z0 at t ¼ T can be obtained by numerically integrating over T the initial-
value problem defined by the linear ordinary differential equations (ODEs) (5.27) with the initial conditions (5.28).
In addition to the integration of the current solution z(t, x0) of (5.16), these two methods for computing ∂z(t, z0) / ∂z0require 2n numerical integrations of 2n-dimensional dynamical systems, which may be computationally intensive for large
systems. However, (5.27) are linear ODEs and their numerical integration is thus less expensive. The numerical cost can be
further reduced if the solution of (5.27) is computed together with the solution of the nonlinear equations of motion in a
single numerical simulation [10].
The sensitivity analysis requires only one additional iteration at each time step of the numerical time integration of the
current motion to provide the Jacobian matrix. The reduction of the computational cost is therefore significant for large-scale
finite element models. In addition, the Jacobian computation by means of the sensitivity analysis is exact. The convergence
troubles regarding the chosen perturbations of the finite-difference method are then avoided. Hence, the use of sensitivity
analysis to perform the shooting procedure represents a meaningful improvement from a computational point of view.
As the monodromy matrix ∂zp(T, zp0) / ∂zp0 is computed, its eigenvalues, the Floquet multipliers, are obtained as a by-
product, and the stability analysis of the NNM motions can be performed in a straightforward manner.
5.5.4 Algorithm for NNM Computation
The algorithm proposed for the computation of NNM motions is a combination of shooting and pseudo-arclength continua-
tion methods, as shown in Fig. 5.9. It has been implemented in the MATLAB environment. Other features of the algorithm
such as the step control, the reduction of the computational burden and the method used for numerical integration of the
equations of motion are discussed in [7].
So far, the NNMs have been considered as branches in the continuation space (zp0, T). An appropriate graphical depictionof the NNMs is to represent them in a frequency-energy plot (FEP). This FEP can be computed in a straightforward
manner: (1) the conserved total energy is computed from the initial conditions realizing the NNM motion; and (2) the
frequency of the NNM motion is calculated directly from the period.
5.6 Nonlinear Modal Analysis
As presented in Sect. 5.3, the nonlinearities are located at WEMS ends. These nonlinearities are activated when
displacements are large enough to hit the mechanical stops. Due to the flexibility of WEMS attachment, large displacements
are observed for the majority of LNMs. However, according to the linear study of Sect. 5.4, the nonlinear investigations are
restricted to four modes.
The energy range of interest for the continuation is determined by the displacements observed at WEMS ends. Indeed,
experimental observation demonstrated that displacements are limited. Therefore, relative displacements larger than the
observed values are not representative of the real physical behavior of the structure.
56 L. Renson et al.
Among the NNMs presented, one can readily distinguish two categories of modes, namely the energy-dependent (e.g.,
Fig. 5.11) and the energy-independent modes (e.g., Fig. 5.10). The latter correspond to the nonlinear extension of linear
modes that do not involve WEMS motion. An example is provided by the ninth mode (Fig. 5.10) where the deformation is
localized at the SASSA. As the energy increases, the modal shape (Fig. 5.5d) presented in the linear study is unchanged and
the mode remains linear.
Figure 5.11 presents the continuation of the first LNM. This LNM correspond to a local WEMS motion and is therefore
sensitive to nonlinearities present in the model. For low energies, the structural behavior remains purely linear and the
resonance frequency does not depend on the energy. However, as energy increases, a steep modification of the frequency-
energy dependence appears. At this transition, one can observe that relative displacements at WEMS ends enter in the
regularization area. Beyond this transition, a plateau appears and interactions between the first LNM and other LNMs are
achieved. The frequency content of the periodic solution evolves with the energy and includes third, fifth, and higher-order
harmonics (up to the 17th order).
Fig. 5.9 Algorithm for NNM
computation
5 Nonlinear Modal Analysis of the Smallsat Spacecraft 57
Figure 5.12 displays the frequency-energy dependence of the third NNM. It highlights the presence of a tongue, revealing
the existence of an 3:1 internal resonance between the third (Fig. 5.5b) and the sixteenth LNM (Fig. 5.13b). The latter
corresponds to an octagonal structure panels mode. Along the tongue (e.g., point b in Fig. 5.12), the modal shape evolves
from the third to the sixteenth LNM and is therefore a special combination of both modes (Fig. 5.13a). This mode has no
linear counterpart and highlights the possibility of interactions between local and global SC modes due to the presence of
nonlinearities.
The frequency-energy dependence of the seventh NNM is presented in Fig. 5.14. The presence of a tongue again
highlights modal interactions between the seventh LNM (Fig. 5.5c) and a higher-order LNM. Here, it is interesting to
observe that the seventh LNM mainly involves a motion of the SASSA. However, due to the nonlinearities of the WEMS,
nonlinear couplings between this SASSA mode and global structural modes are achieved.
10−2 100 102 10437.41
37.412
37.414
37.416
37.418
37.42
37.422
37.424
37.426
37.428
Energy [J]
Freq
uenc
y[H
z]
Fig. 5.10 Frequency-energy
dependence of the
ninth NNM
10−6 10−4 10−2 100 102 104 10610.65
10.66
10.67
10.68
10.69
10.7
10.71
10.72
10.73
10.74
Energy [J]
Freq
uenc
y[H
z]Fig. 5.11 Frequency-energy
dependence of the
first NNM
58 L. Renson et al.
Fig. 5.13 (a) Modal shape of the third NNM combining the third and the sixteenth LNMs (b in Fig. 5.12). (b) Modal shape at the internal
resonance equivalent to the sixteenth LNM (WEMS remains quiescent)(c in Fig. 5.12)
10−5 10−4 10−3 10−2 10−1 100 10128.1
28.15
28.2
28.25
28.3
28.35
28.4
28.45
28.5
Energy [J]
Freq
uenc
y[H
z]
(a) (b)
(c)
Fig. 5.12 Frequency-energy dependence of the third NNM. (a) Low-energy point. (b) Point in the tongue describing the 3:1 internal resonance.
(c) Bifurcation point
10−3 10−2 10−1 100 101 102 10331.58
31.6
31.62
31.64
31.66
31.68
31.7
31.72
31.74
31.76
Energy [J]
Freq
uenc
y[H
z]
Fig. 5.14 Frequency-energy dependence of the seventh NNM
5 Nonlinear Modal Analysis of the Smallsat Spacecraft 59
5.7 Conclusions
In this paper, the fundamental concepts regarding undamped nonlinear normal modes and their numerical computation were
reviewed. A new regularization procedure was presented and revealed to be accurate for the modeling of piecewise linear
restoring forces.
Targeting the computation of the nonlinear modes, a linear modal analysis was presented and some interesting modes
were identified for further investigations. A reduced-order model accurate in the (0–200 Hz) range was employed to reduce
the computational burden.
Finally, the nonlinear normal modes of the spacecraft were presented. Internal resonances highlighted the possibility of
mode interactions between local (WEMS) and global structural modes.
Acknowledgements This paper has been prepared in the framework of the ESA Technology Research Programme study “Advancement of
Mechanical Verification Methods for Non-linear Spacecraft Structures (NOLISS)” (ESA contract No.21359/08/NL/SFe).
The authors would like to thank Dr. Maxime Peeters for all the constructive discussions. The author L. Renson would like to acknowledge the
Belgian National Fund for Scientific Research (FRIA fellowship) for its financial support.
References
1. Vakakis AF, Manevitch LI, Mikhlin YV, Pilipchuk VN, Zevin AA (1996) Normal modes and localization in nonlinear systems. Wiley,
New York
2. Kerschen G, Peeters M, Golinval JC, Vakakis AF (2009) Nonlinear normal modes, part I: a useful framework for the structural dynamicist.
Mech Syst Signal Process 23(1):170–194
3. Lee YS, Kerschen G, Vakakis AF, Panagopoulos PN, Bergman LA, McFarland DM (2005) Complicated dynamics of a linear oscillator with a
light, essentially nonlinear attachment. Phy D-Nonlinear Phenom 204(1–2):41–69
4. Russell AG (2000) Thick skin, faceted, CFRP, monocoque tube structure for smallsats. European conference on spacecraft structures,
materials and mechanical testing, Noordwijk
5. Camarasa P, Kiryenko S (2009) Shock attenuation system for spacecraft and adaptor (SASSA). European conference on spacecraft structures,
materials and mechanical testing, Noordwijk
6. Craig R, Bampton M (1968) Coupling of substructures for dynamic analysis. AIAA J 6:1313–1319
7. Peeters M, Viguie R, Serandour G, Kerschen G, Golinval JC (2009) Nonlinear normal modes, part II: toward a practical computation using
numerical continuation techniques. Mech Syst Signal Process 23(1):195–216
8. Seydel R Practical bifurcation and stability analysis, from equilibirum to chaos, 2nd edn. Springer, New York (1994)
9. Nayfeh AH, Balachandran B (1995) Applied nonlinear dynamics: analytical, computational, and experimental methods. Wiley, New York
10. Br€uls O, Eberhard P (2006) Sensitivity analysis for dynamic mechanical systems with finite rotations. Int J Numer Meth Eng 1:1–29
60 L. Renson et al.
Chapter 6
Filter Response to High Frequency Shock Events
Jason R. Foley, Jacob C. Dodson, and Alain L. Beliveau
Abstract A variety of effects can introduce nonlinearities into filter response when measuring shock signals: small- versus
large-signal frequency response, electronic nonlinearities, spurious electrical noise, etc. This paper examines the analytic
response of ideal filters in response to a variety of analytic shock-like signals, including instantaneous pulses of varying rise
rate, frequency content and duration as well as pulse trains with variable timing and duty cycle. The maximum permitted
slew rate, or instantaneous rate, of the shock signal is shown to be a function of the filter type and order. Instantaneous
slew rate is also discussed as an indicator of impulse response, i.e., spectral frequency content that is higher than the filter
cutoff frequency.
Keywords Shock • High frequency • Slew rate • Filter response • Data acquisition • Signal conditioning • Analog filter •
Digital filter • Impulse response
Nomenclature
d Delta function
s Attenuation/neper frequency
t Pulse width/duration
o Angular/radian frequency
F Phase response (frequency domain)
i Imaginary number
s Complex frequency (Laplace variable)
t Time
x Input function (time domain)
G Frequency response or gain function
H Transfer function (frequency domain)
SR Slew rate
T Period or pulse train timing
X Input function (frequency domain)
Y Output function (frequency domain)
J.R. Foley (*)
Air Force Research Laboratory, AFRL/RWMF, 306 W. Eglin Blvd, Bldg. 432, Eglin AFB, FL 32542-5430, USA
e-mail: [email protected]
J.C. Dodson
Air Force Research Laboratory, AFRL/RWMF, Eglin AFB, FL, USA
A.L. Beliveau
Applied Research Associates, Inc., Niceville, FL, USA
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_6, # The Society for Experimental Mechanics, Inc. 2012
61
6.1 Introduction
Acquisition of valid shock and vibration data requires an in-depth understanding of both the physical system dynamics and
the response of every component of the data acquisition system. For example, sigma-delta (SD) analog-to-digital converters(ADC’s) are commonly used in data acquisition systems for digitization of voltage signals [1, 2]. Signal conditioners are
typically used to isolate, filter, convert (to voltage), and/or amplify the output from sensors [3]. To avoid aliasing and
minimize high frequency noise, low pass filters are routinely included in the systems, either integrated into the signal
conditioners or as standalone components. Additionally, the frequency domain characteristics of cables, interfaces, and other
physical components must be considered in precision measurements. Figure 6.1 shows an idealized and simplified block
diagram in the frequency domain of a typical sensor-to-data-acquisition experiment expanded into the various components.
Highly transient events are common in experimental measurements of the dynamics of structures due to shock and
vibration. These impulsive signals are characterized by step-like high voltage levels with correspondingly large frequency
content which often far exceeds the linear bandwidth of individual electronic components. This has the potential to
subsequently induce nonlinear responses in these components [4]. Understanding and diagnosing any possible nonlinearities
is therefore critical to collecting valid shock data. For example, slew induced distortion [5] is possible if frequency content
and instantaneous rates exceed the linear range of amplifiers in the data acquisition and/or signal conditioning systems. This
paper examines analytic filter models to determine the expected dependence on key parameters, such as maximum slew rate,
of their output due to a variety of inputs. This information is critical for reviewing experimental shock data for validity.
6.2 Theory
In practical shock experimentation, a variety of waveforms can be expected for the physical input (X). An impulse function
[6] is perhaps the most basic descriptor. The analytic form of a finite impulse [7] is
xðtÞ ¼ dðtÞ ¼ 1
2p
ð1�1
eiotdo: (6.1)
Equation (6.1) can be used as a convolution kernel to obtain a variety of other possible input functions; these are
categorized and presented in Table 6.1.
The response yðtÞ of a system due to a time input xðtÞ is defined [8] as the convolution integral
yðtÞ ¼ xðtÞ � hðtÞ ¼ð1�1
x tð Þh t� tð Þdt; (6.2)
where hðtÞ is the impulse response (h t� tð Þ is the time domain response at time t from an instantaneous input at t ¼ t). Otherforms of input, such as a rectangular pulse or step response, can obtained by integrating the impulse response over the
duration of the step.
Physicalinput
X
Sensorresponse
Hs
Cable response
Hc1
Front endresponse
Hf
Cable response
Hc2
Data acq.response
Hd
Digitized outputY
Physicalinput
X
Systemresponse
H
Digitizedoutput
Y
Fig. 6.1 Simplified block diagram of transfer functions involved in practical data acquisition
62 J.R. Foley et al.
Table 6.1 Analytic input functions and nomenclature of waveforms commonly encountered in shock and vibration
Case name
Amplitude
PolaritySmall signal finite Large signal finite Infinite
Impulse
t
x(t)
t0
A
SSF
t
x(t)
t0
A
LSF
t
x(t)
t0
II
t
x(t)
t0
P(ositive) or N(egative)
Impulse train
t
x(t)
t0
A
T
SSFIT
t
x(t)
t0
A
T
LSFIT
t
x(t)
t0
T
IIT
t
x(t)
t0
T
[P(ositive)/N(egative)]1
Pulse
tt0
A
τ
SSFP
tt0
A
τ
LSFP
N/A
t
x(t)t0
A
τ
P(ositive) or N(egative)
Pulse train
t
x(t)
t0
A
τ
T
SSFPT
t
x(t)
t0
A
τ
T
LSFPT
NAt
x(t)
t0
-A
τ
AT
[P(ositive)/N(egative)]1
Step
t
x(t)
t0
A
SSFS
t
x(t)
t0
A
LSFS
N/A
t
x(t)t0
-A
P(ositive) or N(egative)
Multi-step
t
x(t)
t
A1
A2
t
SSFMS
t
x(t)
A1
A2
t1 t2
LSFMS
N/A
t
x(t)A1
A2
t1 t2
P(ositive)/N(egative)
6 Filter Response to High Frequency Shock Events 63
The slew rate of an electronics component (generally amplifiers) is the maximum rate where the device response will
remain linear. The analytic expression for the instantaneous slew rate of a signal y is simply its instantaneous time rate of
change, i.e.,
dy
dt¼ y
0 ðtÞ: (6.3)
Combining (6.1) through (6.3) while using the convolution property [7] of a derivative of the delta function,
dy
dt¼ d
dtdðtÞ � hðtÞ½ � ¼ d
0 ðtÞ � hðtÞ ¼ð1�1
d0t� tð Þh tð Þdt ¼ h
0 ðtÞ; (6.4)
we find the impulse slew rate is indeed the derivative of the impulse response. The impulse response of two commonly used
low-pass filter types are now considered: Butterworth and Chebyshev.
6.3 Butterworth Filter Response
The nth-order Butterworth filter [9] is given by the frequency domain transfer function [10] as
HðsÞ ¼ 1Qnj¼1 s� sj
� � (6.5)
where sj are the filter poles. The corresponding frequency response (or gain) function for the low-pass Butterworth filter withunity gain in the passband is
G oð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH ioð Þj j2
q¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ ooc
� �2nr (6.6)
and the phase response is
F oð Þ ¼ arg H iooc
� �� �: (6.7)
The impulse response of the Butterworth filter is shown in Fig. 6.2a. The pulse response is also shown in Fig. 6.2b for
comparison.
0.010.0102
0.01040.0106
0.0108 0
2
4
6
80
0.2
0.4
0.6
0.8
1
a b
Filter order
Butterworth Filter (fc = 10 kHz) Applied to Finite Impulse
Time [s]
Am
plitu
de
0.010.0102
0.01040.0106
0.0108 0
2
4
6
8
0
0.2
0.4
0.6
0.8
1
Filter order
Butterworth Filter (fc = 10 kHz) Applied to Pulse
Time [s]
Am
plitu
de
Fig. 6.2 (a) Finite impulses and (b) pulses filtered with a Butterworth filter (cutoff at 10 kHz) with number of poles from 1 to 8
64 J.R. Foley et al.
The maximum slew rate is then determined by taking the maximum slope of the impulse response, i.e., the
maximum of (6.4). The maximum allowed slew rate for Butterworth filters is plotted versus cutoff frequency and order
in Fig. 6.3.
6.4 Chebyshev (Type I) Filter Response
The Chebyshev (type I) transfer function is given by
HðsÞ ¼ HnðeÞQnj¼1 s� sj
� � (6.8)
where HnðeÞ is a function of the filter order and the passband ripple, e [10]. The impulse and pulse response is shown in
Fig. 6.4a,b, respectively. The cutoff frequency and order dependence is shown in Fig. 6.5a,b for two different ripple values
(3 dB and 0.75 dB).
6.5 Discussion
The maximum allowable slew rate is proportional to the cutoff frequency, i.e., max h0 ðtÞ / oc. This linear dependence
with cutoff frequency is readily seen in Figs. 6.3 and 6.5. The decrease slew rate with respect to filter order is found by using
a Heaviside expansion [8] of the impulse response of a Butterworth filter, given by
hðtÞ ¼ L�1 HðsÞ½ � ¼Xnr¼1
Krest; (6.9)
102 103 104 105 106 107 108102
103
104
105
106
107
108
109
Filter Cutoff Frequency [Hz]
Max
imum
Allo
wed
Sle
w R
ate
(dx/dt
) [s
-1]
Maximum Allowed Slew Rate vs. Filter Cutoff Frequency for Butterworth Filter
0
12
3
4
5
67
8
Filter Order
Fig. 6.3 Maximum allowed slew rate for Butterworth filters of order 1 through 8
6 Filter Response to High Frequency Shock Events 65
where Kr ¼ s� srs� s1ð Þ s� s2ð Þ . . . s� snð Þ
����s¼sr
: Correspondingly, the slew rate is then given by
h0 ðtÞ ¼
Xnr¼1
sKrest: (6.10)
Therefore, as the pole order increases, the coefficients Kr decrease as a power of s2�n. The dependence of the maximum
rate thus is monotonically decreasing versus filter order r, as shown in Fig. 6.6 below.
Together, the dependence of the maximum slew rate on cutoff frequency and filter order establishes an upper limit to
expected outputs from a given filter design. This upper limit is hypothesized to be a criterion for diagnosing both impulse
inputs (input spectral frequency content higher than the filter cutoff frequency) as well as possible nonlinearities in
measurement systems due to high frequency/high amplitude sensor output in shock experiments.
6.6 Future Work
Experiments to validate the linear and nonlinear response of analog filters in data acquisition components are ongoing. These
experiments will have a schematic layout as shown in Fig. 6.7. Oscilloscopes will generally have a>100� higher sample
rate and bandwidth [11] at the expense of decreased vertical resolution when compared with a data acquisition system (see,
for example, Refs. [12, 13]). The scope is used to capture “truth data” on the output waveform in the pulse generator into an
equivalent impedence data so that the maximum rate generated by the pulse generator can be verified.
Future work also includes analyzing “black box” filters as an inverse parametric identification problem. Full system
models will be implemented to model nonlinear responses, such as slew-rate distortion. Additionally, the filter
characteristics will then be used to estimate the impulsive nature of an observed mechanical response.
6.7 Summary
The analytic response of ideal filters was calculated for a variety of analytic signals typical of shock. The maximum
permitted slew rate of the shock signal was shown to be a monotonic function of the filter type and order. Instantaneous slew
rate was proposed as an indicator for impulse response and possible nonlinear filter response.
0.010.0102
0.01040.0106
0.0108 0
2
4
6
80
0.2
0.4
0.6
0.8
1
a b
Filter order
Chebyshev Type I Filter (fc = 10 kHz) Applied to Impulse
Time [s]
Am
plitu
de
0.010.0102
0.01040.0106
0.0108 0
2
4
6
8-0.2
0
0.2
0.4
0.6
0.8
1
Filter order
Chebyshev Type I Filter (fc = 10 kHz) Applied to Pulse
Time [s]
Am
plitu
de
Fig. 6.4 (a) Finite impulses and (b) 10 ms pulses filtered with a Chebyshev type I filter (cutoff at 10 kHz) with number of poles from 1 to 8
66 J.R. Foley et al.
102
103
104
105
106
107
108
102
103
104
105
106
107
108
109
Filter Cutoff Frequency [Hz]
Max
imum
Allo
wed
Sle
w R
ate
(dx/dt
) [s
-1]
Maximum Allowed Slew Rate vs. Filter Cutoff Frequency for Chebyshev Type I Filter
0
12
3
4
5
67
8
Filter Order
102
103
104
105
106
107
108
102
103
104
105
106
107
108
109
Filter Cutoff Frequency [Hz]
Max
imum
Allo
wed
Sle
w R
ate
(dx/dt
) [s
-1]
Maximum Allowed Slew Rate vs. Filter Cutoff Frequency for Chebyshev Type I Filter
01
2
3
45
6
78
Filter Order
a
b
Fig. 6.5 Maximum allowed slew rate for Chebyshev type I filters of order 1 through 8 with passband ripple of (a) 3 dB and (b) 0.75 dB
6 Filter Response to High Frequency Shock Events 67
Acknowledgements The authors would like to thank AFOSR (Program Manager: Dr. David Stargel) for supporting this research effort via
research task 09RW01COR. Opinions, interpretations, conclusions, equipment selections, and recommendations are those of the authors and are
not necessarily endorsed by the United States Air Force.
References
1. Boser BE, Wooley BA (1988) The design of sigma-delta modulation analog-to-digital converters. Solid-State Circuit IEEE J 23(6):1298–1308
2. van de Plassche R (1978) A sigma-delta modulator as an A/D converter. Circuit Syst IEEE Trans 25(7):510–514
3. Wilson JS (2005) Sensor technology handbook. Elsevier, Amsterdam
4. Walter PL (1978) Limitations and corrections in measuring dynamic characteristics of structural systems. Sandia National Laboratories
technical report, SAND-78-1015
5. Allen P (1978) A model for slew-induced distortion in single-amplifier active filters. Circuit Syst IEEE Trans 25(8):565–572
6. Walter P, Nelson H (1979) Limitations and corrections in measuring structural dynamics. Exp Mech 19(9):309–316
t
x(t)
t0
A
T
PulseGenerator
DAQ
Oscilloscope
Fig. 6.7 Experiment schematic to monitor slew response of DAQ
1 2 3 4 5 6 7 8 9 10104
105
106
107
108
Filter Order
Max
imum
Sle
w R
ate y
'(t)
Fig. 6.6 Maximum slew rate versus filter order for a Butterworth filter (cutoff frequency is 100 MHz)
68 J.R. Foley et al.
7. Bracewell RM (1965) The Fourier transform and its applications. McGraw-Hill, New York, pp 6–7, 244–250
8. Rorabaugh CB (1999) DSP primer. McGraw-Hill, New York
9. Butterworth S (1930) On the theory of filter amplifiers. Exp Wirel Wirel Eng 7:536–541
10. Bateman VI, Hansche BD, Solomon OM (1995), Use of a laser doppler vibrometer for high frequency accelerometer characterizations. Sandia
National Laboratories technical report SAND-95-1041C
11. —— (2010) Digital Phosphor Oscilloscope TDS5034B/TDS5054B/TDS5104B Data Sheet, Tektronix, Beaverton
12. —— (2003) NI PXI-5122 Specifications, National Instruments, Austin
13. —— (2003) NI PXI-6133 Specifications, National Instruments, Austin
6 Filter Response to High Frequency Shock Events 69
Chapter 7
Simplified Nonlinear Modeling Approach for a Bolted
Interface Test Fixture
Charles Butner, Douglas Adams, and Jason R. Foley
Abstract The sensitivity of the response characteristics of a bolted interface to the bolt preload level used in the joint can
often cause the dynamic properties of a system to be difficult to predict. A bolted interface test fixture was fabricated to
investigate the effects of preload changes on the system dynamic response characteristics, and experimental results indicated
that increases in bolt preload led to increases in modal frequency and decreases in modal damping. Furthermore, the system
demonstrated a nonlinear behavior that resulted in the increase in modal frequency due to increases in impact amplitude
when preload levels were low. The experimental results motivated the creation of a simplified low order nonlinear system
model to represent the two dominant modes of the system. A model was used to describe the relationship between static
stiffness and preload to account for the changes in initial bolt preload, and cubic stiffness terms were included to account for
the amplitude dependent nonlinearity that was observed. The resulting model was able to accurately simulate system
frequencies and general trends, but was unable to match some response characteristics of the system due to the lack of force
information for the high amplitude loading used in the experiments.
Keywords Nonlinear dynamics • Bolted interface • Modal frequency • Modal damping
7.1 Introduction and Background
The uncertainty that is introduced into a measurement based on the mounting condition of a sensor is a problem that often
influences the validity of structural vibration tests. Even when a sensor can be bolted to the test specimen, the amount of
preload in this joint can significantly change the response of the sensor. In a particular problem being studied by the Air
Force Research Lab Fuzes Branch, a triaxial sensor mount is attached to a more massive body through a preloaded interface,
and the preload used in the joint is a known source of uncertainty in the measurement. To study this particular interaction, a
test fixture was designed and built to simulate the interface. The fixture design is illustrated in Fig. 7.1.
The first component of the fixture is a large circular plate with a diameter of 460 mm and a thickness of 20 mm.
The second component of the fixture is a smaller square plate with a 180 mm height and width and a 20 mm thickness.
The square plate also contains three spherical standoffs of 20 mm height that localize the contact area between the two plates.
Both plates were machined from 4140 Alloy Steel. The fixture is assembled by bolting the two components together with M-
16 bolts and instrumented load washers to measure the static and dynamic preload in the bolts.
The study of the fixture began with experiments that subjected the bolted interface fixture to impulsive loading. The true
load path of the system being simulated by the fixture passes through the circular plate and into the square plate.
The experimental phase began with a low force amplitude modal test, in which the fixture dynamic behavior was
characterized using modal impact testing. In the second phase of testing, high amplitude impacts were applied to the test
fixture with a Hopkinson bar in order to approach a more realistic loading scenario, since the structure being modeled was
known to undergo very large forces.
C. Butner (*) • D. Adams
Purdue University, Center for Systems Integrity, 1500 Kepner Drive, Lafayette, IN 47905, USA
e-mail: [email protected]
J.R. Foley
Air Force Research Laboratory Munitions Directorate, Fuzes Branch, Eglin Air Force Base, Eglin AFB, FL, USA
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_7, # The Society for Experimental Mechanics, Inc. 2012
71
The results of the experiments conducted on the fixture led to the creation of a nonlinear three degree of freedom system
model that can be used to predict certain response characteristics of the system. The modal testing was used to estimate
modal parameters of the system for the two dominant modes present, while the high amplitude test results were used to
determine a relationship between bolt preload and natural frequency for the system.
Previous work by Adams et al. [1], Butner et al. [2] and Butner in [3] details the past experimental work that was
conducted on this fixture and the interface that it represents. The results of the work demonstrated an amplitude dependent
nonlinearity in the fixture at low preloads that was less severe as preload was increased. The work also noted that forces
where amplified across the interface at high preload levels, and that a preload relaxation event took place during the impacts,
indicating that the system response characteristics could vary due to an impact.
A review article by Ibrahim and Pettit [4] included a discussion of a vast range of bolted joint characteristics including
energy dissipation, bolt relaxation, natural frequency dependence, and bolt modeling approaches. Many uncertainties
associated with a bolted interface were also described. The accepted modeling techniques for bolted joint properties such
as joint stiffness, which is modeled as a constant value in many simplistic approaches, were found to be inadequate as
described by Grosse and Mitchell in [5] as well as in the work by Lehnhoff and Bunyard in [6]. The lack of analytical
relationships to accurately model these properties has led researchers to use experimental data and finite element models to
understand how bolted joints respond for different preload levels and excitation forces. An article written by Moo-Zung Lee
[7] explained how the preload in a bolted joint could change the stiffness properties. When a joint has no preload, an applied
force that attempts to separate the members will be applied entirely to the bolt. In this case, the stiffness of the joint is equal
to the stiffness of the bolt alone. When preload is applied to the bolt, the members are preloaded together, and the joint
stiffness becomes a function of the bolt and member stiffness values. This relationship indicated that a change in bolt preload
could also lead to nonlinearity in the system response. The Budynas and Nisbett text [8] presented an approach to model the
clamped member stiffness by using a conical change in pressure throughout the bolted joint. This approach motivates the use
of a nonlinear cubic stiffness model for joint stiffness.
Studies have also been performed on the changes in modal properties of multibody systems as a function of bolt preload.
A study was performed by Caccese et al. in [9] that attempted to detect loose bolts in a hybrid composite/metal bolted
interface through several approaches; one of which observed changes in modal parameters. The test setup consisted of
16 bolts around the perimeter of a square plate that clamped a composite panel to the square metallic plate. When only one
bolt was loosened slightly, little change in the fundamental frequency of the structure was detected, but when the bolt
was completely loosened, a drop in frequency was detected. The fundamental frequency of the assembly dropped more
quickly when all of the bolts were loosened. This work demonstrated that there was a large change in frequency for the
completely loose bolts case and partially tightened bolt case, but much less of a change between partially tightened bolts and
completely tightened bolts. This result was consistent with the behavior described about bolted joint stiffness in [7]. Stiffness
increases significantly when the bolt is initially preloaded, which results in a large upward shift in fundamental frequency,
but continued tightening results in a smaller change in stiffness, and thus a smaller increase in fundamental frequency.
A study done by Peairs et al. in [10] demonstrated similar behavior. A general trend of increasing modal frequency with an
increase in bolt preload was seen for many modes.
7.2 Experimental Approach and Key Results
A detailed description of the experiments conducted on the test fixture can be found in [2, 3], but since the purpose of this
paper is to describe the modeling approach used to simulate the system, only the experimental results that motivated the
modeling approach will be discussed. As mentioned previously, the experimental component of this research took place in
two phases. The first phase consisted of characterizing the system dynamics through the use of modal impact testing.
Fig. 7.1 Bolted interface test
fixture used for experiment
72 C. Butner et al.
The second phase consisted of applying large amplitude impacts to the fixture by using a Hopkinson bar in order to approach
a more realistic loading scenario to characterize the dynamic response for these higher loading levels. The test configuration
for the modal test and Hopkinson bar test can be seen in Figs. 7.2 and 7.3, respectively.
To characterize the system dynamics initially, four complete modal impact tests were conducted on the fixture.
Two variables were adjusted between the tests. The modal impact tests were conducted with the static preload in the
bolts tuned at levels between 1 kN and 20 kN, and the amplitude range of the impacts were held between 10 lbf and 20 lbf in
one set of tests or between 200 lbf and 300 lbf for a second set of tests. The goal of the test was to characterize the nonlinear
response characteristics due to impact amplitude as the preload in the bolts was adjusted.
A second experiment was conducted using a Hopkinson Bar to impact the test fixture. The Hopkinson bar applies an
impact through a transfer bar. This method was chosen because a one inch diameter aluminum transfer bar could be used in
order to prevent deformation at the impact location on the much harder steel test fixture. These tests were conducted by
suspending the test fixture vertically from an engine hoist and preloading a transfer bar against the test fixture. The bolts were
tightened to a pre-defined torque with a torque wrench, and each load cell’s static value was recorded. The transfer bar was
then impacted by a projectile that was fired by a gas gun. The impact force was stepped up by monitoring the air pressure that
was used to launch the projectile. For this experiment, six PCB 350C02 shock accelerometers were used to measure
acceleration at both ends of each of the three bolts. The load washer dynamic data was also recorded. During this round of
testing, preload levels of hand tight, 50 ft-lb, and fully tight, 100 ft-lb, were tested with impacts at 5 psi, 10 psi, 15 psi, and
20 psi. Due to the excitation type used, impact forces could not be measured in this experiment. The aim of this experiment
was to achieve an excitation that better reflected the realistic loading scenario that was expected.
Fig. 7.2 (a) Instrumented test fixture and (b) impact hammer with load washer signal conditioners
Fig. 7.3 Hopkinson bar test
setup
7 Simplified Nonlinear Modeling Approach for a Bolted Interface Test Fixture 73
The results of the four modal tests revealed several key trends in the system that would later be used in the model
formulation. First, a general trend of increasing natural frequency with increasing bolt preload was seen, which was
consistent with the results in literature. The modal damping in the system was seen to decrease with an increase in preload,
due to the reduction in the slapping and/or slipping across the interface. Damping increased with an increase in impact
amplitude as well. The system was also shown to behave more linearly with respect to impact amplitude as preload was
increased.
The high amplitude Hopkinson bar testing also demonstrated an increase in natural frequency with preload, and helped to
provide a relationship between static bolt preload in ft-lbs with the modal frequencies of interest. This experiment also
demonstrated a force amplification effect across the interface at high preload. These experiments also demonstrated an
amplitude dependence of the system natural frequencies when preload was low, as shown in Fig. 7.4. The natural frequency
increases during an impact event because as the joint is forced together by the impact, a stiffening effect occurs. This effect is
not seen in the higher preload cased because the impact does not overcome the preload force.
7.3 Three Degree of Freedom Nonlinear System Model
The analysis of the experimental results for the test fixture identified several interesting phenomena pertaining to the
effects of preload on the dynamic response of the two coupled bodies. Some of these effects could be qualitatively
explained using a linear system model, such as the amplification effects across the bolted interface and modal behavior
that was observed in the shock test experiments. The changes in natural frequency and damping as a function of the initial
static preload could be modeled by introducing changes in the linear connecting springs and dampers. Some of the
experimental results could not be explained with a linear model. The most prevalent nonlinear effect of interest was the
dependence of the natural frequency on the impact force amplitude. This result was observed most clearly in the shock
test data that was plotted in Fig. 7.4. A low order nonlinear system model was developed to explain the changes in natural
frequency as a function of impact force level. This model would be useful for predicting the behavior of future system
configurations, and gaining a better understanding of the behavior of the system in general for use in interpreting measured
data in operation.
7.3.1 Modeling Approach
The decision to use a low order modeling approach was motivated by the results of the Hopkinson bar shock loading
experiments. In this data, only two modes were dominant in the low frequency range. The reason for this was that the input
force was applied at the center of the circular plate, which was a node of vibration for many of the system modes of vibration.
500 550 600 650 700 750 800 850 9000
2
4
6
8
10
12
14
Frequency (Hz)
Am
plitu
de o
f Spe
ctru
m (
m/s
^2/H
z) 5psi Impact10psi Impact15psi Impact20psi Impact
Fig. 7.4 First mode of vibration for test fixture with hand-tight preload
74 C. Butner et al.
Since the force was being applied to a common node of vibration, it did not excite modes that shared this node to a significant
level. The two dominant modes that were excited had an anti-node at the center of the circular plate. In reality, the load path
of the system that was emulated using the test fixture lies along the edge of the circular plate. It was reasoned that if a
uniform distributed force was applied around the circumference of the circular plate, the same modes would be excited as if
a force was applied at the center of the plate due to the circular symmetric nature of the modes of vibration. The low
frequency modes for which the center of the circular plate was a node of vibration also consisted of non-uniform motion at
the outer edge of the circular plate, while those that had an anti-node at the center of the circular plate had uniform motion
at the edge of the circular plate. Because of these observations, the modeling approach was deemed to be valid for uniform
forces applied around the circular plate’s edge, which was the load path for this structure in practice.
While only two dominant modes were observed in the shock testing data, and many aspects of the system lent themselves
to being represented by a two degree of freedom model, this configuration would not represent the system accurately. In the
data, the first mode that was excited was a flexible body mode of the circular plate and rigid body motion of the square plate.
The second excited mode exhibited asynchronous motion between the two plates as the primary motion. Taking this
observation into account, the system for these two modes of vibration was modeled using three degrees of freedom as shown
in Fig. 7.5. This model included an additional degree of freedom and, therefore, resulted in the presence of a third mode of
vibration. This third mode was tuned to a very low frequency by using soft ground springs and allowing the entire mass of the
test fixture to oscillate as a rigid body. This motion was an accurate representation of the test fixture because the three modes
of vibration allowed for the flexible body mode of the circular plate, the asynchronous mode between the plates, and a low
frequency rigid body mode that simulated the test fixture at it was swung on its flexible supports. Linear, adjustable springs
and dampers were placed between the degrees of freedom in order to allow for system adjustments for different static
preload values. Additional preload would result in increased joint stiffness and decreased damping as discussed in the
literature in [7], as well as in previously described experimental results. Nonlinear elements were also required to allow
the system’s modal frequencies to increase with impact force amplitude. An asymmetric cubic stiffness nonlinearity was
chosen for this purpose.
The decision to use a cubic stiffness term was motivated by several factors. The primary reason for choosing this type of
nonlinearity was the observations made in the experimental data. When the fixture was forced with impacts of increasing
amplitude, the modal frequencies increased. This result could be accomplished with any spring that stiffens with deflection.
In the case of the test fixture, however, the system behaved linearly in cases where the impact amplitude was small or the
preload was high. These cases both corresponded to conditions where deflection across the bolted interfaces was low.
A cubic stiffness term had a small effect for small deflections, which allowed the system response to remain mostly linear for
this case. When the static preload was lowered or the impact amplitude was increased, the increased deflections resulted in
an increase in natural frequency. The cubic stiffness was introduced for increased compression across the bolted interface,
M1 M2 M3
Circular Plate Square Plate
Cubic Springs
Linear Adjustable Springs and Dampers
To Account for Static Preload Changes
x1 x2 x3
f
C1,K1 C2,K2 C3,K3
Fig. 7.5 Three degree
of freedom nonlinear
system model
7 Simplified Nonlinear Modeling Approach for a Bolted Interface Test Fixture 75
however, because it was thought that large impacts momentarily increased the preload between members by forcing them
together, thereby increasing the joint stiffness. It was for these reasons that a cubic stiffness term was considered to be
acceptable.
7.3.2 Determination of System Parameters
The formulation of the system model began with the construction of a linear system of equations that accurately
approximated the system for a given preload value and impact amplitude. This model could then be adjusted to reflect
changes in initial preload and the cubic stiffness terms could be included to account for nonlinear effects. The first step in this
process was to determine the modal mass, stiffness, and damping for each mode from the experimental data. The model was
meant to approximate motion on either side of the bolted interface, so data was used that captured the motion on either side
of the interface. Frequency response functions were used from accelerometers on either side of the bolted interface near one
of the bolts, when applying an impact at the accelerometer location on the circular plate. This measurement approach
provided a driving point and a cross point measurement. While only a driving point measurement was required for the
estimation of modal parameters, the cross point measurement was needed to obtain accurate estimates of the modal vectors.
The lr and Apqr values for the two modes of interest were calculated from the system simultaneously using the Local
Least Squares Algorithm, which is a low order frequency domain technique that operates on a single degree of freedom
assumption. For systems without closely spaced modes, the data near the peak can be considered to behave as a single degree
of freedom system corresponding to the parameters from that particular mode. Nearby modes could contribute residual
values to these peaks and reduce the accuracy of these assumptions, but for the test fixture data this was not the case for the
modes of interest.
Once the lr and Apqr values were obtained from the experimental data and verified for both the driving point and cross
point measurements, the modal mass, stiffness and damping parameters could be estimated. The modal parameters that were
calculated all had small imaginary portions with respect to the real portion, so the values were approximated using only the
real portion for model simplicity. Before these values could be reduced to a system of equations of motion to represent the
system, a rigid body mode was created. For this mode, a modal mass was chosen that was approximately equal to the weight
of the entire fixture. A modal stiffness value was chosen to match the 1.5 Hz mode of the test fixture at it was swung on its
supports. A modal damping value was chosen to be proportional to the other modes. Table 7.1 listed the modal mass,
damping, and stiffness values that were used for the final model. The addition of a third degree of freedom required the
modal vectors to be reformulated. The rigid body mode consisted of all masses moving together. The second mode, which
corresponded to the flexible body mode for the circular plate, was represented by the first and second masses moving relative
to one another, with no deflection in the spring between the second and third masses. The third mode consisted of no
deflection between the first two masses, and asynchronous motion between the second and third masses. The modal vectors
chosen for the final form of the model were listed in Table 7.2.
In order to complete the linear component of the system model, the modal mass, stiffness, and damping matrices were
converted from modal coordinates to absolute coordinates. The modal vector matrix was used to transform between
coordinates. In order verify the accuracy of the linear system of equations, the FRFs were calculated from the equations
and compared to experimental data. The driving point position versus force FRF for this simple linear system was calculated
from the linear system representation by matrix inversion. This measurement was compared to the experimental data as
shown in Fig. 7.6. It was seen that the model approximated the experimental FRF well despite the simplifications that were
made throughout the estimation process. The artificial rigid body mode also matched the low frequency content well. Note
that the system only estimates the two dominant modes from the symmetric loading case, where the loading used to generate
the experimental FRF contains more modal content.
7.3.3 Model Adjustments to Account for Preload and Nonlinear Behavior
The linear system of equations that was formulated for the system were found to be an accurate representation of the
experimental data, but the model in this initial form only had the capacity to recreate one set of test conditions. Previous
examination of experimental data indicated that these two modal frequencies showed the tendency to shift upward with
increases in preload. The damping was also known to reduce as preload was increased. A model was developed to describe
the stiffness increase as a function of initial preload.
76 C. Butner et al.
After reviewing the literature, it was found that the modeling approach for bolted joint stiffness varied from source to
source. While some textbook references modeled bolted joint stiffness as a constant value, finite element modeling
approaches in [2] determined that joint stiffness was a function of applied load. Since no analytical formula for joint
stiffness as a function of bolt preload was readily available, it was decided to examine experimental data for a trend that
would reveal the proper form for the model. The expression for the natural frequency of a single degree of freedom system in
terms of stiffness and mass was given as follows:
on ¼ffiffiffiffiffi
K
M
r
: (7.1)
After solving this expression for the stiffness term K, the following expression was obtained:
K ¼ Mo2n: (7.2)
It was then assumed that the modal mass underwent minimal change with changes in preload; therefore, the stiffness was
found to be proportional to the natural frequency squared.
It was expected that changes in bolt preload would affect both of the modes of vibration of interest in different ways.
While the increase of preload would result in an increase in modal frequency for both modes, the ratio of stiffness change
would not be identical for both modes, since the type of motion exited was different. For this reason an expression was
calculated for the stiffness of the spring that dominated each mode separately. The K2 spring in Fig. 7.5 was the primary
spring that participated (deflected to a large degree) in the first mode of vibration, while the K3 spring dominated the second
mode. The model for joint stiffness was determined in the same manner for each mode. First, the known preload values were
0 500 1000 1500 2000 2500 300010-13
10-12
10-11
10-10
10-9
10-8
10-7
10-6
Frequency (Hz)
|FR
F| (
m/N
)
Fig. 7.6 Comparison of driving point position versus force FRF from experimental data (blue) and three degree of freedom system of equations
(green) (color figure online)
Table 7.1 Modal parameter
estimates used for three degree
of freedom system model
Mode Mr (kg) Cr (N s/m) Kr (N/m)
1 50 624.9 4441.3
2 40.7 312.4 7.385e+08
3 18.3 328.1 5.609e+09
Table 7.2 Modal vectors
used for three degree
of freedom system model
– Mode 1 Mode 2 Mode 3
Mass 1 1.0000 0.8626 �0.2316
Mass 2 1.0000 1.0000 �0.2316
Mass 3 1.0000 1.0000 1.0000
7 Simplified Nonlinear Modeling Approach for a Bolted Interface Test Fixture 77
plotted with their corresponding natural frequency squared for each mode, and a power function was fit to this data. These
power functions where then scaled appropriately for each mode by taking the ratio of the corresponding spring stiffness and
natural frequency from the linear system equations and scaling the power functions by that ratio. The result was a curve that
enabled the spring stiffness estimate to be calculated for any preload value in ft-lbs, where the hand-tight preload was
estimated as 1 ftlb. The stiffness curves for the K2 and K3 springs were plotted in Figs. 7.7 and 7.8, respectively. The shapes
of these curves indicated that when preload was adjusted from a very low value to a mid-range value, the stiffness and natural
frequency changed significantly, but changed little for continued increases in preload. This data was consistent with the
results predicted by Moo-Zung Lee in [7], and was experimentally observed by Caccese et al. in [9]. A decrease in damping
was also seen in the system with increased preload as discussed previously. This trend was not as clear, and the proper
frequency response function data that would be required to develop a curve for the damping values was not available from
the shock testing results, since the input forces could not be measured. For this reason, the damping values were changed
proportionally to the stiffness values, but in the opposite direction. As stiffness was increased, damping was decreased
proportionally because of the nature of the experimentally observed results that indicated the damping decreased with
increases in static preload level.
The final portion of the model that needed to be determined was the cubic stiffness nonlinearity. This nonlinearity was
chosen because of its use in the literature, as well as its ability to allow for the stiffening behavior that was observed
0 50 100 1502
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4 x 1010
Preload (ft-lb)
Con
nect
ing
Spr
ing
K2
Stif
fnes
s (N
/m)
Fig. 7.7 K2 spring stiffness versus preload level
0 50 100 1502
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4 x 109
Preload (ft-lb)
Con
nect
ing
Spr
ing
K3
Stif
fnes
s (N
/m)
Fig. 7.8 K3 spring stiffness versus preload level
78 C. Butner et al.
experimentally. The goal of this nonlinear term was to produce an increase in stiffness for either very small initial preloads or
very large impacts. This nonlinearity was a result of the impact force that enabled the motions of the masses on either side of
the bolt to overcome the initial preload across the interface, and effectively increased the stiffness in the joint. For this
reason, a cubic spring was implemented that only contributed stiffness to the system when the relative motion between the
two bodies it connected deflected in a way that caused a compression of the spring. When the compression was relaxed,
the cubic spring would not cause a stiffness lower than the initial joint stiffness. The model set the contribution of the cubic
stiffness to zero when elongation of the joint was detected in the numerical simulation.
A larger force resulted in a larger deflection, which would make the effect of a cubic spring more evident. The stiffness
coefficients for the nonlinear terms were tuned so that small deflections would not result in a detectable effect from the cubic
spring and the system would behave linearly. On the other hand, a large deflection would cause an appreciable effect from the
nonlinear spring. When examining the response of the linear system, it was determined that relative deflection values between
the masses had overlapping regions between high and low preload levels for the given impact amplitude range. Since the cubic
stiffness nonlinearity should have only affected the low preload level substantially for the impact amplitude range being tested,
it was determined that a scale factor was needed to achieve this effect. Since the system became more resistant to separation at
higher preloads, it was decided to model this scale factor inversely to the stiffness model. The lowest preload level tested in the
model as 1 ft-lb. For this value the relative deflections between masses was scaled by one, or held the same. As preload was
increased slightly, the scale factor decreased quickly in magnitude, but decreased less for further increases in preload.
This scale factor allowed for the system to be more resistant to nonlinear behavior at higher preloads.
7.3.4 Numerical Solution Approach for Nonlinear Modal
With the addition of the cubic stiffness terms into the model, the simple solution methods that were utilized previously were
no longer sufficient. Newmark’s method was chosen to solve the nonlinear form of the model. The solution procedure was
taken from the Geradin and Rixen textbook [11]. This method consisted of the calculation of the displacement, velocity, and
acceleration vectors at each time step for a given forcing function. The estimates were adjusted through an iterative
procedure until a specified convergence criterion had been achieved. In the model, the forcing function that was used was
an impulse that lasted for only one time step. This was an idealized approximation of a shock impact. The peak amplitude
was scaled accordingly. Once the displacement, velocity, and acceleration time histories had been calculated, FRFs were
created and compared. Preloads of 1 ft-lb, 50 ft-lb, and 100 ft-lb were simulated, with force levels of 2,500 lbf, 5,000 lbf,
7,500 lbf, and 10,000 lbf. The force estimates used where incremented proportionally to the levels used in the shock testing,
but the actual force inputs where unknown.
7.3.5 Results of Model Simulation
Once the model solution had been obtained for all of the test parameters, the resulting FRFs were examined to determine if
the model accurately represented the behavior of the test fixture. First, the changes in natural frequency and damping with
preload value were examined. The results for all three preload levels and the 2,500 lbf impact were plotted in Fig. 7.9.
This force was low enough to minimize the nonlinear effects for the low preload case. It was seen in the plot that the model
contained the same frequencies that were observed in the experimental data for each preload level. A trend of increasing
natural frequencies with preload was also evident in the plot. The damping reduction was also evident, as the peaks became
more narrow and higher in amplitude as preload was increased.
After it was determined that the model properly reflected changes in modal frequency and damping due to changes in
initial preload, the FRFs were examined to determine if the cubic stiffness nonlinearity did a proper job of modeling the
system response to large forces at low preload levels. Figure 7.10 shows the FRFs for all impact amplitudes for the low
preload level zoomed in on mode one. It was seen that the nonlinearity was obviously excited as the FRFs were not all the
same. For a linear system, the FRFs would not change as a function of impact amplitude. The peaks shifted upward in
frequency, while they also broadened and decreased in amplitude. This indicated that the modal frequency increased as
expected from the shock testing results, and the damping increased as seen in the earlier results from the modal tests.
The only issue with these results was the peak distortion and harmonics introduced by the cubic stiffness term. This was not
seen in the experimental data, but all other aspects of the model forced response matched the experimental data.
Once the low preload results from the model were determined to be sufficiently accurate, the 50–100 ftlb cases were
examined. Figure 7.11 shows the FRFs for all impact amplitudes at the 50 ftlb preload level, also zoomed in on mode one.
7 Simplified Nonlinear Modeling Approach for a Bolted Interface Test Fixture 79
It can be seen from these plots that the model response was nearly linear for the higher preload ranges for the impact
amplitudes that were tested. This result was consistent with what was noted in experimental results. The resistance to
separation caused by the high preload levels suppressed the effects of the nonlinear cubic stiffness terms.
Once the performance of the model was tested, the results of the simulation were compared to the experimental data from
the shock testing experiments. Since accurate input force information was not available for the shock input tests, FRFs where
not available for comparison. For this reason, the acceleration spectra from the model and shock experiments where
compared. Figure 7.12 shows the hand tight preload level for all impact air pressures used. This data is what revealed the
increasing natural frequency with increasing impact amplitude behavior for high amplitude shock loading. When the model
acceleration traces from the driving point measurements for each of the impact amplitudes were first examined, it was
determined that the responses were much more lightly damped than those seen in the experimental data. This is because
despite the fact that the shock data trends where used to tune the nonlinear aspects of the model, the base linear
characteristics where taken from a low amplitude modal test. The slapping damping mechanism was not activated for
very low impact amplitudes, so damping behavior was expected to differ for high amplitude tests. The model damping was
increased in order to achieve a similar frequency range to those seen in the experimental data. The result of the increased
damping model is shown in Fig. 7.13. With the increase in damping, the general shapes of both the model results and the data
matched very well. In both cases, a common backbone is shared by the different responses, but the increased amplitude
0 500 1000 1500 2000 2500 300010-12
10-11
10-10
10-9
10-7
10-6
Frequency (Hz)
|FR
F| (
m/N
)10-8
Fig. 7.9 Model position FRFs for hand-tight preload (blue), 50 ftlb preload (green), and 100 ftlb preload (red) for the 2,500 lbf impact (color
figure online)
600 605 610 615 620 625
10-8
10-7
Frequency (Hz)
|FR
F|(
m/N
)
Fig. 7.10 First mode of model position FRFs for 2,500 lbf impact (blue), 5,000 lbf impact (green), 7,500 lbf impact (red), and 10,000 lbf impact
(cyan) for the hand-tight preload level (color figure online)
80 C. Butner et al.
640 650 660 670 680 690 700 710 720
10-8
10-7
Frequency (Hz)
|FR
F| (
m/N
)
Fig. 7.11 First mode of model position FRFs for 2,500 lbf impact (blue), 5,000 lbf impact (green), 7,500 lbf impact (red), and 10,000 lbf impact
(cyan) for the 50 ftlb preload level (color figure online)
500 550 600 650 700 750 800 850 9000
2
4
6
8
10
12
14
Frequency (Hz)
Am
plitu
de o
f Spe
ctru
m (
m/s
^2/H
z)
5psi Impact10psi Impact15psi Impact20psi Impact
Fig. 7.12 Accelerometer one frequency spectra for hand tight preload level and all impact amplitudes
500 550 600 650 700 750 800 850 9000
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Frequency (Hz)
Am
plitu
de o
f Spe
ctru
m (
m/s
^2/H
z)
Fig. 7.13 Driving point frequency spectra from model with increased damping for hand tight preload level and 2,500 lbf (blue) 5,000 lbf (green)7,500 lbf (red) and 10,000 lbf (cyan) impact amplitudes (color figure online)
7 Simplified Nonlinear Modeling Approach for a Bolted Interface Test Fixture 81
led to a larger frequency excitation range. This confirmed that the cubic stiffness nonlinearity was fairly accurate.
The major difference in the plots however was the difference in scale. The experiential data was excited to acceleration
levels that were two orders of magnitude above the model results.
The large difference in amplitude was the result of several contributing factors. The first of these was that the model only
consisted of two modes, where the experimental data had contributions from many more high frequency, and weakly excited
low frequency modes of vibration. Each of these modes contributed a small residual that increased the amplitude of the
spectra. This alone cannot explain such a large difference. The model results could be scaled to similar levels to those seen in
the experiments by simply reducing the values of the mass, stiffness, and damping matrices by two orders of magnitude.
Reducing these matrices is equivalent to increasing the force levels by two orders of magnitude, so the large difference in
values could be explained partially by inaccurate forcing levels. The force levels used in the model where chosen to be close
to the approximate force estimations from the shock testing. The force estimates where known to be inaccurate due to system
nonlinearity, but another important factor in the accuracy of the estimates was how the training data was taken for these
approximations. The impacts where applied with a modal hammer to the fixture through a transfer bar. This means that the
predictions made using this data were for the force applied to the transfer bar, not the force applied to the circular plate.
The transfer bar introduced its own frequency response function and dynamics, so these two forces were not equal.
The transfer bar was necessary however to avoid damaging the fixture. The forces applied to the fixture could have been
much larger than the estimates indicated. Finally, it is known that the system dynamic properties changed with impact
amplitude range. This change in dynamics could have also resulted in changes to the coefficient matrices, which could have
also resulted in the change in response amplitude.
7.4 Conclusions
After the analysis of the nonlinear system model was complete, it was determined that the model performance was sufficient
at approximating the system response. However, the model was limited in several ways. The low order approach was based
on the assumption that the excitation was applied uniformly to the outer edge of the circular plate, or to the center of the
circular plate. This was only the case if the loads were applied exactly as desired in the use of the actual interface being
simulated in this research. The peak distortion and harmonics that were seen in the model results were also an undesirable
effect, but these effects were minimal if the excitation amplitude was not extremely large. One effect of the system that was
not included in this model was the preload relaxation effect. While this effect was known to exist, it was not known for
certain how this affected the system during an impact. It would certainly lead to a lower initial preload for the system
subsequent to an impact, but it was unknown how this would affect the modal response during a single impact. The primary
reason for this lack of information was that the shock data consisted of a very short time window length and lacked sufficient
frequency resolution to differentiate between frequency components from early and late portions of the response time
history. The model also resulted in inaccurate damping and response amplitude levels when compared to the shock data. This
was due to the differences in system dynamics for the shock data, and the data used to create the model. The model was
sufficient for many realistic simulations, and could be used to reasonably predict system response characteristics for
conditions that have yet to be tested.
References
1. Adams DE, Yoder N, Butner CM, Bono R, Foley J, Wolfson J (2010) Transmissibility analysis for state awareness in high bandwidth structures
under broadband loading conditions. In: Proceedings of IMAC XXVII, Jacksonville
2. Butner CM, Adams DE, Foley JR (2011) Understanding the effect of preload on the measurement of forces transmitted across a bolted
interface. In: Proceedings of IMAC XXIX, Jacksonville
3. Butner CM (2011) Investigation of the effects of bolt preload on the dynamic response of a bolted interface. Thesis, Purdue University
4. Ibrahim RA, Pettit CL (2005) Uncertainties and dynamic problems of bolted joints and other fasteners. J Sound Vib 279(3–5):857–936
5. Grosse IR, Mitchell LD (1990) Nonlinear axial stiffness characteristics of axisymmetric bolted joints. J Mech Des 112(3):442–449
6. Lehnhoff TF, Bunyard BA (2001) Effects of bolt threads on the stiffness of bolted joints. J Press Vessel Technol 123(2):161–165
7. Kenneth JK, Moo-Zung L (2010) Modeling the effects of bolt preload
8. Budynas RG, Nisbett JK (2008) Shigley’s mechanical engineering design, 8th edn. McGraw-Hill, New York
9. Caccese V, Mewer R, Vel SS (2004) Detection of bolt load loss in hybrid composite/metal bolted connections. Eng Struct 26(7):895–906
10. Inman DJ, Peairs DM, Park G (2001) Investigation of self-monitoring and self-healing bolted joints. In: Proceedings of 3rd international
workshop on structural health monitoring, Stanford, pp 430–439
11. Gradin M (1997) Mechanical vibrations. Wiley, New York
82 C. Butner et al.
Chapter 8
Transmission of Guided Waves Across Prestressed Interfaces
Jacob C. Dodson, Janet Wolfson, Jason R. Foley, and Daniel J. Inman
Abstract Prestresses can be both applied and/or environmentally generated and change the dynamics of structures. The
preload can change due environmental factors, such as temperature variations, and operational excitation including
impulsive loading. This study uses a novel Hopkinson bar configuration, the Preload Interface Bar, and investigates the
effect of prestress on the stress wave propagation across interfaces between the incident and transmission bars. The non-ideal
interface with a partial gap is experimentally investigated and analyzed. The preloaded partial gap causes generation of
flexural waves at the interface. The interface is modeled as a preload dependent elastic joint in which flexural and
longitudinal waves can be transmitted and reflected. The analytical and experimental energy transfer and mode conversion
from reflection and transmission will be examined in the spectral domain. The analysis shows that as the preload increases
the interface becomes stiffer and converges to a near-perfect 1-D interface.
8.1 Nomenclature
x longitudinal degree of freedom r reflection amplitude ratio Superscripts
y vertical degree of freedom t transmission efficiency A extensional strain
k wavenumber x reflection efficiency B bending stain
o angular frequency K joint stiffness Total total
A cross sectional area �K normalized joint stiffness transmission/reflection
h height u(x) axial displacement
c wave velocity v(x) vertical displacement Subscripts
E modulus of elasticity F(x) axial force L longitudinal
I area moment of inertia V (x) shear force B bending
r mass density M(x) moment n near field
n Poisson’s ratio y(x) angle of rotation 1 motion/force of bar 1
U wave amplitude s(x, t) stress 2 motion/force of bar 2
t transmission amplitude ratio, time exx(x, t) axial strain S motion/force of
interfacial spring
x property in x direction
y property in y direction
y property in y direction
J.C. Dodson (*)
Department of Mechanical Engineering, Virginia Polytechnic Institute and State University,
310 Durham Hall, Mail Code 0261 Blacksburg, VA
e-mail: [email protected]
J. Wolfson • J.R. Foley
Air Force Research Laboratory AFRL/RWMF, 306 W. Eglin Blvd., Bldg. 432, Eglin AFB, FL 32542–5430,
D.J. Inman
Department of Aerospace Engineering, University of Michigan Ann Arbor, MI 48109,
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_8, # The Society for Experimental Mechanics, Inc. 2012
83
8.2 Introduction
To predict the dynamics of a complex structure the dynamics of the individual components and the interfaces between them
must be understood. In an environment which prestresses the mechanical system and its components, the preloaded
interfaces are a crucial part that is not currently understood. Initial investigations on how the preload affects transmission
are very recent [1]. Older research shows that preload induces nonlinear effects on the response of structures such as
beams [2]. Due to the complexities of interfaces, they are often assumed to be ideal and they are modeled with the
compatibility conditions of the displacements and stresses being equal on both sides of the interface. While this approach
simplifies the problem it does not address lack of understanding of interfacial dynamics.
In the cases where non-ideal, or partial, interfaces occur a common approach is to use a contact stiffness model which
assumes an elastic force between the interfaces [3, 4]. Alternatively another method of modeling the partial interface
problem is to model the initial gap at the interface. Barber et al. examined the dynamics of an initial gap which closes
during the wave transmission between two semi-infinite media [5]. Daehnke and Rossmanith also addressed the case in
which a gap is initially open but the wave displacement closes the gap and allow transmission for the highest amplitude
part of the pulse [4]. Both papers generate piecewise displacement and force relations. While this method may be more
accurate in the time domain, modeling of the angled gap between two finite bars becomes difficult quickly and may be
addressed in a later paper. For this paper we will only use the contact stiffness method to model the preloaded imperfect
interface.
The longitudinal and shear waves propagating over various types of interfaces or joints in semi-infinite medium have
been examined in the geophysics field for waves due to mining explosives and seismic activity [4]. Daehnke and Rossmanith
discuss the reflection, transmission, and conversion of longitudinal and shear waves at; ideal interfaces, interfaces with
normal and shear stiffness values, and prestressed interfaces among others. The preload dependent stiffness values have been
empirically determined for various types of rocks, the authors hypothesize that as the preload increases the joints become
stiffer allowing more energy to propagate across the interface.
Guided waves are waves constructed of the reflections and refractions of the longitudinal and shear waves off the stress-
free boundaries of a finite structure. Guided waves propagate along the length of the structure and can be described by the
motion of the waves. Two of the primary wave modes are longitudinal and flexural, or bending. The mechanism for
converting and transmitting the wave modes across the interfaces in bars, beams, and plates are similar to the plane wave
interaction with the interfaces in a semi-infinite medium. The primary difference is the extra degree of freedom given to the
finite structures. In the 2-D semi-infinite medium the interface has only two directions it is able to move, while in the finite
bar, beam, and plate the interfaces at the end of the structure can also rotate. When Daehnke and Rossmanith modeled the
interface with stiffness values, they used a transverse and normal stiffness to transmit the shear and longitudinal
components [4]. Leung and Pinnington have investigated the transmission and reflection of incident longitudinal and
flexural waves on a right angle joint of two plates using linear spring and dashpot models utilizing three degrees of freedom:
axial displacement, vertical displacement and rotational motion [6, 7]. In order to accurately model the system, we have to
capture both axial and flexural motion of the joint. We will follow Leung and Pinnington’s approach and model the interface
as a parallel combination of linear and rotational springs whose values are preload dependent and relate to the longitudinal
and flexural modes in the bars.
This paper discusses the modeling and experimentation conducted on a imperfect interface under various preloads. The
novel Hopkinson bar configuration used, the Preload Interface Bar, has been previously discussed in an investigation
examining the effect of grease on the interface on the stress wave propagation across interfaces between the incident and
transmission bars [8]. The preload affects the transmission of axial waves and generation of flexural waves at the interface.
The interface is modeled with a contact stiffness model with three degrees of freedom and attempts to capture the dynamics
present in the experiment.
8.3 Theory
8.3.1 Axial Wave Propagation
The propagation of strain pulses along a uniform right cylindrical bar is a well-studied area of mechanics [9] that provides a
starting point for data analysis. The partial differential equation for elementary rod theory is given as [10, 11]
84 J.C. Dodson et al.
E@2uðx; tÞ@x2
� r@2uðx; tÞ
@t2¼ 0 (8.1)
where E is the elastic modulus and r is the density of the bar material. For an arbitrary function h(x, t), the time-space
Fourier transform is h(k, o) and is defined by
hðk;oÞ ¼Z 1
�1
Z 1
�1hðx; tÞe�iðkxþotÞdxdt (8.2)
and the inverse is given by
hðx1; tÞ ¼ 1
4p2
Z 1
�1
Z 1
�1hðk;oÞeiðkxþotÞdkdo (8.3)
where x is the spacial dimension, k is the spatial wavenumber, o is the angular frequency, and t is time. Applying the time-
space Fourier transform (8.2) to the elementary rod equation (8.1) yields
ð�Ek2 þ ro2Þuðk;oÞ ¼ 0: (8.4)
Solving for k give us the wavenumber relation for the bar
kbar ¼ oc
(8.5)
where c is the longitudinal elastic wave speed in a bar and is defined as
c ¼ffiffiffiE
r
s: (8.6)
In the development of the elementary rod differential equation (8.1) the assumed displacement relation is
�uðx; y; tÞ ¼ uðx; tÞ (8.7)
where �u(x, y, t) is the total longitudinal displacement field and u(x, t) is the longitudinal displacement at the midpoint of the
beam (y ¼ 0). For a longitudinal wave traveling down the bar the associated axial, or extensional, strain ExxA(x, t) can be
written as
EAxxðx; tÞ ¼@uðx; tÞ@x
: (8.8)
8.3.2 Flexural Wave Propagation
The propagation of bending, or flexural, waves can be typically modeled as satisfying the Euler-Bernoulli Beam or
Timoshenko Beam equation [10]. For simplicity of the argument we will use the Euler-Bernoulli Beam differential equation
given as
EI@4vðx; tÞ@x4
þ rA@2vðx; tÞ
@t2¼ 0 (8.9)
8 Transmission of Guided Waves Across Prestressed Interfaces 85
Applying the time-space Fourier transform (8.2) to the elementary rod equation (8.9) yields
ðEIk4 � rAo2Þvðk;oÞ ¼ 0 (8.10)
solving for the wavenumber gives
k2 ¼ �ffiffiffiffiffiffiffiffiffiffiffio2rAEI
r(8.11)
In the development of the Euler-Bernoulli beam differential equation (8.9) the assumed displacement relation is
�uðx; y; tÞ ¼ �y@vðx; tÞ@x
(8.12)
�vðx; y; tÞ ¼ vðx; tÞ (8.13)
where �vðx; y; tÞ is the total vertical displacement field and v(x, t) is the vertical displacement at the midpoint of the beam
y ¼ 0. The associated axial strain due to bending (ExxB) is
EBxxðx; y; tÞ ¼@�uðx; y; tÞ
@x¼ �y
@2vðx; tÞ@x2
(8.14)
8.3.3 Separation of Measured Strain
Using measured strain pairs on the surfaces of the beam we can decouple the strain due to bending and the strain due to
extension. If we assume a coupled elementary rod and Euler-Bernouilli beam model then our axial displacement would be
�uðx; y; tÞ ¼ uðx; tÞ � y@vðx; tÞ@x
(8.15)
and using the definitions of axial strain due to longitudinal and flexural waves from (8.8) and (8.14) our axial strain on the top
and bottom of the beam would be
Exxðx; h2; tÞ ¼ EAxxðx; tÞ � EBxxðx; tÞ (8.16)
Exxðx;� h
2; tÞ ¼ EAxxðx; tÞ þ EBxxðx; tÞ: (8.17)
Adding the measured strain on top (8.16) and bottom (8.17) will cancel the bending strain, while subtraction of the measured
strains will cancel the axial strain. From the strain measurements on the top and bottom of the beam we can decouple the
strain due to axial motion and the strain due to flexural motion. The decoupled stains can be written as
EAxxðx; tÞ ¼Exxðx; h2 ; tÞ þ Exxðx;� h
2; tÞ
2(8.18)
EBxxðx; tÞ ¼Exxðx;� h
2; tÞ � Exxðx; h2 ; tÞ2
: (8.19)
8.3.4 Elastic Joint Model
Leung and Pinnington have investigated the transmission and reflection of incident longitudinal and flexural waves on a right
angle joint using linear spring and dashpot models [6, 7]. This analysis sets up the problem in the same manner and models
86 J.C. Dodson et al.
the interface with elastic springs. The functional form of the transmitted and reflected waves are assumed and the amplitude
components are solved for using the force relations at the interface. Here we model the non-ideal interface as an elastic joint
with three degrees of freedom: axial displacement, vertical displacement and rotation. The model of the joint and the force
diagram can be seen in Fig. 8.1a, b, respectively.
In this analysis the elementary rod theory is assumed for axial motion and the Euler-Bernoulli beam theory is assumed for
transverse motion and rotation. The longitudinal and flexural waves are assumed to travel uncoupled along the semi-infinite
bars and coupling only takes place at the joint. We assume an longitudinal wave is incident into the joint.
Consider a longitudinal incident wave of amplitudeU(o) propagating into the joint in bar 1. The axial displacement in the
frequency domain u(x, o) is composed of the incident wave and the reflecting wave and can be written using the longitudinal
displacement as
u1ðx;oÞ ¼ UðoÞðe�ikbarðoÞx þ rLðoÞeikbarðoÞxÞ (8.20)
where i ¼ ffiffiffiffiffiffiffi�1p
, kbar(o) is the corresponding bar wavenumber as a function of frequency, and rL(o) is the amplitude of the
reflected longitudinal component. The transmitted longitudinal wave can be written as
u2ðx;oÞ ¼ UðoÞtLðoÞe�ikbarðoÞx (8.21)
where tL(o) is the amplitude of the transmitted longitudinal component. The reflected flexural wave can be written using the
vertical displacement as
v1ðx;oÞ ¼ UðoÞðrBðoÞeik1ðoÞx þ rnðoÞek2ðoÞxÞ (8.22)
where k1(o) and k2(o) are the corresponding beam wavenumbers as a function of frequency, rB(o) is the amplitude of the
reflected flexural component, and rn(o) is the amplitude of the reflected near field evanescent flexural component. The
transmitted flexural wave is
v2ðx;oÞ ¼ UðoÞðtBðoÞe�ik1ðoÞx þ tnðoÞe�k2ðoÞxÞ (8.23)
tB(o) is the amplitude of the transmitted flexural component, and tn(o) is the amplitude of the transmitted near field
evanescent flexural component. Using the force diagram in Fig. 8.1b the force, moment and displacement relations for the
elementary rod and Euler-Bernoulli beam can be written as [10]
M1ðx;oÞ ¼ EI@2v1ðx;oÞ
@x2M2ðx;oÞ ¼ EI
@2v2ðx;oÞ@x2
(8.24)
V1ðx;oÞ ¼ �EI@3v1ðx;oÞ
@x3V2ðx;oÞ ¼ �EI
@3v2ðx;oÞ@x3
(8.25)
F1ðx;oÞ ¼ �EA@u1ðx;oÞ
@xF2ðx;oÞ ¼ �EA
@u2ðx;oÞ@x
(8.26)
y1ðx;oÞ ¼ @v1ðx;oÞ@x
y2ðx;oÞ ¼ @v2ðx;oÞ@x
: (8.27)
∞ ∞
u1 u2v2v1
V2V1
VSFSMSF1 F2
θ1 θ2M1 M2
Ka
b
y
Kx
Kθ
∞ ∞
1
21
2
Fig. 8.1 The modeled
interface (a) spring models (b)
force diagram at the interface
of the bars
8 Transmission of Guided Waves Across Prestressed Interfaces 87
At the joint between the two rods we assume three linear elastic springs, which correspond to the three degrees of freedom in
the joint. The force and moment boundary conditions are applied at the joint (x ¼ 0) and are
M1ð0;oÞ ¼ M2ð0;oÞ ¼ �Kyðy2ð0;oÞ � y1ð0;oÞÞ (8.28)
V1ð0;oÞ ¼ V2ð0;oÞ ¼ �Kyðv2ð0;oÞ � v1ð0;oÞÞ � Kyxðu2ð0;oÞ � u1ð0;oÞÞ (8.29)
F1ð0;oÞ ¼ F2ð0;oÞ ¼ �Kxðu2ð0;oÞ � u1ð0;oÞÞ � Kxyðv2ð0;oÞ � v1ð0;oÞÞ (8.30)
where Ky, Ky, Kx, Kxy, and Kyx are the rotational, shear, axial, and cross stiffness values. Using (8.20)–(8.30) and assuming
that the area (A), elastic modulus (E) and area moment of inertia (I) are the same for both bars the unknown reflection and
transmission coefficients rB, rL, tB, and tL can be solved for. For each frequency the unknown coefficients can be written as
rL ¼ � kbarðk21k22 þ 2ik1 �Ky þ 2k2 �KyÞk21k
22ðkbar þ 2i�KxÞ þ k1ð2ikbar �Ky � 4�Kx
�Ky þ 4�Kxy�KyxÞ þ 2k2ðkbar �Ky þ 2ið�Kx
�Ky � �Kxy�KyxÞÞ
(8.31)
tL ¼ � 2ik21k22�Kx þ 4ik2ð�Kx
�Ky � Kxy�KyxÞ þ k1ð�4�KxKy þ 4�Kxy
�KyxÞk21k
22ðkbar þ 2i�KxÞ þ k1ð2ikbar �Ky � 4�Kx
�Ky þ 4�Kxy�KyxÞ þ 2k2ðkbar �Ky þ 2ið�Kx
�Ky � �Kxy�KyxÞÞ
(8.32)
rB ¼ �2k22kbar�Kyx
ð�ik1 þ k2Þðk21k22ðkbar þ 2i�KxÞ þ k1ð2ikbar �Ky � 4�Kx�Ky þ 4�Kxy
�KyxÞ þ 2k2ðkbar �Ky þ 2ið�Kx�Ky � �Kxy
�KyxÞÞÞ(8.33)
tB ¼ �rB (8.34)
where the dimensionless terms �Kx ¼ Kx
AE ,�Ky ¼ Ky
EI ,�Kyx ¼ Kyx
EI , and�Kxy ¼ Kxy
AE are the stiffness ratios. The ratios �Ky and �Kyx are
the ratios of the respective interfacial stiffness to the bending stiffness of the beam (EI) while �Kx and �Kxy are the ratios of the
respective interfacial stiffness to the axial stiffness of the bar (AE). Note that Ky does not appear in the solution so we can set
Ky ¼ 0. Note that the cross stiffness term Kyx governs the axial to flexural wave conversion. The preload has not been
explicitly modeled in the interface, the stiffness values �Kx, �Ky, �Kyx, and �Kxy are a function of the preload on the bars and will
be empirically determined.
The transmission and reflection efficiency are related measures with respect to the vibrational power [6, 7]. For materials
of the same cross-section and material properties the transmission efficiency t and the reflection efficiency x are
t ¼ transmitted vibrational power
incident vibrational power(8.35)
x ¼ reflected vibrational power
incident vibrational power: (8.36)
For the given problem we can write the transmission and reflection efficiencies for the transmitted and reflected longitudinal
waves as
tLðoÞ ¼ jtLðoÞj2 xLðoÞ ¼ jrLðoÞj2 (8.37)
and the efficiencies for the bending or flexural waves as
tBðoÞ ¼ jtBðoÞj2 xBðoÞ ¼ jrBðoÞj2 (8.38)
To test the solution we will examine a few simple cases. If we assume no connections between the rods choose �Kx ¼ �Ky ¼�Kyx ¼ �Kxy ¼ 0 then the transmission and reflection efficiencies become
xLðoÞ ¼ 1 tLðoÞ ¼ 0 xBðoÞ ¼ 0 tBðoÞ ¼ 0
88 J.C. Dodson et al.
for all frequencies o, which are the efficiencies for the ideal free end of the rod, full axial reflection and no transmission or
flexural conversion. If we assume only axial motion through a joint choose �Ky ¼ �Kyx ¼ �Kxy ¼ 0 and let �Kx ! 1 then the
transmission and reflection efficiencies become
xLðoÞ ¼ 0 tLðoÞ ¼ 1 xBðoÞ ¼ 0 tBðoÞ ¼ 0
for all frequencies o, which are the efficiencies for a stiff joint at the intersection of the rods, we have full axial transmission
and no reflection or flexural conversion. One more thing to note is that the transmitted and reflected efficiencies of the
flexural wave are always equal
xBðoÞ ¼ tBðoÞ (8.39)
this implies the ability to generate transmitted and reflected flexural waves of equal amplitudes at the interface that will
propagate away from the interface in both bars.
8.3.5 Total Transmission and Reflection
The reflection and transmission efficiencies were developed to be a function of frequency. In order to compare the
transmission and reflection of various preloads, the total transmission and reflection efficiency will be used and is the sum
of the efficiencies over frequency and can be written as
tTotal ¼Po transmitted vibrational powerPo incident vibrational power
(8.40)
xTotal ¼Po reflected vibrational powerPo incident vibrational power
: (8.41)
The efficiencies tTotal and xTotal are the total energy transmitted or reflected across the interface.
8.4 Apparatus
At the Air Force Research Laboratory, Munitions Directorate, Fuzes Branch (ARFL/RWMF), Dr. Jason Foley, Dr. Alain
Beliveau, and Dr. Janet Wolfson have developed the Preload Interface Bar (PIB) located at Eglin AFB [8]. The preload
interface apparatus puts two bars in compression and introduces a dynamic signal to propagate across the interface. The PIB
setup can be seen in Fig. 8.2a.
Interface
Hydraulic PressBackstop
Transmission Bar4' Steel Bar - 1.5 Ø
Incident Bar8' Steel Bar - 4' at 1.5Ø + 4' at 1.25" Ø Strain Gages
24" 18" 18"12"
Dynamicloading
ImpactHammer
a
b
Fig. 8.2 The preload interface bar (a) experimental setup (b) static and dynamic loading on the bar
8 Transmission of Guided Waves Across Prestressed Interfaces 89
Both incident and transmission bars in the preload interface bar setup are hardened AISI 1566 steel (r ¼ 7.8 g/cm3,
E ¼ 210 GPa, n ¼ 0.29). The transmission bar is 4 ft long and has a diameter of 1.5 in., while the incident bar is 8 ft long
with two 4 ft sections. One section has a diameter of 1.5 in. and the other has a diameter of 1.25 in.. This second bar is set up
such that the smaller diameter runs through a backstop. The two 1.5 in. diameter sections are compressed against each other
using a hydraulic ram. This causes the interface to be under a compressional load. A dynamic load is imparted on the bar on
the 1.25 in. diameter section and it propagates through the backstop, into the 1.5 in. diameter section and across the
interface. Because the bars were not match finished, with no preload the ends of the bars are aligned, but not square with one
another. This can be seen in Fig. 8.3.
8.4.1 Instrumentation
Load cells, both dynamic and static are used to input and record the forces imparted on the preload interface bar. The
dynamic response of the incident and transmission bars were recorded using strain gages along the bars at various preload
values. We also used pressure sensitive film to monitor the interface between the bars as the preload changes.
Multiple uniaxial strain gages aremounted on both the incident and transmission bar along the two 4 ft sections that are under
preload. The axial distribution of the gages allows the tracking of the stress wave propagating through the system. The gages
locations are measured from the backstop and are located at 24, 42, 54, and 72 in.. The 24 and 72 in. locations have four strain
gages placed axially on the bar, one every 90 ∘ . The 42 and 54 in. locations only have two gages distributed axially, one every
180 ∘ . The axial distribution of the gages is to use the diametrically opposed pairs to allow bending and extensional cancellation
for the observation of either only the axial or flexural mode respectively as shown in (8.18) and (8.19). Uniaxial semiconductor
strain gages were used in this setup for their fast response time ( � 10 ns) and correspondingly large bandwidth ( � 10 MHz).
The gage length is typically � 1 mm and the gage factor is around 150 [12], providing orders-of-magnitude of improvement in
sensitivity over typical foil gages with gage factors of 2. One common disadvantage of semiconductor gages is a strong
temperature dependence, this is not a concern for the dynamic tests run due to the short duration ( � 10 ms).
Two different load cells were used in the experiment, the first was a dynamic impact hammer and the second was a static
load cell. A PCB model 086C04 impact hammer was used to record the impulse force input to the system. A static load cell
was used to determine the amount of preload the hydraulic press put on the system. Tests were run with the preload from the
hydraulic press at 0–5,500 lb force. Also medium range (1,400–7,000 psi) Fujifilm Prescale pressure sensitive film was used
to monitor the loading of the interface with increments of 1,000 lb of preload on the system up to 5,000 lb.
The data acquisition system used was a National Instruments PXIe-1065 chassis with a NI PXIe-8130 Controller using
two PXI 6133 cards. Triggering is based on the analog channel with the reference force hammer. Signal conditioning and
sensor power is accomplished via a Precision Filter 28000 chassis with 28144A Quad-Channel Wideband Transducer
Conditioner with voltage and current excitation cards. The data from the strain gages and force hammer was recorded with a
sampling rate of 2.5 Mhz and Precision Filter system run with open bandwidth (� 800 kHz).
8.5 Results and Discussion
The several sets of tests were conducted using the preload interface bar. Sets of five tests were run with the hydraulic press at
15 different preload values ranging from 0 to 5,500 lb force. Assuming the conservation of mechanical energy over the
interface, the incoming longitudinal wave can be transmitted or reflected, and in doing so part of it may be converted to a
flexural wave.
Fig. 8.3 Partial gap between
the aligned bars with no
preload
90 J.C. Dodson et al.
The ends of the bars are aligned, but not square with one another. Pressure sensitive film was used to investigate how
uniform the load distribution is over the cross-section of the bar. The film was put between the bars at increments of 1,000 lb
pf preload. The medium range (1,400–7,000 psi) pressure sensitive film was used. The dark pink has a stress of 7,000 psi or
greater, while the white areas have stress values of less than 1,400 psi. The varying color saturation between white and dark
pink corresponds to a linear scale of stress values between 1,400 and 7,000 psi. The film can be seen in Fig. 8.4. Due to the
type of finishing used on the ends of the bars, the the interface is only loaded in the outer 0.25 in. annulus of the bar as can be
seen in Fig. 8.4a–e. As the preload increases the gap closes, this is evident with the pressure sensitive film.Experimental
stress transmission and reflection coefficients were calculated and can be seen in Fig. 8.5. At low preloads (0–2,000 lb),
when the interface is partially open it can be seen that flexural waves are generated at the interface and transmitted and
reflected at near equal amplitudes. A point of interest is that the amplitude of the flexural transmission and reflection
efficiencies (tBtotal and xB
total) are around 3, much greater than the theoretical maximum of 1. This indicates that we are not
accounting for energy that is contributing to the flexural waves. We hypothesize that this unaccounted energy may be due to
the preload on the system, the incident energy used was only the incident longitudinal wave. Over the entire preload range
the longitudinal transmission of energy increases as the preload increases due to the stiffening of the interface. The
transmission acts much similar to a smoothed step function, only allowing a partial longitudinal transmission until the
critical preload range, between 1,000 and 2,000 lb, is reached which then the interface is closed and allows full transmission
of the longitudinal wave and no generation of the flexural wave. Notice that the change from generating flexural waves to
transmission of pure longitudinal waves is gradual over the preload range of 1,000–2,000 lb.
Time histories of the axial and flexural strain at representative low and high preload values of 100 and 4,500 lb can be
seen in Fig. 8.6. The longitudinal wave has a reflection at the interface for when the preload is 100 lb but no reflection in the
4,500 lb preload case. There is close to perfect axial transmission for a preload of 4,500 lb, and a small flexural wave that
travels the length of the bar generated with the impact hammer. When the preload is 100 lb a flexural wave is generated at
the interface and superimposed on the small flexural wave, we know this because the gages at 42 and 54 in. first see the large
flexural wave and then the gages at 24 and 72 in. see it close to 200 ms later.
Fig. 8.4 Pressure sensitive
film at the interface at various
preloads: (a) 1,000 lb
(b) 2,000 lb (c) 3,000 lb
(d) 4,000 lb (e) 5,000 lb
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500
0
0.5
1
1.5
2
2.5
3
Preload [lbs]
Tra
nsm
ission
and
Ref
lect
ion
Effic
ienc
ies τ
L
Total
ξL
Total
τB
Total
ξB
Total
Fig. 8.5 Experimental total
transmission and reflection
efficiencies as a function of
preload
8 Transmission of Guided Waves Across Prestressed Interfaces 91
Using (8.31)–(8.34) we constructed a numerical model and tried to match the transmission and reflection efficiencies to
the experimental data. A nonlinear least squares algorithm was used to minimize the error between the experimental data and
theory on the frequency interval of 0–10 kHz. The upper frequency bound of 10 kHz was chosen to allow for enough
frequency excitation amplitude to give us meaningful experimental values, the excitation amplitude dies off around 15 kHz.
We used 500 randomly generated initial dimensionless stiffness values for each preload value to find the stiffness values thatminimize the error. The stiffness values for each preload value can be seen in Fig. 8.7.
For the low preload regime (0–1,000 lb) the transverse stiffness ratio �Ky is at the upper bound (10,000), the transverse
cross stiffness ratio �Kyx is gradually increasing with preload, the longitudinal cross-stiffness ratio �Kxy remains constant
1200 1400 1600 1800 2000 2200-20
-10
0
10
20Longitudinal Strain time histories, Interface Bar - 4 ft Steel: Preload 100 lb
Time t [μs] Time t [μs]
Time t [μs] Time t [μs]
Mic
rost
rain
e(t)
[μm
/m]
24-0042-0054-0072-00
1200 1400 1600 1800 2000 2200-20
-10
0
10
20Flexural Strain time histories, Interface Bar - 4 ft Steel: Preload 100 lb
Mic
rost
rain
e(t)
[μm
/m]
1200 1400 1600 1800 2000 2200-20
-10
0
10
20Longitudinal Strain time histories, Interface Bar - 4 ft Steel: Preload 4500 lb
Mic
rost
rain
e(t)
[μm
/m]
24-0042-0054-0072-00
1200 1400 1600 1800 2000 2200-20
-10
0
10
20Flexural Strain time histories, Interface Bar - 4 ft Steel: Preload 4500 lb
Mic
rost
rain
e(t)
[μm
/m]
a b
Fig. 8.6 Strain response to impulse hammer with a preload of (a) 100 lb (b) 4,500 lb
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 55000
500
1000
1500
2000
Preload [lb]
Optimized Stiffness Values
Dim
ension
less
Stiffne
ss
Ky
Kx
Kxy
Kyx
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 55000
2000
4000
6000
8000
10000
Preload [lb]
Optimized Stiffness Values
Dim
ension
less
Stiffne
ss
Ky
Kx
Kxy
Kyx
a b
Fig. 8.7 Numerical dimensionless stiffness values calculated as a function of preload (a) full value range (b) closer view of dimensionless
stiffness ratios in the range from 0 to 2,000
92 J.C. Dodson et al.
around 80, and the longitudinal stiffness ratio �Kx is low ( < 10). In the transition preload regime (1,000–2,000 lb) the
transverse stiffness ratio �Ky and the longitudinal cross-stiffness ratio �Kxy drops to a near 0 values, the transverse cross
stiffness ratio �Kyx jumps to the upper bound (10,000), and the longitudinal stiffness ratio �Kx increases with preload to around
500. For the high preload regime ( � 2,000 lb) the only value that changes is the longitudinal stiffness ratio �Kx which
generally increases as the preload in the system increases. This implies that the interface is stiffing with the preload and
agrees with the general trend of Daehnke and Rossmanith’s empirical-based model [4].
The transmission and reflection efficiencies of 100 and 4,500 lb preloads are shown in Fig. 8.8. The dimensionless stiffness
values used to generate the analytical model at 100 lb preload shown in Fig. 8.8a are: �Kx ¼ 8, �Ky ¼ 10; 000, �Kxy ¼ 76 and,�Kyx ¼ 1; 420. For the 4,500 lb preload shown in Fig. 8.8b the dimensionless stiffnesses are: �Ky ¼ 0, �Kx ¼ 930, �Kxy ¼ 0,�Kyx ¼ 10; 000. Themodeling of the efficiencies for the 4,500 lb of preloadmatches verywellwith a least square error residual of
0.26, this is due to the near perfect interface created by the preload over the frequency range of 0–10 kHz. The model behavior
for preloads below 2,000 lb do not match the experimental data very well, the least square error residual is 30.34. Themismatch
of the model and the experimental data can be seen in Fig. 8.8a. This is due to the fidelity of the model. The elementary rod
theory and Euler-Bernoulli beam theory were assumed in the model development, these models do have the correct degrees of
freedom, but do not accurately capture the spectral dynamics of the interface. The model moderately captures the general trend,
but needs to be refined to more accurately capture the dynamics for the cases where preload is less than 2,000 lb.
8.6 Conclusion
The effect of the partial gap interface on the transmission, reflection, and conversion of waves was experimentally and
analytically analyzed. It was experimentally shown that at low preloads (0–2,000 lb) the partial gap generated near-equal
amplitude transmitted and reflected flexural waves, and a small amount of the energy was longitudinally transmitted. At high
1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
Frequency [kHz]
Transmission and Reflection Efficiencies Interface Bar - 4 ft Steel: Preload 100 lb
ξ L
TheoryExperiment
1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
Frequency [kHz]
ξ L
1 2 3 4 5 6 7 8 9 100
1
2
Frequency [kHz]
τ B
1 2 3 4 5 6 7 8 9 100
1
2
Frequency [kHz]
ξ B
ξ Lξ L
τ Bξ B
1 2 3 4 5 6 7 8 9 100
1
1.5
0.5
0
1
1.5
0.5
Frequency [kHz]
Transmission and Reflection Efficiencies Interface Bar - 4 ft Steel: Preload 4500 lb
TheoryExperiment
1 2 3 4 5 6 7 8 9 10Frequency [kHz]
1 2 3 4 5 6 7 8 9 100
1
2
Frequency [kHz]
1 2 3 4 5 6 7 8 9 100
1
2
Frequency [kHz]
a b
Fig. 8.8 Transmission and reflection efficiencies with a preload of and stiffness coefficients of (a) 100 lb with �Kx ¼ 8, �Ky ¼ 10; 000, �Kxy ¼ 76,�Kyx ¼ 1; 420 (b) 4,500 lb �Ky ¼ 0, �Kx ¼ 930, �Kxy ¼ 0, �Kyx ¼ 10; 000
8 Transmission of Guided Waves Across Prestressed Interfaces 93
preloads ( � 2,000 lb) the system allowed near 100% londitudinal transmission and no flexural wave generation at the
interface. An analytical model was developed modeling the partial gap interface as an elastic stiffness with cross terms. The
analytical model did not adequately predict the low preload transmission and reflection due to lack of spectral fidelity, but
did capture the high preload transmission. The model shows that as the preload increases above 2,000 lb, the stiffness of the
interface increases. Future work includes refining the model to accurately caputure the spring gap behavior at low preloads.
This analysis gives us some insight to see how preloaded intefaces act under dynamic loading, taking us one step closer to
fully understanding a complex prestressed mechanical system.
Acknowledgements J. Dodson would like to acknowledge support from the Department of Defense SMART (Science, Mathematics, And
Research Transformation) scholarship program. The authors also wish to thank the Air Force Office of Scientific Research (PM: Dr. David Stargel)
for supporting this project. Opinions, interpretations, conclusions and recommendations are those of the authors and are not necessarily endorsed
by the United States Air Force.
References
1. Adams D, Yoder N, Butner C, Bono R, Foley J, Wolfson J (2011) Transmissibility analysis for state awareness in high bandwidth structures
under broadband loading conditions. Nonlinear modeling and applications, vol. 2. In: Proulx T (ed) Vol. 11 of Conference proceedings of the
society for experimental mechanics series, Springer, New York, pp 137–148
2. Chattopadhyay S (1993) Dynamic response of preloaded joints. J Sound Vib 163(3):527–534
3. Offterdinger K, Waschkies E (2004) Temperature dependence of the ultrasonic transmission through electrical resistance heated imperfect
metal-metal interfaces. NDT & E Int 37(5):361–371
4. Daehnke A, Rossmanith H.P (1997) Reflection and refraction of plane stress waves at interfaces modelling various rock joints. Fragblast: Int J
Blasting Fragm 1(2):111–231
5. Barber J, Comninou M, Dundurs J (1982) Contact transmission of wave motion between two solids with an initial gap. Int J Solids Struct 18
(9):775–781
6. Leung R, Pinnington R (1990) Wave propagation through right-angled joints with compliance-flexural incident wave. J Sound Vib 142
(1):31–48
7. Leung R, Pinnington R (1992) Wave propagation through right-angled joints with compliance: longitudinal incidence wave. J Sound Vib 153
(2):223–237
8. Foley JR, Dodson JC, McKinion CM, Luk VK, Falbo GL (2010) Split Hopkinson bar experiments of preloaded interfaces. In: Proceedings of
the IMPLAST 2010 conference, SEM, 2010
9. Kolsky H, (1963) Stress waves in solid. Clarendon Press, Oxford
10. Doyle JF (1997) Wave propagation in structures : spectral analysis using fast discrete Fourier transforms, 2nd edn. Springer, New York
11. Inman DJ (2007) Engineering Vibration, Pearson, Upper Saddle River
12. Kulite Semiconductor Products, Kulite Strain Gage Manual, 2001
94 J.C. Dodson et al.
Chapter 9
Equivalent Reduced Model Technique Development
for Nonlinear System Dynamic Response
Louis Thibault, Peter Avitabile, Jason R. Foley, and Janet Wolfson
Abstract The dynamic response of structural systems commonly involves nonlinear effects. Often times, structural
systems are made up of several components, whose individual behavior is essentially linear compared to the total
assembled system. However, the assembly of linear components using highly nonlinear connection elements or contact
regions causes the entire system to become nonlinear. Conventional transient nonlinear integration of the equations of
motion can be extremely computationally intensive, especially when the finite element models describing the components
are very large and detailed.
In this work, the Equivalent Reduced Model Technique (ERMT) is developed to address complicated nonlinear contact
problems. ERMT utilizes a highly accurate model reduction scheme, the System Equivalent Reduction Expansion Process
(SEREP). Extremely reduced order models that provide dynamic characteristics of linear components, which are
interconnected with highly nonlinear connection elements, are formulated with SEREP for the dynamic response evaluation
using direct integration techniques. The full-space solution will be compared to the response obtained using drastically
reduced models to make evident the usefulness of the technique for a variety of analytical cases.
Keywords Nonlinear analysis • Forced response • Linear components for nonlinear analysis • Reduced order modeling •
Modal analysis
Nomenclature
Symbols
½Xn� Full set displacement vector
½Xa� Reduced set displacement vector
½Xd� Deleted set displacement vector
Ma½ � Reduced mass matrix
Mn½ � Expanded mass matrix
Ka½ � Reduced stiffness matrix
Kn½ � Expanded stiffness matrix
Ua½ � Reduced set shape matrix
Un½ � Full set shape matrix
Ua½ �g Generalized inverse
L. Thibault (*) • P. Avitabile
Structural Dynamics and Acoustic Systems Laboratory, University of Massachusetts Lowell, One University Avenue,
Lowell, MA 01854, USA
e-mail: [email protected]
J.R. Foley • J. Wolfson
Air Force Research Laboratory, Munitions Directorate, Fuzes Branch, Eglin Air Force Base, 306 W. Eglin Blvd., Bldg 432,
Eglin AFB, FL 32542-5430, USA
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_9, # The Society for Experimental Mechanics, Inc. 2012
95
T½ � Transformation matrix
TU½ � SEREP transformation matrix
½p� Modal displacement vector
M½ � Physical mass matrix
C½ � Physical damping matrix
K½ � Physical stiffness matrix
½F� Physical force vector
½€x� Physical acceleration vector
½ _x� Physical velocity vector
½x� Physical displacement vector
a Parameter for Newmark integration
b Parameter for Newmark integration
Dt Time step
U12½ � Mode contribution matrix
9.1 Introduction
Nonlinear response analysis typically involves significant computation, especially if the system matrices for the full
analytical model are used to obtain the forced nonlinear response solution. Due to the significant computational resources
required for these types of nonlinear problems, the analyst may often be unable to investigate specific nonlinear scenarios in
depth, particularly if the nonlinear elements are characterized with a set of performance characteristics related to tempera-
ture, preload, deflection, etc. Thus, there is significant motivation to develop several reduced order models that can
accurately predict nonlinear system response with substantially reduced computation time.
A particular area of interest is the dynamic response of systems with nonlinear connections. These systems are typically
made up of several components, whose individual behavior is essentially linear compared to the total assembled system.
Local regions where component interconnections exist cause the entire system to become nonlinear. The components that
make up the system may be linear but the response of the system is nonlinear due to the nature of the nonlinear component
interconnection. The technique employed in this paper, the Equivalent Reduced Model Technique (ERMT), was developed
to address this class of nonlinear problem.
ERMT [1] is implemented in this work using the System Equivalent Reduction Expansion Process (SEREP) [2], which
allows for the formulation of a dramatically reduced model that accurately preserves the full analytical model dynamics with
very few degrees of freedom (DOFs). Discrete nonlinear connection elements are then assembled into the reduced model in
the local regions where component interconnections occur. Using the reduced models that are developed with ERMT in
conjunction with direct integration techniques allows computationally efficient forced response solutions to be obtained.
These techniques can also be easily extended to experimental components if the system matrices are updated using any of the
direct model updating techniques such as those identified in [3].
This approach was first presented by Avitabile and O’Callahan [4] where a detailed overview of the applicable theory was
provided, along with a simple analytical example. Friswell et al. [5] looked at reducing models with local nonlinearities
using several different reduction schemes for a periodic solution. Lamarque and Janin [6] looked at modal superposition for
simple single-DOF and two-DOF systems with impact and concluded that modal superposition had limitations due to
difficulties in developing the general formulas with the nonlinear impacts. Ozguven and Kuran [7] converted nonlinear
Ordinary Differential Equations (ODE’s) into a set of nonlinear algebraic equations, which could be reduced by using linear
modes. This technique was found to provide the best reduction in computation time when the structure was excited at a
forcing frequency that corresponded to a resonance of the structure. An alternative approach that has been studied uses
Nonlinear Normal Modes (NNMs) which are formulated by Ritz vectors [8, 9]. This approach seeks to extend the concept of
linear orthogonal modes to nonlinear systems.
This paper presents the analytical time response results of a cantilevered beam system subjected to an input force pulse.
Four cases are studied: single beam with no contact, single beam with single contact, two beams with single contact, and two
beams with two contacts. For each of these cases, two types of contact stiffness are considered, a soft contact representing a
rubber/isolation material, and a hard contact representing a metal on metal contact. For all cases, the time response results of
the full-space model will be used as the reference solution.
96 L. Thibault et al.
9.2 Theory
9.2.1 Equivalent Reduced Model Technique (ERMT)
TheEquivalentReducedModel Technique (ERMT) is based on concepts related tomodel reduction,which are summarized herein.
9.2.1.1 General Reduction Techniques
Model reduction is typically performed to reduce the size of a large analytical model to develop a more efficient model for
further analytical studies. Most reduction or condensation techniques affect the dynamic characteristics of the resulting
reduced model. Model reduction is performed for a number of reasons, but the technique is used primarily as a mapping
technique for expansion. In general, a relationship between the full set of finite element DOFs and the reduced set of DOFs
needs to be formed as
Xnf g ¼ Xa
Xd
� �¼ T½ � Xaf g (9.1)
The ‘n’ subscript denotes the full set of finite element DOFs, the ‘a’ subscript denotes the active set of DOFs (sometimes
referred to as master DOFs), and the subscript ‘d’ denotes the deleted DOFs (sometimes referred to as omitted DOFs); the
[T] transformation matrix relates the mapping between these two sets of DOFs.
The reduced mass and stiffness matrices are related to the full-space mass and stiffness matrices using congruent matrix
operations as
Ma½ � ¼ T½ �T Mn½ � T½ � and Ka½ � ¼ T½ �T Kn½ � T½ � (9.2)
What is most important in model reduction is that the eigenvalues and eigenvectors of the original system are preserved as
accurately as possible in the reduction process. If this is not maintained then the matrices are of questionable value.
The eigensolution is then given by
Ka½ � � l Ma½ �½ � Xaf g ¼ 0f g (9.3)
Because reduction schemes such as Guyan Condensation [10] and Improved Reduced System Technique [11] are based
primarily on the stiffness of the system, the eigenvalues and eigenvectors will not be exactly reproduced in the reduced
model. However, the System Equivalent Reduction Expansion Process (SEREP) [2] exactly preserves the eigenvalues
and eigenvectors in the reduced model.
9.2.1.2 System Equivalent Reduction Expansion Process (SEREP)
The SEREP modal transformation relies on the partitioning of the modal equations representing the system DOFs relative to
the modal DOFs using
Xnf g ¼ Xa
Xd
� �¼ Ua
Ud
� �pf g (9.4)
Using a generalized inverse, this can be manipulated to give
pf g ¼ Ua½ �T Ua½ �� ��1
Ua½ �T Xaf g ¼ Ua½ �g Xaf g (9.5)
which is then used to relate the ‘n’ DOFs to the ‘a’ DOFs as
Xnf g ¼ Un½ � Ua½ �g Xaf g ¼ TU½ � Xaf g (9.6)
with
TU½ � ¼ Un½ � Ua½ �g (9.7)
9 Equivalent Reduced Model Technique Development for Nonlinear System Dynamic Response 97
Equation (9.7) represents the SEREP transformation matrix that is used in the reduction of the finite element mass and
stiffness matrices as described in (9.2).
SEREP relies heavily on a “well developed” finite element dynamic model from which an ‘n’ dimensional eigensolution
is obtained. In addition, the quality of the SEREP reduced model depends on the selection of active ‘a’ DOFs used in the
formulation of the generalized matrix inverse of the ‘a’ DOFs modal vector partition. Both conditions affect the rank and
matrix conditioning required to define a good SEREP transformation matrix needed to develop well-behaved reduced system
matrices. The reduced models developed in this work are for the equivalent condition, where the number of modes used is
equal to the number of DOFs retained in the reduction process.
Since the transformation matrix is formed from the eigenvectors of the full-space finite element model, the reduced
matrices preserve the eigenvalues and eigenvectors of the full-space model. This implies that any collection of desired
eigenvectors can be retained in an exact sense for the reduced model. This fact is significant in terms of the development of
efficient models from large finite element models used for forced response studies, especially those that contain discrete
nonlinear effects that are typical in joints and connections for many structural systems.
An additional model reduction approach [12] can be used where a highly accurate reduced model is formulated by using
Guyan reduction in conjunction with analytical model improvement. The advantages of this technique are that the desired
mode shapes are preserved in an exact sense while the fully ranked and well-conditioned system matrices obtained from
Guyan reduction are maintained in the process.
9.2.1.3 Mode Contribution Identification
To ensure that the reduced models are minimally affected by modal truncation when assembling system models using
multiple reduced component models, the contribution of component modes to the assembled system is computed using
Mode Contribution ¼ UAn
� UB
n
� � �TMAB
n
� UAB
n
� (9.8)
where [UA] and [UB] are the unmodified component modal matrices that are organized into a partitioned matrix, [MAB] is the
modified system mass matrix, and [UAB] is the modified system modal matrix. The mode contribution matrix is computed
for all possible system configurations using full-space component models.
The resulting mode contribution matrix is the key to identifying the necessary set of modal vectors to accurately obtain
the final modified system modes. This is similar to using the [U12] matrix that is computed in Structural Dynamic
Modification (SDM) [13] to identify the contributions of component modes in the assembled system modes. The [U12]
matrix contains the scaling coefficients needed to form the final modified set of modal vectors [U2] from the initial
unmodified set of modal vectors [U1]. Figure 9.1 illustrates how the [U12] matrix is used in forming the final modified set
of modes [U2], where ‘m’ modes of the [U12] matrix are used, and ‘n-m’ modes are excluded.
Fig. 9.1 Mode contribution identification using [U12] matrix from SDM
98 L. Thibault et al.
9.2.1.4 Response Analysis Technique
The reduced component mass and stiffness matrices described in (9.2) are used in a normal system assembly to connect the
linear components with highly nonlinear connection elements. Nonlinear direct integration of the equations of motion is then
performed to obtain the system response. The ERMT process is shown schematically in Fig. 9.2.
9.2.1.5 Newmark Direct Integration Technique
In this work, the Newmark Method [14] is used to perform the direct integration of the equations of motion for the ERMT
solution process. From the known initial conditions for displacement and velocity, the initial acceleration vector is computed
using the equation of motion and the applied forces as
€~x0 ¼ M½ ��1 ~F0 � C½ � _~x0 � K½ �~x0� �
(9.9)
where€~x0 ¼ initial acceleration vector_~x0 ¼ initial velocity vector
x0 ¼ initial displacement vector~F0 ¼ initial force vector
Choosing an appropriate Dt, a, and b, the displacement vector is
~xiþ1 ¼ 1
a Dtð Þ2 M½ � þ baDt
C½ � þ K½ �" #�1
~Fiþ1 þ M½ � 1
a Dtð Þ2~xi þ1
aDt_~xi þ 1
2a� 1
�€~xi
!(
þ C½ � baDt
~xi þ ba� 1
�_~xi þ b
a� 2
�Dt2€~xi
�)(9.10)
FORMULATE PHYSICALDATA BASE
PERFORM NUMERICAL INTEGRATION FOR NEXT ΔT
CHECK FOR GAPS ORNONLINEARITY
ANY CHANGE IN THE CURRENT LINEAR PHYSICAL STATE?
SUBSTITUTE M, C, K MATRICES FOR APPLICABLE
CONTACT STATE
DETERMINE FORCE AND/ORINITIAL CONDITIONS
NO YES
Fig. 9.2 Schematic for ERMT process
9 Equivalent Reduced Model Technique Development for Nonlinear System Dynamic Response 99
The values chosen for a and b were ¼ and ½, respectively. This assumes constant acceleration and the integration process is
unconditionally stable, where a reasonable solution will always be reached regardless of the time step used. However, the time
step should be chosen such that the highest frequency involved in the system response can be characterized properly to avoid
numerical damping in the solution. The time step should be chosen to be at least 10 times smaller than the period of the highest
frequency involved in the system response. The time step used for the analytical cases studied in this paper was 0.0001 s.
Following the displacement vector calculation, the acceleration and velocity vectors are computed for the next time
step using
_~xiþ1 ¼ _~xi þ 1� bð ÞDt€~xi þ bDt€~xiþ1 (9.11)
€~xiþ1 ¼ 1
a Dtð Þ2 ~xiþ1 �~xið Þ � 1
aDt_~xi � 1
2a� 1
�€~xi (9.12)
This process is repeated at each time step for the duration of the time response solution desired.
9.2.2 Time Response Correlation Tools
In order to quantitatively compare two different time solutions, two correlation tools were employed: The Modal Assurance
Criterion (MAC) and the Time Response Assurance Criterion (TRAC).
9.2.2.1 Modal Assurance Criterion (MAC)
The Modal Assurance Criterion (MAC) [15] is widely used as a vector correlation tool. In this work, the MAC was used to
correlate all DOF at a single instance in time. The MAC is written as
MACij ¼X1if gT X2j
� � 2X1if gT X1if g�
X2j� T
X2j� h i (9.13)
where X1 and X2 are displacement vectors. MAC values close to 1.0 indicate strong similarity between vectors, where
values close to 0.0 indicate minimal or no similarity.
9.2.2.2 Time Response Assurance Criterion (TRAC)
The Time Response Assurance Criterion (TRAC) [3] quantifies the similarity between a single DOF across all instances in
time. The TRAC is written as
TRACji ¼X1j� T
X2if gh i2
X1j� T
X1j� h i
X2if gT X2if g� (9.14)
where X1 and X2 are displacement vectors. TRAC values close to 1.0 indicate strong similarity between vectors, where
values close to 0.0 indicate minimal or no similarity.
9.3 Model Description and Cases Studied
This section presents the analytical models developed as well as the cases studied. The full-space time solution is used as the
reference solution for all cases.
100 L. Thibault et al.
9.3.1 Linear Component Models: Beam A and Beam B
Two planar element beam models were generated using MAT_SAP [16], which is a finite element modeling (FEM) program
developed for MATLAB [17], and were used for all of the cases studied. Figure 9.3 shows the two beams assembled into
the linear system, where the red points are the active DOFs in the reduced order models, and the black arrow denotes the
force pulse input location (DOF 105). Note that 3 in. of each beam are clamped for the cantilevered boundary condition
that was applied.
Table 9.1 lists the characteristics of the beam models and Table 9.2 lists the natural frequencies for the first 10 modes of
each beam component model. The mode shapes for the unmodified beam components are provided in Appendix A. Damping
was assumed 1% of critical damping for all unmodified component modes as well as for all modified system modes in all of
the cases studied.
The force pulse input to the system is an analytic force pulse designed to be frequency band-limited, exciting modes up to
1,000 Hz while minimally exciting higher order modes. Using this force pulse, the number of modes involved in the response
can be determined easily, as modes above 1,000 Hz can be considered to have negligible participation in the response.
Modes 1–5 will be primarily excited in Beam A, while modes 1–4 will be primarily excited in Beam B. Figure 9.4 shows the
analytical force pulse in the time and frequency domain.
The full-space linear component models were reduced down to ‘a’ and ‘aa’ space using SEREP. The active DOFs and
modes retained in the reduced component models are listed in Table 9.3. Note that only translational DOF were used in the
reduced component models.
F
Full ‘n’ Space
‘a’ Space
‘aa’ Space
F
F
Beam A
Beam A
Beam A
Beam B
Beam B
Beam B
Fig. 9.3 Schematic of linear
beam models with force pulse
input location
Table 9.1 Beam model
characteristicsProperty Beam A Beam B
Length (in.) 18 16
Width (in.) 2 4
Thickness (in.) 0.123 0.123
# of elements 72 64
# of nodes 73 65
# of DOF 146 130
Node spacing (in.) 0.25 0.25
Material Aluminum Aluminum
Mass density (lb/in.3) 2.54E�4 2.54E�4
Young’s modulus (Msi) 10 10
9 Equivalent Reduced Model Technique Development for Nonlinear System Dynamic Response 101
9.4 Case 1: Single Beam with No Contact
For the first case, the system is a single beam (Beam A) with no contact. This system is linear and there is no change in the
state of the system, where the number of modes needed is only a function of the input force spectrum. Based on the input
force spectrum seen previously in Fig. 9.4 and the natural frequencies for Beam A in Table 9.2, five modes are needed to
accurately compute the system time response. Therefore, the smallest reduced model that can be used to prevent the effects
of mode truncation is five DOF to maintain the SEREP condition, as discussed previously. To determine if this assumption is
accurate, the system response was plotted in the time and frequency domain for ‘aa’ space with comparison to the full ‘n’
space solution, as shown in Fig. 9.5. The response for the ‘a’ space model was not shown, due to the accurate results
observed from the ‘aa’ space model response.
Table 9.2 Natural frequencies
of unmodified beam component
models
Mode # Beam A Beam B
1 12.91 22.62
2 84.12 141.56
3 252.34 396.60
4 519.59 776.92
5 806.16 1,284.71
6 1,256.55 1,918.28
7 1,682.96 2,678.33
8 2,201.36 3,563.89
9 2,755.52 4,572.70
10 3,510.01 5,707.04
0 500 1000 1500-110
-105
-100
-95
-90
-85
-80
-75
-70
-65
-60
Frequency (Hz)
dB F
orce
(lb
f)
FFT of Analytical Force Pulse
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-20
-15
-10
-5
0
5
Time (sec)
For
ce (
lbf)
Analytical Time Domain Force Pulse
Fig. 9.4 Analytical force pulse in the time domain (left) and frequency domain (right)
Table 9.3 Active DOFs and modes retained in the reduced component models
Model # of DOF Retained modes Active DOF
Beam A – ‘a’ space 17 1–17 33, 41, 49, 57, 65, 73, 81, 89, 97, 101, 105, 113, 121, 129, 137, 141, 143
Beam A – ‘aa’ space 5 1–5 57, 81, 101, 105, 141
Beam B – ‘a’ space 14 1–14 3, 5, 9, 17, 25, 33, 41, 45, 49, 57, 65, 73, 81, 89
Beam B – ‘aa’ space 5 1–5 5, 25, 45, 65, 89
102 L. Thibault et al.
The system response in Fig. 9.5 shows that the reduced model is able to accurately capture the response of the system due
to the inclusion of the modes primarily excited by the input force pulse. To quantify the similarity of the reduced model
results with the full-space solution, the MAC and TRAC were computed between the full model and the reduced model time
responses, which were then averaged, as listed in Table 9.4. The solution time for each model is also listed to show the
decrease in computation time when reduced models are used.
Table 9.4 provides further confirmation that the ‘aa’ space reduced model is sufficient for accurately computing the time
response for this particular case. In addition, the solution time for the full-space model is over 30 s in contrast to the reduced
‘aa’ space model, which is less than 1 s. ERMT was shown in this case to provide significant reduction in computation time
for a linear system.
This first case demonstrates that an accurate time response solution can be obtained using a drastically reduced model
with limited number of modes, if the primary modes excited by the structure are retained in the reduced model. However, if
the retained modes are selected incorrectly, an accurate time response will not be obtained, regardless of how many
additional modes and active DOFs are retained in the reduced model.
The following cases will show the application of ERMT for contact situations, which causes the system to become
nonlinear. In addition, both soft and hard contacts will be studied to show the effect that different contact stiffness has on the
accuracy of the reduced model results. The soft stiffness case will be studied first, as this contact stiffness is unlikely to excite
a higher frequency range than the input force pulse applied to the models.
9.5 Case 2: Soft Contact
9.5.1 Case 2-A: Single Beam with Single Soft Contact
This case consists of the tip of Beam A coming into contact with a fixed object once the beam has displaced a known gap
distance of 0.05 in., as shown in Fig. 9.6 for the full-space and reduced space models.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Time (sec)
Dis
plac
emen
t (in
)DOF 141 Time Response - Single Beam No Contact
n Space-146 DOF
aa Space-5 DOF
0 500 1000 1500-200
-180
-160
-140
-120
-100
-80
-60
-40
Frequency (Hz)
dB D
ispl
acem
ent (
in)
FFT of DOF 141 Time Response - Single Beam No Contact
n Space-146 DOF
aa Space-5 DOF
Fig. 9.5 Comparison of ‘n’ and ‘aa’ spacemodels for single beamwith no contact for DOF 141 – time response (left) and FFT of time response (right)
Table 9.4 Average MAC and TRAC for reduced models and solution times for single beam with no contact
Model # of DOF Average MAC Average TRAC Solution time (s)
‘n’ space 146 1 1 34.6
‘a’ space 17 0.99999687 0.99999999 1.9
‘aa’ space 5 0.99993281 0.99999992 0.7
9 Equivalent Reduced Model Technique Development for Nonlinear System Dynamic Response 103
The contact is represented as an additional spring stiffness that is added when the contact location on Beam A closes the
specified gap distance. This is performed by exchanging the physical system matrices from the initial no contact state to
the new contact state until the contact is separated. The contact stiffness of 10 lb/in. will be used to represent a soft contact,
which is typically seen in a damper or isolation mount. For the purposes of ERMT, the spring stiffness was applied to the
full-space physical model, prior to model reduction. In addition, the contact is represented as a spring stiffness in only the
translational DOF, and not in the rotational DOF as well. The contact location is at DOF 141 of Beam A. Table 9.5 lists the
first ten natural frequencies of the system with the soft contact stiffness applied. Figure 9.7 shows the mode contribution
matrix, which indicates the unmodified component modes that participate in the modified system modes. The mode
contributions are computed using full-space models so that modal truncation is not a concern. The various box colors
indicate the amount that each of the unmodified component modes contributes to a modified system mode; the actual
contribution ranges for each color are shown. The mode shapes for the modified system are provided in Appendix A.
Table 9.5 and Fig. 9.7 show that the addition of the soft spring has a pronounced effect on the first two lower order modes,
with modes 3 and higher remaining relatively unaffected. Figure 9.7 indicates that both modes 1 and 2 of the unmodified
component are needed to obtain either mode 1 or 2 of the modified system. The frequencies for modes 3 and higher of the
unmodified component are minimally affected for the modified system. Therefore, additional unmodified component mode
shapes are not needed to form the modified system modes above mode 3 (as indicated by the red main diagonal). For a soft
contact, the input force pulse primarily dominates the frequency spectrum excited. Therefore, the modes required to
accurately predict the system response are modes 1–5 for the unmodified component as well as for the modified system.
To confirm this, the system response was plotted in the time and frequency domain for ‘aa’ space with comparison to the full
‘n’ space solution, as shown in Fig. 9.8. The response for the ‘a’ space model was not shown, due to the accurate results
observed from the ‘aa’ space model response.
Table 9.5 Natural frequencies
for single beam with single soft
contact
Mode # Unmodified Single soft contact
1 12.91 26.09
2 84.12 86.02
3 252.34 252.54
4 519.59 519.65
5 806.16 806.18
6 1,256.55 1,256.55
7 1,682.96 1,682.96
8 2,201.36 2,201.36
9 2,755.52 2,755.53
10 3,510.01 3,510.02
Full ‘n’ Space
‘a’ Space
‘aa’ Space
F
F
F
Fig. 9.6 Diagram of single
beam with single contact for
full-space and reduced space
models
104 L. Thibault et al.
The system response in Fig. 9.8 shows that the reduced model accurately captures the response of the system due to the
inclusion of the modes primarily excited by the input force pulse. To quantify the similarity of the reduced model results with
the full-space solution, the MAC and TRAC were computed between the full model and the reduced model time responses,
which were then averaged, as listed in Table 9.6. The solution time for each model is also listed to show the decrease in
computation time when reduced models are used.
Table 9.6 provides further confirmation that the ‘aa’ space reduced model is sufficient for accurately computing the time
response for this particular case. In addition, the solution time for the full-space model is over 30 s in contrast to the reduced
‘aa’ space model, which is less than 1 s. ERMT was shown in this case to provide significant reduction in computation time
for a nonlinear system that consists of a single component with a single soft contact.
Table 9.6 Average MAC and TRAC for reduced models and solution times for single beam with single soft contact
Model # of DOF Average MAC Average TRAC Solution time (s)
‘n’ space 146 1 1 34.8
‘a’ space 17 0.99999687 0.99999999 1.9
‘aa’ space 5 0.99993256 0.99999979 0.7
0 500 1000 1500-200
-150
-100
-50
Frequency (Hz)
dB D
ispl
acem
ent (
in)
FFT of DOF 141 Time Response - Single Beam Single Soft Contact
n Space-146 DOFaa Space-5 DOF
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
-0.1
-0.05
0
0.05
0.1
0.15
Time (sec)
Dis
plac
emen
t (in
)
DOF 141 Time Response - Single Beam Single Soft Contact
n Space-146 DOFaa Space-5 DOF
Fig. 9.8 Comparison of ‘n’ and ‘aa’ space models for single beam with single soft contact for DOF 141 – time response (left) and FFT of
time response (right)
MODEModified System Mode Shapes
Single 10 lb/in Contact
1 2 3 4 5 6 7 8 9 10
Unm
odifie
d B
eam
A M
ode
Sha
pes 1
2
3
4
5
6
7
8
9
10
Bar Color Min Value Max Value
Black 0.005 0.1
Blue 0.1 0.2
Green 0.2 0.3
Cyan 0.3 0.5
Magenta 0.5 0.7
Yellow 0.7 0.9
Red 0.9 1.0
Fig. 9.7 Mode contribution matrix for single beam with single soft contact
9 Equivalent Reduced Model Technique Development for Nonlinear System Dynamic Response 105
This second case demonstrated that the mode contribution matrix should be examined to determine the number of
unmodified component modes that are needed to obtain the modified system modes of interest. The input force pulse may
excite only a few modified system modes, but several unmodified component modes may be required to produce the
modified system modes. Failure to include the contributing modes will result in an incorrect time response, regardless of how
many additional modes and active DOFs are retained in the reduced model. When the mode contribution matrix is examined
and the correct unmodified component modes are used in the reduced model, an accurate nonlinear time response solution
can be obtained using a drastically reduced model.
9.5.2 Case 2-B: Two Beams with Single Soft Contact
This case consists of the tip of Beam A coming into contact with Beam B once the specified gap distance of 0.05 in. between
Beams A and B is closed, as shown in Fig. 9.9 for the full-space and reduced space models.
The same soft contact stiffness of 10 lb/in. was used to represent the contact of the beams as explained in Case 2-A.
The contact occurs at DOF 141 of Beam A and DOF 45 of Beam B. Table 9.7 lists the natural frequencies for the modified
system as well as for the unmodified components. The mode shapes for the modified system are provided in Appendix A.
Figure 9.10 shows the mode contribution matrix used for identifying the unmodified component modes that contribute in
the modified system modes.
Due to the contact between the two beams, the first two modes from both unmodified components are needed to obtain the
first mode of the modified system, which increases the number of modes needed in the reduced models. Based on the input
force pulse, which excites up to 1,000 Hz, the first 10 modes of the modified system are expected to be excited, which can be
seen in Table 9.7. Examining Fig. 9.10 shows that for the first 10 modes of the modified system, the first 6 modes of Beam A
and the first 4 modes of Beam B are needed to obtain the first 10 modes of the modified system. For a soft contact, the input
force pulse primarily dominates the frequency spectrum excited. Therefore, the modes required to accurately predict the
system response are those observed in the mode contribution matrix, as stated previously. To show the effects of mode
truncation, the system response was plotted in the time and frequency domain for ‘a’ space and ‘aa’ space with comparison
to the full ‘n’ space solution in Fig. 9.11, where the ‘aa’ space model is missing a key contributing mode in the modified
system response – mode 6 of Beam A.
The ‘a’ space model response in Fig. 9.11 can be observed to have very good correlation with the full-space solution;
however, the ‘aa’ space model response indicates that not including mode 6 of Beam A degrades the results obtained.
Full ‘n’ Space
‘a’ Space
‘aa’ Space
F
F
F
Beam A
Beam A
Beam A
Beam B
Beam B
Beam B
Fig. 9.9 Diagram of two
beams with single contact for
full-space and reduced space
models for configuration 1
106 L. Thibault et al.
To quantify the similarity of the reduced model results with the full-space solution, the MAC and TRAC were computed
between the full model and the reduced model time responses, which were then averaged, as listed in Table 9.8a. The solution
time for each model is also listed to show the decrease in computation time when reduced models are used.
MODEModified System Mode Shapes -10 lb/in Contact -Configuration 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Unm
odifie
d B
eam
A M
ode
Sha
pes
1
2
3
4
5
6
7
8
9
10
11
Unm
odifie
d B
eam
B M
ode
Sha
pes 1
2
3
4
5
6
7
8
9
10
Bar Color Min Value Max Value
Black 0.005 0.1
Blue 0.1 0.2
Green 0.2 0.3
Cyan 0.3 0.5
Magenta 0.5 0.7
Yellow 0.7 0.9
Red 0.9 1.0
Fig. 9.10 Mode contribution matrix for two beams with single soft contact for configuration 1
Table 9.7 Natural frequencies for two beams with single soft contact for configuration 1
Mode # Two beams, single soft contact
Unmodified components
Beam A Beam B
1 20.35 12.91 22.62
2 29.58 84.12 141.56
3 86.01 252.34 396.60
4 142.43 519.59 776.92
5 252.54 806.16 1,284.71
6 396.71 1,256.55 1,918.28
7 519.65 1,682.96 2,678.33
8 777.00 2,201.36 3,563.89
9 806.18 2,755.52 4,572.70
10 1,256.55 3,510.01 5,707.04
11 1,284.80 – –
12 1,682.96 – –
13 1,918.28 – –
14 2,201.36 – –
15 2,678.39 – –
16 2,755.53 – –
17 3,510.02 – –
18 3,563.89 – –
19 3,948.55 – –
20 4,572.72 – –
9 Equivalent Reduced Model Technique Development for Nonlinear System Dynamic Response 107
To improve the ‘aa’ space model results, the addition of DOF 143 and mode 6 from Beam A was included in the reduced
model. From the mode contribution matrix in Fig. 9.10, mode 6 of Beam A is the primary contributing mode that is needed to
obtain mode 10 of the modified system. The reduced system response that uses 11 DOF and 11 modes is plotted in the time
and frequency domain with comparison to the full ‘n’ space solution in Fig. 9.12.
The ‘aa’ space model time response in Fig. 9.12 shows significant improvement in the correlation with the full-space
model when an additional DOF and key contributing mode of Beam A is included in the reduced model. In Table 9.8b, the
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
-0.1
-0.05
0
0.05
0.1
0.15
Time (sec)
Dis
plac
emen
t (in
)DOF 141 Time Response - Two Beam Single Soft Contact
n Space-276 DOF
a Space-31 DOF
0 500 1000 1500-200
-150
-100
-50
Frequency (Hz)
dB D
ispl
acem
ent (
in)
FFT of DOF 141 Time Response - Two Beam Single Soft Contact
n Space-276 DOF
a Space-31 DOF
0 500 1000 1500-200
-150
-100
-50
0
50
Frequency (Hz)
dB D
ispl
acem
ent (
in)
FFT of DOF 141 Time Response - Two Beam Single Soft Contact
n Space-276 DOF
aa Space-10 DOF
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-4
-3
-2
-1
0
1
2
3
4
5
6
Time (sec)
Dis
plac
emen
t (in
)
DOF 141 Time Response - Two Beam Single Soft Contact
n Space-276 DOF
aa Space-10 DOF
Fig. 9.11 Comparison of ‘n’, ‘a’, and ‘aa’ space models for two beams with single soft contact for DOF 141 – time response (top) and FFT of time
response (bottom)
Table 9.8 Average MAC and TRAC for reduced models and solution times for two beams with single soft contact
Model # of DOF Average MAC Average TRAC Solution time (s)
(a) Incorrect number of modes used in ‘aa’ space model
‘n’ space 276 1 1 87.9
‘a’ space 31 0.99999686 0.99999952 4.8
‘aa’ space 10 0.67818808 0.10422630 1.1
(b) Suitable number of modes used in ‘aa’ space model
‘n’ space 276 1 1 87.9
‘a’ space 31 0.99999686 0.99999952 4.8
‘aa’ space 11 0.99996327 0.99999980 1.1
108 L. Thibault et al.
MAC is improved from 0.678 to 0.999 and the TRAC is improved from 0.104 to 0.999 with essentially no change in
the solution time when DOF 143 and mode 6 from Beam A are included in the reduced model. This shows that significant
errors can result when key contributing component modes are not included in the assembled system for reduced models.
The improved MAC and TRAC results further confirm that the 11 DOF reduced model is sufficient for accurately computing
the time response for this particular case. In addition, the solution time for the full-space model is approximately 90 s in
contrast to the ‘aa’ space reduced model, which is only 1 s. ERMT was shown in this case to provide significant reduction
in computation time for a nonlinear multiple component system with a single soft contact while maintaining a highly
accurate time response solution.
9.5.3 Case 2-C: Two Beams with Multiple Soft Contacts
This case consists of Beam A coming into contact with Beam B in three different configurations at two possible contact
locations with a specified gap of 0.05 in., as shown in Fig. 9.13. Note that each system is a potential configuration of the two
components depending on the relative displacements of the two beams; no contact is also a possible configuration.
The same soft contact stiffness of 10 lb/in. was used to represent the contact of the beams as explained in Case 2-A.
The contact for configuration 1 occurs at DOF 141 of Beam A and DOF 45 of Beam B. The contact for configuration
2 occurs at DOF 101 of Beam A and DOF 5 of Beam B. The contact for configuration 3 occurs when both contacts for
configurations 1 and 2 are closed simultaneously. Table 9.9 lists the natural frequencies for the modified system
configurations as well as for the unmodified components. The mode shapes for the modified system configurations are
provided in Appendix A. Figure 9.14 shows the mode contribution matrices used for identifying the unmodified component
modes that participate in the modified system modes for the three possible system configurations.
For this case, three potential modified system configurations exist, which results in three separate mode contri-
bution matrices. Based on the input force pulse, which excites up to 1,000 Hz, the first 10 modes of the modified system
for all three configurations are expected to be excited, which can be seen in Table 9.9. Examining Fig. 9.14 indicates that for
the first 10 modes of the modified system for all three configurations, the first six modes of Beam A and the first four modes
of Beam B are needed to obtain the first 10 modes of the modified system for all potential configurations. For a soft contact,
the input force pulse primarily dominates the frequency spectrum excited. Therefore, the unmodified component modes
required to accurately predict the system response are those observed in the mode contribution matrices, as stated previously.
Table 9.9 and Fig. 9.14 also show that not only does the additional spring stiffness affect the number of unmodified
component modes needed, but the location of the spring affects the combination of modes needed for each configuration as
well. Depending on whether the spring is at the tip of Beam A or the tip of Beam B, the mode shapes and frequencies change
0 500 1000 1500-200
-150
-100
-50
Frequency (Hz)
dB D
ispl
acem
ent (
in)
FFT of DOF 141 Time Response - Two Beam Single Soft Contact
n Space-276 DOF
aa Space-11 DOF
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
-0.1
-0.05
0
0.05
0.1
0.15
Time (sec)
Dis
plac
emen
t (in
)DOF 141 Time Response - Two Beam Single Soft Contact
n Space-276 DOF
aa Space-11 DOF
Fig. 9.12 Comparison of ‘n’ and ‘aa’ space models for two beams with single soft contact for DOF 141 – time response (left) and FFT of time
response (right)
9 Equivalent Reduced Model Technique Development for Nonlinear System Dynamic Response 109
noticeably. For example, the second mode of Beam B is needed to obtain the first mode of the modified system for
configurations 1 and 3, but is not needed in configuration 2. To show the effects mode truncation has on the solution, the
system response was plotted in the time and frequency domain for ‘a’ space and ‘aa’ space with comparison to the full ‘n’
space solution in Fig. 9.15, where the ‘aa’ space model is missing a key contributing mode in the modified system response –
mode 6 of Beam A.
The ‘a’ space model response in Fig. 9.15 can be observed to have very good correlation with the full-space solution;
however, the ‘aa’ space model response indicates that not including mode 6 of Beam A degrades the results obtained.
To quantify the similarity of the reduced model results with the full-space solution, the MAC and TRAC were computed
between the full model and the reduced model time responses, which were then averaged, as listed in Table 9.10a.
The solution time for each model is also listed to show the decrease in computation time when reduced models are used.
Table 9.9 Frequencies for two beams with multiple soft contact configurations
Mode #
Modified system – soft contacts Unmodified components
Config. 1 Config. 2 Config. 3 Beam A Beam B
1 20.35 14.78 21.08 12.91 22.62
2 29.58 33.04 39.23 84.12 141.56
3 86.01 85.71 87.24 252.34 396.60
4 142.43 142.98 143.82 519.59 776.92
5 252.54 252.61 252.82 806.16 1,284.71
6 396.71 396.96 397.07 1,256.55 1,918.28
7 519.65 519.61 519.68 1,682.96 2,678.33
8 777.00 777.04 777.12 2,201.36 3,563.89
9 806.18 806.28 806.30 2,755.52 4,572.70
10 1,256.55 1,256.56 1,256.56 3,510.01 5,707.04
11 1,284.80 1,284.76 1,284.85 – –
12 1,682.96 1,682.99 1,682.99 – –
13 1,918.28 1,918.29 1,918.30 – –
14 2,201.36 2,201.37 2,201.37 – –
15 2,678.39 2,678.34 2,678.39 – –
16 2,755.53 2,755.53 2,755.53 – –
17 3,510.02 3,510.02 3,510.03 – –
18 3,563.89 3,563.89 3,563.89 – –
19 3,948.55 3,948.54 3,948.55 – –
20 4,572.72 4,572.70 4,572.72 – –
Configuration 1
Configuration 2
Configuration 3
Beam A
Beam A
Beam A
Beam B
Beam B
Beam B
Fig. 9.13 Diagram of two
beams with multiple contacts
for configurations 1, 2, and 3
110 L. Thibault et al.
0 500 1000 1500-200
-150
-100
-50
0
50
Frequency (Hz)
dB D
ispl
acem
ent
(in)
FFT of DOF 141 Time Response - Two Beam Two Soft Contacts
n Space-276 DOF
aa Space-10 DOF
0 500 1000 1500-200
-150
-100
-50
Frequency (Hz)
dB D
ispl
acem
ent
(in)
FFT of DOF 141 Time Response - Two Beam Two Soft Contacts
n Space-276 DOF
a Space-31 DOF
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
-0.1
-0.05
0
0.05
0.1
0.15
Time (sec)
Dis
plac
emen
t (in
)
DOF 141 Time Response - Two Beam Two Soft Contacts
n Space-276 DOF
a Space-31 DOF
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-4
-3
-2
-1
0
1
2
3
4
5
6
Time (sec)
Dis
plac
emen
t (in
)DOF 141 Time Response - Two Beam Two Soft Contacts
n Space-276 DOF
aa Space-10 DOF
Fig. 9.15 Comparison of ‘n’, ‘a’, and ‘aa’ space models for two beams with multiple soft contacts for DOF 141 – time response (top) and FFT of
time response (bottom)
MODEModified System Mode Shapes -10 lb/in Contact - Configuration 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Unm
odifi
ed B
eam
A M
ode
Shap
es
1234567891011
Unm
odifi
ed B
eam
B M
ode
Shap
es 12345678910
Bar Color Min Value Max ValueBlack 0.005 0.1Blue 0.1 0.2Green 0.2 0.3Cyan 0.3 0.5Magenta 0.5 0.7Yellow 0.7 0.9Red 0.9 1.0
MODEModified System Mode Shapes -10 lb/in Contact - Configuration 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Unm
odifi
ed B
eam
A M
ode
Shap
es
1234567891011
Unm
odifi
ed B
eam
B M
ode
Shap
es 12345678910
MODEModified System Mode Shapes - 10 lb/in Contact - Configuration 3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Unm
odifi
ed B
eam
A M
ode S
hape
s 1234567891011
Unm
odifi
ed B
eam
B M
ode S
hape
s 12345678910
Fig. 9.14 Mode contribution matrix for two beams with soft contacts for configurations 1, 2, and 3
As seen in Case 2-B, the mode contribution matrix provides a good understanding of the effects mode truncation has on
the solution obtained. Therefore, to improve the ‘aa’ space results, the addition of DOF 143 and mode 6 from Beam A was
included in the reduced model. From the mode contribution matrices in Fig. 9.14, mode 6 of Beam A is the primary
contributing mode that is needed to obtain mode 10 of the modified system in all three configurations. The reduced system
response that uses 11 DOF and 11 modes is plotted in the time and frequency domain with comparison to the full ‘n’ space
solution in Fig. 9.16.
The ‘aa’ space model time response in Fig. 9.16 shows significant improvement in the correlation with the full-space
model when an additional DOF and key contributing mode of Beam A is included in the reduced model. In Table 9.10b, the
MAC is improved from 0.678 to 0.999 and the TRAC is improved from 0.104 to 0.999 with essentially no change in
the solution time when DOF 143 and mode 6 from Beam A are included in the reduced model. The improved MAC and
TRAC results further confirm that the 11 DOF reduced model is sufficient for accurately computing the time response for
this particular case. In addition, the solution time for the full-space model is approximately 90 s in contrast to the ‘aa’
space reduced model, which is only 1 s. ERMT was shown in this case to provide significant reduction in computation
time for a nonlinear multiple component system with multiple soft contacts while maintaining a highly accurate time
response solution.
The system response for this case is the same as in Case 2-B, which only has a single potential contact location, rather
than two as in this case. In the cases to follow where hard contact stiffness is studied, the second contact location does come
into contact, however for the soft contact stiffness, contact did not occur at all of the possible locations. This case was
included to maintain continuity with the hard contact stiffness cases studied in following section. The soft contact models
analyzed in Case 2 demonstrate how the mode participation matrix can be used to identify the component modes that are
needed to generate accurate time response solutions using reduced models. Case 2 also shows that ERMT can provide highly
accurate results with significantly reduced computation time for nonlinear systems with soft contacts.
0 500 1000 1500-200
-150
-100
-50
Frequency (Hz)
dB D
ispl
acem
ent (
in)
FFT of DOF 141 Time Response - Two Beam Two Soft Contacts
n Space-276 DOF
aa Space-11 DOF
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
-0.1
-0.05
0
0.05
0.1
0.15
Time (sec)
Dis
plac
emen
t (in
)
DOF 141 Time Response - Two Beam Two Soft Contacts
n Space-276 DOF
aa Space-11 DOF
Fig. 9.16 Comparison of ‘n’ and ‘aa’ space models for two beams with multiple soft contacts for DOF 141 – time response (left) and FFT of time
response (right)
Table 9.10 Average MAC and TRAC for reduced models and solution times for two beams with multiple soft contacts
Model # of DOF Average MAC Average TRAC Solution time (s)
(a) Incorrect number of modes used in ‘aa’ space model
‘n’ space 276 1 1 89.7
‘a’ space 31 0.99999686 0.99999952 4.9
‘aa’ space 10 0.67818808 0.10422630 1.1
(b) Suitable number of modes used in ‘aa’ space model
‘n’ space 276 1 1 89.7
‘a’ space 31 0.99999686 0.99999952 4.9
‘aa’ space 11 0.99996327 0.99999980 1.1
112 L. Thibault et al.
9.6 Case 3: Hard Contact
For the soft contact stiffness models discussed in Case 2, the contact stiffness was soft enough that the contact did not excite
a frequency bandwidth beyond the input force spectrum. Thus, the number of modes needed in the time response was
controlled by the forcing function, and the contributing unmodified component modes needed to obtain the modified system
modes were based on the mode contribution matrix. As long as the frequency bandwidth excited by the contact stiffness is
below the highest frequency excited by the input force pulse, the procedure for identifying the number of modes needed
works well, as in Case 2. However, for a harder contact stiffness situation, there is a possibility that the contact can excite a
frequency bandwidth beyond the input spectrum. Under this scenario, the number of unmodified component modes needed
in the modified system response would not be a function of the input spectrum, but of the contact stiffness. In order to
examine this possible scenario in detail, the same cases were studied as in Case 2, but with 1,000 lb/in. contact stiffness.
9.6.1 Case 3-A: Single Beam with Single Hard Contact
This case consists of the tip of Beam A coming into contact with a fixed object once the beam has displaced a known gap
distance of 0.05 in., as explained previously in Case 2-A. Table 9.11 lists the first 10 natural frequencies of the modified
system with the hard contact stiffness applied, while Fig. 9.17 shows the mode contribution matrix. The mode shapes for the
modified system are provided in Appendix A.
Table 9.11 and Fig. 9.17 indicate that the first 6 modes of the unmodified component are needed to obtain the first five
modes of the modified system. This is expected, as the mode shapes with the harder spring attached look less like the original
model and therefore requires more mode mixing to form the modified system modes. The frequencies of the unmodified
component for mode 6 and higher are minimally affected for the modified system, which do not require additional
unmodified component mode shapes to form the modified system modes (as indicated by the red main diagonal). Therefore,
six active DOF and the first six modes of the unmodified component are needed to accurately obtain the modified system
MODEModified System Mode Shapes
1000 lb/in Contact1 2 3 4 5 6 7 8 9 10
Unm
odifie
d B
eam
A M
ode
Sha
pes 1
2
3
4
5
6
7
8
9
10
Bar Color Min Value Max Value
Black 0.005 0.1
Blue 0.1 0.2
Green 0.2 0.3
Cyan 0.3 0.5
Magenta 0.5 0.7
Yellow 0.7 0.9
Red 0.9 1.0
Fig. 9.17 Mode contribution
matrix for single beam with
single hard contact
Table 9.11 Natural frequencies
for single beam with single hard
contact
Mode # Unmodified Single hard contact
1 12.91 66.68
2 84.12 223.55
3 252.34 326.04
4 519.59 529.68
5 806.16 808.16
6 1,256.55 1,256.73
7 1,682.96 1,682.97
8 2,201.36 2,201.52
9 2,755.52 2,755.76
10 3,510.01 3,510.46
9 Equivalent Reduced Model Technique Development for Nonlinear System Dynamic Response 113
response. In order to confirm this, the system response was plotted in the time and frequency domain for ‘a’ space and ‘aa’
space with comparison to the full ‘n’ space solution in Fig. 9.18. The response for the ‘aa’ space model is shown to illustrate
when a key contributing mode (mode 6 of Beam A) is not included in the reduced model for the modified system.
The ‘a’ space model response in Fig. 9.18 can be observed to have very good correlation with the full-space solution;
however, in contrast to Case 2-A the ‘aa’ space model response indicates that not including mode 6 of Beam A degrades the
results obtained. The first six modes of the unmodified component are needed to obtain the first five modes of the modified
system as indicated in Fig. 9.17, which were not able to be adequately represented using only modes 1–5 for the ‘aa’ space
model. Even though the input force spectrum was limited primarily to the first five modes of the unmodified component, the
first six modes are needed for the modified system. The hard contact does not appear to contribute significant energy beyond
1,000 Hz when contact occurs, which shows that the number of modes needed to accurately compute the time response for
the reduced models is governed by the forcing function and not the contact stiffness for this case. To quantify the similarity
0 500 1000 1500-160
-140
-120
-100
-80
-60
-40
-20
0
Frequency (Hz)
dB D
ispl
acem
ent (
in)
FFT of DOF 141 Time Response - Single Beam Single Hard Contact
n Space-146 DOF
aa Space-5 DOF
0 500 1000 1500-160
-140
-120
-100
-80
-60
-40
-20
0
Frequency (Hz)
dB D
ispl
acem
ent (
in)
FFT of DOF 141 Time Response - Single Beam Single Hard Contact
n Space-146 DOF
a Space-17 DOF
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
Dis
plac
emen
t (in
)DOF 141 Time Response - Single Beam Single Hard Contact
n Space-146 DOF
a Space-17 DOF
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
Dis
plac
emen
t (in
)
DOF 141 Time Response - Single Beam Single Hard Contact
n Space-146 DOF
aa Space-5 DOF
Fig. 9.18 Comparison of ‘n’, ‘a’, and ‘aa’ space models for single beam with single hard contact for DOF 141 – time response (top) and FFT of
time response (bottom)
Table 9.12 Average MAC and TRAC for reduced models and solution times for single beam with single hard contact
Model # of DOF Average MAC Average TRAC Solution time (s)
‘n’ space 146 1 1 35.1
‘a’ space 17 0.99999686 0.99999980 1.9
‘aa’ space 5 0.99854646 0.98781940 0.7
114 L. Thibault et al.
of the reduced model results with the full-space solution, the MAC and TRAC were computed between the full model and
the reduced model time responses, which were then averaged, as listed in Table 9.12. The solution time for each model is
also listed to show the decrease in computation time when reduced models are used.
The system response for the hard contact is observably more nonlinear compared to the soft contact case, where the
modes of the system are more difficult to identify in the FFT of the response. The FFT of the time response for both the soft
and hard contact is shown in Fig. 9.19 for comparison.
0 500 1000 1500-200
-180
-160
-140
-120
-100
-80
-60
-40
-20
0
Frequency (Hz)
dB D
ispl
acem
ent (
in)
FFT of DOF 141 Time Response - Single Beam Single Contact
Hard Contact
Soft Contact
Fig. 9.19 Comparison of the
FFT for single beam with
single soft and hard contact
Table 9.13 Natural frequencies for two beams with single hard contact for configuration 1
Mode # Two beams, single hard contact
Unmodified components
Beam A Beam B
1 21.25 12.91 22.62
2 67.74 84.12 141.56
3 127.04 252.34 396.60
4 236.02 519.59 776.92
5 338.31 806.16 1,284.71
6 424.26 1,256.55 1,918.28
7 532.04 1,682.96 2,678.33
8 786.59 2,201.36 3,563.89
9 809.25 2,755.52 4,572.70
10 1,256.69 3,510.01 5,707.04
11 1,294.66 – –
12 1,682.97 – –
13 1,918.72 – –
14 2,201.53 – –
15 2,683.90 – –
16 2,755.78 – –
17 3,510.46 – –
18 3,564.14 – –
19 3,948.79 – –
20 4,575.19 – –
9 Equivalent Reduced Model Technique Development for Nonlinear System Dynamic Response 115
The favorable MAC and TRAC results in Table 9.12 further confirm that the 17 DOF ‘a’ space model is sufficient for
accurately computing the time response for this particular case. In addition, the solution time for the full-space model is over
30 s in contrast to the reduced ‘a’ space model, which is less than 2 s. This case showed that an accurate time response
solution can be obtained using ERMT with drastically reduced models for a single component with a single hard contact.
9.6.2 Case 3-B: Two Beams with Single Hard Contact
This case consists of the tip of Beam A coming into contact with Beam B once the specified gap distance of 0.05 in. between
Beams A and B is closed, as discussed previously in Case 2-B. The same hard contact stiffness of 1,000 lb/in. was used to
represent the contact of the beams as explained in Case 3-A. Table 9.13 lists the natural frequencies for the modified system
as well as for the unmodified components. Figure 9.20 shows the mode contribution matrix for identifying the unmodified
component modes that contribute in the modified system modes. The mode shapes for the modified system are provided
in Appendix A.
Based on the input force pulse, which excites up to 1,000 Hz, the first 10 modes of the modified system are expected to be
excited, which can be seen in Table 9.13. Examining Fig. 9.20 shows that for the first 10 modes of the modified system, the
first six modes of Beam A and the first seven modes of Beam B are needed to obtain the first 10 modes of the modified
system. For a soft contact, the input force pulse primarily dominates the frequency spectrum excited, however this may not
always be true for a hard contact scenario. The modes required to accurately predict the system response are provided in the
mode contribution matrix; however, there may be additional higher order modes excited by the hard contact that cannot be
identified based solely on the input force spectrum. For hard contact situations, care must be taken when generating reduced
models such that higher order modes that participate in the system response are included. To show the effects of severe mode
truncation, the system response was plotted in the time and frequency domain for ‘a’ space and ‘aa’ space with comparison
to the full ‘n’ space solution in Fig. 9.21, where the ‘aa’ space model is missing several key contributing modes in the
modified system response.
Note that in contrast to the soft contact, the hard contact excited modes well above the 1,000 Hz bandwidth of the input
force pulse (up to approximately 3,000 Hz). Accordingly, the number of unmodified component modes identified using the
mode contribution matrix to accurately obtain the modified system time response is no longer governed by the input force
spectrum, but by the frequency bandwidth excited by the contact stiffness impact. To quantify the similarity of the reduced
model results with the full-space solution, the MAC and TRAC were computed between the full model and the reduced
model time responses, which were then averaged, as listed in Table 9.14a. The solution time for each model is also listed to
show the decrease in computation time when reduced models are used.
MODEModified System Mode Shapes -1000 lb/in Contact -Configuration 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Unm
odifi
ed B
eam
A M
ode
Shap
es
1234567891011
Unm
odifi
ed B
eam
B M
ode
Shap
es 12345678910
Bar Color Min Value Max ValueBlack 0.005 0.1Blue 0.1 0.2Green 0.2 0.3Cyan 0.3 0.5Magenta 0.5 0.7Yellow 0.7 0.9Red 0.9 1.0
Fig. 9.20 Mode contribution matrix for two beams with single hard contact for configuration 1
116 L. Thibault et al.
The transient portion of the ‘a’ space model response in Fig. 9.21 can be observed to have very good correlation with the
full-space solution. However, in contrast to Case 3-A the hard contact appears to contribute noticeable energy beyond
the 1,000 Hz input force spectrum, which was not anticipated when identifying the unmodified component modes that
contribute in the modified system modes. In Fig. 9.21, the ‘aa’ space model produced less desirable results due to severe
mode truncation. The ‘a’ space model provides very good results for the initial transient portion of the system response,
0 500 1000 1500-300
-250
-200
-150
-100
-50
0
50
Frequency (Hz)
dB D
ispl
acem
ent (
in)
FFT of DOF 141 Time Response - Two Beam Single Hard Contact
n Space-276 DOF
aa Space-10 DOF
0 500 1000 1500-300
-250
-200
-150
-100
-50
0
Frequency (Hz)
dB D
ispl
acem
ent (
in)
FFT of DOF 141 Time Response - Two Beam Single Hard Contact
n Space-276 DOF
a Space-31 DOF
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-2
0
2
4
6
8
10
12
14
Time (sec)
Dis
plac
emen
t (in
)
DOF 141 Time Response - Two Beam Single Hard Contact
n Space-276 DOF
aa Space-10 DOF
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Time (sec)
Dis
plac
emen
t (in
)DOF 141 Time Response - Two Beam Single Hard Contact
n Space-276 DOF
a Space-31 DOF
Fig. 9.21 Comparison of ‘n’, ‘a’, and ‘aa’ space models for two beams with single hard contact for DOF 141 – time response (top) and FFT of time
response (bottom)
Table 9.14 Average MAC and TRAC for reduced models and solution times for two beams with single hard contact
Model # of DOF Average MAC Average TRAC Solution time (s)
(a) MAC and TRAC for full time response (0–0.4 s)
‘n’ space 276 1 1 87.9
‘a’ space 31 0.99684042 0.97628620 4.9
‘aa’ space 10 0.40375102 0.12895379 1.1
(b) MAC and TRAC for transient portion of time response (0–0.2 s)
‘n’ space 276 1 1 87.9
‘a’ space 31 0.99995692 0.99958379 4.9
‘aa’ space 10 0.45859557 0.11002554 1.1
9 Equivalent Reduced Model Technique Development for Nonlinear System Dynamic Response 117
however degraded correlation can be observed after the initial transient. This is due to higher order modes that are missing in
the reduced model not being excited until a high force impact occurs at the hard stiffness contact location. This behavior will
be different depending on the location and level of the input excitation, due to the nonlinear characteristics of the system.
The number of modes included in the ‘a’ space model is sufficient when only the forces pulse is considered. However, due to
the hard contact, more modes are needed to accurately compute the system response beyond the initial transient portion.
To observe the high frequency content excited by the hard contact, the FFT of the system response is shown in Fig. 9.22 over
a 5,000 Hz band with comparison to the soft contact case.
Often times for nonlinear systems, the initial transient portion of the time response is of interest, not the entire solution.
When the ‘a’ space solution is evaluated for only the transient portion of the response (the first 0.2 s), the MAC improves
from 0.997 to 0.999 and the TRAC improves from 0.976 to 0.999, as shown in Table 9.14b.
The ‘a’ space model for the two beam system with a single hard contact produced very good results for the initial transient
portion of the response, which is generally the portion of the system response of interest for nonlinear systems. However, less
desirable results were obtained using the ‘aa’ space model, due to not enough modes being included to obtain the modified
system response. As seen in Fig. 9.20, the first 10 modes of the modified system requires the first six modes of Beam A and
the first seven modes of Beam B, which were not able to be adequately represented using the ‘aa’ space model. The number
of modes included in the ‘a’ space model is sufficient when only the force pulse is considered. However, due to the hard
contact stiffness, more modes are needed to accurately compute the system response beyond the initial transient portion.
This shows that throughout the nonlinear system time response, hard contacts can potentially excite higher frequencies than
the input excitation. Therefore, a reduced model with greater than 31 DOF and 31 modes is needed to obtain an accurate time
response solution beyond the initial transient for this case. The favorable MAC and TRAC results in Table 9.14b show that
the 31 DOF ‘a’ space model is sufficient for accurately computing the initial transient portion of the time response for this
particular case. In addition, the solution time for the full-space model is almost 90 s in contrast to the reduced ‘a’ space
model, which is approximately 5 s.
ERMT was shown in this case to provide significant reduction in computation time for a nonlinear multiple component
system with a single hard contact while maintaining a highly accurate initial transient time response solution. However, care
must be taken when generating reduced models for nonlinear systems with hard contact stiffness, due to the higher
frequencies that can potentially be excited in the system.
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-300
-250
-200
-150
-100
-50
0
Frequency (Hz)
dB D
ispl
acem
ent
(in)
FFT of DOF 141 Time Response - Two Beam Single Contact
Hard Contact
Soft Contact
Fig. 9.22 Comparison of the
FFT for two beams with single
soft and hard contact
118 L. Thibault et al.
9.6.3 Case 3-C: Two Beams with Multiple Hard Contacts
This case consists of BeamAcoming into contact with BeamB in three different configurationswith hard contact stiffness at two
possible contact locations with a specified gap distance of 0.05 in., as discussed previously in Case 2-C. Table 9.15 lists the
natural frequencies for the modified system as well as for the unmodified components. Figure 9.23 shows the mode contribution
matrices used for identifying the unmodified component modes that participate in the modified system modes for the three
possible system configurations. The mode shapes for the modified system configurations are provided in Appendix A.
Table 9.15 and Fig. 9.23 show that for the input force pulse applied, ten systemmodes are excited and modes 1–8 of Beam
A and modes 1–7 of Beam B are needed. However, Case 3-B showed that additional modes beyond the input excitation
spectrum are activated by the hard contact stiffness. Therefore, larger reduced models are needed for the inclusion of
Bar Color Min Value Max Value
Black 0.005 0.1
Blue 0.1 0.2
Green 0.2 0.3
Cyan 0.3 0.5
Magenta 0.5 0.7
Yellow 0.7 0.9
Red 0.9 1.0
MODEModified System Mode Shapes - 1000 lb/in Contact - Configuration 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Unm
odifie
d B
eam
A M
ode
Sha
pes
1
2
3
4
5
6
7
8
9
10
11
Unm
odifie
d B
eam
B M
ode
Sha
pes 1
2
3
4
5
6
7
8
9
10
MODEModified System Mode Shapes - 1000 lb/in Contact - Configuration 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Unm
odifie
d B
eam
A M
ode
Sha
pes
1
2
3
4
5
6
7
8
9
10
11
Unm
odifie
d B
eam
B M
ode
Sha
pes 1
2
3
4
5
6
7
8
9
10
MODEModified System Mode Shapes - 1000 lb/in Contact - Configuration 3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Unm
odifie
d B
eam
A M
ode Sha
pes
1
2
3
4
5
6
7
8
9
10
11
Unm
odifie
d B
eam
B M
ode
Sha
pes 1
2
3
4
5
6
7
8
9
10
Fig. 9.23 Mode contribution matrix for two beams with hard contacts for configurations 1, 2, and 3
Table 9.15 Natural frequencies for two beams with multiple hard contact configurations
Mode #
Modified system – hard contacts Unmodified components
Config. 1 Config. 2 Config. 3 Beam A Beam B
1 21.25 15.50 36.22 12.91 22.62
2 67.74 69.52 113.69 84.12 141.56
3 127.04 119.79 228.57 252.34 396.60
4 236.02 234.14 312.68 519.59 776.92
5 338.31 325.85 339.40 806.16 1,284.71
6 424.26 467.09 504.56 1,256.55 1,918.28
7 532.04 527.49 532.41 1,682.96 2,678.33
8 786.59 786.04 790.33 2,201.36 3,563.89
9 809.25 828.78 833.74 2,755.52 4,572.70
10 1,256.69 1,257.17 1,257.22 3,510.01 5,707.04
11 1,294.66 1,290.03 1,300.10 – –
12 1,682.97 1,686.00 1,686.01 – –
13 1,918.72 1,920.05 1,920.48 – –
14 2,201.53 2,202.67 2,202.84 – –
15 2,683.90 2,678.85 2,684.38 – –
16 2,755.78 2,756.01 2,756.28 – –
17 3,510.46 3,511.34 3511.79 – –
18 3,564.14 3,563.98 3564.23 – –
19 3,948.79 3,948.54 3948.79 – –
20 4,575.19 4,572.70 4575.19 – –
9 Equivalent Reduced Model Technique Development for Nonlinear System Dynamic Response 119
additional modes that participate in the modified system response to obtain an accurate time response solution. The system
response was plotted in the time and frequency domain for ‘a’ space and ‘aa’ space with comparison to the full ‘n’ space
solution in Fig. 9.24, where the ‘aa’ space model is missing several key contributing modes in the modified system response.
The transient portion of the ‘a’ space model response in Fig. 9.24 can be observed to have very good correlation with the
full-space solution. As seen in Case 3-B, the hard contact contributes significant energy beyond the 1,000 Hz input force
spectrum. In Fig. 9.24, the ‘aa’ space model produced less desirable results, due to severe mode truncation. The ‘a’ space
model provides very good results for the initial transient portion of the response, however degraded correlation is observed
after the initial transient. The number of modes included in the ‘a’ space model is sufficient when only the forces pulse is
considered. However, due to the hard contact, more modes are needed to accurately compute the system response beyond the
initial transient portion, which was discussed in Case 3-B. To observe the high frequency content excited by the hard contact,
the FFT of the system response is shown in Fig. 9.25 over a 5,000 Hz band with comparison to the soft contact case.
As observed in Case 3-B, the frequency excited by the hard contact stiffness was well above the bandwidth of the input
force pulse. The mode contribution matrix provided a good indication of the unmodified component modes that participate
in the modified system modes for the soft contact stiffness scenario. However, higher order modes beyond the input force
spectrum are excited by the hard contact stiffness and the modes identified in the mode contribution matrix are governed by
the frequency bandwidth excited by the hard contact impact. To quantify the similarity of the reduced model results with the
full-space solution, the MAC and TRAC were computed between the full model and the reduced model time responses,
which were then averaged, as listed in Table 9.16a. The solution time for each model is also listed to show the decrease in
computation time when reduced models are used.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Time (sec)
Dis
plac
emen
t (in
)DOF 141 Time Response - Two Beam Two Hard Contacts
n Space-276 DOF
a Space-31 DOF
0 500 1000 1500-250
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0
50
Frequency (Hz)
dB D
ispl
acem
ent
(in)
FFT of DOF 141 Time Response - Two Beam Two Hard Contacts
n Space-276 DOF
aa Space-10 DOF
0 500 1000 1500-250
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0
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dB D
ispl
acem
ent
(in)
FFT of DOF 141 Time Response - Two Beam Two Hard Contacts
n Space-276 DOF
a Space-31 DOF
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-10
-5
0
5
10
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20
25
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Dis
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emen
t (in
)
DOF 141 Time Response - Two Beam Two Hard Contacts
n Space-276 DOF
aa Space-10 DOF
Fig. 9.24 Comparison of ‘n’, ‘a’, and ‘aa’ space models for two beams with multiple hard contacts for DOF 141 – time response (top) and FFT
of time response (bottom)
120 L. Thibault et al.
Often times for nonlinear systems, the initial transient portion of the time response is of interest, not the entire solution.
When the ‘a’ space solution is evaluated for only the transient portion of the response (the first 0.2 s), the MAC improves
from 0.890 to 0.999 and the TRAC improves from 0.574 to 0.999, as shown in Table 9.16b.
The ‘a’ space model for the two beam system with multiple hard contacts produced very good results for the initial transient
portion of the response, which is generally the portion of the system response of interest for nonlinear systems. However, less
desirable results were obtained using the ‘aa’ space model, due to not enough modes being included to obtain the modified
system response. As seen in Fig. 9.23, the first 10modes of the modified system requires the first eight modes of BeamA and the
first sevenmodes ofBeamB,whichwere not able to be adequately represented using the ‘aa’ spacemodel. The number ofmodes
included in the ‘a’ space model is sufficient when only the forces pulse is considered. However, due to the hard contact, more
modes are needed to accurately compute the system response beyond the initial transient portion, as discussed in Case 3-B.
Therefore, a reduced model with greater than 31 DOF and 31 modes is needed to obtain an accurate time response solution
beyond the initial transient for this case. The favorable MAC and TRAC results in Table 9.16b show that the 31 DOF ‘a’ space
model is sufficient for accurately computing the initial transient portion of the time response for this particular case. In addition,
the solution time for the full-space model is almost 90 s in contrast to the reduced ‘a’ space model, which is approximately 5 s.
ERMT was shown in this case to provide significant reduction in computation time for a nonlinear multiple component
system with multiple hard contacts where significant component interaction occurs. In addition, a highly accurate initial
transient time response solution was obtained using drastically reduced models.
Comparing the soft and hard contact cases shows that the number of modes needed in the reduced models can be
predicted accurately as long as the input spectrum bandwidth defines the frequency range that is excited, as seen in the soft
Table 9.16 Average MAC and TRAC for reduced models and solution times for two beams with hard contacts
Model # of DOF Average MAC Average TRAC Solution time (s)
(a) MAC and TRAC for full time response (0–0.4 s)
‘n’ space 276 1 1 88.5
‘a’ space 31 0.88955510 0.57445998 4.9
‘aa’ space 10 0.45974452 0.07460348 1.1
(b) MAC and TRAC for transient portion of time response (0–0.2 s)
‘n’ space 276 1 1 88.5
‘a’ space 31 0.99995692 0.99958379 4.9
‘aa’ space 10 0.53251049 0.24627142 1.1
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-250
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-50
0
Frequency (Hz)
dB D
ispl
acem
ent
(in)
FFT of DOF 141 Time Response - Two Beam Multiple Contact
Hard Contact
Soft Contact
Fig. 9.25 Comparison of the FFT for two beams with multiple soft and hard contacts
9 Equivalent Reduced Model Technique Development for Nonlinear System Dynamic Response 121
contact cases. However, once this condition is no longer true, determining the number of modes needed requires additional
knowledge of the frequency bandwidth excited by the contacts ahead of time. Care must be taken when generating reduced
models for nonlinear systems with hard contact stiffness, due to the higher frequencies that can potentially be excited in the
system. For both soft and hard contact cases, ERMT was shown to provide a substantial reduction in computation time while
maintaining highly accurate time response solutions, which demonstrates the usefulness of this technique.
9.7 Time Step Selection Effect on Solution
For the time step of 0.0001 s used in the cases studied, Raleigh Criteria and Shannon’s Sampling Theorem state that the
maximum frequency range that can be observed is 5 kHz, which is well above the 1,000 Hz frequency bandwidth excited by
the input force pulse. Although this time resolution may be fine enough to accurately capture the time response for the linear
system, this resolution may not be adequate when the response becomes nonlinear. In addition, the contact stiffness of the
nonlinear system also affects the frequency bandwidth excited, as discussed previously in Case 3. Since the time step chosen
affects the instance in time that contact occurs, the system may respond differently depending on whether the time duration
of the impact is short (soft contact) or long (hard contact). For a soft contact, the system remains in contact for a longer time
duration, and the time response should be relatively consistent regardless of the time step chosen. For the hard contact,
however, the system may experience high frequency impact chatter, which may not be captured when a coarse time step is
used. The system response was computed for the single beam with soft and hard contact using a time step of 0.00005 s and
the previously used time step of 0.0001 s, which are compared in Fig. 9.26 for the first 0.1 s of the time response solutions.
The left plot in Fig. 9.26 shows that for the soft contact, the smaller time step minimally effected the time response
solution. The results compare very well for the soft contact, due to the system remaining in contact with the soft spring for a
long time duration. For the hard contact in the right plot of Fig. 9.26, the system comes into contact and then immediately
bounces off. Since the time step directly affects the acceleration, there is a divergence in the solution immediately after the
first contact occurs. Although this study shows that the time step selected was not sufficiently small enough for the hard
contact cases, the time step of 0.0001 s was used for all of the previous cases in order to demonstrate the main principles and
computational advantages of ERMT. Further study is needed to determine the time step required when the contact stiffness
dictates the spectral energy content that participates in the nonlinear system response.
9.8 Comparison to Large Finite Element Models
ERMT was shown to provide substantial reduction in computation time for the simple beam models studied in this paper.
The computational advantages of the technique would be further emphasized when used for a detailed finite element model,
which are typically found in industry. Figure 9.27 shows a very detailed model of a helicopter/missile/wing configuration.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-0.1
0
0.1
0.2
0.3
0.4
Time (sec)
Dis
plac
emen
t (in
)
Single Beam Single Hard Contact Response at DOF 141, Sampling Rate Comparison
dt1 = 0.0001 sec
dt2 = 0.00005 sec
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Time (sec)
Dis
plac
emen
t (in
)Single Beam Single Soft Contact Response at DOF 141, Sampling Rate Comparison
dt1 = 0.0001 sec
dt2 = 0.00005 sec
Fig. 9.26 Comparison of system response using different time steps for soft contact (left) and hard contact (right)
122 L. Thibault et al.
These detailed models are used in dynamic simulations to compute the transient dynamics that occur during a missile
firing under a variety of conditions. Due to the system being very complex with many detailed components, the time
response solutions require significant computation (on the order of hours to days). However, utilizing reduced order models
of the various components allows for substantial reduction in computation time. In addition, multiple system configurations
can be efficiently studied in detail to provide the analyst with the wealth of information needed to improve the system design.
9.9 Conclusion
A computationally efficient technique – Equivalent Reduced Model Technique (ERMT) – that utilizes reduced linear
component models assembled with discrete nonlinear connection elements to perform nonlinear forced response analysis is
presented. Four cases of increasing complexity are studied. The mode contribution matrix is used to identify the modes
required to form accurate reduced order models. The technique was shown to yield accurate results when compared to the
reference solution and provided significant improvement in computational efficiency for the analytical nonlinear forced
response cases studied.
9.10 Future Work
Future workwill be performed using ERMT to demonstrate the usefulness of the technique with experimental data validation.
For comparison to experimental data, several additional considerations are needed to yield accurate results. First, the
underlying linear system will need to be a highly accurate model in order to predict the system time responses correctly.
Care will need to be taken in modeling and updating the component models to test data to ensure that both the model and
measurements are reflective of the physical structure. Second, the damping will have to be measured experimentally for the
linear system for as many modes as possible in order to have high correlation in the time domain. In addition, efforts will be
needed to determine the correct damping for the physical system models, as the damping may change once the component(s)
are in the state of contact. Third, the stiffness of the contact will have to be determined to accurately model the system and
compute the time response. In addition, the impact force from the beam contact may need to be accounted for in the analytical
models, where this additional force input to the structure may potentially affect the time response. Finally, variation in the
time step used was found to affect the results obtained and therefore, additional work is needed to remedy this item of concern.
Acknowledgements Some of the work presented herein was partially funded by Air Force Research Laboratory Award No. FA8651-10-1-0009
“Development of Dynamic Response Modeling Techniques for Linear Modal Components”. Any opinions, findings, and conclusions or
recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the particular funding agency.
The authors are grateful for the support obtained.
Fig. 9.27 Detailed FEM of
helicopter/missile/wing
configuration
9 Equivalent Reduced Model Technique Development for Nonlinear System Dynamic Response 123
Appendix A: Component and System Mode Shapes
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Unmodified Mode Shapes – Beam B
1 – 22.62 Hz 2 –141.56 Hz 3 – 396.60 Hz
4 –776.92 Hz 5 –1284.71 Hz 6 –1918.28 Hz
7 –2678.33 Hz 8 –3563.89 Hz 9 –4572.70 Hz
10 –5707.04 Hz 11 – 6956.91 Hz 12 –8324.25 Hz
Unmodified Mode Shapes –Beam A
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4 –519.59 Hz 5 –806.16 Hz 6 –1256.55 Hz
7 –1682.96 Hz 8 –2201.36 Hz 9 –2755.52 Hz
10 –3510.01 Hz 11 –3948.54 Hz 12 –5076.34 Hz
Fig. A.1 Mode shapes for unmodified components A and B
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Single Beam Single Contact Mode Shapes –10 lb/in Spring
1 – 26.09 Hz 2 – 86.02 Hz 3 – 252.54 Hz
4 – 519.65 Hz 5 – 806.18 Hz 6 – 1256.55 Hz
7 – 1682.96 Hz 8 – 2201.36 Hz 9 – 2755.53 Hz
10 – 3510.02 Hz 11 – 3948.55 Hz 12 – 5076.34 Hz
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Single Beam Single Contact Mode Shapes –1000 lb/in Spring
1 – 66.68 Hz 2 – 223.55 Hz 3 – 326.04 Hz
4 – 529.68 Hz 5 – 808.16 Hz 6 – 1256.73 Hz
7 – 1682.97 Hz 8 – 2201.52 Hz 9 – 2755.76 Hz
10 – 3510.46 Hz 11 – 3948.79 Hz 12 – 5076.36 Hz
0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18
0 2 4 6 8 10 12 14 16 180 2 4 6 8 10 12 14 16 18
0 2 4 6 8 10 12 14 16 18
0 2 4 6 8 10 12 14 16 18
0 2 4 6 8 10 12 14 16 180 2 4 6 8 10 12 14 16 18
0 2 4 6 8 10 12 14 16 180 2 4 6 8 10 12 14 16 180 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18
0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18
0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18
0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18
0 2 4 6 8 10 12 14 16 18
Fig. A.2 Mode shapes for single beam with single soft (left) and single hard (right) contact
124 L. Thibault et al.
0 5 10 15 20 25
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Multiple Beam Contact Mode Shapes - Configuration 1 – 10 lb/in Spring
1 – 20.35 Hz 2 – 29.58 Hz 3 – 86.01 Hz
4 – 142.43 Hz 5 – 252.54 Hz 6 – 396.71 Hz
7 – 519.65 Hz 8 – 777.00 Hz 9 – 806.18 Hz
10 – 1256.55 Hz 11 – 1284.80 Hz 12 – 1682.96 Hz
0 5 10 15 20 25 0 5 10 15 20 250 5 10 15 20 250 5 10 15 20 25
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Multiple Beam Contact Mode Shapes – Configuration 2 – 10 lb/in Spring
1 – 14.78 Hz 2 – 33.04 Hz 3 – 85.71 Hz
4 – 142.98 Hz 5 – 252.61 Hz 6 – 396.96 Hz
7 – 519.61 Hz 8 – 777.04 Hz 9 – 806.28 Hz
10 – 1256.56 Hz 11 – 1284.76 Hz 12 – 1682.99 Hz
0 5 10 15 20 250 5 10 15 20 250 5 10 15 20 25
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Multiple Beam Contact Mode Shapes – Configuration 3 – 10 lb/in Spring
1 – 21.08 Hz 2 – 39.23 Hz 3 – 87.24 Hz
4 – 143.82 Hz 5 – 252.82 Hz 6 – 397.07 Hz
7 – 519.68 Hz 8 – 777.12 Hz 9 – 806.30 Hz
10 – 1256.56 Hz 11 – 1284.85 Hz 12 – 1682.99 Hz
Fig. A.3 Mode shapes for two beam system for multiple soft contact configurations
9 Equivalent Reduced Model Technique Development for Nonlinear System Dynamic Response 125
0 5 10 15 20 25-60
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Multiple Beam Contact Mode Shapes – Configuration 3 – 1000 lb/in Spring
1 – 36.22 Hz 2 – 113.69 Hz 3 – 228.57 Hz
4 – 312.68 Hz 5 – 339.40 Hz 6 – 504.56 Hz
7 – 532.41 Hz 8 – 790.33 Hz 9 – 833.74 Hz
10 – 1257.22 Hz 11 – 1300.10 Hz 12 – 1686.01 Hz
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Multiple Beam Contact Mode Shapes – Configuration 1 – 1000 lb/in Spring
1 – 21.25 Hz 2 – 67.74 Hz 3 – 127.04 Hz
4 – 236.02 Hz 5 – 338.31 Hz 6 – 424.26 Hz
7 – 532.04 Hz 8 – 786.59 Hz 9 – 809.25 Hz
10 – 1256.69 Hz 11 – 1294.66 Hz 12 – 1682.97 Hz
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60
Multiple Beam Contact Mode Shapes – Configuration 2 – 1000 lb/in Spring
1 – 15.50 Hz 2 – 69.52 Hz 3 – 119.79 Hz
4 – 234.14 Hz 5 – 325.85 Hz 6 – 467.09 Hz
7 – 527.49 Hz 8 – 786.04 Hz 9 – 828.78 Hz
10 – 1257.17 Hz 11 – 1290.03 Hz 12 – 1686.00 Hz
Fig. A.4 Mode shapes for two beam system for multiple hard contact configurations
126 L. Thibault et al.
References
1. Avitabile P, O’Callahan JC, Pan EDR (1989) Effects of various model reduction techniques on computed system response. In: Proceedings of
the seventh international modal analysis conference, Las Vegas, February 1989
2. O’Callahan JC, Avitabile P, Riemer R (1989) System equivalent reduction expansion process. In: Proceedings of the seventh international
modal analysis conference, Las Vegas, February 1989
3. Van Zandt T (2006) Development of efficient reduced models for multi-body dynamics simulations of helicopter wing missile configurations.
Master’s thesis, University of Massachusetts Lowell, 2006
4. Avitabile P, O’Callahan JC (2009) Efficient techniques for forced response involving linear modal components interconnected by discrete
nonlinear connection elements. Mech Syst Signal Process 23(1):45–67, Special Issue: Non-linear Structural Dynamics
5. Friswell MI, Penney JET, Garvey SD (1995) Using linear model reduction to investigate the dynamics of structures with local non-linearities.
Mech Syst Signal Process 9(3):317–328
6. Lamarque C, Janin O (2000) Modal analysis of mechanical systems with impact non-linearities: limitations to a modal superposition. J Sound
Vib 235(4):567–609
7. Ozguven H, Kuran B (1996) A modal superposition method for non-linear structures. J Sound Vib 189(3):315–339
8. Al-Shudeifat M, Butcher E, Burton T (2010) Enhanced order reduction of forced nonlinear systems using new Ritz vectors. In: Proceedings of
the twenty-eighth international modal analysis conference, Jacksonville, February 2010
9. Rhee W Linear and nonlinear model reduction in structural dynamics with application to model updating. PhD dissertation, Texas Technical
University
10. Guyan RJ (1965) Reduction of stiffness and mass matrices. AIAA J 3(2):380
11. O’Callahan JC (1989) A procedure for an improved reduced system (IRS) model. In: Proceedings of the seventh international modal analysis
conference, Las Vegas, February 1989
12. Marinone T, Butland A, Avitabile P (2012) A reduced model approximation approach using model updating methodologies. In: Proceedings of
the thirtieth international modal analysis conference, Jacksonville, February 2012
13. Avitabile P (2003) Twenty years of structural dynamic modification – a review. Sound Vib Mag 37:14–27
14. Rao S (2004) Mechanical vibrations, 4th edn. Prentice Hall, New Jersey, pp 834–843
15. Allemang RJ, Brown DL (2007) A correlation coefficient for modal vector analysis. In: Proceedings of the first international modal analysis
conference, Orlando, February 2007
16. O’Callahan J (1986) MAT_SAP/MATRIX, A general linear algebra operation program for matrix analysis. University of Massachusetts
Lowell, 1986
17. MATLAB (R2010a) The MathWorks Inc., Natick, M.A.
9 Equivalent Reduced Model Technique Development for Nonlinear System Dynamic Response 127
Chapter 10
Efficient Computational Nonlinear Dynamic Analysis
Using Modal Modification Response Technique
Tim Marinone, Peter Avitabile, Jason R. Foley, and Janet Wolfson
Abstract Generally, structural systems contain nonlinear characteristics in many cases. These nonlinear systems require
significant computational resources for solution of the equations of motion. Much of the model, however, is linear where the
nonlinearity results from discrete local elements connecting different components together. Using a component mode
synthesis approach, a nonlinear model can be developed by interconnecting these linear components with highly nonlinear
connection elements.
The approach presented in this paper, the Modal Modification Response Technique (MMRT), is a very efficient technique
that has been created to address this specific class of nonlinear problem. By utilizing a Structural Dynamics Modification
(SDM) approach in conjunction with mode superposition, a significantly smaller set of matrices are required for use in the
direct integration of the equations of motion. The approach will be compared to traditional analytical approaches to make
evident the usefulness of the technique for a variety of test cases.
Keywords Nonlinear analysis • Forced nonlinear response • Linear components for nonlinear analysis • Modal analysis •
Mode superposition
Nomenclature
Symbols
M½ � Physical mass matrix
C½ � Physical damping matrix
K½ � Physical stiffness matrix�M1½ � Modal mass matrix for state 1
D �M12½ � Modal mass change matrix�K1½ � Modal stiffness matrix for state 1
D �K12½ � Modal stiffness change matrix
DM12½ � Physical mass change matrix
DK12½ � Physical stiffness change matrix
U½ �g Generalized Inverse
U1½ � Mode shapes for state 1
U12½ � Mode contribution matrix
U2½ � Mode shapes for state 2
T. Marinone (*) • P. Avitabile
Structural Dynamics and Acoustic Systems Laboratory, University of Massachusetts Lowell, One University Avenue,
Lowell, MA 01854, USA
e-mail: [email protected]
J.R. Foley • J. Wolfson
Air Force Research Laboratory, Munitions Directorate, Fuzes Branch, Eglin Air Force Base, 306 W. Eglin Blvd, Bldg 432,
Eglin AFB, FL 32542-5430, USA
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_10, # The Society for Experimental Mechanics, Inc. 2012
129
Ff g Physical force
€p1f g Modal acceleration
_p1f g Modal velocity
p1f g Modal displacement
€x1f g Physical acceleration
_x1f g Physical velocity
x1f g Physical displacement
10.1 Introduction
Generally, any nonlinear response analysis involves significant computation, especially if the full analytical model system
matrices are used for the forced response problem. These nonlinearities can be broadly broken down into two categories:
global and local, with significant effort expended on research of both. Due to the significant computational time required for
these nonlinear cases, the analyst may often be unable to investigate the nonlinearities in depth, especially if a set of
performance characteristics related to temperature, preload, deflection, etc. characterize the nonlinear connection elements.
Thus, there is significant motivation to develop a set of reduced order models that can accurately predict nonlinear response
at a substantially reduced computation time.
One area of interest involves the dynamic response of systems with nonlinear connections. These systems are typically
linear, but the introduction of the local nonlinearity causes the system to become highly nonlinear. The approach taken in this
paper is to treat the system as a linear system, but to model the nonlinear component as a change in the linear system by
utilizing a Structural Dynamic Modification. Accordingly, direct integration schemes in conjunction with mode superposi-
tion are employed in order to maximize the efficiency of the model.
This approach was first presented by Avitabile and O’Callahan [1] where a detailed overview of the theory and a simple
analytical example were provided. Friswell et al. [2] looked at reducing models with local nonlinearities with several
different reduction schemes for a periodic solution. Lamarque and Janin [3] looked at modal superposition using a 1-DOF
and 2-DOF system with impact and concluded that modal superposition had limitations due to difficulties in modeling
impact. Ozguven [4] converted non-linear ODE’s into a set of non-linear algebraic equations, which could be reduced by
using linear modes. This technique was found to provide the best reduction in computational time when the structure was
excited at a forcing frequency that corresponded to a resonance of the structure. An alternative approach that has been
studied uses Non-linear Normal Modes (NNMs) which are formulated by Ritz vectors [5, 6]. This approach seeks to extend
the concept of linear orthogonal modes to nonlinear systems.
This paper presents the analytical time results of a beam system subjected to a force impulse. Four cases are studied:
single beam with no contact, single beam with contact, multiple beams with single contact, and multiple beams with multiple
contacts. For each of these cases two types of contact stiffness will be studied; a soft contact representing a rubber/isolation
material, and a hard contact representing a metal on metal contact. For all cases, the time results of the full space model will
be used as a reference.
10.2 Theory
10.2.1 Modal Modification Response Technique (MMRT)
The MMRT technique is based on the Structural Dynamic Modification process and mode superposition method. From the
modal data base of an unmodified system, structural changes can be explored using the modal transformation to project
changes from physical space to modal space of the unmodified system; this results in a heavily coupled set of equations
which are drastically less than those of the physical model and can be written as
. ..
�M1
. ..
2664
3775þ D �M12½ �
2664
3775 €p1f g þ
. ..
�K1
. ..
2664
3775þ D �K12½ �
2664
3775 p1f g ¼ 0½ � (10.1)
where
130 T. Marinone et al.
D �M12½ � ¼ U1½ �T DM12½ � U1½ �D �K12½ � ¼ U1½ �T DK12½ � U1½ �
(10.2)
The solution of the modified set of modal equations in modal space produces an eigensolution and the resulting
eigenvectors of this are the [U12] matrix; this contains the scaling coefficients necessary to form the final modal vectors
from the starting modal vectors. This [U12] matrix is the key to identifying the necessary set of modal vectors to accurately
predict the final modified set of modes [7]. Figure 10.1 illustrates the contribution of the [U12] matrix in forming the final
modified set of modes [U2], where “m” modes of the [U12] matrix are used, and “n-m” modes are excluded.
Mode superposition is executed in a piecewise linear fashion depending on the “state” of the nonlinear connection
element. Once the linear state changes, a structural modification is performed to update the characteristics of the system
along with updated initial conditions to proceed on with the numerical integration. This equation is written as
IAm� �
IBm� �� �
€pA� �€pBf g
� þ
O2A
m
h iO2B
m
h i24
35þ U½ �T KTIE½ � U½ �
24
35 pA
� �pBf g
� ¼ UA½ �T f A
� �UB½ �T f Bf g
( )(10.3)
where the initial conditions for the updated state are
pf gj ¼ U½ �gj xf gði�1Þ
_pf gj ¼ U½ �gj _xf gði�1Þ (10.4)
and the generalized inverse can be written as either a pseudo inverse or mass weighted inverse
U½ �gj ¼ Uj
� �TUj
� �h i�1
Uj
� �Tor
U½ �gj ¼ �Mj
� �Uj
� �TMj
� � (10.5)
Figure 10.2 shows the schematic of this technique where the state of the system is checked at each time step by computing
the physical response from mode superposition.
10.2.2 Direct Integration of the Equations of Motion
The direct integration of the equations of motion used here are that of the Newmark [8] method commonly used.
From the known initial conditions for displacement and velocity, the initial acceleration is
€~x0 ¼ M½ ��1 ~F0 � C½ � _~x0 � K½ �~x0 �
(10.6)
Fig. 10.1 Schematic for SDM process using [U12]
10 Efficient Computational Nonlinear Dynamic Analysis Using Modal Modification Response Technique 131
where
€~x0 ¼ initial acceleration vector_~x0 ¼ initial velocity vector
x0 ¼ initial displacement vector~F0 ¼ initial force vector
Choosing an appropriate Dt, a and b (the values chosen for the analytical case studies were 0.0001 s, 0.25 and 0.5 in order
to satisfy sampling parameters and to give constant acceleration), the displacement vector is
~xiþ1 ¼ 1
a Dtð Þ2 M½ � þ baDt
C½ � þ K½ �" #�1
~Fiþ1 þ M½ � 1
a Dtð Þ2~xi þ1
aDt_~xi þ 1
2a� 1
� €~xi
!(
þ C½ � baDt
~xi þ ba� 1
� _~xi þ b
a� 2
� Dt2€~xi
� (10.7)
With the displacement vector known, the acceleration and velocity vectors are
_~xiþ1 ¼ _~xi þ 1� bð ÞDt€~xi þ bDt€~xiþ1 (10.8)
€~xiþ1 ¼ 1
a Dtð Þ2 ~xiþ1 �~xið Þ � 1
aDt_~xi � 1
2a� 1
� €~xi (10.9)
Normal rules regarding integration of the equations of motion are utilized here and are not further discussed.
10.2.3 Time Response Correlation Tools
In order to quantitatively compare two different time solutions, two correlation tools (TRAC and MAC) will be used [9]. Xniand Xnj are the two compared displacement vectors.
TRAC (Time Response Assurance Criterion) – Correlates single DOF across all instances in time.
TRACji ¼Xnj� �T
Xnif gh i2
Xnj� �T
Xnj� �h i
Xnif gT Xnif g� � (10.10)
Formulate Modal Data Base
Determine Force and/orInitial Conditions
Compute Modal Force andModal Initial Conditions
Perform Numerical Integrationfor Next ΔT
Compute Physical ResponseFrom Mode Superposition
Check For Gaps orNonlinearity
Any Change in the CurrentLinear Modal State?
Perform Structural Dynamic Modification to
Reflect Changes
Update Modal Data Base
No Yes
Fig. 10.2 Schematic
for MMRT
132 T. Marinone et al.
MAC (Modal Assurance Criterion) – Correlates all DOF at single instance in time.
MACij ¼Xnif gT Xnj
� �� �2Xnif gT Xnif g� �
Xnj� �T
Xnj� �h i (10.11)
Both MAC and TRAC values close to 1.0 indicate strong similarity between vectors, where values close to 0.0 indicate
minimal or no similarity.
10.3 Model Description and Cases Studied
This section presents the analytical models developed as well as the cases studied. The full-space time solution is used as the
reference solution for all cases.
10.3.1 Model: Beam A and Beam B
Two planar element beam models created using MAT_SAP [10] (a FEM program developed for MATLAB [11]) are used
for all the cases studied. Figure 10.3 shows the two beams assembled into the linear system, where the red points are the
accelerometer measurements, and the blue location is the point of force applied to the system (point 14). Note that 3 in. of
each beam are clamped for the cantilevered boundary condition.
Table 10.1 lists the beam characteristics, while Table 10.2 lists the natural frequencies for the first 10 modes.
The force pulse is an analytic force pulse designed to be frequency band-limited, exciting modes up to 1,000 Hz while
minimally exciting the higher order modes. Using this force pulse, the number of modes that will be involved in the response
can be determined easily, as modes above 1,000Hz can be considered of negligible importance in the response.Modes 1–5will
be primarily excited in BeamA, while modes 1–4 will be primarily excited for BeamB. Figure 10.4a, b show the force pulse in
the time and frequency domain. Damping was assumed 1% of critical damping for all modes (both unmodified and system).
10.3.2 Case 1: Single Beam with No Contact
For the first case, the system is a single beam (Beam A) with no contact. As this system is linear and because there is no SDM
performed, the number of modes needed is only a function of the input excitation. Based on the input force spectrum seen
+
Beam A Beam B
F1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1716
Fig. 10.3 Schematic of impact and response points for beam system
Table 10.2 Model beam frequencies
Mode # 1 2 3 4 5 6 7 8 9 10
Beam A 12.91 84.12 252.34 519.59 806.16 1,256.55 1,682.96 2,201.36 2,755.52 3,510.01
Beam B 22.62 141.56 396.6 776.92 1,284.71 1,918.28 2,678.33 3,563.89 4,572.7 5,707.04
Table 10.1 Model beam characteristics
Beam
Length
(in.)
Width
(in.)
Thickness
(in.)
Num.
of elements
Num.
of nodes
Num.
of DOF
Node spacing
(in.) Material
Density
(lb/in.3)
Young’s modulus
(Msi)
A 18 2 0.123 72 73 146 0.25 Aluminum 2.54E-04 10
B 16 4 0.123 64 65 130 0.25 Aluminum 2.54E-04 10
10 Efficient Computational Nonlinear Dynamic Analysis Using Modal Modification Response Technique 133
previously in Fig. 10.4b, the time response needs five modes. In order to determine if this assumption is accurate, the FFT of the
response of systemwas plotted using an increasing number ofmodes and is compared to the full solution (using all 146modes) in
Fig. 10.5.
The FFT in Fig. 10.5 shows that modes 1–3 have a large magnitude, in contrast to modes 4 and 5, which are minimally
excited. Figure 10.5 further shows that a much smaller mode set is required for the time response, as the force pulse has only
excited the first few modes of the system. Based on these results, a minimum of three modes should be used in order to
approximate the full space model accurately. In order to confirm this, the time response at point 14 will be computed using
modes 1 through 5 and will be compared to the full time solution in Fig. 10.6a–e. In addition, Fig. 10.6f will be computed
using all modes (excluding mode 1) to show the effect if a key mode (one with significant magnitude) is not included.
As seen in Fig. 10.6a, b, using only one or two modes is not sufficient to accurately reproduce the full space model, as the
input force spectrum excited additional higher order modes. Once the third mode was added, however, the response overlays
almost perfectly and the further addition of modes 4 and 5 have a negligible effect. Finally, even though 145 of the 146 modes
were used in Fig. 10.6f, the exclusion of mode 1 prevents the accurate reconstruction of the full time response. In order to
further show the effect of mode truncation, the MAC and TRAC of the time responses were averaged as listed in Table 10.3.
Table 10.3 indicates that TRAC is more sensitive than MAC for determining the accuracy of time responses, as MAC is
weighted by DOFs with large responses. Table 10.3 also provides further confirmation that the first three beam modes are the
most critical modes for this time response, as the TRAC improves significantly when adding modes 2 and 3, but has a
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-20
-15
-10
-5
0
5
Time (sec)
For
ce (
lbf)
Analytical Time Domain Force Pulse
0 500 1000 1500-110
-105
-100
-95
-90
-85
-80
-75
-70
-65
-60
Frequency (Hz)
dB F
orce
(lb
f)
FFT of Force PulseAnalytical Time Domain Force Pulse
Fig. 10.4 Analytical force pulse in the time (a) and frequency (b) domain
0 500 1000 1500-200
-150
-100
-50dB
Res
pons
e (g
)
Frequency (HZ)
Analytical FFT of Single Beam With No Contact at Point 14
1 Mode2 Modes3 Modes4 Modes5 Modes146 Modes
Fig. 10.5 FFT of time
response for single beam no
contact using varying number
of modes
134 T. Marinone et al.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Response at Pt. 14 due to Impact at Pt.14 for Single Beam With No Contact
Time [s]
Dis
plac
emen
t [in
]
146 Modes
1 Mode
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Response at Pt. 14 due to Impact at Pt.14 for Single Beam With No Contact
Time [s]
Dis
plac
emen
t [in
]
146 Modes
2 Modes
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Response at Pt. 14 due to Impact at Pt.14 for Single Beam With No Contact
Time [s]
Dis
plac
emen
t [in
]
146 Modes
3 Modes
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Response at Pt. 14 due to Impact at Pt.14 for Single Beam With No Contact
Time [s]
Dis
plac
emen
t [in
]
146 Modes
4 Modes
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Response at Pt. 14 due to Impact at Pt.14 for Single Beam With No Contact
Time [s]
Dis
plac
emen
t [in
]
146 Modes
5 Modes
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Response at Pt. 14 due to Impact at Pt.14 for Single Beam With No Contact
Time [s]
Dis
plac
emen
t [in
]
146 Modes
No Mode 1
a b
c d
e f
Fig. 10.6 Comparison of full solution results for single beam with no contact to results using modes 1 (a), 1–2 (b), 1–3 (c), 1–4 (d), 1–5 (e) and all
modes but mode 1 (f)
10 Efficient Computational Nonlinear Dynamic Analysis Using Modal Modification Response Technique 135
minimal improvement when adding the higher order modes. Finally, the solution time for the full system is close to 100 s in
contrast to the modal model, which is less than 5 s. MMRT is shown to have noticeable improvements in computational time
even for the linear system.
In order to demonstrate that the number of modes needed is controlled by the forcing function, the same beam
configuration was run where the analytical force input was changed to excite the first 250 Hz. As fewer modes are excited
over this new frequency bandwidth, the response is affected by fewer modes. Comparing the previous results for using one
and two modes in Table 10.3, the TRAC improves to 0.826 when using one mode, and improves to 0.993 when using two
modes. These results show that decreasing the bandwidth of the forcing function reduces the effect of mode truncation, as the
impact spectrum does not distribute energy to the higher frequency modes.
This first case demonstrated that an accurate time solution could be obtained using a limited number of modes, if the
primary modes excited by the structure are included in the modal database. If the wrong selection of modes is used, the
analyst will not obtain the correct time response, regardless of how many other modes are used.
The following cases will show the application of the MMRT when there is contact and the system becomes nonlinear.
In addition, both soft and hard contacts will be studied to show the effect of different contact stiffness on the accuracy of the
modal models. The soft stiffness case will be studied first, as this contact stiffness is unlikely to excite a high frequency range.
10.4 Case A: Soft Contact
10.4.1 Case A-2: Single Beam with Contact
This case consists of the tip of Beam A coming into contact with a fixed object once the beam has displaced a known gap
distance (0.05 in.) shown in Fig. 10.7.
The contact is represented as an additional spring stiffness that is added as soon as Beam A closes the gap. The contact
stiffness of 10 lb/in. will be used to represent a soft contact, such as typically seen in a damper or isolation mount. For the
purposes of MMRT, the spring stiffness was applied to the original model as a SDM. In addition, the contact is only
represented as a spring stiffness in the translational DOF, not in the rotational DOF as well. Table 10.4 lists the frequency
values of the SDM system, while Fig. 10.8 shows the [U12] matrix, or the contribution of the original modes in the SDM (a
highlighted box indicates that the mode of the original model contributes to the SDM model with a magnitude greater than
1.2% – the actual range for each color is shown below).
Table 10.4 and Fig. 10.8 shows that the addition of the spring has a pronounced effect on the lower order modes, with higher
ordermodes remaining relatively unaffected. The soft spring has a noticeable effect on the first twomodes, asmodes 3 and up are
approximately at the same frequency and require no additional mode shape to be added to form the SDM modal matrix (as
indicated by the redmain diagonal). FromCase 1 (single beamwith no contact), themodes needed in theMMRT process are the
modes which are excited by the input force spectrum. For the forcing function used, these should be the first three modes of the
system at a minimum, with additional higher order modes having minimal effect. Examining Fig. 10.8 shows that only modes 1
and 2 are used in the SDM process, so using only three modes should provide a high correlation to the full response solution.
In order to confirm this, the time response at point 14 will be computed using modes 1 through 5 and will be compared to
the full time solution in Fig. 10.9a–e for the soft spring. In addition, Fig. 10.9f will show the time response when mode 3 is
not included when forming the SDM modal database.
As the first three modes of the SDM database (the primary modes excited by the forcing function) only required the first
three modes of the unmodified modal database with the soft spring, three modes were adequate to represent the full system
model with a high degree of accuracy. As seen previously, the addition of modes 4 and 5 had minimal effect due to the input
force spectrum drops off at the higher frequencies. Figure 10.9f showed that the exclusion of mode 3 from the modal
database had a small negative effect. As mode 3 did not contribute to any SDM modes other than mode 3, however, the
response was reasonable comparable to the full space solution (average TRAC ¼ 0.884).
Table 10.3 Average MAC
and TRAC versus # of modes
used for single beam
no contact solution
# of modes Average MAC Average TRAC Solution time (s)
1 0.9439 0.8098 1.67
2 0.9942 0.9044 2.13
3 0.9985 0.9862 2.81
4 0.9986 0.9876 3.52
5 0.9994 0.9991 4.51
146 1.0000 1.0000 99.25
136 T. Marinone et al.
TheMAC and TRAC of the time responses were averaged as listed in Table 10.5 to show the effect of the additional modes.
As explained previously, the first three modes were sufficient to approximate the soft spring solution with a high degree of
accuracy. Finally, using MMRT causes a drastic reduction in solution time as seen comparing the solution time from the full
solution to the time using only a few modes.
This second case demonstrated that the analyst should examine the [U12] matrix to determine the number of modes that
are used in the SDMmodal. Even though the forcing input may only excite a few modes, the SDMmodal database may need
additional modes. Failure to include these modes will produce an incorrect time response regardless of how many other
modes are used.
10.4.2 Case A-3: Multiple Beams with Single Contact
This case consists of the tip of Beam A coming into contact with Beam B once Beam A has displaced a known gap distance
(0.05 in.) shown in Fig. 10.10.
The model uses the same soft stiffness contact of 10 lb/in. to represent the contact of the beams as explained in Case A-2.
Table 10.6 lists the frequency values of the SDM system along with which beam is excited, while Fig. 10.11 shows the
contribution of the original modes in the SDM.
Due to the contact between the two beams, the first mode of the SDM system now consists of modes from both beams,
which increases the number of modes needed for the SDM modal database. Based on the force input, which excites up to
1,000 Hz, the first 10 modes of the system should be excited. Examining Fig. 10.11 shows that for the first 10 modes of the
Table 10.4 Frequencies
for single beam with contact
for soft spring
Mode # Unmodified Soft spring – SDM
1 12.91 26.09
2 84.12 86.02
3 252.34 252.54
4 519.59 519.65
5 806.16 806.18
6 1,256.55 1,256.55
7 1,682.96 1,682.96
8 2,201.36 2,201.36
Fig. 10.8 Mode contribution matrix for single beam with soft single contact
Fig. 10.7 Diagram of single
beam with contact
10 Efficient Computational Nonlinear Dynamic Analysis Using Modal Modification Response Technique 137
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
a b
c d
e f
0.1Response at Pt. 14 due to Impact at Pt.14 for Single Beam with Soft Contact
Time [s]
Dis
plac
emen
t [in
]
146 Modes1 Mode
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1Response at Pt. 14 due to Impact at Pt.14 for Single Beam with Soft Contact
Time [s]
Dis
plac
emen
t [in
]
146 Modes2 Modes
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1Response at Pt. 14 due to Impact at Pt.14 for Single Beam with Soft Contact
Time [s]
Dis
plac
emen
t [in
]
146 Modes3 Modes
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1Response at Pt. 14 due to Impact at Pt.14 for Single Beam with Soft Contact
Time [s]
Dis
plac
emen
t [in
]
146 Modes4 Modes
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1Response at Pt. 14 due to Impact at Pt.14 for Single Beam with Soft Contact
Time [s]
Dis
plac
emen
t [in
]
146 Modes5 Modes
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1Response at Pt. 14 due to Impact at Pt.14 for Single Beam with Soft Contact
Time [s]
Dis
plac
emen
t [in
]
146 ModesNo Mode 3
Fig. 10.9 Comparison of full solution results for single beam with soft contact to results using modes 1 (a), 1–2 (b), 1–3 (c), 1–4 (d), 1–5 (e) and
all modes but mode 3 (f)
138 T. Marinone et al.
SDM system, the soft spring modal database requires modes 1 through 6 of Beam A and modes 1 through 4 of Beam B.
Note that the number of modes used in the time response is controlled by the frequency bandwidth of the input forcing
function, while the number of modes used in the SDM database is controlled by the [U12] matrix.
The time solution using the expected number of modes in the time response (10) with the required number of modes for
the SDM (6 from Beam A and 4 from Beam B) will be computed and compared to the full solution in Fig. 10.12a.
In addition, the effect of including incorrect modes in the time response will be shown by using the same modes in the time
response, where the SDM modal database contains all modes of Beam A but no modes of Beam B in Fig. 10.12b.
Figure 10.12a shows that including the necessary modes (6 from Beam A and 4 from Beam B) needed to form the [U2]
matrix based on the number of modes (10) excited by the input forcing function is enough to accurately reproduce the time
solution. If not all of the necessary modes are included as seen in Fig. 10.12b where no modes from Beam B were included,
using all of the modes of Beam A is not sufficient to accurately compute the response.
In contrast to the model of the single beam, more than five modes were needed in order to obtain an accurate time solution
of the system, due to the need for mode shapes from both beams. The MAC and TRAC of the time responses were averaged
as listed in Table 10.7.
Table 10.5 Average MAC
and TRAC versus # of modes
used for single beam contact
solution
# of modes
Soft
Average MAC Average TRAC Solution time (s)
1 0.8816 0.2884 1.68
2 0.9670 0.8858 1.98
3 0.9910 0.9699 2.54
4 0.9659 0.9677 3.78
5 0.9893 0.9860 4.63
Full 1.0000 1.0000 186.25
Fig. 10.10 Diagram
of multiple beam with
single contact
Table 10.6 Frequencies for
multiple beams with single
contact for soft spring
Mode # Unmodified SDM – soft spring
1 Beam A – 12.91 20.35
2 Beam B – 22.62 29.58
3 Beam A – 84.12 86.01
4 Beam B – 141.56 142.43
5 Beam A – 252.34 252.54
6 Beam B – 396.6 396.71
7 Beam A – 519.59 519.65
8 Beam B – 776.92 777.00
9 Beam A – 806.16 806.18
10 Beam A – 1,256.55 1,256.55
11 Beam B – 1,284.71 1,284.80
12 Beam A – 1,682.96 1,682.96
13 Beam B – 1,918.28 1,918.28
14 Beam A – 2,201.36 2,201.36
15 Beam B – 2,678.33 2,678.39
16 Beam A – 2,755.52 2,755.53
17 Beam A – 3,510.01 3,510.02
18 Beam B – 3,563.89 3,563.89
19 Beam A – 3,948.54 3,948.55
20 Beam B – 4,572.70 4,572.72
10 Efficient Computational Nonlinear Dynamic Analysis Using Modal Modification Response Technique 139
The third case demonstrated that the mode shapes of both components affect the modeling of nonlinear systems.
The forcing function bandwidth governs the number of modes needed to compute the time response and may be different
from the number of modes needed to compute the SDMmodal database. The [U12] matrix is critical in order to determine the
number of modes needed to form the system mode shapes, and the exclusion of modes used in the matrix can noticeably
degrade the correlation. For the soft spring contact, using the predicted number of mode shapes based on the forcing function
and [U12] matrix provided an accurate analytical time solution.
Bar Color Min. ValueMaxValue
Black 0.0120 0.0200
Blue 0.0200 0.0500
Green 0.0500 0.1000
Cyan 0.1000 0.2000
Magenta 0.2000 0.5000
Yellow 0.5000 0.8000
Red 0.8000 1.0000
System Mode Shapes -Soft Spring
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Bea
m A
Unm
odifie
d M
ode
Sha
pes
12345678910
Bea
m B
Unm
odifie
d M
ode
Sha
pes
12345678910
Fig. 10.11 Mode contribution matrix for multiple beams with single contact for soft spring
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Response at Pt. 14 due to Impact at Pt.14 for Multiple Beams with Soft Single Contact
Time [s]
Dis
plac
emen
t [in
]
10 Modes in Time Response (146 Beam A and 0 Beam B - U2)276 Modes
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Response at Pt. 14 due to Impact at Pt.14 for Multiple Beams with Soft Single Contact
Time [s]
Dis
plac
emen
t [in
]
10 Modes in Time Response (6 Beam A and 4 Beam B - U2)276 Modes
a b
Fig. 10.12 Comparison of results using 10 modes in the time response with the [U2] matrix formed from 6 Beam A and 4 Beam B modes (a) and
146 Beam A and 0 Beam B modes (b) for multiple beam single contact with soft spring
Table 10.7 Average MAC and TRAC versus # of modes used for multiple beam single contact solution
Soft spring
# of modes Average MAC Average TRAC Solution time (s)
10 – time (6 Beam A + 4 Beam B) – [U2] 0.9998 0.9916 6.78
10 – time (146 Beam A + 0 Beam B) – [U2] 0.8146 0.6167 6.78
276 1.0000 1.0000 1137.74
140 T. Marinone et al.
10.4.3 Case A-4: Multiple Beams with Multiple Contacts
This case consists of the tip of Beam A coming into contact with the tip of Beam B shown in Fig. 10.13a–c. Note that each
system is a potential configuration of the beam depending on the relative displacements of the two beams.
The model uses the same soft spring stiffness value as described in Case A-3. Because the system has multiple possible
configurations, multiple SDMs are required in order to represent the different possible configurations. The modal database
must therefore contain all of the necessary modes for all of the configurations in order to obtain an accurate model.
Table 10.8 lists the frequency values of the SDM system, while Fig. 10.14 shows the contribution of the original modes in
the SDM.
Table 10.8 and Fig. 10.14 shows that not only does the stiffness of the spring affect the number of modes needed, but the
location of the spring affects the modes as well. Depending on whether the spring is at the tip of Beam A or tip of Beam B,
the mode shapes and frequencies change noticeably. For example, the second mode of Beam A is used for the first mode of
the soft spring SDM for configurations 1 and 3, but is not needed in configuration 2. Thus, if only configuration 2 came into
play this mode could be neglected in the modal database; due to also using configurations 1 and 3 that require this second
mode, this mode should be used in the database.
Examining Fig. 10.14 shows that for a forcing function which excites the first 10 modes of the system, the soft spring
configuration requires modes 1–6 of Beam A and modes 1–4 of Beam B. Figure 10.15 compares the full solution to the
solution using 10 modes for the time solution with 6 modes from Beam A and 4 modes from Beam B for the [U2] matrix for
the soft spring.
a b c
Fig. 10.13 Diagram of multiple beam with multiple contact for configurations 1(a), 2 (b) and 3 (c)
Table 10.8 Frequencies for
multiple beams with single
contact for soft spring Mode # Unmodified
Soft spring
Config. 1 Config. 2 Config. 3
1 Beam A – 12.91 20.35 14.78 21.08
2 Beam B – 22.62 29.58 33.04 39.23
3 Beam A – 84.12 86.01 85.71 87.24
4 Beam B – 141.56 142.43 142.98 143.82
5 Beam A – 252.34 252.54 252.61 252.82
6 Beam B – 396.6 396.71 396.96 397.07
7 Beam A – 519.59 519.65 519.61 519.68
8 Beam B – 776.92 777 777.04 777.12
9 Beam A – 806.16 806.18 806.28 806.3
10 Beam A – 1,256.55 1,256.55 1,256.56 1,256.56
11 Beam B – 1,284.71 1,284.80 1,284.76 1,284.85
12 Beam A – 1,682.96 1,682.96 1,682.99 1,682.99
13 Beam B – 1,918.28 1,918.28 1,918.29 1,918.30
14 Beam A – 2,201.36 2,201.36 2,201.37 2,201.37
15 Beam B – 2,678.33 2,678.39 2,678.34 2,678.39
16 Beam A – 2,755.52 2,755.53 2,755.53 2,755.53
17 Beam A – 3,510.01 3,510.02 3,510.02 3,510.03
18 Beam B – 3,563.89 3,563.89 3,563.89 3,563.89
19 Beam A – 3,948.54 3,948.55 3,948.54 3,948.55
20 Beam B – 4,572.70 4,572.72 4,572.70 4,572.72
10 Efficient Computational Nonlinear Dynamic Analysis Using Modal Modification Response Technique 141
As seen in Case 3, the [U12] matrix provides a good understanding of the soft spring model. Even with a more complex
model with multiple possible configurations, the computational savings gained from the MMRT technique are substantial.
This case used all of the lessons learned from the previous three cases in order to compute an accurate time solution using
MMRT with a subset of the modes and demonstrate the usefulness of the technique on a complex structure with multiple
potential interactions.
10.5 Case B: Hard Contact
For the soft stiffness contact cases shown above, the contact stiffness was soft enough that the contact did not excite a
frequency bandwidth beyond the excitation force spectrum. Thus, the forcing function controls the number of modes needed
in the time response, and the number of modes needed in the SDM modal database is based on those modes. As long as the
excitation bandwidth due to the contact stiffness impact is below the input bandwidth, the procedure to identify the number
of modes needed works well as seen in Cases A-2 through A-4.
For a harder stiffness contact case, however, there is a possibility that the contact stiffness would excite a frequency
bandwidth beyond the excitation force spectrum. Under this scenario, the number of modes needed in the time response
would be a function of not only the forcing function, but also of the contact stiffness. In order to examine this possible
scenario in detail, the contact stiffness of 1,000 lb/in. is used for the same cases.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Bea
m A
Unm
odifie
d M
ode
Sha
pes
Bea
m B
Unm
odifie
d M
ode
Sha
pes
System Mode Shapes - Soft Spring - Configuration 1 System Mode Shapes - Soft Spring - Configuration 2 System Mode Shapes - Soft Spring - Configuration 3
1.00000.8000Red
0.80000.5000Yellow
0.50000.2000Magenta
0.20000.1000Cyan
0.10000.0500Green
0.05000.0200Blue
0.02000.0120Black
Max Value
Min. ValueBar Color
1.00000.8000Red
0.80000.5000Yellow
0.50000.2000Magenta
0.20000.1000Cyan
0.10000.0500Green
0.05000.0200Blue
0.02000.0120Black
Max Value
Min. ValueBar Color
Fig. 10.14 Mode contribution matrix for multiple beams with multiple contacts for soft spring
276 1 1 915.21
Soft Spring
# of Modes
0.9884 0.9774 11.7510 – Time
(6 Beam A + 4 Beam B) – [U2]
AverageMAC
AverageTRAC
SolutionTime (s)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1Response at Pt. 14 due to Impact at Pt.14 for Multiple Beams with Soft Multiple Contact
Time [s]
Dis
plac
emen
t [in
]
10 Modes in Time Response (6 Beam A and 4 Beam B - U2)276 Modes
Fig. 10.15 Comparison of results in the time domain for multiple beams with multiple contacts with soft spring
142 T. Marinone et al.
10.5.1 Case B-2: Single Beam with Contact
This case consists of the tip of Beam A coming into contact with a fixed object once the beam has displaced a known gap
distance (0.05 in.) as explained previously in Case A-2. Figure 10.16 shows the [U12] matrix and lists the natural frequencies
of the first eight modes.
Figure 10.16 shows that in order to accurately obtain the first five system modes, the first five modes of the unmodified
system are required. This is expected, as the mode shapes with the harder spring attached look less like the original model
and therefore require more modes in order to form the SDM. As Fig.10.16 shows that modes 1–5 of the unmodified system
are involved in the first three modes of the SDM, five modes should be used to provide a high correlation to the full response
solution. In order to confirm this, the time response at point 14 will be computed using 1 through 5 modes and compared to
the full time solution in Fig. 10.17a–e for the hard spring. In addition, Fig. 10.17f will show the time response when mode 3
is not included when forming the SDM modal database.
In contrast to Case A-2, using the hard spring produced poor results using modes 1–4. As seen in Fig. 10.16, modes 2 and
3 of the SDM database require modes 1–5 of the unmodified modal database and were not able to be adequately represented
using only modes 1–4. Even though the forcing function was limited primarily to the first three modes of the system, the first
3 modes of the SDM required more than three modes of the unmodified modal database. Figure 10.17f showed that the
exclusion of mode 3 from the modal database decreased the accuracy of the results. Examining Fig. 10.16 shows that mode 3
had a substantial contribution to SDM modes 1–4, and thus the SDM modes could not be fully represented without the
missing mode. The response is poor compared to the full space solution (average TRAC ¼ 0.439).
In order to show the effect of the modes, the MAC and TRAC of the time responses were averaged as listed in Table 10.9.
As the hard spring solution required five modes of the unmodified modal database, the response remained poor until all
five of the modes were included as seen in the sudden increase in TRAC (0.6615–0.9949).
Finally, the FFT of the time response for both the soft and hard spring is shown in Fig. 10.18. Note that the hard spring
FFT is more nonlinear in contrast to the soft spring, where the main frequencies of the system can still be identified clearly.
The hard spring does not contain significant energy beyond 1,000 Hz, showing that for this case, the forcing function and not
the contact stiffness govern the number of modes needed in the time response.
10.5.2 Case B-3: Multiple Beams with Single Contact
This case consists of the tip of Beam A coming into contact with the tip of Beam B once Beam A has displaced a known gap
distance (0.05 in.) as explained previously in Case A-3. Table 10.10 lists the frequency values of the SDM system along with
which beam is excited, while Fig. 10.19 shows the [U12] matrix.
As seen in Case A-3, the forcing function excited the first 10 modes of the system. Examining Fig. 10.19 shows that for
the first 10 modes of the SDM system, the hard spring modal database requires modes 1–6 of Beam A and modes 1–5 of
Beam B. The time solution using the expected number of modes in the time response (10) with the required number of modes
for the SDM (6 from Beam A and 5 from Beam B) will be computed and compared to the full solution in Fig. 10.20.
The expected number of modes used for the time response (10) with the modes used in the SDM modal database (6 from
Beam A and 5 from Beam B), does not produce an accurate solution as seen in Fig. 10.20. In contrast, Case A-3 with the soft
Mode #
1 12.91 66.681
12
3
4
5
6
78
2 3 4 5 6 7 8 9 10
223.55
326.04
529.68
808.16
84.12
252.34
519.59
806.16
1256.55 1256.73
Bea
m A
Unm
odifie
d M
ode
Shap
es
1682.96 1682.97
2201.36 2201.52
2
3
4
5
6
7
8
UnmodifiedHard Spring-
SDMSystem Mode Shapes-Hard Spring
Bar Color
Black 0.0120 0.02000.0500
0.10000.2000
0.5000
0.8000
1.0000
0.0200
0.0500
0.1000
0.2000
0.5000
0.8000
Blue
GreenCyan
Magenta
Yellow
Red
Min.Value
Max.Value
Fig. 10.16 Mode contribution matrix for single beam with single contact for soft (a) and hard (b) spring
10 Efficient Computational Nonlinear Dynamic Analysis Using Modal Modification Response Technique 143
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35Response at Pt. 14 due to Impact at Pt.14 for Single Beam with Hard Contact
Time [s]
Dis
plac
emen
t [in
]
146 Modes1 Mode
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35Response at Pt. 14 due to Impact at Pt.14 for Single Beam with Hard Contact
Time [s]
Dis
plac
emen
t [in
]
146 Modes2 Modes
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6Response at Pt. 14 due to Impact at Pt.14 for Single Beam with Hard Contact
Time [s]
Dis
plac
emen
t [in
]
146 Modes3 Modes
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35Response at Pt. 14 due to Impact at Pt.14 for Single Beam with Hard Contact
Time [s]
Dis
plac
emen
t [in
]
146 Modes4 Modes
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35Response at Pt. 14 due to Impact at Pt.14 for Single Beam with Hard Contact
Time [s]
Dis
plac
emen
t [in
]
146 Modes5 Modes
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6Response at Pt. 14 due to Impact at Pt.14 for Single Beam with Hard Contact
Time [s]
Dis
plac
emen
t [in
]
146 ModesNo Mode 3
a b
c d
e f
Fig. 10.17 Comparison of full solution results for single beam with hard contact to results using modes 1 (a), 1–2 (b), 1–3 (c), 1–4 (d), 1–5 (e) and
all modes but mode 3 (f)
144 T. Marinone et al.
spring did produce an accurate solution using the same mode set as seen in Fig. 10.12a. Figure 10.21 shows the time response
using 23 modes for the time response, where 20 modes from Beam A and 15 modes from Beam B formed the SDM database.
Figure 10.21 shows that additional modes are needed beyond the expected number of modes (based only on the input
force) in order to compute the accurate time response of the hard spring system. To investigate why using the expected
number of modes did not produce an accurate time response for the hard spring stiffness, the FFT of the time response for
both springs is shown in Fig. 10.22.
Note that in contrast to the soft spring, the hard spring contact excited modes above 1,000 Hz (almost up to 3,000 Hz),
which is above the input force spectrum. Accordingly, the number of modes needed in the time response is no longer
governed by the forcing function bandwidth, but by the contact stiffness bandwidth. This explains why Fig. 10.21 required
additional modes in order to obtain an accurate answer, as the contact stiffness was exciting modes that were not included
when the first 10 modes of the system formed the database. Since these unused higher order modes were excited by the
contact stiffness, they needed to be included to obtain a correct time solution. In order to show the effect of the modes, the
MAC and TRAC of the time responses were averaged as listed in Table 10.11.
10.5.3 Case B-4: Multiple Beams with Multiple Contacts
This case consists of the tip of Beam A coming into contact with the tip of Beam B once the relative displacements of the
beams are within a specified contact tolerance (0.001 in.) as described previously in Case A-4. Table 10.12 lists the
frequency values of the SDM system, while Fig. 10.23 shows the contribution of the original modes in the SDM.
Figure 10.23 shows that the hard spring configuration requires modes 1–6 of Beam A and modes 1–5 of Beam B. As Case
B-3 showed that additional modes beyond the modes excited by the forcing function are required when using the hard spring,
only the time solution where accurate results are obtained is shown. Using the number of modes based on the forcing
Table 10.9 Average MAC
and TRAC versus # of modes
used for single beam contact
solution
# of modes
Hard
Average MAC Average TRAC Solution time (s)
1 0.8672 0.3591 1.65
2 0.8586 0.5056 2.29
3 0.8569 0.5479 3.01
4 0.8935 0.6615 3.73
5 0.9992 0.9949 4.27
Full 1.0000 1.0000 298.91
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-220
-200
-180
-160
-140
-120
-100
-80
-60
-40
-20
Frequency (Hz)
dB R
espo
nse
(g)
Analytical FFT at Point 14 - Single Beam with Contact
Soft SpringHard Spring
Fig. 10.18 FFT comparison
between the soft and hard
spring for single beam with
contact
10 Efficient Computational Nonlinear Dynamic Analysis Using Modal Modification Response Technique 145
function provides an average TRAC of 0.3625, showing that there are an insufficient number of modes used as expected.
Figure 10.24 compares the full solution to the solution using 23 modes for the time solution with 20 modes from Beam A and
15 modes from Beam B for the [U2] matrix for the hard spring.
As seen in Cases A-3 and B-3, the [U12] matrix provides a good understanding of the soft spring model, but a poor
understanding of the hard spring model. In order to show the effect of the modes, the MAC and TRAC of the time responses
were averaged as listed in Table 10.13.
Finally, the FFT of the time response was computed for both the soft and hard spring shown in Fig. 10.25.
As seen in Case B-3, the hard contact has excited modes above the forcing function bandwidth of 1,000 Hz. Since the
contact stiffness now governs the number of modes needed in the time response, more than the 10 modes excited by the
forcing function are required.
Comparing the soft and hard spring cases shows that the number of modes needed can be predicted accurately as long as
the forcing function bandwidth defines the frequency range that is excited as seen in the soft spring cases. Once that
condition is no longer true, however, determining the number of modes needed requires some knowledge of the frequency
Table 10.10 Frequencies
for multiple beams with single
contact for hard spring
Mode # Unmodified SDM – hard spring
1 Beam A – 12.91 21.25
2 Beam B – 22.62 67.74
3 Beam A – 84.12 127.04
4 Beam B – 141.56 236.02
5 Beam A – 252.34 338.31
6 Beam B – 396.6 424.26
7 Beam A – 519.59 532.04
8 Beam B – 776.92 786.59
9 Beam A – 806.16 809.25
10 Beam A – 1,256.55 1,256.69
11 Beam B – 1,284.71 1,294.66
12 Beam A – 1,682.96 1,682.97
13 Beam B – 1,918.28 1,918.72
14 Beam A – 2,201.36 2,201.53
15 Beam B – 2,678.33 2,683.90
16 Beam A – 2,755.52 2,755.78
17 Beam A – 3,510.01 3,510.46
18 Beam B – 3,563.89 3,564.14
19 Beam A – 3,948.54 3,948.79
20 Beam B – 4,572.70 4,575.19
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201234567891012345678910
Bea
m A
Unm
odifie
d M
ode
Sha
pes
Bea
m B
Unm
odifie
d M
ode
Sha
pes
System Mode Shapes - Hard Spring
Bar Color Min. ValueMax Value
Black 0.0120 0.0200
Blue 0.0200 0.0500
Green 0.0500 0.1000
Cyan 0.1000 0.2000
Magenta 0.2000 0.5000
Yellow 0.5000 0.8000
Red 0.8000 1.0000
Fig. 10.19 Mode contribution matrix for multiple beams with single contact for hard spring
146 T. Marinone et al.
bandwidth of contact ahead of time. For both cases, however, the analyst can obtain an accurate modal solution with
significant computational savings provided the used modes satisfy the force spectrum and the contact stiffness spectrum.
10.6 Contact Time Step Study
For the chosen Dt of 0.0001 s, Raleigh Criteria and Shannon’s Sampling Theorem state that the maximum frequency range
that can be observed is 5 kHz, well above the forcing function frequency range. Although this time resolution may be fine
enough to accurately capture the time response for the linear system, this resolution may not be adequate when the response
becomes nonlinear. Since the Dt chosen affects when the system first comes into contact, the system may respond differently
depending on whether the impact is slow or abrupt (i.e. soft or hard spring). For a slow spring, the system remains in contact
for a longer period of time, and the time response should remain consistent regardless of the time step chosen. For the hard
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.05
0
0.05
0.1
0.15Response at Pt. 14 due to Impact at Pt.14 for Multiple Beams with Hard Single Contact
Time [s]
Dis
plac
emen
t [in
]
10 Modes in Time Response (6 Beam A and 5 Beam B - U2)276 Modes
Fig. 10.20 Comparison of
results using 10 modes in the
time response with the [U2]
matrix formed from 6 Beam A
and 5 Beam B modes for
multiple beam single contact
with hard spring
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Response at Pt. 14 due to Impact at Pt.14 for Multiple Beams with Hard Single Contact
Time [s]
Dis
plac
emen
t [in
]
23 Modes in Time Response (20 Beam A and 15 Beam B - U2)276 Modes
Fig. 10.21 Comparison of
results using 23 modes in the
time response with the [U2]
matrix formed from 20 Beam
A and 15 Beam B modes for
multiple beam single contact
with hard spring
10 Efficient Computational Nonlinear Dynamic Analysis Using Modal Modification Response Technique 147
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-240
-220
-200
-180
-160
-140
-120
-100
-80
-60
Frequency (Hz)
dB R
espo
nse
(g)
Analytical FFT at Pt. 14 - Multiple Beam with Single Contact
Soft SpringHard Spring
Fig. 10.22 FFT comparison
between the soft and hard
spring for multiple beam with
single contact
Table 10.11 Average MAC and TRAC versus # of modes used for multiple beam single contact solution
Hard spring
# of modes Average MAC Average TRAC Solution time (s)
10 – time (6 Beam A + 5 Beam B) – U2 0.7888 0.5232 6.87
23 – time (20 Beam A + 15 Beam B) – U2 1.0000 0.9988 17.11
276 1.0000 1.0000 1240.13
Table 10.12 Frequencies for
multiple beams with single
contact for soft and hard springs Mode # Unmodified
Hard spring
Config. 1 Config. 2 Config. 3
1 Beam A – 12.91 21.25 15.5 36.22
2 Beam B – 22.62 67.74 69.52 113.69
3 Beam A – 84.12 127.04 119.79 228.57
4 Beam B – 141.56 236.02 234.14 312.68
5 Beam A – 252.34 338.31 325.85 339.4
6 Beam B – 396.6 424.26 467.09 504.56
7 Beam A – 519.59 532.04 527.49 532.41
8 Beam B – 776.92 786.59 786.04 790.33
9 Beam A – 806.16 809.25 828.78 833.74
10 Beam A – 1,256.55 1,256.69 1,257.17 1,257.22
11 Beam B – 1,284.71 1,294.66 1,290.03 1,300.10
12 Beam A – 1,682.96 1,682.97 1,686.00 1,686.01
13 Beam B – 1,918.28 1,918.72 1,920.05 1,920.48
14 Beam A – 2,201.36 2,201.53 2,202.67 2,202.84
15 Beam B – 2,678.33 2,683.90 2,678.85 2,684.38
16 Beam A – 2,755.52 2,755.78 2,756.01 2,756.28
17 Beam A – 3,510.01 3,510.46 3,511.34 3,511.79
18 Beam B – 3,563.89 3,564.14 3,563.98 3,564.23
19 Beam A – 3,948.54 3,948.79 3,948.54 3,948.79
20 Beam B – 4,572.70 4,575.19 4,572.70 4,575.19
148 T. Marinone et al.
spring, however, the system may experience a high frequency impact chatter, which may not be seen with a large time step.
Figure 10.26a, b show the time response for the first 0.1 s for the soft and hard spring respectively, where a time step of
0.00005 s and the previously used time step of 0.0001 s are compared.
Figure 10.26a shows that for the soft spring system, the reduced time step had minimal effect on the time solution. Since
the system remains in contact with the soft spring for an extended period, the results compare very well. For the hard spring
in Fig. 10.26b, the system comes into contact and then immediately bounces off. Since the time step duration directly affects
the acceleration, there is a divergence in the solution.
Although this study shows that the chosen time step was not sufficiently fine for the hard spring, the time step of 0.0001 s
was used for all of the previous cases in order to demonstrate the main principles of MMRT. Further study is required to
determine the required time step when the contact stiffness dominates the response.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
System Mode Shapes - Hard Spring - Configuration 1 System Mode Shapes - Hard Spring - Configuration 2 System Mode Shapes - Hard Spring - Configuration 3
sepahSedo
Mde ifido
m nU
Ama e
Bsep ahS
ed oM
de ifid omn
UB
ma eB
1.00000.8000Red
0.80000.5000Yellow
0.50000.2000Magenta
0.20000.1000Cyan
0.10000.0500Green
0.05000.0200Blue
0.02000.0120Black
Max Value
Min. ValueBar Color
1.00000.8000Red
0.80000.5000Yellow
0.50000.2000Magenta
0.20000.1000Cyan
0.10000.0500Green
0.05000.0200Blue
0.02000.0120Black
Max Value
Min. ValueBar Color
Fig. 10.23 Mode contribution matrix for multiple beams with multiple contacts for hard spring
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
Response at Pt. 14 due to Impact at Pt.14 for Multiple Beams with Hard Multiple Contact
Time [s]
Dis
plac
emen
t [in
]
23 Modes in Time Response (20 Beam A and 15 Beam B - U2)276 Modes
Fig. 10.24 Comparison of
results in the time domain for
multiple beams with multiple
contacts with hard spring
Table 10.13 Average MAC and TRAC versus # of modes used for multiple beams multiple contact solution
Hard spring
# of modes Average MAC Average TRAC Solution time (s)
23 – time (20 Beam A + 15 Beam B) – U2 0.9999 0.9962 17.11
276 1.0000 1.0000 1196.69
10 Efficient Computational Nonlinear Dynamic Analysis Using Modal Modification Response Technique 149
10.7 Comparison to Large DOF Models
Although the computational time savings using MMRT is significant, the effect is not dramatically seen until comparing to a
large FEM model as typically seen in industry. Figure 10.27 shows the FEM of a missile rack system with a fine mesh
resolution.
Analysts use this model in dynamic simulations to compute the transient dynamics during a missile firing under a variety
of condition. As the system is a complex system with many detailed components, the time response models require
significant computation time (typically days). The computational time can be dramatically reduced by using the mode
shapes and frequencies of the various components. This allows analysts to study multiple configurations in detail, providing
further information in order to improve the design.
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-220
-200
-180
-160
-140
-120
-100
-80
-60
-40
Frequency (Hz)
dB R
espo
nse
(g)
Analytical FFT at Pt. 14 - Multiple Beam with Multiple Contact
Soft SpringHard Spring
Fig. 10.25 FFT comparison
between the soft and hard
spring for multiple beam with
multiple contact
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1Response at Pt. 14 due to Impact at Pt.14 for Single Beam with Soft Contact
Time [s]
Dis
plac
emen
t [in
]
dt = 0.0001 sdt = 0.00005 s
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3Response at Pt. 14 due to Impact at Pt.14 for Single Beam with Hard Contact
Time [s]
Dis
plac
emen
t [in
]
dt = 0.0001 sdt = 0.00005 s
a b
Fig. 10.26 Comparison of results with two different time steps for soft (a) and hard (b) contact
150 T. Marinone et al.
10.8 Conclusion
A proposed technique – Modal Modification Response Technique (MMRT) – for computing the time response of a nonlinear
system is described. For linear systems that are connected by a nonlinear local connection, mode superposition and structural
dynamic modification can be used to approximate the nonlinear response of the system at significant computational savings.
An analytical study of this technique is shown using a linear beam systemwith contact due to impact. Four cases of increasing
complexity are studied; observations on the number of modes needed for each case aremade based on the forcing function and
SDM mode contribution matrix. For all cases, MMRT yields accurate results with substantially less computation time.
10.9 Further Work
Further work will be done using this technique to demonstrate the usefulness of this approach. For comparison to
experimental results, several additional items will need to be studied in order to yield accurate results.
First, the underlying linear system will need to be a very high accuracy model in order to accurately predict the time
responses. Care will need to be taken in modeling and updating the model to test data to ensure that both the model and
measurements are reflective of the physical structure. Second, the damping will have to be measured experimentally for the
linear system for as many modes as possible in order to have high time correlation. In addition, efforts will be needed to
determine the correct damping for the SDM models, as the damping may change once the beam(s) are in contact. Third, the
stiffness of the contact will have to be determined in order to accurately calculate the SDM. In addition, the impact force of
the beam contact may need to be included in the analytical model as this is an additional force input to the structure that
affects the time response. Finally, variation in the time step used was found to affect the results obtained and will require
further study.
Acknowledgements Some of the work presented herein was partially funded by Air Force Research Laboratory Award No. FA8651-10-1-0009
“Development of Dynamic Response Modeling Techniques for Linear Modal Components”. Any opinions, findings, and conclusions or
recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the particular funding agency.
The authors are grateful for the support obtained.
Fig. 10.27 Typical large
FEM model
10 Efficient Computational Nonlinear Dynamic Analysis Using Modal Modification Response Technique 151
Appendix A: Component and System Mode Shapes
0 2 4 6 8 10 12 14 16-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16-60
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0
20
40
60
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20
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60
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60
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0 2 4 6 8 10 12 14 16-60
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0 2 4 6 8 10 12 14 16-60
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0 2 4 6 8 10 12 14 16-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16-60
-40
-20
0
20
40
60
Unmodified Mode Shapes – Beam B
1 – 22.62 Hz 2 – 141.56 Hz 3 – 396.60 Hz
4 – 776.92 Hz 5 – 1284.71 Hz 6 – 1918.28 Hz
7 – 2678.33 Hz 8 – 3563.89 Hz 9 – 4572.70 Hz
10 – 5707.04 Hz 11 – 6956.91 Hz 12 – 8324.25 Hz
Unmodified Mode Shapes – Beam A
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
601 - 12.91 Hz 2 – 84.12 Hz 3 – 252.34 Hz
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
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0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
4 – 519.59 Hz 5 – 806.16 Hz 6 – 1256.55 Hz
7 – 1682.96 Hz 8 – 2201.36 Hz 9 – 2755.52 Hz
10 – 3510.01 Hz 11 – 3948.54 Hz 12 – 5076.34 Hz
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
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40
60
0 2 4 6 8 10 12 14 16 18-60
-40
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0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
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0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
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0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
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0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
Single Beam Single Contact Mode Shapes – 10 lb/in Spring
1 – 26.09 Hz 2 – 86.02 Hz 3 – 252.54 Hz
4 – 519.65 Hz 5 – 806.18 Hz 6 – 1256.55 Hz
7 – 1682.96 Hz 8 – 2201.36 Hz 9 – 2755.53 Hz
10 – 3510.02 Hz 11 – 3948.55 Hz 12 – 5076.34 Hz
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
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0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
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60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
0 2 4 6 8 10 12 14 16 18-60
-40
-20
0
20
40
60
Single Beam Single Contact Mode Shapes – 1000 lb/in Spring
1 – 66.68 Hz 2 – 223.55 Hz 3 – 326.04 Hz
4 – 529.68 Hz 5 – 808.16 Hz 6 – 1256.73 Hz
7 – 1682.97 Hz 8 – 2201.52 Hz 9 – 2755.76 Hz
10 – 3510.46 Hz 11 – 3948.79 Hz 12 – 5076.36 Hz
152 T. Marinone et al.
0 5 10 15 20 25-60
-40
-20
0
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40
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-40
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20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
Multiple Beam Contact Mode Shapes - Configuration 1 – 10 lb/in Spring
1 – 20.35 Hz 2 – 29.58 Hz 3 – 86.01 Hz
4 – 142.43 Hz 5 – 252.54 Hz 6 – 396.71 Hz
7 – 519.65 Hz 8 – 777.00 Hz 9 – 806.18 Hz
10 – 1256.55 Hz 11 – 1284.80 Hz 12 – 1682.96 Hz
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
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-40
-20
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-40
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60
0 5 10 15 20 25-60
-40
-20
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60
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-40
-20
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-40
-20
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-40
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-40
-20
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60
0 5 10 15 20 25-60
-40
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40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
Multiple Beam Contact Mode Shapes – Configuration 2 – 10 lb/in Spring
1 – 14.78 Hz 2 – 33.04 Hz 3 – 85.71 Hz
4 – 142.98 Hz 5 – 252.61 Hz 6 – 396.96 Hz
7 – 519.61 Hz 8 – 777.04 Hz 9 – 806.28 Hz
10 – 1256.56 Hz 11 – 1284.76 Hz 12 – 1682.99 Hz
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
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40
60
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-40
-20
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20
40
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-40
-20
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20
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60
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-40
-20
0
20
40
60
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-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
Multiple Beam Contact Mode Shapes – Configuration 3 – 10 lb/in Spring
1 – 21.08 Hz 2 – 39.23 Hz 3 – 87.24 Hz
4 – 143.82 Hz 5 – 252.82 Hz 6 – 397.07 Hz
7 – 519.68 Hz 8 – 777.12 Hz 9 – 806.30 Hz
10 – 1256.56 Hz 11 – 1284.85 Hz 12 – 1682.99 Hz
10 Efficient Computational Nonlinear Dynamic Analysis Using Modal Modification Response Technique 153
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
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60
0 5 10 15 20 25-60
-40
-20
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40
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0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
Multiple Beam Contact Mode Shapes – Configuration 3 – 1000 lb/in Spring
1 – 36.22 Hz 2 – 113.69 Hz 3 – 228.57 Hz
4 – 312.68 Hz 5 – 339.40 Hz 6 – 504.56 Hz
7 – 532.41 Hz 8 – 790.33 Hz 9 – 833.74 Hz
10 – 1257.22 Hz 11 – 1300.10 Hz 12 – 1686.01 Hz
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
Multiple Beam Contact Mode Shapes – Configuration 1 – 1000 lb/in Spring
1 – 21.25 Hz 2 – 67.74 Hz 3 – 127.04 Hz
4 – 236.02 Hz 5 – 338.31 Hz 6 – 424.26 Hz
7 – 532.04 Hz 8 – 786.59 Hz 9 – 809.25 Hz
10 – 1256.69 Hz 11 – 1294.66 Hz 12 – 1682.97 Hz
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
0 5 10 15 20 25-60
-40
-20
0
20
40
60
Multiple Beam Contact Mode Shapes – Configuration 2 – 1000 lb/in Spring
1 – 15.50 Hz 2 – 69.52 Hz 3 – 119.79 Hz
4 – 234.14 Hz 5 – 325.85 Hz 6 – 467.09 Hz
7 – 527.49 Hz 8 – 786.04 Hz 9 – 828.78 Hz
10 – 1257.17 Hz 11 – 1290.03 Hz 12 – 1686.00 Hz
154 T. Marinone et al.
References
1. Avitabile P, O’Callahan J (2009) Efficient techniques for forced response involving linear modal components interconnected by discrete
nonlinear connection elements. Mech Syst Signal Process 23(1):45–67
2. Friswell MI, Penney JET, Garvey SD (1995) Using linear model reduction to investigate the dynamics of structures with local non-linearities.
Mech Syst Signal Process 9(3):317–328
3. Lamarque C, Janin O (2000) Modal analysis of mechanical systems with impact non-linearities: limitations to a modal superposition. J Sound
Vib 235(4):567–609
4. Ozguven H, Kuran B (1996) A modal superposition method for non-linear structures. J Sound Vib 189(3):315–339
5. Al-Shudeifat M, Butcher E, Burton T (2010) Enhanced order reduction of forced nonlinear systems using new Ritz vectors. In: Proceedings of
the IMAC-XXVIII. 2010. Print
6. Rhee W (2000) Linear and nonlinear model reduction in structural dynamics with application to model updating. PhD dissertation. Texas Tech
University
7. Avitabile P (2003) Twenty years of structural dynamic modification – a review. Sound Vib 37(1):14–27
8. Rao S (2004) Mechanical vibrations, 4th edn. Prentice Hall, New Jersey, pp 834–843
9. Van Zandt T (2006) Development of efficient reduced models for multi-body dynamics simulations of helicopter wing missile configuration.
Master’s thesis, University of Massachusetts Lowell, April 2006
10. O’Callahan J (1986) MAT_SAP/MATRIX, A general linear algebra operation program for matrix analysis. University of Massachusetts
Lowell, 1986
11. MATLAB (R2010a) The MathWorks Inc., Natick, MA
10 Efficient Computational Nonlinear Dynamic Analysis Using Modal Modification Response Technique 155
Chapter 11
Spectral Domain Force Identification of Impulsive Loading
in Beam Structures
Pooya Ghaderi, Andrew J. Dick, Jason R. Foley, and Gregory Falbo
Abstract In this paper, an identification method is presented for calculating impulsive loads from the propagating
mechanical wave which are produced in beam-like structures. This method uses a spectral finite element method (SFEM)
model of a segment of the structure to calculate force information from the measured response. The SFEMmodel is prepared
from the Euler-Bernoulli beam equation in the frequency domain. The method is studied using simulated response data and
then applied to data collected from an experimental system. Excellent performance is observed for nominal conditions and a
parametric study is performed to determine how different factors affect accuracy. Factors studied include structure size,
loading location, and loading duration. When limited to only acceleration data, the use of finite differencing methods to
obtain the required slope response information is determined to provide the most significant source of error in the identified
force information.
Keywords SFEM • Structural finite element method • Impact force identification
11.1 Introduction
Direct measurement of the impact force applied to amechanical structure is not always possible due to the nature of the impact
or the complexity of the structure. Research efforts have focused on developing indirect methods to identify the applied
impact force. Inverse methods are a common technique for calculating the impact force from a measured response and an
accurate model of the system. Impact force identification has been studied for applications in various fields such as gas pipes
[1], composite structures [2], and health monitoring [3].
Numerous methods have been developed using inverse methods for impact force identification. Some common
techniques for force identification are the deconvolution method [4], state variable formulation [5], the sum of weighted
accelerations [6] and the spectral element method [7]. The popular method is the deconvolution technique which uses an
assumption of linear behavior of the response to allow for the application of the convolution integral in order to determine
the system response. When using this method, the applied force is obtained by extracting the impact force from the
relationship between the response and the convolution integral of the impact force and the system’s impulse response.
This technique has been applied in both the time domain (e.g. [4]) and the spectral domain (e.g. [8]).
In the work of Doyle [4], a time domain deconvolution technique was developed in order to experimentally obtain
dynamic contact laws. Response behavior was monitored by using strain gauges mounted onto a beam-like structure and the
impulsive load was applied by using a pendulous ball. Chang and Sun [9] calculated the applied impact force by using an
experimental Green’s function and the time domain signal deconvolution. The reconstructed force was found independent of
the location of the sensor on a composite beam-like structure.
P. Ghaderi • A.J. Dick (*)
Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX 77005, USA
e-mail: [email protected]
J.R. Foley
Air Force Research Laboratory, Eglin Air Force Base, FL, USA
G. Falbo
LMS Americas, Inc, Detroit, MI, USA
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_11, # The Society for Experimental Mechanics, Inc. 2012
157
The deconvolution technique has also been used in the spectral domain for solving force identification problems. Due to
the ease of the deconvolution calculation in the frequency domain, this form of the force identification method was more
easily implemented. In the work of Doyle [8], a spectral domain deconvolution method was developed for experimentally
determining contact laws. The process of determining the impact force by using experimental strain data was based on the
relationship between the contact force and the strain at different locations on the test structure. The contact force was
obtained in the spectral domain and extra preprocessing was done in order to prepare the experimental data for the force
identification process. Due to the realistic boundary conditions of the experimental system, reflections were present in the
collected data. In order to remove the reflections, the signal was appended with the theoretical solution of the infinite beam.
To address the periodic nature of the fast Fourier transform (FFT), zero padding was also used to improve the quality of the
identified force.
According to the complexity of a structure, finite element techniques are more compatible than analytical methods for
predicting a response. By implementing the finite element method in a spectral domain, a powerful tool was developed [10].
The spectral finite element method (SFEM) is highly effective for modeling wave propagation problems and used for
structural impact analysis. The SFEM is used in structural mechanics for applications such as force detection [7] and damage
detection [11, 1212]. In the work of Doyle [13], an FFT based SFEM was used to study wave propagation problems in
beams. In the work of Mitra and Gopalakrishnan [7], the discrete wavelet transform was used in the implementation of the
SFEM in a wavelet domain. In this work, the accuracy and computational costs of the wavelet domain SFEM were
determined to outperform an equivalent frequency domain SFEM when performing force identification in beams. However,
applying the wavelet transform to the system model requires rigorous mathematics while the frequency domain implemen-
tation is more straight-forward.
In this study, a frequency domain SFEM is used for identifying the impulsive force applied to beam-like structures.
Acceleration data from multiple locations on the structure are used to calculate displacement and slope information at these
positions. A subset of the response, both spatially and temporally, is identified in order to avoid reflections. In the spectral
domain, the inverse method is applied to the response data by using a simplified model which corresponds to the spatial
subset of the data to calculate the impulsive force. The simplified model utilizes semi-infinite elements or “throw-off”
elements in order to satisfy the lack of reflection observed in the subset of the response data.
The remainder of the paper is organized as follows. The specific test system on which the performance of the force
identification method is studied is presented in Sect. 2. The impulse force identification procedure is described in Sect. 3. The
numerical implementation and results of a parametric study are detailed in Sects. 4 and 5, respectively. Experimental
verification is presented in Sect. 6. Concluding remarks are included in Sect. 7.
11.2 Test System
The performance of the force identification method is studied both numerically and experimentally on a beam structure.
The experimental structure is a 6 ft (1.83 m) long aluminum beam with a square cross-section and width of 1 in. (25.4 mm) as
shown in Fig. 11.1. The structure is instrumented with 22 mounted linear accelerometers distributed evenly along its length.
The acceleration data is collected by the sampling frequency of 16.384 kHz. The beam is suspended from bungee cords in
Fig. 11.1 Photograph of test
system
158 P. Ghaderi et al.
order to provide free-free boundary conditions. An impact hammer is used for applying the impulsive force and the force is
measured by a force transducer for comparison with identified force information.
Acceleration data is initially obtained from numerical simulations with a full SFEM model of this system. Material
properties are selected based on the aluminum material of the test structure. Structural damping properties are determined by
comparing experimental and simulated responses. Section 6 describes the application of the force identification method
using experimentally obtained acceleration data.
11.3 Impulse Force Identification
In this study, a method using SFEM is developed to identify an impulse force applied to a beam structure. The acceleration
data from the test structure is transformed into the spectral domain and is converted to displacement. The slope information
required for the identification process is calculated by using a finite difference method. Zero-padding is applied to the
response data in order to address the assumption of periodicity intrinsic to the FFT and improve the final results. A spectral
finite element beam model is used to obtain the stiffness matrix for a segment of the beam around the location where the
impulse force is applied. The frequency domain response data and the stiffness matrix from the simplified model are used to
calculate the force in the frequency domain. The force identification procedure is illustrated in Fig. 11.2. Detailed
descriptions of each step are presented in the following subsections.
11.3.1 Acceleration Data
The process starts with acceleration data from multiple locations along the length of the beam system. The corresponding
deflections of the system are calculated by integrating the acceleration data. The integration of the acceleration is performed
in the frequency domain in order to improve accuracy and ensure compatibility with the frequency domain representation of
the system. The integration procedure is performed by dividing the spectral domain acceleration data by the imaginary form
of the frequency.
11.3.2 Deflection and Slope Calculation
Since the acceleration data only provides information regarding the translational motion at discrete points along the
structure, it is necessary to calculate the corresponding slope response at each location. This is accomplished by using a
Pade scheme finite differencing method [14]. By utilizing the relationship which the Pade scheme provides between the
position and slope of points along the structure, the rotation at each of the nodes is calculated. Due to the nature of the Pade
scheme, higher accuracy is achieved near the center of the beam and the slope information near the ends of the structure
suffer from lower accuracy. Therefore, this error in the slope calculations might influence the force identification procedure
which will be discussed further in Sect. 5.3.
Fig. 11.2 Force calculation
algorithm
11 Spectral Domain Force Identification of Impulsive Loading in Beam Structures 159
11.3.3 Signal Conditioning for the Force Calculations
The force identification procedure uses the propagating wave directly after the impact to identify the applied impulsive
force. Because the FFT algorithm assumes periodicity and because the time series deflections and rotations are truncated to
contain only the propagating signals, zero padding is applied in order to address convergence issues [8].
11.3.4 Spectral Finite Element Model
The wave propagation starts directly after the impact force is applied to the beam and it propagates out from this location
until it reaches the ends of the structure and is reflected. In this force identification method, the impulse force is calculated
from a subset of the data. This subset of data corresponds to a reduction of both the spatial and temporal dimensions around
the location where the propagated wave originated, as illustrated by Fig. 11.3.
The data subset is then used with a simplified model of the system to calculate the desired force information. The
simplified model corresponds to the segment of the original system resulting from the reduction of the spatial dimension.
This model also uses semi-infinite or throw-off elements, which allows the propagating wave to pass through a node without
any reflections. The throw-off elements are added at each end of the simplified model to eliminate reflections at the boundary
conditions. Although the simplified SFEM model differs considerably from the full SFEM model used for the simulations,
the response predicted by the simplified model agrees very well with the local response behavior of the full model as
illustrated in Fig. 11.4. The beam SFEM model is derived from the Euler-Bernoulli beam equation. The spectral finite
element model for the beams is detailed in the work by Doyle [15].
X (m)
time
(ms)
0 0.5 1 1.50
0.5
1
1.5
-2000
-1000
0
1000
2000Fig. 11.3 Experimentally
observed wave propagation
(acceleration) in a free-free
beam and the boxed area
corresponds to data selected
for analysis with simplified
model
X (m)
time
(ms)
0 0.1 0.2 0.3 0.4 0.50
0.2
0.4
0.6
0.8
1
1.2
1.4
-500
0
500
1000
1500Fig. 11.4 Simulated
mechanical wave propagation
(acceleration) using simplified
model with throw-off
elements at the ends to
eliminate reflection
160 P. Ghaderi et al.
11.4 Numerical Implementation
Results obtained by applying the force identification method described in the previous section to response behavior from
numerical simulations are presented in this section. The procedure is first applied to response behavior by using true
deflection and slope information produced by the simulation. This eliminates the potential error introduced by applying the
finite difference method to calculate slope information.
The calculated force information corresponds to the node locations which were included in the subset of data analyzed.
This initial work is focused on characterizing point forces which are applied at the location of one of the accelerometers. The
force values calculated from the simulated response are plotted in Fig. 11.5. The simulated beam is 18 ft (5.49 m) long
aluminum beam with a square cross-section and a width of 1 in. (25.4 mm). This length corresponds to three times that of the
beam described in Sect. 2. The increased length allows for a greater amount of time before the reflected wave returns to the
segment of the beam from which the subset of data is selected for analysis.
The force identified from the simulated response agrees very well with the true applied force. In order to quantify the
accuracy of the force identification method, a Root Mean Square (RMS) error is calculated by comparing the identified force
values to the true values used in the simulation. For the nominal case presented in Fig. 11.5, the RMS error is 2.74 N at
the location of loading during the impulse and averages 2.17 N for five locations considered over the remainder of the 2 ms
presented. This error is quite small when compared with the 600 N peak value of the impulsive load. The time series of the
applied force used in the simulation is based on the type of loading that the impact hammer produces. This loading data
includes an impulsive load with 0.4 ms duration.
11.5 Parametric Study
The geometric and loading properties utilized in the previous section were selected in order to provide nominal conditions
for performing force characterization. In this section, these properties are varied in order to study how they influence the
ability to accurately calculate the applied impulsive load. This includes varying the length of the structure, the effect of slope
calculation on the force identification results, the location of the applied force, and the duration of the impulsive load.
11.5.1 Structure Length
In order to study the influence of the length of the structure on the accuracy of the calculate force information, the process is
applied to response behavior simulated for a 6 ft (1.83 m) long beam. This length corresponds to the length of the test system
described in Sect. 2. All other conditions are maintained and the simulated displacement and slope information is used in the
force calculation. This allows for the separation of the influence of structure length and slope calculation on the accuracy of
the identified force information.
0 0.5 1 1.5 2-100
0
100
200
300
400
500
600
Time ms
For
ce (
N)
Neighboring nodesNeighboring nodesDetected forceNeighboring nodesNeighboring nodesActual force
Fig. 11.5 Force information
calculated for five nodes using
a simulated response for an
18 ft (5.49 m) long beam
11 Spectral Domain Force Identification of Impulsive Loading in Beam Structures 161
The force information calculated for the shorter structure is plotted in Fig. 11.6. The force information agrees very well
with the true applied force. The RMS error is 4.01 N at the loading location during the impulse and averages 1.98 N over the
remainder of the 2 ms for this configuration. The calculated force values for the neighboring nodes have small non-zero
values, deviation from the true zero values. This is believed to result due to the shorter length of time before the reflected
wave returns to the position on the structure where the load was applied. This time between the initial propagating wave and
the reflection and how it is influenced by changing the structure size is illustrated in Fig. 11.7.
11.5.2 Slope Calculated by Using Pade Scheme
As the response of the structure in the experimental setup is measured by using standard accelerometers, only deflection data
can be calculated directly from the measured acceleration. The slope information is calculated by using a Pade scheme with
the simulated deflection data. The accuracy of the finite difference slope calculation is illustrated in Fig. 11.8 where the
magnitude of the difference between the simulated values and values calculated by using the Pade scheme is plotted against
time and position. The effect of including the finite difference calculation on the accuracy of the calculated force information
is illustrated in Fig. 11.9. The RMS error for this case is 31.7 N at the loading location during the impulse and averages
6.09 N during the remainder of the 2 ms. By using the finite differencing method to calculate the slope information, the error
in the identified impulsive force is significantly increased. The RMS error value for the force information after the impulse is
also increased by calculating the slope information but to a lesser extent.
0 0.5 1 1.5 2-100
0
100
200
300
400
500
600
Time (ms)
For
ce (
N)
Neighboring nodesNeighboring nodesDetected forceNeighboring nodesNeighboring nodesActual force
Fig. 11.6 Force information
calculated for five nodes using
a simulated response for a 6 ft
(1.83 m) long beam
0 2 4 6-2000
0
2000
Time (ms)
acc.
(m
/s2 )
0 2 4 6-2000
0
2000
Time (ms)
acc.
(m
/s2 )
Fig. 11.7 Acceleration time
series for the beam position
where the impulsive load is
applied for (top) a 18 ft
(5.49 m) beam and (bottom)an 6 ft (1.83 m) long beam
162 P. Ghaderi et al.
11.5.3 Loading Location
As discussed in the Sect. 3, the central differencing equations are of a higher order than those used for calculating derivatives
at the ends of the structure. All of the calculated force information previously presented corresponds to impulsive loading
near the center of the structure. In order to study how the reduced accuracy of the calculated slope information near the ends
of the structure influences the calculated force information, response behavior is simulated than analyzed when the
impulsive load is applied near the end of the structure. This force information is presented in Fig. 11.10. The RMS error
for the identified impulsive load, which is 31.2 N, is not significantly affected by this change is location. However, by
moving closer to the end, the decreased accuracy in the slope information is observed to significantly influence the period of
time after the impulse, increasing the RMS error to 94.3 N.
In addition to the reduced accuracy of the calculated slope information, by moving the position of loading toward the end
of the beam there is less time before the reflected waves return to this location. As a result, large discrepancies are observed
in the calculated force information after 1 ms of time has passed. While changes to the geometric or material properties of the
structure can increase this time, the limited accuracy of the calculated slope data is still expected to influence the
performance of the force characterization procedure.
11.5.4 Loading Duration
In order to further study the performance of the force identification procedure, the duration of the applied load is varied.
A system response is simulated for an impulsive load with twice the original duration. The simulation is performed with the
X (m)
time
(ms)
0 0.5 1 1.50
0.5
1
1.5
1
2
3
4
5
6
7
8
9
x 10-4Fig. 11.8 Magnitude of
difference between simulated
slope data and slope
information calculated from
displacement data versus time
and position
0 0.5 1 1.5 2-100
0
100
200
300
400
500
600
Time (ms)
For
ce (
N)
Neighboring nodesNeighboring nodesDetected forceNeighboring nodesNeighboring nodesActual force
Fig. 11.9 Force information
calculated from simulation
data when load is applied
at the middle of the beam
11 Spectral Domain Force Identification of Impulsive Loading in Beam Structures 163
model of the 18 ft (5.49 m) long beam and the simulated slope data is used to calculate the force information. This force
information is presented in Fig. 11.11 and good agreement is seen. The RMS errors for the impulse and the period after the
pulse are 6.54 and 1.29 N, respectively. Although the duration of the impulsive load is increased, the time for the mechanical
wave to reflect back to the loading location remains the same. The longer duration of the load only decreases the time
between the initial wave and the reflection by the 0.4 ms that the duration of the load is extended.
11.6 Experimental Verification
Experimental data is collected from the system described in Sect. 2. The force information obtained by applying the force
identification method to the acceleration data is presented in Fig. 11.12. While the location of the loading is successfully
identified along with the qualitative characteristics of the impulsive load, errors exist between the force data measured with
the force transducer of the impact hammer and the calculated values. The RMS errors for the impulse and the period shown
in the figure after the impulse are 67.3 and 106.2 N, respectively. When compared to the force information calculated for
nominal conditions, the experimental structure is shorter, the load is applied away from the center, and the slope response
information is calculated by using the finite differencing method. Each of these differences contribute to the reduced
accuracy of the calculated for information. Based on the results of the parametric study, the main source of error is believed
to be due to the use of the calculated slope information.
0 0.5 1 1.5 2-200
0
200
400
600
800
Time (ms)
For
ce (
N)
Neighboring nodesNeighboring nodesDetected forceNeighboring nodesNeighboring nodesActual force
Fig. 11.10 Force information
calculated from simulation
data when load is applied near
one end of the beam
0 1 2 3-100
0
100
200
300
400
500
600
Time (ms)
For
ce (
N)
Neighboring nodesNeighboring nodesDetected forceNeighboring nodesNeighboring nodesActual force
Fig. 11.11 Force information
calculated from simulation
data for impulsive loading
with twice duration
164 P. Ghaderi et al.
11.7 Concluding Remarks
In this study, the impulse force identification procedure using the SFEM for a beam structure is presented. The procedure is
applied to simulation and experimental data for propagating mechanical waves resulting from the application of an impulse
force. Excellent agreement is observed for nominal conditions. However, different sources of error are identified through a
parametric study. These sources of error are observed to influence the identification process when it is applied to
experimental data. The most significant source of error is determined to be the use of a finite difference method to calculate
slope response information. Significant improvement in the accuracy of the identified force information would be achieved
by the direct measurement of the slope information, possible through the use of coupled accelerometer–gyro sensors.
Acknowledgements The support of this work from the Air Force Research Laboratory under Cooperative Agreement FA8651-10-2-0006 and
from the Air Force Office of Scientific Research under Grant FA9550-11-1-0108 is gratefully acknowledged.
References
1. Kim M-S, Lee S-K, Kim S-J (2008) Identification of impact force on the gas pipe based on analysis of acoustic wave. Int J Mod Phys B
22(9–11):1039–1044
2. Yan G, Zhou L (2009) Impact load identification of composite structure using genetic algorithms. J Sound Vib 319(3–5):869–884
3. Hu N, Fukunaga H (2005) A new approach for health monitoring of composite structures through identification of impact force. J Adv Sci
17(1–2):82–89
4. Doyle JF (1984) An experimental method for determining the dynamic contact law. J Exp Mech 24(1):10–16
5. Hollandworth PE, Busby HR (1989) Impact force identification using the general inverse technique. Int J Impact Eng 8(4):315–322
6. Bateman VI, Carne TG, Gregory DL, Attaway SW, Yoshimura HR (1991) Force reconstruction for impact tests. J Vib Acoust 113(2):192–200
7. Mitra M, Gopalakrishnan S (2005) Spectral formulated wavelet finite element for wave propagation and impact force identification in
connected 1-D waveguides. Int J Solids Struct 42(16–17):4695–4721
8. Doyle JF (1984) Further developments in determining the dynamic contact law. J Exp Mech 24(4):265–270
9. Chang C, Sun CT (1989) Determining transverse impact force on a composite laminate by signal deconvolution. J Exp Mech 29(4):414–419
10. Patera AT (1984) A spectral element method for fluid dynamics: laminar flow in a channel expansion. J Comput Phys 54(3):468–488
11. Ostachowicz WM (2007) Damage detection of structures using spectral finite element method. J Comput Struct 86(3–5):454–462
12. Krawczuk M (2002) Application of spectral beam finite element with a crack and iterative search technique for damage detection. J Finite Elem
Anal Des 38(6):537–548
13. Doyle JF, Farris TN (1990) Spectrally formulated element for wave propagation in 3-D frame structures. Int J Anal Exp Modal Anal
5(4):223–237
14. Lele SK (1992) Compact finite difference schemes with spectral-like resolution. J Comput Phys 103(1):16–42
15. Doyle JF (1997) Wave propagation in structures: spectral analysis using fast discrete Fourier transforms. Springer, New York
0 0.2 0.4 0.6 0.8 1-400
-200
0
200
400
600
800
1000
Time (ms)
For
ce (
N)
Neighboring nodes
Neighboring nodes
Detected force
Neighboring nodes
Neighboring nodes
Actual force
Fig. 11.12 Force information
calculated from experimental
data
11 Spectral Domain Force Identification of Impulsive Loading in Beam Structures 165
Chapter 12
Free-Pendulum Vibration Absorber ExperimentUsing Digital Image Processing
Richard Landis, Atila Ertas, Emrah Gumus, and Faruk Gungor
Abstract Using image processing and analysis, the dynamic behavior of the beam-free-pendulum system under low
and high sinusoidal excitation was investigated. The system responses were investigated experimentally in the neighborhood
of primary resonance condition. The results exhibited autoparametric interaction between the beam and the free pendulum
when the primary resonance condition was satisfied. Experiments were conducted for two different pendulum weights
under two different shaker forcing amplitudes, and the results were compared. Experimental data were obtained by sweeping
between the frequencies that contain the resonance condition under investigation. The results of experiments for
different beam-tip mass and pendulum mass ratios indicate that more powerful absorption action can be achieved when
the smaller mass ratios are used.
Keywords Nonlinear vibration • Autoparametric vibration absorber • Internal resonance • Free pendulum absorber
12.1 Introduction
There is widespread interest in pendulum modeling and the use of the pendulum as a vibration absorber. This interest ranges
from the dynamics of Josephson’s Junction in solid state physics [1] to the rolling motion of ships [2] and the rocking motion
of buildings and structures under earthquakes [3]. Vibration mitigation has found extensive usage in aerospace structures,
civil engineering structures, and mechanical machinery. A comprehensive survey on vibration suppression devices was
given by Sun et al. [4]. They reviewed the current developments in passive absorbers, adaptive absorbers, and active
absorbers. Passive tuned vibration absorbers are also referred to as dynamic vibration absorbers [5] or tuned mass dampers.
Much of the analytical work done on the inverted spherical pendulum for undamped systems was done by Lowenstern
[6], Hemp and Sethna [7] and by Moran [8] for damped systems. These three papers dealt with the stability criteria of the
inverted spherical pendulum. Recently, Ertas and Garza investigated the dynamics and bifurcations of an impacting
spherical pendulum with large angle and parametric forcing. The pendulum system was studied with nine different bobs
and two different base configurations with an external frequency of 24.6–24.9 Hz. Comparative analysis was performed at
low and high Coulomb damping values for the inverted, impacting pendulum [9].
Passive and active vibration absorbers were used by many researchers to reduce the vibration level of flexible structures.
Miwa et al. reported the case involving an active mass damper (AMD) system installed on the roof of a building to investigate
the vibration characteristics of multi-story houses built on soft ground near vibration sources such as railways and expressways
[10]. Muller et al. studied the modeling and control techniques of an active vibration isolation system. They compared
experimental findings with simulated data and discussed the results [11]. Holt and Singh investigated the active/passive
vibration control of continuous systems by zero assignment. They reported that the results of their study would lead to the
developments in the control strategies for complex structures and implementation of piezoelectric actuators and sensors for
vibration control [12]. Viguie and Kerschen, used the concept of nonlinear energy sink (NES) to reduce the vibration level of
multi-degree-of-freedom linear structures. They reported that this approach requires the development of an efficient NES
design procedure. Their research presented such a procedure based upon bifurcation analysis using the softwareMatCont [13].
R. Landis • A. Ertas (*) • E. Gumus • F. Gungor
Mechanical Engineering Department, Texas Tech University, Lubbock, TX 79409, USA
e-mail: [email protected]
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_12, # The Society for Experimental Mechanics, Inc. 2012
167
Many mechanical structures can be modeled as flexible beams with tip mass attached along the span. For example,
as a model for an “autoparametric” vibration absorber, Haxton and Barr [14] studied a flexible column with tip mass fixed to
a heavy block undergoing parametric vibration. The Autoparametric Vibration Absorber is a device designed to absorb
the energy from the primary mass (main mass) at conditions of combined internal and external resonance. Autoparametric
resonance is a special case of parametric vibration and is said to exist if the condition at the internal resonance and
external resonance are met simultaneously due to external force [15–18].
Autoparametric resonance may occur in nonlinear systems with two or more degrees of freedom, and if normal mode
frequencies of the corresponding linearized systems are governed by the linear relationship
Xn
i¼1
kioi ¼ 0
where ki are integers, n is the number of degrees of freedom, and oi are the system’s natural frequencies. When the linear
spring–mass–damper system is coupled to a pendulum, the resulting system possesses quadratic nonlinearities due to inertial
coupling with the rotational motion of the pendulum. If the system has quadratic nonlinearity in two degree of freedom, then
system internal resonance occurs when
o2 ¼ O and o2 ¼ 2o1
where O is the excitation frequency,o1 is the lower mode frequency, and o2 is the higher mode frequency. This is called 1:2
internal resonance condition and leads to nonlinear modal coupling between two modes.
The Autoparametric Vibration Absorber has received considerable attention since the mid-1980s, and researchers
published many interesting papers. Struble and Heinbockel [19] used asymptotic methods to investigate the energy transfer
observed by Selvin [20]. Struble [21] used the perturbation method for a parametrically excited pendulum and obtained
resonant solutions when the excitation frequency was nearly twice the natural frequency of the system. Hatwal et al. [22]
investigated nonlinear vibrations of a spring-mass-damper system with a parametrically excited pendulum. The harmonic
balance method was used to solve the system response. Performance of the system as an autoparametric vibration absorber
was studied. Cuvalci and Ertas [23] investigated a simple pendulum as a vibration absorber for flexible structures. Through
experimental and theoretical studies, they reported that a simple pendulum can be effectively used as a vibration absorber for
flexible structures.
Baja et al. [24] studied forced, weakly nonlinear oscillations of a two degree-of-freedom autoparametric vibration
absorber system at resonant excitations. The authors used a method of averaging to obtain first-order approximations
to the response of the system. They studied the bifurcation when the pendulum was lucked and also observed that the
coupled-mode response can undergo Hopf bifurcation to limit cycle motions when the two linear modes are mistuned away
from the exact internal resonance condition. Vyas and Bajaj [25] studied the dynamics of a resonantly excited single-degree-
of-freedom linear system coupled to an array of non-linear autoparametric vibration absorbers. They investigated
the stability and bifurcations of equilibria of the averaged equations. In their paper, the effect of various parameters on
the performance of the Autoparametric Vibration Absorber (AVA) is discussed. A nonlinear adaptive vibration absorber to
control the vibrations of flexible structures at a two-to-one internal resonance was investigated by Ashour and Nayfeh [26].
Through a proposed model, energy was exchanged between the structure and the controller and, near resonance, the
structure’s response almost diminished.
Authors of this paper, as many other distinguished researchers, investigated the possibility of the pendulum for the
vibration absorber [27–32]. In these studies, a digital angular measurement system for measuring the full 360� angular
displacement of the pendulum was used [33]. At that time, there was no reliable device available to measure the full
pendulum response. In this paper, image processing technique was used for measuring the angular displacement of the
pendulum to analyze the nonlinear dynamic response of an Autoparametric Vibration Absorber of a free-pendulum system.
12.2 Experimental Setup and Procedure
The experiments were performed to obtain the frequency response curves for the different pendulum weights (one ball and
two balls) under two different shaker forcing amplitudes. For the detail dynamics of the free pendulum, the experiments were
conducted at the specific forcing frequencies and time histories to plot the FFTs and the phase plains. The complete experimental
168 R. Landis et al.
system is shown in Fig. 12.1. The model is excited by the shaker table, the amplitude and frequency of which controlled by the
vibration control system.
The experimental model used in this research study has a beam with appendages consisting of a mass-free-pendulum
system. A high-speed camera was used to capture the position of the beam/tip-mass/free pendulum system through the
duration of the experiment. The beam/tip-mass/free pendulum system consists of a flexible beam that is rigidly clamped at
the base. The tip of the beam consists of an appendage, which consists of a lumped mass and a free pendulum. Both systems
are shown in Figs. 12.2 and 12.3, respectively.
The experiments were performed for 1.00 Hz, 2.00 Hz, 3.00 Hz, 3.50 Hz, 3.75 Hz, 3.85 Hz, 3.90 Hz, 3.95 Hz, 4.00 Hz,
4.05 Hz, 4.10 Hz, 4.25 Hz, 4.30 Hz, 4.40 Hz, 4.50 Hz, and 5.00 Hz for both the one and two ball free pendulum systems.
Experimental data were obtained by sweeping between the frequencies (o) that contain the resonance condition
under investigation obean=pendulum�system ¼ 4:0 Hz and: opendulum ¼ 2:0 Hz� �
. The data (response amplitudes of beam
and the free pendulum) were recorded, and it was assumed that the system reached a steady state within each increment
of the frequency. In other words, for all frequency sweep experiments, at each frequency interval, the system was allowed to
dwell for 60 s in order to reach stability before the data were taken. In the experiment, detuning was set to be
opendulum
obeam=pendulum�system¼ 0:5
Experiments were performed for two amplitudes of f0 ¼ 1:88 mm excitation and f0 ¼ 2:3 mm for non-impact
and for impact cases, respectively. Since test procedures can vary, experiments should begin in a specific manner to
avoid the inclusion of too many variables into the experimental test, hence causing uncertainty in the decision-making of
the test results. The beam material and length, the tip mass, the damping coefficient of the beam, and the pendulum were
taken to be constant for all experiments. As mentioned before, the experiment was investigated for two main parameters:
the excitation amplitude and the free pendulum weight.
Fig. 12.1 Experimental setup
Fig. 12.2 One ball beam/tip-mass/free pendulum system
12 Free-Pendulum Vibration Absorber Experiment Using Digital Image Processing 169
For beam and free pendulum responses, measurement at a high speed image collection of up to 500 frames per second
with a 1280 � 1024 image resolution camera was used. Due to the frame grabber limitations of 1 GB of memory,
the maximum recording time for all the experimental cases was approximately 26 s at a frame rate of 200 frames per
second. The experiment duration was long enough to observe all possible dynamics of the system under consideration.
The collected frames were captured into memory and then stored as tiff files onto the hard drive. Then the files were
converted to JPG format RealJPEG Pro and analyzed using MatlabR2006r2 basic and imagery toolboxes.
Many of the techniques of digital image processingwere developed in the early 1960s. Since then, image processing has been
used for satellite imaging, medical imaging, videophone, character recognition and many other applications. However, the cost
of image processingwas quite highwith the computing equipment of that era. In the 1970s, digital image processing proliferated
when the cost of computers decreased and they became faster. Recently, emerging video processing techniques demonstrated
their potential applications in engineering. In order to effectively use image screening and analysis techniques for beam/
tip-mass/free pendulum experiments, camera and environment setup and verification was critical in reducing or eliminating
image distortion, misalignment, tonal misadjustment and image processing complexity between sequential image pairs.
12.3 Results and Discussions
The experimental analysis was performed for three cases, and the following diagrams are plotted for detailed dynamics:
Frequency response curves.
Time history.
FFT.
Phase plain.
12.3.1 Case-I
In this case, the mass ratio of the beam/tip-mass and the pendulum mass (one ball) was taken to be approximately 1/18 and
the shaker excitation amplitude was set approximately to f0 ¼ 1:88 mm. The natural frequency of the beam/free pendulum
system was set approximately to obeam system ¼ 4:0 Hz and to maintain the condition of autoparametric interaction,
the frequency of the pendulum was set approximately to opendulum ¼ 2:0 Hz. As shown in Fig. 12.4, the first experiment
was performed to investigate the beam response curves without the pendulum being in action. For this experiment, two balls
Fig. 12.3 Two balls beam/
tip-mass/free pendulum
system
170 R. Landis et al.
that will be used as a damper later were locked on the tip mass (housing). The forcing frequency was increased in a
stepwise manner until the necessary regions of interest were covered. Figure 12.4 demonstrates that the maximum beam
response occurred at the excitation frequency of 4.05 Hz. Observing Fig. 12.4, the beam was oscillating approximately 25�
peak-to-peak when the primary resonance condition occurred, namely when the natural frequency of the system reached
approximately obeam system ¼ 4:05. Note that when the pendulum was lucked on the tip mass (housing track), the natural
frequency of the system turned out to be slightly higher than the tuning frequency of 4.00 Hz (see Fig. 12.4). It was difficult
to adjust the exact tuning ratio. To create Fig. 12.4, experiments were performed at 3.00 Hz, 3.50 Hz, 3.75 Hz, 3.85 Hz,
3.90 Hz, 3.95 Hz, 4.00 Hz, 4.05 Hz, 4.10 Hz, 4.25 Hz, 4.50 Hz, and 5.00 Hz for the lucked beam/tip-mass/pendulum system.
When an absorbing pendulum was free in the housing track and the frequency of the free pendulum was tuned to one half
of the beam-mass system frequency, the free pendulum had a sudden amplitude increase at the start of the autoparametric
region whereas the beam response decreased to approximately 0.9� at the first response peak (Fig. 12.5a, point P1) and 2.25�
Fig. 12.4 Experimental
frequency response curves
without absorber
Frequency (Hz)
ϕ (d
eg)
Frequency (Hz)
Autoparametric Region
Complete EnergyExchange
θ (d
eg)
a
b
Fig. 12.5 Experimental
frequency response curves
(a) beam/tip-mass response
with absorber, (b) free-pendulum response for
f0 ¼ 1.88 mm and mass
ratio ¼ 1/18 (one ball)
12 Free-Pendulum Vibration Absorber Experiment Using Digital Image Processing 171
at the second response peak of the beam (Fig. 12.5a, point P2). The motion of the beam/tip-mass was almost diminished and
the whole energy was absorbed by the free pendulum when the primary resonance case was reached (Fig. 12.5a, b). In other
words, the complete energy transfer between two modes occurred when the beam/tip-mass frequency was twice the free
pendulum frequency and the forcing frequency was equal to beam/tip-mass frequency. To create Fig. 12.5a, b, experiments
were performed at 3.00 Hz, 3.50 Hz, 3.75 Hz, 3.85 Hz, 3.90 Hz, 3.95 Hz, 4.00 Hz, 4.05 Hz, 4.10 Hz, 4.25 Hz, Hz, 4.30 Hz,
4.35 Hz, 4.40 Hz, 4.50 Hz, and 5.00 Hz with the shaker excitation amplitude of f0 ¼ 1:88 mm.
Second peak observed in the beam/tip-mass response was due to the beating phenomenon and will be explained later in
this section. As shown in Fig. 12.5b, point A was the starting point of the autoparametric region. Energy transfer from beam
to pendulum continued to increase until point B and then took the shape of a bathtub curve, which consists of three
responses, namely, a decreasing response followed by an approximately constant response and then an increasing response
until point C. The beating phenomenon was observed when the excitation frequency reached 4.25 Hz (between points C and
C0). After point C’ until point D (frequencies at 4.30–4.35 Hz), the beating phenomenon ceased to exist. Points B and C on
the pendulum response curve were important as they define the complete energy exchange region. As shown in this figure,
the autoparametric region ended at point E.
From Fig. 12.6a, it is evident that the oscillation of the beam was reduced to almost zero and the complete energy was
absorbed by the pendulum. Some insight into the characteristics of the limit-cycle oscillation can be obtained by examining
the phase portrait of pendulum as shown in Fig. 12.6f. At the neighborhood of the tuning frequency (3.95 Hz), the pendulum
was having purely sinusoidal oscillation and exhibited elliptical phase orbit. Beam and pendulum FFT diagrams showed
strong peaks where the beam frequency was twice the pendulum frequency (see Fig. 12.6c,d). Complete autoparametric
interaction between beam and pendulum and one to two frequency relationship obeam ¼ 2opendulum and Oforcing ¼ obeam
� �
60a
e f
b c
d
g
40
20
Deg
ree
Deg
ree
Time (sec)
Theta (degree) vs Time (sec) Phi (degree) vs Time (sec) FFT of the Beam
FFT of the Pendulum
Phi vs Theta
Wp (Hz)
Wb (Hz)
Phi
dPhi
/dt
The
taf(
Wp)
f(W
b)
dthe
ta/d
t
PhiTheta
Theta dot Vs Theta Phi dot vs Phi
Time (sec)
0
-20
-40
-60
60
40
20
0
-20
-40
-600
800
600
400
200
-200
-400
-800
-600
-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
2500
2000
1500
1000
500
-500
-1000
-1500
-2000
-2500-50 -40 -30 -20 -10 0 10 20 30 40 50 -50
-0.6
-0.4
-0.2
0
0.2
1.4
0.6
0.8
1
-40 -30 -20 -10 0 10 20 30 40 50
0
1
0
1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.51
1.52
2.53
3.54
4.5
0
10
20
30
40
50
60
6x102
Fig. 12.6 Detail beam/tip-mass/pendulum system dynamics for the forcing frequency of 3.95 Hz, f0 ¼ 1:88 mm
172 R. Landis et al.
between beam and pendulum can also be verified by the phase plane shown in Fig. 12.6g. The loop shown in this figure is the
evidence for one to two frequency relationships between beam and pendulum.
The data taken during the experiment at 4.25 Hz frequency are shown in Fig. 12.7. An FFT was performed to give
the spectra observed in Fig. 12.7c,d. Figure 12.7 shows an interesting behavior of the beam/tip-mass/-pendulum system.
Since the forcing frequency of 4.25 Hz was close enough to the natural frequency of the system (approximately 4.0 Hz),
a phenomenon known as beating occurred. As shown in Fig. 12.7c, besides the peaks with periods around 4.25 Hz
which correspond with the shaker excitation, a peak with a period around 3.95 Hz was observed. As seen from the FFT,
the fluctuating beam response having peaks at these two mode frequencies yield the beating phenomenon. Although the
mode frequency of 4.25 Hz within the complete energy exchange region, Fig. 12.7g shows a very distinct pattern instead of
the regular loop, indicating the o1 ¼ 2o2 relationship. As seen from Fig. 12.7a, the amplitude of the beam response built up
and then diminished in a regular pattern. Figure 12.7e shows that the beam data was scattered. This is not only because beam
oscillations do not exactly repeat themselves from one cycle to the next but also because of the beating effect. Figure 12.7f
shows that the pendulum was having multi-limit-cycle with the response of approximately 60�, which was larger than
the pendulum response cases in the neighborhood of exact autoparametric interaction region (around 3.95–4.0 Hz).
Of course, this response increased due to beating effect.
The beating phenomenon may occur in machines, flexible structures, twin-engine propeller airplanes, electric power
houses, bells, human ears, guitar strings, etc. In some cases, the beating phenomenon is desirable, but in some cases,
it may not be desirable. For this application, complete energy exchange region shown in Fig. 12.5 should be free from
beating action as much as possible. In this design application, the beating phenomenon is an undesirable effect because
energy can be transferred from the secondary system (i.e., free pendulum) back to the primary system (beam). Hence, the
investigation of frequency of the beating phenomenon existence and associated harmonics within the design range of energy
exchange is important.
10a
e f
b c
d
g
8
6
4
2
Deg
ree
Deg
ree
Time (sec)
Theta (degree) vs Time (sec) Phi (degree) vs Time (sec) FFT of the Beam
FFT of the Pendulum
Phi vs Theta
Wp (Hz)
Wb (Hz)
Phi
dPhi
/dt
The
ta
f(W
p)f(
Wb)
dthe
ta/d
t
PhiTheta
Theta dot Vs Theta Phi dot vs Phi
Time (sec)
0
-2
-4
-8
-40
-10
60
40
20
0
-20
-40
-600
800
600
400
200
-200
-400
-800
-600
-2.5 -2 -1.5 -1 -0.5 0 1 1.5 2 2.50.5
2500
2000
1500
1000
500
-500
-1000
-1500
-2000
-2500-80 -60 -40 -20 0 20 40 60
-80
2.5
1.5
1
0.5
-0.5
-1
-1.5
-2
-2.5
0
2
-60 -40 -20 0 20 40 50
00
1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 100
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.51
1.52
2.5
33.5
44.5
0
10
20
30
40
50
60
6x102
Fig. 12.7 Detail beam/tip-mass/pendulum system dynamics for the forcing frequency of 4.25 Hz, f0 ¼ 1:88 mm
12 Free-Pendulum Vibration Absorber Experiment Using Digital Image Processing 173
12.3.2 Case II
The aim of this part of the experimentation was to assess and compare two experimental models, namely, with one ball
and two balls (mass ratio of the beam-tip-mass and the pendulum mass (two balls) is taken to be approximately 2/17)
by experimenting with the same parameters used in Case I. Namely, the shaker excitation amplitude was set
approximately to f0 ¼ 1:88 mm. The natural frequency of the beam/free pendulum system was approximately
obeam system ¼ 4:00 Hz, and to maintained the condition of autoparametric interaction, the frequency of the pendulum
was set approximately to opendulum ¼ 2:00 Hz. Figure 12.8 compares one and two ball free pendulum models responses
over the interval from 3.00 Hz to 5.0 Hz. The broken line represents the two ball free pendulum model results while the
solid line represents the one ball free pendulum model results. In comparing both models, the following observations
were made. As seen from Fig. 12.8, the autoparametric region for the one ball free pendulum model started at point A1
and died at point E1 while the autoparametric region for the two ball free pendulum model started at A2 and died at E2.
The delay in the start of the two ball free pendulum models was observed because larger mass required more energy
level to swing. In both models, the energy level continued to increase as both systems approached the exact
autoparametric energy level shown by points B1 and B2. At these points, the autoparametric interaction condition
(obeamsystem ¼ 2 � opendulum) was satisfied. In comparing the one and two ball free pendulum responses, the two ball
system had a smaller angle of swing than one ball system since the larger mass absorbed more energy without having
bigger angle of swing. From Fig. 12.8, it is clear that both pendulum model responses exhibited quite different energy
exchange trends between points B and C. Both models showed the beating phenomenon exactly at the same frequency
of 4.25 Hz.
Figure 12.9 shows the comparison of experimental frequency response curves for one ball free pendulum system for
the excitation amplitude of, f0 ¼ 1:88 mm. As shown in this figure, the beam beating phenomenon was observed when the
excitation frequency reached 4.25 Hz. This figure clearly shows that the beating was distorted and ceased to exist at
frequency 4.30 Hz and frequency 4.35 Hz, respectively. However, at these frequencies, because of the effect of the beating,
the pendulum response still fluctuated with high amplitude. At the frequency of 4.40 Hz, the pendulum oscillation almost
died out. Beam FFT plots shown in Fig. 12.9 are another indication of how the beating phenomenon ceased out. When
forcing frequency reached 4.30 Hz, the peak that causes the beating phenomenon at the natural frequency of the system
disappeared (see Fig. 12.9c for 4.35 Hz).
Figure 12.10 shows the comparison of experimental frequency response curves for the two ball free pendulum system
for the excitation amplitude of f0 ¼ 1:88 mm. As in the case of the one ball model, this model also experienced the beating
phenomenon exactly at the same excitation frequency of 4.25 Hz. It is interesting to note that immediately after the
beating frequency of 4.25 Hz, the effect of the beating disappeared and, consequently, the pendulum oscillation died out
at the excitation frequency of 4.30 Hz. This shows that an absorber with a larger mass can eliminate the region C01 – D1
shown in Fig. 12.8.
ϕ (d
eg)
Frequency (Hz)
Fig. 12.8 Comparison of experimental frequency response curves for one ball and two balls free pendulum systems for f0 ¼ 1:88 mm
174 R. Landis et al.
Fig. 12.9 Comparison of experimental frequency response curves for one ball free pendulum systems for f0 ¼ 1:88 mm
12 Free-Pendulum Vibration Absorber Experiment Using Digital Image Processing 175
12.3.3 Case III
The objective of this part of the experimentation was to observe the response of the beam and free pendulum when the shaker
excitation amplitude was set approximately to f0 ¼ 2:3 mm peak to peak. As in the previous cases, the natural frequency
of the beam/free pendulum system was set approximately to obeam system ¼ 4:00 Hz, and to maintain the condition of
autoparametric interaction, the frequency of the pendulum was set approximately to opendulum ¼ 2:00 Hz. Many researchers
investigated the use of impact dampers to achieve efficient structural damping [34–40]. They have studied the effect of
certain parameters and their control on the damping effect of the vibration absorber. In this case, by setting a higher shaker
excitation amplitude, the phenomenon of impact was created to investigate whether impact would change the characteristics
of the pendulum damping and the system response behavior. In this case, the experiments were performed by using one and
two pendulum balls at the forcing frequency of 4.05 Hz, 4.25 Hz, and 4.30 Hz. Figure 12.11 shows the position of the free
pendulum when impact occurs. Considering the reference points in image processing calculations, the impact angle for the
one ball case was approximately 65� with � 2� error possibility. In the two ball free pendulum case, the impact angle was
54� with � 3� error. Our observations revealed that the clear impact case occurred at 4.05 Hz for both one ball and two ball
beam pendulum systems. A comparison of the response behavior of the one ball beam pendulum system for excitation inputs
1.88 mm and 2.3 mm is given in Fig. 12.12. Figure 12.12 clearly reveals that the impact of the pendulum did not reduce the
Fig. 12.10 Comparison of experimental frequency response curves for two ball free pendulum systems for f0 ¼ 1:88 mm
176 R. Landis et al.
Fig. 12.11 Position of impact angles for free pendulum
Fig. 12.12 Comparison of one ball beam pendulum system response for excitation amplitude of 1.88 mm and 2.3 mm at 4.05 Hz
oscillation of the beam; instead it increased the response amplitude considerably. When the system was excited with
1.88 mm, the beam’s maximum response was approximately 3� peak to peak. Whereas, when the excitation amplitude
increased to 2.3 mm, the beam response amplitude increased to 49� peak to peak (see Fig. 12.12). When the system was
excited with lower amplitude, it had strong periodic motion and satisfied the autoparametric interaction for vibration
absorption (see Fig. 12.12c, d for amplitude of 1.88 mm). However, the higher excitation amplitude caused many irregular
frequencies as it lost its periodicity (see Fig. 12.12c, d for amplitude of 2.3 mm).
Figure 12.13 compares the response behavior of the two ball pendulum system with excitation amplitudes of 1.88 mm and
2.3 mm at 4.05 Hz. This figure illustrates that many distinct frequencies existed in the system response when the excitation
amplitude was 2.3 mm. Also, a comparison of Fig. 12.13a for both amplitudes (no impact at 1.88 mm excitation and impact
at 2.3 mm excitation) supports that the impact of the pendulum increased beam oscillation. However, at a higher excitation
amplitude, the pendulum with larger mass created autoparametric interaction, hence coupling between two modes.
This coupling can only reduce the maximum beam oscillation amplitude from 49� to 37.9�.More insight into the characteristics of the impact response of the system can be obtained by examining Fig. 12.14.
As shown in Fig. 12.14, the ball impacted at point A when the pendulum angle reached 54.8�. During this time, the beam was
swinging upward. It is interesting to note that after the impact, the ball remained approximately at the same position until it
reached point B. Basically, the downward swing of the beam was holding the impacted ball at the top of the pendulum track
Fig. 12.13 Comparison of two ball beam pendulum system response for excitation amplitude of 1.88 mm and 2.3 mm at 4.05 Hz
178 R. Landis et al.
from point A to B until it reached its bottom position. After point B, the ball traveled with very high velocity to the other side
of the track and impacted at point C. Similarly, it remained at the same position until it reached point D. Note the separation
at angle �50.5� between the two balls, which creates a valley in the ball response.
12.4 Conclusions
Using image processing and analysis, the dynamic behavior of beam free pendulum system under low and high sinusoidal
excitation was investigated for the external resonance condition of Oexcitation ¼ obeam with the primary resonance
obeam ¼ 2opendulum. The system responses were investigated experimentally in the neighborhood of primary resonance
condition. The results exhibited autoparametric interaction between the beam and the free pendulum when the primary
resonance condition was satisfied. The results of experiments for different beam-tip mass and pendulum mass ratios
indicates that more powerful absorption action can be achieved when the smaller mass ratios are used. For this model,
either one ball or two ball free pendulum, creating impact will not reduce the oscillation of the beam; instead it will increase
the response amplitude of the beam. Although image processing is relatively new in vibration measurements and analysis,
it is a cost effective and valuable tool.
Acknowledgment We wish to thank Dr. Murat M. Tanik for his assistance and help.
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180 R. Landis et al.
Chapter 13
Suppression of Regenerative Instabilities by Means of Targeted
Energy Transfers
A. Nankali, Y.S. Lee, and T. Kalmar-Nagy
Abstract Regenerative effects in machining arise from the fact that the cutting force exerted on a tool is influenced not only
by the current position but also by that in the previous revolution. Hence, the equation of motion for the tool appears as a delay
differential equation, and the regenerative instability results in steady-state periodic motions, called limit cycle oscillations.
We study targeted energy transfers for suppressing regenerative instabilities by applying a nonlinear energy sink (NES) to a
single-degree-of-freedom machine tool model. A series of bifurcation analysis by means of numerical continuation
techniques demonstrate that there are three distinct suppression mechanisms; that is, recurrent burstouts and suppressions,
and partial and complete suppressions of regenerative instabilities.We characterize each suppressionmechanism numerically
by means of wavelet and Hilbert transforms and analytically by means of the complexification-averaging (CX-A) technique.
Furthermore, we extend the CX-A analysis to perform asymptotic analysis by introducing a reduced-order model and
partitioning slow-fast dynamics. The resulting singular perturbation analysis yields parameter conditions and regions for
the three suppression mechanisms, which exhibit good agreement with the bifurcations sets obtained from numerical
continuation methods. The results will help design NESs for passively controlling regenerative instabilities in machine tools.
Keywords Targeted energy transfer • Regenerative instability • Nonlinear energy sink • Machine tool dynamics
13.1 Introduction
We study targeted energy transfers for suppressing regenerative instabilities in a single-degree-of-freedom (SDOF) machine
tool model by coupling an ungrounded nonlinear energy sink (NES). Regenerative effects in machining arises from the fact
that the cutting force exerted on a tool is influenced not only by the current position but also by that in the previous
revolution. Hence, the equation of motion for the tool appears as a delay differential equation, which renders even an SDOF
dynamical system to be infinite-dimensional (See, for example, Dombovari et al. [1] and Nayfeh and Nayfeh [2] for recent
studies on machine tool dynamics).
Kalmar-Nagy et al. [3] analytically proved, by means of center manifold theorem (See, e.g., [4]), the existence of
subcritical Hopf bifurcation in an SDOF machine tool model with the regenerative cutting force being retained up to the
cubic order. Furthermore, practical stability limit in turning process was investigated by considering contact loss issues in
the regenerative cutting force [5], which can predict stable, steady-state periodic tool vibrations (or limit cycle oscillations –
LCOs). Such LCOs would create adverse effects on machining quality, and various passive and active means have been
considered to improve machining stability boundary (e.g., see [6–11]). In particular, direct use or variations of linear/
nonlinear tuned mass damper (TMD [8, 9]) are probably the most popular approach to passive chatter suppression.
However, even if the TMD is initially designed (tuned) to eliminate resonant responses near the eigenfrequency of a
primary system, the mitigating performance may become less effective over time due to aging of the system, temperature or
A. Nankali • Y.S. Lee (*)
Department of Mechanical and Aerospace Engineering, New Mexico State University, 1040 S. Horseshoe St, Las Cruces, NM 88003, USA
e-mail: [email protected]; [email protected]
T. Kalmar-Nagy
Department of Aerospace Engineering, Texas A&M University, College Station, TX 77845, USA
e-mail: [email protected]
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_13, # The Society for Experimental Mechanics, Inc. 2012
181
humidity variations and so forth, thus requiring additional adjustment or tuning of parameters. It is only recently that
passively controlled spatial (hence dynamic) transfers of vibrational energy in coupled oscillators to a targeted point where
the energy eventually localizes were studied by utilizing an NES (See Vakakis et al. [12] for the summary of up-to-date
developments); and this phenomenon is simply called targeted energy transfer (TET). The NES is basically a device that
interacts with a primary structure over broad frequency bands; indeed, since the NES possesses essential stiffness non-
linearity, it may engage in (transient) resonance capture [13] with any mode of the primary system. It follows that an NES
can be designed to extract broadband vibration energy from a primary system, engaging in transient resonance with a set of
most energetic modes. In particular, Lee et al. [14–16] applied an ungrounded NES to an aeroelastic system, and numerically
and experimentally demonstrated that a well-designed NES can even completely eliminate aeroelastic instability. Three
suppression mechanisms were identified; that is, recurrent burstouts and suppressions, intermediate and complete elimina-
tion of self-excited instability in the aeroelastic system. Such mechanisms were investigated by means of bifurcation
analysis and complexification-averaging (CxA) technique [17].
In this workwe present study of TETmechanisms in suppressing regenerative chatter instability in a turning process. For this
purpose, we first review nonlinear dynamics of a SDOFmachine tool model; then, perform a linear stability analysis to explore
the effects of NES parameters on the occurrence of Hopf bifurcation (i.e., stability boundary on the plane of cutting depth and
rotational speed of a workpiece). To properly understand the suppressionmechanisms that appear similar to those in the previous
aeroelastic applications, numerical bifurcation analysis is performed by utilizing DDEBIFTOOL [18]. The CxA technique is
also utilized for analytical study of TET mechanisms (i.e., resonance captures). Finally, we perform asymptotic analysis for
regenerative instability suppression, which reveals the domain of attractions for the three suppression mechanisms [19, 20].
13.2 Dynamics of an SDOF Tool Model
We review of an SDOF machine tool dynamics model [3, 5], and then introduce an ungrounded NES to the machine tool
model. First, neglecting the NES in Fig. 13.1a, we can write the equation of motion for the SDOF machine tool as
€xþ 2zon _xþ o2nx ¼ �DFx=m (13.1)
where on is the (linearized) natural frequency; z ¼ c=ð2monÞ, the damping factor; and DFx, the cutting force variation. The
cutting force Fxð f Þ is frequently modeled as a power law by curve-fitting data obtained from quasisteady cutting tests [3, 5];
that is, we write
Fxð f Þ ¼ 0 for f � 0;Fxð f Þ ¼ Kwf a for f>0 (13.2)
where a is the cutting force exponent (a ¼ 0:75will be used in this work); w, the cutting width; f , the chip thickness; K, atest-related parameter assumed to be constant.
Tool
Workpiece
+
NES
5
4.5
4
3
3.5
2.5
2
1.5
1
0.5
00.5 1 1.5
Unstable region
a b
Stable region
2 2.5 3cs ms
m
k
ks
x
yc
n = 4n = 3
n = 2
n = 1
pp m
in
u3
Ω
Ω
Fig. 13.1 Machine tool model coupled to an ungrounded NES (a) and stability chart (b)
182 A. Nankali et al.
The cutting force variation can be expressed as
DFxð f Þ ¼ Fx � Fxð f0Þ ¼ �Fxðf0Þ for Df � � f0Kwð f a � f a0 Þ for Df>� f0
�(13.3)
where f0 refers to the nominal chip thickness at steady-state cutting; and Df , the chip thickness variation, which can be
expressed as Df ¼ f � f0 ¼ xðtÞ � xðt� tÞ � x� xt where t ¼ 2p=O is the delay or the period of revolution of the
workpiece with constant angular velocity O. Then, (13.1) can be rewritten as
€xþ 2zon _xþ o2nx ¼
k1f0ma
for Df � � f0
k1f0ma
1� f=f0ð Þa½ � for Df>� f0
8>><>>: (13.4)
where the cutting force coefficient k1 is introduced, k1 ¼ @Fx=@f jf¼f0¼ aKwf a�1
0 , which is the slope of the power-law curve
at the nominal chip thickness. Introducing the rescaling introduced in [5], we finally obtain the fully-nondimensional
equations of motion as
€xþ 2z _xþ x ¼pð2� aÞ
3afor Dx� 2� a
3
pð2� aÞ3a
1� 1� 3
2� aDx
� �a� �for Dx<
2� a3
:
8>><>>: (13.5)
where p ¼ k1=ðmo2nÞ is the nondimensional chip thickness; and Dx ¼ min xt � x; ð2� aÞ=3ð Þ due to the multiple regenera-
tive effects (or the contact loss for periods longer than a revolution of the workpiece). Since contact loss occurs when the
amplitude of tool vibrations is sufficiently large [5], a permanent contact can be assumed for sufficiently small amplitudes.
Taylor-expanding the cutting force about the nominal chip thickness and retaining the nonlinear terms up to the cubic order,
we obtain the equation of motion for a permanent contact model
€xþ 2z _xþ x ¼ pDx� pdðDx2 þ Dx3Þ (13.6)
where Dx ¼ xt � x and d ¼ 32ða� 1Þ=ð2� aÞ. Figure 13.1b depicts the stability chart in ðp;OÞ domain for a ¼ 0:75 so that
d ¼ 0:3, and z ¼ 0:1 so that pmin ¼ 2zð1þ zÞ ¼ 0:22 (see, e.g., [3]).
13.3 Suppression of Regenerative Instability by Means of TET
13.3.1 Nonlinear Energy Sink and Stability Enhancement
Now we apply an ungrounded nonlinear energy sink (NES) to the SDOF machine tool model (cf. Fig. 13.1a). Then, the
nondimensional equations of motion in state-vector form can be written as
_x ¼ Axþ Rxt þ fðx; xtÞ (13.7)
where x ¼ fx1; x2; x3; x4gT , x1 ¼ x; x2 ¼ y; x3 ¼ _x; x4 ¼ _y;
A ¼
0 0 1 0
0 0 0 1
�1� p 0 �2ðzþ z1Þ 2z10 0 2z1=ò �2z1=ò
26664
37775; R ¼
0 0 0 0
0 0 0 0
p 0 0 0
0 0 0 0
26664
37775; f ¼
0
0
�Cðx1 � x2Þ3 þ pd½ x1 � x1tð Þ2 � x1 � x1tð Þ3��Cðx2 � x1Þ3=ò
8>>><>>>:
9>>>=>>>;
where ò;C, and z1 respectively denote the mass ratio, coupling stiffness, and the damping factor of the NES.
13 Suppression of Regenerative Instabilities by Means of Targeted Energy Transfers 183
Assuming and substituting the solution of (13.7) to be xðtÞ ¼ expðltÞX, then we obtain the eigenvalue problem typical for
a delay-differential system.
lI� A� Re�lt� �X ¼ 0 (13.8)
where I is an identity matrix. For a nontrivialX, we derive the characteristic equation as |lI � A � e�ltR| ¼ 0. Substitution
l ¼ jo where j2 ¼ �1 and separation of real and imaginary parts yield
�2ðzþ z1 þ z1=òÞo2 þ 2z1ð1þ pÞ=ò ¼ ð2z1p=òÞ cosotþ po sinot� o3 þ ð1þ pþ 4zz1=òÞo¼ po cosot� ð2z1p=òÞ sinot (13.9)
By squaring and summing both sides of the two equations in (13.9), we obtain
pðoÞ ¼ GðoÞ=FðoÞ (13.10)
where the numerator and denominator can be written as
GðoÞ ¼ o6 � 2ð1� 2z21 � 2z21=ò2 � 4z21=ò� 4zz1 � 2z2Þo4 þ ð1� 8z21=ò
2 � 8z21=òþ 16z2z21=ò2Þo2 þ 4z21=ò
2
FðoÞ ¼ 2o4 � 2ð1� 4z21=ò2 � 4z21=òÞo2 � 8z21=ò
2(13.11)
Also, noting that 1� cosot ¼ 2sin2ðot=2Þ and sinot ¼ 2 sinðot=2Þ cosðot=2Þ, we rearrange (13.9) as
�2ðzþ z1 þ z1=òÞo2 þ 2z1=ò ¼ 2pR sinðot=2Þ cosðot=2þ fÞ� o3 þ ð1þ 4zz1=òÞo ¼ �2pR sinðot=2Þ sinðot=2þ fÞ (13.12)
where R ¼ ½ð2z1=òÞ2 þ o2�1=2 and f ¼ tan�1½2z1=ðòoÞ�. Then, we compute
tan ot=2þ fð Þ ¼ ½o3 � ð1þ 4zz1=òÞo�=½�2ðzþ z1 þ z1=òÞo2 þ 2z1=ò�KðoÞ (13.13)
Since t ¼ 2p/O, the rotational speed O can be derived as
OðoÞ ¼ po=½npþ tan�1KðoÞ � f� (13.14)
where n is the order of the lobe in the stability chart.
Figure 13.2 (left) depicts the changes of the stability boundary by varying the mass ratio ò and fixing the other two NES
parameters. Also, stability enhancement due to the application of anNES can bemeasured by directly calculating the point-wise
shift amount as Dp ¼ (p0 � p)/p � 100 (%), where p and p0 denote the values at the stability boundary with respect to each
5
100
50
Δp (%
)pp m
in pp m
inΔp
(%
)
10−1 10−1100 100
0
4
3
2
No NESNo NES=0.02 =0.1
Ω Ω
=0.2 =0.3 =0.6ζ1=0.02 ζ1=0.06 ζ1=0.1 ζ1=0.2
1
0
5
100
50
0
4
3
2
1
0
Fig. 13.2 Stability charts: (left) effects of NES mass ratio (ò) for z1 ¼ 0:1 and C ¼ 0:5; (right) effects of ENS damping factor (z1) for ò ¼ 0:1 andC ¼ 0:5
184 A. Nankali et al.
O without and with an NES, respectively. Upward shift of the stability boundary occurs more significantly near the valley than
near the cusp points of the lobes, which will be useful in practical applications of chatter suppression. The shifting amount of the
transition curve does not appear to be significant with a small NES mass (about 5% improvement near valley of the lobes);
however, the upward shift becomes increasing monotonically as the mass ratio increases. The ranges of the eigenfrequencies at
the transition curves tend to become lower as the mass ratio increases; and above certain mass ratio the eigenfrequency intervals
are shifted upward (cf. ò ¼ 0:6).The changes of stability boundary with respect to the NES damping factor with the other NES parameters fixed are
depicted in Fig. 13.2 (right). Although by increasing z1 the transition curve is shifted upward, stability enhancement by
increasing the damping factor is less prominent compared to that by increasing the NES mass. Rather, if z1 increases toohigh, Dp decreases, from which we conclude that adding more damping to a system does not always result in delay
occurrence of Hopf bifurcation. The eigenfrequency ranges on the transition curves shift upward as z1 increases initially, butthen shift downward for higher damping factors. We remark that such delay occurrence of Hopf bifurcations due the NES
parameters is absolutely independent of the magnitude of the essential nonlinearity C in the NES. This is because (13.10) and
(13.14) do not contain any terms associated with the essential nonlinearity. Indeed, this behavior was already observed in the
bifurcation analysis considered in Lee et al. [16] for understanding of the robustness enhancement of aeroelastic instability
suppression by means of SDOF and MDOF NESs.
The evaluation of overall stability enhancement can be discussed in terms of the amount of the upward shift at the valleys
of the lobes (i.e., the minimum value of pðoÞwhich is independent of the lobe order). We introduce the following quantity as
a measure of such shift at the valleys of the transition curve.
Dpmin ¼ ðp0min � pminÞ=pmin � 100ð%Þ (13.15)
where pmin ¼ 2zð1þ zÞ and p0min are the minimum values of pðoÞ without and with an NES, respectively.
Figure 13.3 depicts the contour plot of Dpmin with respect to ðò; z1Þ, which clearly illustrates that the optimal stability
enhancement (i.e., maxDpmin) occurs at a certain nonlinear relation between ò and z1 (the thick line which can be approximated
as the function, z1 � 0:35òq where q ¼ 1=1:3 from minimizing the mean square errors between the two curves).
13.3.2 Bifurcation Analysis and TET Mechanisms
We apply the numerical continuation technique for delay-differential equations (DDEBIFTOOL [18]) to study bifurcation
behaviors of the trivial equilibrium and the limit cycles. For example, Fig. 13.4a compares the bifurcation diagram for the
tool amplitudes by means of numerical continuation for the case when no NES is applied (thick dashed line) and when an
NES is involved (solid lines). As in the previous aeroelastic applications [14], three distinct TET mechanisms are identified
1250(%)
200
150
100
50
0
0.8
0.9
0.7
0.6
0.5ζ 1
0.4
0.3
0.2
0.1
00 0.2
max Δpmin
0.4 0.6 0.8 1
Fig. 13.3 NES mass ratio and
damping factor for optimal
stability enhancement
13 Suppression of Regenerative Instabilities by Means of Targeted Energy Transfers 185
in suppressing regenerative chatter instability; that is, recurrent burstouts and suppression, intermediate and complete
elimination of regenerative instability (cf. See Fig. 13.4b for typical time history for each suppression mechanism).
The first suppression mechanism is characterized by a recurrent series of suppressed burstouts of the tool response,
followed by eventual complete suppression of the regenerative instabilities. The beating-like (quasiperiodic) modal
interactions observed during the recurrent burstouts turn out to be associated with Neimark-Sacker bifurcations of a periodic
solution (cf. Fig. 13.4a) and to be critical for determining domains of robust suppression [16]. To investigate a more detail of
this mechanism, Fig. 13.5 depicts the displacements of both the tool and NES and their wavelet transforms. Also, rigorous
energy exchanges between the two modes are evidenced in Fig. 13.5, through which a series of 1:1 transient resonance
captures and escapes from resonance occurs.
The second suppression mechanism is characterized by intermediate suppression of LCOs, and is commonly observed
when there occurs partial LCO suppression. The initial action of the NES is the same as in the first suppression mechanism.
Targeted energy transfer to the NES then follows under conditions of 1:1 TRC, followed by conditions of 1:1 PRC where the
tool mode attains constant (but nonzero) steady-state amplitudes. We note that, in contrast to the first suppression
mechanism, the action of the NES is nonrecurring in this case, as it acts at the early phase of the motion stabilizing the
tool and suppressing the LCO. The third suppression mechanism involves energy transfers from the tool to the NES through
1
0.8
a b
0.6
0.4
Dis
plac
emen
ts
0.2
Tool amplitudew/o NES NES amplitude
NS2
H
NS1
LPC2
LPC1
Tool amplitudew/ NES
0
1 2 3p/pmin
4 5 6 0−0.2
−0.5
0.5
−1
1
−1
0.2
0x(t)
x(t)
x(t)
1
0
0
50 100 150Time, (s)
200
3rd Suppression Mechanism
2nd Suppression Mechanism
1st Suppression Mechanism
250 300
Fig. 13.4 (a) Bifurcation diagram for the tool and NES amplitudes for (13.6) (O ¼ 2:6; e ¼ 0:2; z1 ¼ 0:1;C ¼ 0:5): H, LPC and NS denote Hopf,
limit point cycle, and Neimark-Sacker bifurcation points, respectively. (b) Typical tool displacements for the three suppression mechanisms
1
−1
−1
1
0
0
0 00
1
2
3
40
1
2
3
4
50
NE
S
Fre
quen
cy
Too
l
100 150 100Time Time
200 200250 300 300
Fig. 13.5 Displacements and their wavelet transform spectra for a typical first suppression mechanism
186 A. Nankali et al.
nonlinear modal interactions during 1:1 RCs. The displacements for the tool and the NES exhibit exponential decrement
finally resulting in complete elimination of LCOs.
In order to analytically study the underlying TETmechanisms,we employ theCxAmethod first introduced byManevitch [17].
We introduce the new complex variables in the following.
C1ðtÞ ¼ _xðtÞ þ joxðtÞ � ’1ðtÞe jot;C2ðtÞ ¼ _yðtÞ þ joyðtÞ � ’2ðtÞe jot (13.16)
where j2 ¼ �1. Then, denoting by ðÞ the complex conjugate, we can express the original real variables in terms of the new
complex ones
xðtÞ ¼ 1
2joðC1 �C
1Þ ¼1
2joð’1e
jot � ’1e
�jotÞ; xðt� tÞ ¼ 1
2joð’1ðt� tÞe joðt�tÞ � ’
1ðt� tÞe�joðt�tÞÞ
_xðtÞ ¼ 1
2ðC1 þC
1Þ ¼1
2ð’1e
jot þ ’1e
�jotÞ; €xðtÞ ¼ ð _’1 þ jo’1Þe jot � jo2ð’1e
jot þ ’1e
�jotÞ(13.17)
and similar expressions can be obtained for the NES variables. Substituting into the equations of motion and averaging out
the fast dynamics over e jot, we obtain a set of two complex-valued modulation equations governing the slow-flow dynamics,
_’ ¼ Fð’; ’tÞ (13.18)
where ’ ¼ f’1; ’2gT . Expressing the slow-flow amplitudes in polar form, ’kðtÞ ¼ akðtÞe jbkðtÞ, where akðtÞ; bkðtÞ 2 R;k ¼ 1; 2, we obtain the set of real-valued slow-flow equations such that
_a1 ¼ f1ða1; a2;fÞ; _a2 ¼ f2ða1; a2;fÞ; _f ¼ gða1; a2;fÞ (13.19)
where f � b1 � b2.Figure 13.6 directly compares the approximate and exact solutions of the tool displacement for the case of the first
suppression mechanism, which demonstrates a good agreement; furthermore, the non-time-like patterns (i.e., formation of
multiple loops) of the phase difference f depicts that the underlying TET mechanism involves a series of 1:1 transient
resonance captures and escapes from resonance.
Finally, we note that the numerical and analytical studies for TET mechanisms above are valid only for vibrations with
small amplitudes (i.e., before contact loss occurs); in particular, the permanent contact model with truncated nonlinear terms
cannot predict any stable steady-state periodic vibrations of high amplitudes. That is, the truncated nonlinearity in the
regenerative cutting force will not predict the existence of a saddle-node bifurcation point right after contact loss occurs.
The details of machine tool dynamics can be found in [5], where stable periodic motions are predicted. By applying the
ungrounded NES, we can still observe the three distinct TET mechanisms, as depicted in Fig. 13.7. Similar arguments can be
made for nonlinear modal interactions between the tool and NES as in the case of permanent contact.
1
0
0 50 100 150
direct numerical simulation analytical approximation
Time, t Phase difference f
f•
200 250 300 350 −2.5 −1.5 −0.5−2 0−1
−0.5
0.5T
ool D
ispl
acem
ent
1.5
0.5
−0.5
−1
1
0
−1
Fig. 13.6 Comparison of the approximate solution from the CxA analysis with the (numerically) exact solution for a typical first suppression
mechanism (on the left); demonstration of non-time-like phase difference (on the right)
13 Suppression of Regenerative Instabilities by Means of Targeted Energy Transfers 187
13.4 Asympotic Analysis of Regenerative Instability Suppression
In this section, we perform an asymptotic analysis [19, 20] for the three suppression mechanisms to estimate the domains of
attraction in the parameter space. For this purpose, we introduce the coordinate transformation
v ¼ xþ ey;w ¼ x� y (13.20)
where v and w are the physical quantities for the center of mass (with a factor of 1 + e) and the relative displacement,
respectively. Then, the equations of motion (13.7) become
€vþ 2z_vþ e _w1þ e
þ vþ ew1þ e
¼ pvt þ ewt
1þ e� vþ ew
1þ e
� �
€wþ 2z_vþ e _w1þ e
þ 2z1 _wþ vþ ew1þ e
þ 4
31þ eð Þw3 ¼ p
vt þ ewt
1þ e� vþ ew
1þ e
� � (13.21)
where proper rescaling conditions are applied [19]. Since a single fast-frequency dominates for the three suppression
mechanisms, we introduce the complexification similar to (13.16).
_vðtÞ þ jovðtÞ � ’1ðtÞe jot; _wðtÞ þ jowðtÞ � ’2ðtÞe jot (13.22)
Substituting into (13.21) and performing averaging over the fast component ejot, we obtain the slow-flow equation
_’1 ¼ F1ð’1; ’2; ’1t; ’2tÞ; _’2 ¼ F2ð’1; ’2; ’1t; ’2tÞ (13.23)
Introducing polar form to the slow variables, ’1 ¼ V exp½jy1�; ’2 ¼ W exp½jy2�, we can derive real-valued slow-flow
dynamics
_V ¼ eF1ðV;W;Vt;Wt;fÞ; _W ¼ F2ðV;W;Vt;Wt;fÞ;_f ¼ G1ðV;W;Vt;Wt;fÞ þ eG2ðV;W;Vt;Wt;fÞ
(13.24)
2
−2
−1
−10 50 100 150 200 250
Tool Displacements
w/o NES
1st Supp Mech
2nd Supp Mech
3rd Supp Mech
w/ NES
Time (s)
300 350 400 450 500
0
1
1
0
0x(t)
x(t)
x(t)
Fig. 13.7 Typical time
responses for the three
suppression mechanisms
for the contact loss model
in (13.5)
188 A. Nankali et al.
where V;W 2 R;f ¼ y1 � y2;F1;F2;G1;G2 ¼ Oð1Þ, and the details of the right-hand sides are omitted because of their
complexity. For sufficiently small e, we can write (13.24) as
_V ¼ OðeÞ ) V ¼ VðetÞ; _W ¼ F2ðV;W;Vt;Wt;fÞ;_f ¼ G1ðV;W;Vt;Wt;fÞ þ OðeÞ
((13.25)
Considering the equilibrium points of (13.24), we can derive the slow-invariant manifold (SIM) for equilibrium and super
slow-flow (SSF) dynamics, respectively, as
SIM : F2ðV0;W0;f0Þ ¼ 0 and G1ðV0;W0;f0Þ ¼ 0 ) HðV0;W0Þ ¼ 0
SSF : @V=@ðetÞ ¼ F1ðV0;W0Þ(13.26)
The intersections between SIM and SSF equations provide the number of equilibrium points for the slow-flow dynamics
(13.24) and their stability [19]. Figure 13.8 presents a result for asymptotic analysis for a typical first suppression mechanism,
where amplitude modulations can be observed due to relaxation oscillations (strongly-modulated response [19]).
13.5 Concluding Remarks
Suppression of regenerative instability in a single-degree-of-freedom (SDOF) machine tool model was studied by means of
one-way, passive, broadband targeted energy transfers (TETs). Two models were considered for the tool dynamics:
Permanent contact model and contact loss model. Stability and bifurcation analysis were carried out for both models.
An ungrounded nonlinear energy sink (NES) was coupled to the SDOF tool, by which biased energy transfers from the tool
to the NES. Shifts of the stability boundary (i.e., Hopf bifurcation point) with respect to chip thickness were examined for
various NES parameter conditions. It was shown that there should be an optimal value of damping for a fixed mass ratio to
shift the stability boundary for stably cutting more material off by increasing chip thickness. Also, magnitude of NES
nonlinear stiffness does not have any effect on stability boundary while increasing mass ratio improves stability. The limit
cycle oscillation (LCO) due to the regenerative instability in a tool model which appeared as being subcritical for permanent
contact model were (locally) eliminated or attenuated at a fixed rotational speed of a workpiece (i.e., a delay period) by TETs
to the NES. Contact loss model depicted supercritical LCOs at relatively high displacement of the tool. Utilizing NES for
contact loss model shifted bifurcation diagram of tool displacement such a way to improve stability. Three suppression
mechanisms have been identified as was in the previous aeroelastic applications, and each suppression mechanism was
investigated numerically by time histories of displacements, and wavelet transforms and instantaneous modal energy
exchanges. The analytical means by complexification-averaging technique showed that resonance captures are the underlying
dynamical mechanism for TETs.
3
2.5
= 0.02; p = 0.7
SIM : H (V0,W0) = 0
SSF : F1(V0,W0) = 0
1.5
0.5
2
1
00 2 4
x(t)
6 8 0 200 400TimeW0
2( t)
V02 (
t)
600 800 1000
1
0.5
−0.5
−1
0
Fig. 13.8 Asymptotic analysis of the first suppression mechanism and corresponding tool response
13 Suppression of Regenerative Instabilities by Means of Targeted Energy Transfers 189
Acknowledgments This material is based upon work supported by the National Science Foundation under Grant Numbers CMMI-0928062 (YL)
and CMMI-0846783 (TK).
References
1. Dombovari Z, Barton DAW, Wilson RE, Stepan G (2011) On the global dynamics of chatter in the orthogonal cutting model. Int J Non Lin
Mech 46:330–338
2. Nayfeh AH, Nayfeh NA (2011) Analysis of the cutting tool on a lathe. Nonlinear Dyn 63:395–416
3. Kalmar-Nagy T, Stepan G, Moon FC (2001) Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations. Nonlinear
Dyn 26:121–142
4. Namachchivaya NS, Van Roessel HJ (2003) A center-manifold analysis of variable speed machining. Dyn Syst 18(3):245–270
5. Kalmar-Nagy T (2009) Practical stability limits in turning (DETC2009/MSNDC-87645). In: ASME 2009 international design engineering
technical conferences and computers and information in engineering conference, San Diego, California, 30 August–2 September, 2009
6. Lin SY, Fang YC, Huang CW (2008) Improvement strategy for machine tool vibration induced from the movement of a counterweight during
machining process. Int J Mach Tool Manuf 48:870–877
7. Khasawneh FA, Mann BP, Insperger T, Stepan G (2009) Increased stability of low-speed turning through a distributed force and continuous
delay model. J Comput Nonlin Dyn 4:041003-1-12
8. Moradi H, Bakhtiari-Nejad F, Movahhedy MR (2008) Tunable vibration absorber design to suppress vibrations: an application in boring
manufacturing process. J Sound Vibrat 318:93–108
9. Wang M (2011) Feasibility study of nonlinear tuned mass damper for machining chatter suppression. J Sound Vibrat 330:1917–1930
10. Ganguli A (2005) Chatter reduction through active vibration damping. Ph.D. dissertation, Universite Libre de Bruxelles
11. Ast A, Eberhard P (2009) Active vibration control for a machine tool with parallel kinematics and adaptronic actuator. J Comput Nonlin Dyn
4:0310047-1-8
12. Vakakis AF, Gendelman O, Bergman LA, McFarland DM, Kerschen G, Lee YS (2008) Passive nonlinear targeted energy transfer in
mechanical and structural systems: I and II. Springer, Berlin/New York
13. Arnold VI (ed) (1988) Dynamical systems III. Encyclopaedia of mathematical sciences. Springer, Berlin/New York
14. Lee YS, Vakakis AF, Bergman LA, McFarland DM, Kerschen G (2007) Suppression of aeroelastic instability by means of broadband passive
targeted energy transfers, part I: Theory. AIAA J 45(3):693–711
15. Lee YS, Kerschen G, McFarland DM, Hill WJ, Nichkawde C, Strganac TW, Bergman LA, Vakakis AF (2007) Suppression of aeroelastic
instability by means of broadband passive targeted energy transfers, part II: Experiments. AIAA J 45(10):2391–2400
16. Lee YS, Vakakis AF, Bergman LA, McFarland DM, Kerschen G (2008) Enhancing robustness of aeroelastic instability suppression using
multi-degree-of-freedom nonlinear energy sinks. AIAA J 46(6):1371–1394
17. Manevitch L (2001) The description of localized normal modes in a chain of nonlinear coupled oscillators using complex variables. Nonlinear
Dyn 25:95–109
18. Engelborghs K, Luzyanina T, Roose D (2002) Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL. ACM
Trans Math Softw 28(1):1–21
19. Gendelman OV, Vakakis AF, Bergman LA, McFarland DM (2010) Asymptotic analysis of passive nonlinear suppression of aeroelastic
instabilities of a rigid wing in subsonic flow. SIAM J Appl Math 70(5):1655–1677
20. Gendelman OV, Bar T (2010) Bifurcations of self-excitation regimes in a van der Pol oscillator with a nonlinear energy sink. Physica D
239:220–229
190 A. Nankali et al.
Chapter 14
Force Displacement Curves of a Snapping Bistable Plate
Alexander D. Shaw and Alessandro Carrella
Abstract Bistable structures are characterised by rich dynamics, because they necessarily include regions of negative
stiffness between their stable configurations. The presence of this region results in marked nonlinear behaviour, with
a response that can range from periodically stable to chaotic. However, this region can also be exploited to tailor
force-displacement curves.
A possible application of an ad-hoc load deflection curve is a vibration isolator with High Static Low Dynamic Stiffness
(HSLDS). The idea is to confer to the isolator a low dynamic stiffness whilst the high static stiffness maintains a high load
bearing capacity. Therefore coupling this apparatus with mass on a conventional anti-vibration mount demonstrates that
it can be used to reduce the natural frequency of this system, thereby increasing the isolation region of the mount.
This study presents the load-displacement curve of a bistable composite plate, which is loaded transversely at its centre,
whilst its corners are free to rotate and move laterally. Both numerical and experimental results are presented, and it is shown
that the response is highly directional and hysteretic, and that the force is also influenced by velocity as well as displacement.
The geometrical sources of these effects are considered.
Keywords Bistable • Nonlinear • Composite • Isolator
14.1 Introduction
Bistable composite plates can occupy two different stable configurations, between which they may ‘snap’ when forced [1].
They are attracting considerable interest in the field of morphing structures, and for their potential use in actuators, due to
their ability to form multiple shapes with no ongoing power consumption [2, 3]. They also are attracting interest from
dynamics researchers, as their multiple potential wells can lead to highly nonlinear and chaotic responses to excitation [4].
A consequence of having two stable configurations is a region of negative transverse stiffness occurring between these states;
this region is encountered during a snap between one stable state and another. This negative stiffness can be exploited to tailor
force displacement curves, in particular to create a High Static LowDynamic Stiffness (HSLDS) anti-vibrationmount. This is a
device that has high static stiffness, to provide it with good load bearing capacity. However, near its equilibrium point it features
a region of low stiffness, which therefore lowers the natural frequency and increases the isolation region of the mount [5].
In order to create an HLSDS mount exploiting the negative stiffness of a composite bistable plate, we must investigate
the transverse force displacement curve of such plates. This work presents an experimental and numerical investigation into
the force displacement curve of a thermally formed bistable plate, extending the findings of a previous study by Potter et al.
[7] to consider the effect of repeated and reversed displacement cycles. The study shows that the response is rate dependant,
direction dependant and therefore highly hysteretic.
A.D. Shaw (*)
Advanced Composites Centre for Innovation and Science (ACCIS), University of Bristol, Queens Building,
University Walk, Bristol BS8 1TR, UK
e-mail: [email protected]
A. Carrella
Faculty of Engineering, Bristol Laboratory for Advanced Dynamic Engineering (BLADE), LMS International,
Queens Building, University Walk, Bristol BS8 1TR, UK
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_14, # The Society for Experimental Mechanics, Inc. 2012
191
14.2 Experimental Study
14.2.1 Experimental Design
The plate was made from eight plies of IM7/8552 Intermediate Modulus Carbon Fibre pre-preg in the stacking sequence
[0�4; 90�4]T . It was vacuum bagged using a tool plate on both sides, to ensure the most symmetrical resin distribution.
It was cured using the manufacturers recommended cycle in an autoclave at 180�C for 2 h. The ‘flat’ plate dimensions are
280 � 280 mm, in line with the fibre direction. A 5 mm hole was drilled at the centre, to accommodate the bolt for the load
tester. Holes 12 mm in diameter were drilled 10 mm in from each corner, to accommodate pivot joints described below.
To restrict the degree to which moisture absorption could affect the material properties, the plate was stored in a sealed
cupboard with desiccant whenever it was not in use.
The plate was supported by its corners, on apparatus designed to provide boundary conditions which do not restrain
the snap through. These corner boundary conditions are vertical pins i.e. they allow free out of plane rotation, zero vertical
displacement and free lateral translation. To achieve the first of these conditions, the corners were fitted with spherical
bearings that were bonded into holes drilled through the plate, that permitted pivoting in any direction. By providing a small
angle to the bearing casing relative to the plate, these bearings allowed unrestricted motion to the full range of angles that
the corners of the plate would adopt between each of its stable states. To simultaneously achieve the second and third
boundary conditions, the bearings were mounted on tall slender steel posts (250 mm long, 3 mm in diameter), which were
rigidly attached to an adjustable base. Vertically these provided stiffness greatly in excess of the plate’s transverse stiffness.
Horizontally, the posts acted as soft cantilever springs. The maximum lateral displacement of the corners is approximately
5 mm so modelling the posts as a simple cantilever shows the maximum horizontal reaction of the corners would be of the
order of 0.75 N. The vertical deflection caused by this motion can also be seen to be negligible.
The entire apparatus was placed on the moving base of an Instron 1341 load tester, with the centre of the plated bolted to a
1 kN load cell. Therefore raising and lowering the base effectively applied displacements to the centre of the plate, allowing
the load displacement curve to be measured. The complete apparatus is shown in Fig. 14.1.
The displacement cycle applied to the plate consisted of increasing displacement at constant velocity over a range
including both stable positions of the plate, then returning at the same velocity to the starting point. This was performed
twice, at a velocity of 100 mm/min.
Fig. 14.1 Experimental
method
192 A.D. Shaw and A. Carrella
14.3 Results
Figure 14.2 shows the experimental force displacement curve. As can be seen, the force shows steady progression with
displacement, interrupted by two sudden changes in force, as shown by steep negative gradients followed by transients
on the graph. These correspond to sudden changes of the shape of the plate, from and to a ‘half snap’ configuration, as shown
in Fig. 14.3. These shape changes occur at different displacements, depending on the direction of the motion.
14.4 Numerical Analysis
14.4.1 Model
The numerical model was created in Abaqus 6.10, using the Dynamic/Implicit solver using nonlinear geometry to allow
the bistable properties to resolved and the quasi-static option to provide appropriate amounts of artificial dissipation
automatically. The mesh was constructed from four-node shell elements constructed with the appropriate composite lay
up. Figure 14.4 shows how the mesh was biased to provide greater resolution near the edges, where stress gradients are seen
to be higher. Vertically pinned, laterally free boundary conditions are applied the locations of the centres of the plate pivot
joints. At the centre, lateral translation and rotation about the vertical axis are restricted, and the vertical displacement is
controlled throughout the simulation.
Fig. 14.2 Experimental force displacement curve, two loading cycles at 100 mm/min. Arrows show direction of motion around cycle
Fig. 14.3 Upward snap sequence, showing initial, half snap and final configurations
14 Force Displacement Curves of a Snapping Bistable Plate 193
The layup and initial geometry of the plate was as described for the experimental design, using lamina properties as
described in Table 14.1. Small eccentricities of 0.25 and 0.5 mm in x and y respectively were added to the position of the
central point, to allow asymmetric shapes to form. It was found that varying these dimensions over a range similar to that
expected for manufacturing errors in the physical experiment did not have a major effect on results found.
The solver follows the following steps:
1. The initial step defines the flat plate at cure temperature.
2. The plate is cured to room temperature (20�C) and adopts one of its bistable shapes. A nominal load is applied to the
centre to ensure that the configuration resolved is consistently the same one of the two possible states.
3. The nominal load is removed.
4. The centre of the plate is displaced to starting position, 40 mm vertically above the original horizontal plate plane.
5. The centre vertical displacement is varied in a linear ramp to �40 mm, over a period of 100 s.
6. The previous step is reversed; the vertical displacement is varied in a linear ramp to 40 mm over 100 s.
14.5 Results
Figure 14.5 shows the results of the FEA simulation. It shows the snap through region characterised by multiple stages,
separated by three sudden shape-change events. Again, the displacement at which these events occur varies with the
direction of motion. Figure 14.6 shows the total strain energy reported by Abaqus against displacement, and it can be seen
that shape change events coincide with sharp drops in the strain energy.
Fig. 14.4 Mesh used for
FEA. Dots indicate whereboundary conditions are
applied
Table 14.1 Assumed lamina properties for IM7/8552 CFRP. Properties taken from
manufacturer’s data, with transverse isotropy used to estimate through-thickness quantities.
t is an average taken from multiple plate measurements
In plane Young’s Modulus, fibre direction, E1 164 GPa
In plane Young’s Modulus, transverse direction, E2 12 GPa
In plane Poisson ratio, n12 0.3
In plane shear modulus, G12 4.6 GPa
Through thickness shear modulus, fibre direction, G13 4.6 GPa
Through thickness shear modulus, transverse direction, G23 4.1 GPa
Ply thickness, t 0.122 mm
194 A.D. Shaw and A. Carrella
14.6 Discussion
14.6.1 Comparison of Experiment and Numerical Results
The numerical and physical experiments show some qualitative similarities, but some key differences remain. Firstly,
the experimental graph is asymmetrical, in that the minima near 10 mm displacement is smaller in magnitude and shows
smoother curves that the maxima near �5 mm displacement. It is believed this is because the plain bearing joints were
necessarily bonded in place when the plate was in one of its stable states. Hence when the plate is flipped to its other state, the
fillet of glue forms a residual stress that applies moments to the plate, reducing the snapping force and distorting the force
displacement graph. Secondly, maximum force is much smaller in the physical experiment than in FEA. It is thought this
is explained by the ingress of moisture from ambient humidity; atmospheric effects on such a magnitude are described
by Etches et al. [6]. Finally, over much of the graph where there is no obvious influence due to shape changes, there
is a small difference due to the direction of travel present in the experimental work that is not apparent in the FEA.
This is attributed to a small amount of frictional or viscous force within the experimental apparatus.
Fig. 14.5 Numerical force displacement curve, arrows indicate path followed
Fig. 14.6 Internal strain energy against displacement. Arrow indicates path followed when displacement moves from negative to positive.
Dashed lines show the assumed projection of the four continuous curves that intersect to form graph
14 Force Displacement Curves of a Snapping Bistable Plate 195
Despite these shortcomings, the positive peak and much of the negative stiffness region of Fig. 14.2 suggest that
FEA captures many of the snapping plate, qualitatively if not quantitatively.
14.6.2 Validity of Quasi-static Assumption
A conventional force-displacement graph implicitly assumes that the system concerned is quasi-static; that motion
is sufficiently slow that dynamic effects may be neglected, and that each point on the graph therefore defines a point of
static equilibrium. For most of Fig. 14.2 and Fig. 14.5 this assumption is met; even where a negative slope is present, the
plate is held in a static equilibrium by the restraint at the centre. However, in the immediate vicinity of shape changes
the quasi-static assumption is invalid, because the plate is performing a rapid motion between a state that has become
unstable, and a new stable state, and is then subsequently oscillating for a short while after this change.
14.6.3 History Dependence; Irreversibility and Hysteresis
The system described is history dependent; for example the force at zero displacement in Fig. 14.5 may be positive if
displacement has moved directly from the negative equilibrium point, or negative if the opposite motion has just been
performed. Related to this is that the shape-change events are irreversible; once a shape change has occurred, returning
directly to the displacement just before the shape change will return to a different centre reaction force and plate shape than
previously existed.
To return to a given shape and displacement after a shape change, displacement must retract further until the opposite
shape change occurs, and then be advanced to the original position. However, this process forms a hysteresis loop that will
permanently dissipate energy input into the system.
14.6.4 Total Strain Energy
Figure 14.6 shows that the total strain energy of the plate can be used to model the behaviour. If we consider the
strain energy/displacement graph as the intersection of four curves (as arbitrarily extended in the figure) associated with
each shape, the plate generally follows the path that has the lowest total strain energy. However, when two energy curves
intersect, the graph follows its original course briefly before jumping down to the new configuration. This delay is explained
qualitatively below.
At the intersection, the two different shapes offer exactly the same total strain energy. Therefore there is no energy
gradient inducing any transformation from one to the other, indeed the intermediate states may demand higher strain energy.
Therefore, the initial configuration must progress until it has higher energy than the new configuration, and it becomes
unstable. Furthermore, since the initial speed of the shape change may be very slow due to a low energy gradient, the rate
of the controlling displacement may bias the apparent displacement at shape change, even at apparently quasi-static
displacement rates.
14.7 Conclusions and Future and Ongoing Work
This work has shown the transverse force-displacement curve for a thermally formed bistable composite plate, using both
numerical and experimental methods. It has demonstrated that the negative stiffness region is not smooth, but punctuated by
a number of unstable shape change events, which are irreversible and lead to a history dependant, hysteretic force
displacement curve. Theses shape changes are related to drops in strain energy of the plate.
Ongoing work is investigating dynamic effects on this curve, applying high frequency deformations to the plate and
investigating its effects when coupled to conventional vibration mounts. Future work will also develop mathematical shape
functions for the stable and half snapped configurations, which allow the shape to be modelled for any given displacement.
It will also investigate other fabrication techniques for bistable plates.
196 A.D. Shaw and A. Carrella
References
1. Dano M-L, Hyer MW (1998) Thermally-induced deformation behavior of unsymmetric laminates. Int J Solids Struct 35(17):2101–2120
2. Daynes S, Weaver PM, Trevarthen JA (2011) A morphing composite air inlet with multiple stable shapes. J Intell Mater Syst Struct
22(9):961–973
3. Diaconu CG, Weaver PM, Mattioni F (2008) Concepts for morphing airfoil sections using bi-stable laminated composite structures. Thin Wall
Struct 46(6):689–701
4. Arrieta A, Neild S, Wagg D (2009) Nonlinear dynamic response and modeling of a bi-stable composite plate for applications to adaptive
structures. Nonlinear Dyn 58:259–272
5. Ibrahim RA (2008) Recent advances in nonlinear passive vibration isolators. J Sound Vibrat 314(3–5):371–452
6. Etches J, Potter K, Weaver P, Bond I (2009) Environmental effects on thermally induced multistability in unsymmetric composite laminates.
Compos Part A Appl Sci Manuf 40(8):1240–1247, Special issue: 15th French national conference on composites - JNC15
7. Potter K, Weaver P, Seman AA, Shah S (2007) Phenomena in the bifurcation of unsymmetric composite plates. Compos Part A Appl Sci Manuf
38(1):100–106
14 Force Displacement Curves of a Snapping Bistable Plate 197
Chapter 15
Characterization of a Strongly Nonlinear Vibration Absorber
for Aerospace Applications
Sean A. Hubbard, Timothy J. Copeland, D. Michael McFarland, Lawrence A. Bergman,
and Alexander F. Vakakis
Abstract We consider the identification of a nonlinear energy sink (NES) designed to limit the vibration of an aircraft wing
by attracting and dissipating energy before a transient response can build into a limit-cycle oscillation (LCO). The device
studied herein is the prototype of an NES intended to be mounted at the tip of a scale-model wing, housed in a winglet, and
capable of interacting dynamically with the wing over a broad frequency range. Because the stiffness of the NES is
essentially nonlinear (i.e., its force-displacement relation is nonlinearizable), it cannot be regarded as a perturbation of a
linear system. Furthermore, the action of the NES requires the presence of some amount of damping, here assumed to be
viscous. Both the nonlinear stiffness and the linear viscous damping have been evaluated using the restoring force surface
method (RSFM), and found to be repeatable across trials and across builds of the system. These findings are summarized and
used in simulations of the NES attached to the wing. The simulations are then compared to experiments (ground vibration
tests), revealing good agreement of transient responses and of frequency-energy dependence, the latter revealed by wavelet
transforms of the computed and measured time series.
Keywords Limit-cycle oscillations • LCO • Aircraft structures • Nonlinear energy sink • NES
15.1 Introduction
One of the most-developed applications of targeted energy transfer (TET) for vibration suppression is the stabilization of
limit-cycle oscillations (LCOs) of aircraft structures. While such self-excited oscillations would seem to be steady-state
phenomena, detailed analysis has shown the onset of LCO in a typical-section (rigid-airfoil) model in subsonic flow to be
characterized by a transient 1:1 resonance between the wing’s heave and pitch motions, followed by a sustained 3:1
resonance. If the initial 1:1 resonance is interrupted, the higher-energy 3:1 resonance, and thus the LCO, may be prevented
entirely. A properly designed passive attachment—a strongly nonlinear vibration absorber—can achieve this by coupling to
small, flow-induced motions of the primary structure early in its response.
Similar events lead to LCO in more realistic conditions, such as a plate-like structure in transonic flow, although the
response frequencies may vary more and more aeroelastic modes may participate. The vibration absorber now cannot be
tuned to a specific frequency, but must be inherently broadband. A nonlinear energy sink (NES), incorporating a small mass,
S.A. Hubbard
Aerospace Engineering Department, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
e-mail: [email protected]
T.J. Copeland (*)
m+p international, inc. 271 Grove Avenue, Bldg G, Verona, NJ 07044, USA
e-mail: [email protected]
D.M. McFarland • L.A. Bergman
Aerospace Engineering Department, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
e-mail: [email protected]; [email protected]
A.F. Vakakis
Mechanical Science and Engineering Department, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
e-mail: [email protected]
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_15, # The Society for Experimental Mechanics, Inc. 2012
199
an essentially nonlinear (i.e., nonlinearizable) coupling stiffness, and a (typically linear, viscous) damping element is ideal for
this application, and we consider such a device here in prototype form, as designed a fabricated for use with a uniform-plate
mock-up of a wind-tunnel model wing. This NES has a single degree of freedom, represented relative rotation of the NES
mass with respect to its mounting point at the wingtip. An essentially nonlinear spring provides restoring torque, and
dissipation occurs in the bearing supporting the mass and in the connections to the springs. The focus of this paper is on
the characterization of these dynamic elements, especially the nonlinear springs, to show that they can be accurately designed
and reliably and repeatably installed.
15.2 Rotary Nonlinear Energy Sink Design and Assembly
Figure 15.1 shows an exploded view of the NES components and how they are assembled, along with a view of the
assembled NES.The backing plate and anchor blocks were machined from 6061 aluminum alloy, and the shaft, anchor
clamps, NES mass, and posts were all machined out of steel. The wires used were straight stock steel wire, and the rotary
bearing was a standard type deep groove ball bearing typically used in automotive applications. The reverse side of the plate
features two ribs which fit over and bolt onto the tip of the uniform-thickness aluminum plate swept wing. The NES has
several features which allow for it to be examined under numerous configurations. The plate allows for one of the anchor
Fig. 15.1 (a) Exploded view
of NES components;
(b) Assembled NES
200 S.A. Hubbard et al.
blocks to be bolted into one of three positions which control the span of the wires, or the blocks can be moved to several
different positions on the circumference of the plate causing the direction of the wire span to rotate. The NES mass features
threaded holes that can be used to add mass, increasing its mass moment of inertia. The wire diameter is only limited by the
diameter of the wire channel in the anchor clamp, and the gaps and positioning of the posts are adjustable to some extent.
The aluminum alloy backing plates serves as a frame upon which the remaining components can be assembled. It also
provides the interface for the device to be attached to the plate wing. The plate is approximately 6 in. in diameter and is
shaped as such to accommodate several configurations. The pattern of 0.5 in holes through the plate serve no purpose other
than to remove unnecessary mass. As their names indicate, the anchor blocks and clamps provide the required boundary
conditions for the wire ends.
When the assembled device in Fig. 15.1b is free to oscillate, or active, the NES mass is allowed to pivot about the shaft.
As the mass pivots, the posts make contact with the wires causing them to be displaced transversely near the center of the
span. As mentioned already, this will produce a nonlinear response from the wires which will force the NES mass in the
reverse direction. This is the configuration that was considered for all that follows.
Some information important for the analyses that follow is listed here. The mass of the fully assembled device shown in
Fig. 15.1b is 0.687 kg, and its mass moment of inertia about the shaft is approximately 1. 27 �10 � 3kgm2 when the NES
mass is locked in place. The mass moment of inertia of the NES mass about the shaft is 2. 2 �10 � 4kgm2.
When assembled, the NES mass is allowed to pivot about the bearing such that the posts are in contact with the piano
wires which are clamped at both ends. The most critical stage of the assembly is the installation of the wires which must be
straight and without tension or compression. When assembled properly, the NES experiences negligible resistance to
rotation from the springs when it is close to its neutral position. Effective performance of the NES requires essentially-
nonlinear stiffness (i.e., nonlinearizable stiffness). As the angle of rotation increases, however, the reaction force due the the
wires, or nonlinear springs, scales with the displacement cubed due the geometric nonlinearity of the displaced wires.
Additional steps were taken to ensure that close clearances were achieved and impacts between various components,
especially the springs and posts, were limited so as to avoid their effects on the dynamics of the system.
15.3 Identification of System Parameters
To confirm that the NES design succeeded in achieving an essential stiffness nonlinearity, several tests were conducted to
estimate the properties of the system. These tests also provided data critical for the modeling of the interaction between the
NES and scale-model wing. The goal of the identification of the NES was to estimate the stiffness and damping of the
system. Each parameter was tested using two independent approaches, one static and one dynamic, which allowed for some
additional understanding of the properties of the device beyond the estimation of the parameters.
15.3.1 Static Test
The static stiffness approach was the most direct method employed for estimating the NES stiffness. It consisted of applying
a known torque and measuring the angular displacement. To accomplish this, the fully assembled NES was bolted to the
testing jig and then clamped in its upright position using a milling vice. Next, the NES mass was locked in its “zero” or
resting position using the locking bolt. In this context, the “zero” position refers to the NES mass position in which the wires
provide no restoring torque. For this test, it was assumed that the assembly of the NES and installation of the wires succeeded
in aligning the zero position with the position of the locked NES mass. Finally, an aluminum block was clamped onto the
plate so that it was offset from the NES mass and aligned parallel to the unloaded wires. This served as a reference position
from which measurements could be taken during testing. The setup is shown in Fig. 15.2.With this configuration, a known
torque could be applied by attaching one end of a string to a post on the NES mass and suspending a mass at the other end
over a pulley. With a torque applied, the displacement of some designated position on the NES mass could be measured with
respect to the reference block. This process was repeated by incrementing the mass and measuring displacement until some
designated maximum torque or displacement was achieved. Then the process was reversed and the displacement
measurements repeated as the torque was unloaded. Upon completion of the loading and unloading of the NES, the process
was repeated so as to induce displacements in the direction opposite that of the previously applied torque.
The data collected from this test can be used to evaluate the stiffness of the NES by assuming some form of the stiffness
function and then estimating the parameters of that function using a least-squares fit. This data was particularly useful for
evaluating the symmetry or asymmetry of the stiffness because of the separate data obtained for positive and negative rotations.
15 Characterization of a Strongly Nonlinear Vibration Absorber for Aerospace Applications 201
15.3.2 Dynamic Test
The restoring force surface method (RFSM), originally developed by Masri and Caughey [5], was used to simultaneously
estimate stiffness and damping. This method consists of measuring the dynamic input and response of a single degree of
freedom system and using that data to estimate the unknown system parameters. A basic explanation follows by considering
(15.1) where m is the mass, €x is the acceleration, pt is the external force in time, all known.
m€xþ f _x; xð Þ ¼ p tð Þ (15.1)
If f _x; xð Þ is some unknown function of the displacement and velocity, x and _x, then the equation of motion can be
rearranged to give
f _x; xð Þ ¼ p tð Þ � m€x|fflfflfflfflfflffl{zfflfflfflfflfflffl}restoringforce
:(15.2)
Fig. 15.2 Experimental setup
for static stiffness test
Fig. 15.3 Experimental setup
for RFSM test
202 S.A. Hubbard et al.
If the form of the function f _x; xð Þ is known or assumed, then the parameters of the function may be estimated using a least
squares fit to the restoring force. Typically, amean-squared error of less than 5% is considered an accurate fit. The background
and details of the method are discussed by Kerschen and Golinval [2], Kerschen et al. [3, 4].
Figure 15.3 shows a photo of the experimental setup for the RFSM test. Similar to the static stiffness test already
described, the NES was bolted to the testing jig which was then clamped to the working surface in an upright position using a
milling vise. A PCB triaxial accelerometer was attached to the upper surface of the NES mass such that one axis of
measurement would intersect the center of rotation. This alignment positioned one of the other measurements axes in such a
way that it would detect only the tangential acceleration of the NES mass. With the accelerometer in place, the distance
between the center of rotation of the NES mass and the center of the transducer was measured. It was assumed that the
accelerometer detects the acceleration at its center, so this measurement allowed the angular acceleration to be determined
from the measured tangential acceleration.
Due to the difficulty of accurately measuring excitation of the rotary system, we consider only the transient response
in the parameter identification. Thus, the restoring force in (15.2) consists only of the inertia term. In practice, the NES
was excited using a PCB impact hammer. The hammer and triaxial accelerometer were powered by a VibPilot data
acquisition system from m+p international. The acquisition system was configured to trigger upon the detection of a
tangential acceleration. Data was sampled at 1,024 Hz for a period of 4 s beginning approximately 0.8 s before the
trigger. The impact hammer force pulse was recorded so that the forced and transient portions of the response could
be separated.
15.4 Results
Before estimating the NES parameters, we assume that the system can be modeled as
f ðx; _xÞ ¼ c _xþ klinxþ knlsgn ðxÞjxja; (15.3)
i.e., viscous dissipation with linear and exponential stiffness components. Weseek to find the values of the dissipation and
stiffness coefficients and the unknownexponent. Using RFSM, all of these parameters can be identified simultaneously.Only
the stiffness terms can be determined using the static test. To demonstratethe consistency of the system, the device was
assembled, tested and identified, andthen disassembled three times. The cycles will be referred to as cases one, two,and
three, respectively, and all results are for systems using 0.025 in. diameter steelwires.
15.4.1 Static Test Results
Data collected from static stiffness tests are summarized in Fig. 15.4a for each case, and Fig. 15.4b shows the result of fitting
f(x) ( _x has been omitted here) to the data.In each case the data indicate that the stiffness when the displacement is zero is
negligible; i.e., essentially nonlinear stiffness has been achieved. Table 15.1 summarizes the results of the parameter
identification with and without the linear-stiffness term.The difference in error between the two models is negligible,
indicating that the exponential term alone is adequate for modeling the system.
−0.1 −0.05 0 0.05 0.1−300
−200
−100
0
100
200
300
Displacement (rad)
Tor
que
(N m
m)
case 1case 2case 3
−0.1 −0.05 0 0.05 0.1−300
−200
−100
0
100
200
300
Displacement (rad)
Tor
que
(N m
m)
a b
Fig. 15.4 (a) Summary of data collected during static-stiffness tests; (b) f(x) fit to static-stiffness data in case three
15 Characterization of a Strongly Nonlinear Vibration Absorber for Aerospace Applications 203
15.4.2 Dynamic Test Results
Figure 15.5 shows one set of data collected using the procedure outlined by for the restoring force surface method. The solid
line represents the portion of the transient response that was considered when estimating the system parameters. Figure 15.6
shows the resulting transient velocity and displacement and Fig. 15.7 shows the measured and reconstructed restoring
torque. This particular data set was collected as part of the identification of NES assembly case two, but is qualitatively
similar to the other data sets collected.
For each of the three cases described in the static stiffness results section, four or five sets of data similar to those shown in
Fig. 15.5 were collected and processed. The system parameters were estimated from the combination of all of the data
simultaneously. This approach was selected over estimating parameters for each trial and then averaging the results for a
number of reasons. First, performing the identification on all trials simultaneously gave more weight to the trials which
collected more useful data. Otherwise, a trial which collected 1 s of useful data would be of the same importance as a trial
that collected 2 s of useful data. More significantly though, it has been shown that the coefficients of the stiffness function
are extremely sensitive to changes in the exponent. This does not allow for a simple linear averaging technique if the
exponent is different for each trial. Thus, considering all of the data sets simultaneously ensures that the best combination of
exponents and coefficients will be determined.
The results of the stiffness parameter estimation are listed in Table 15.2. Once again, the results support the observation
that the linear component of the NES stiffness is insignificant. There was no appreciable reduction in mean squared error
when the coefficient of the linear term, klin, was included in the estimation. Also, when klin was included, it was severalorders of magnitude smaller than the coefficient of the nonlinear term.
Results from RFSM tests for the stiffness in each case were relatively similar (see Fig. 15.8a), indicating that the system
is repeatable. As should be expected, the estimated stiffness was independent of the method used. Figure 15.8b demonstrates
this for case three.The RFSM test also provided an estimate for viscous damping coefficient, although dissipation in the
system is not exclusively viscous. Nonetheless, from case three, c was estimated to be 1.65Nmms/rad. This value is used in
subsequent simulations.
Table 15.1 Summary of the
identified stiffness parameters
using the static stiffness method
fkx ¼ knlsgn (x)jxja f k xð Þ ¼ klinxþ knlsgn xð Þ xj ja
Case knl Nm
rada
� �a MSE(%) klin Nmm
rad
� �knl Nm
rada
� �a MSE(%)
1 193 3.10 3.24 2 193 3.10 3.24
2 695 3.77 1.35 195 3081 4.51 1.23
3 486 3.74 2.11 101 962 4.01 2.07
0 0.5 1 1.5 2 2.5 3 3.5 4−5000
0
5000
10000
15000
Time (s)
Acc
eler
atio
n( ra
ds2
)−5
0
5
10
15
20
For
ce (
mV
)Time historyTransient response
a
b
Fig. 15.5 Typical RFSM data
set
204 S.A. Hubbard et al.
1 1.5 2 2.5 3−20
−10
0
10
20
Time (s)
Vel
ocity
( rad s
)1 1.5 2 2.5 3
−0.2
−0.1
0
0.1
0.2
Time (s)
Dis
plac
emen
t(r
ad)
a
b
Fig. 15.6 Example of
experimentally determined
time history: (a) Velocity;
(b) Displacement
1 1.5 2 2.5 3−1000
−500
0
500
1000
Time (s)
Res
tori
ng T
orqu
e (N
mm
)
MeasuredReconstructed
Fig. 15.7 Measured and
reconstructed restoring torque
Table 15.2 RFSM estimated
stiffness parameters fx ¼ knlsgn (x)jxja f xð Þ ¼ klinxþ knlsgn xð Þ xj ja
Case knl Nm
rada
� �a MSE(%) klin Nmm
rad
� �knl Nm
rada
� �a MSE(%)
1 77 2.75 2.32 153 237 3.28 1.94
2 58 2.67 3.93 130 184 3.21 3.52
3 260 3.43 4.17 75 354 3.61 4.06
−0.1 0 0.1
−400
−200
0
200
400
Displacement (rad)
Res
tori
ng T
orqu
e (N
mm
)
Case1Case2Case3
−0.1 0 0.1
−400
−200
0
200
400
Displacement (rad)
Res
tori
ng T
orqu
e (N
mm
)
RFSMSS
a bFig. 15.8 (a) RFSM
estimated stiffness curves;
(b) Static and RFSM
estimated stiffness curves
for case three
15 Characterization of a Strongly Nonlinear Vibration Absorber for Aerospace Applications 205
15.5 Experimental and Computational Model Validation
The NES was designed to be attached to the tip of a scale-model wing so that its effects could be studied, with the goal of
predicting and observing targeted energy transfer (TET), the one-way transfer of energy from a primary system to a
nonlinear attachment where it is dissipated. The model wing discussed herein is swept with a semispan of 1.35 m and
uniform-thickness aluminum 6061 alloy; it is pictured in Fig. 15.9.A computational model of the wing and NES was
developed using thin-plate finite-elements to which the NES equations of motion were coupled. Details of the computational
model are given by Hubbard [1]. With the NES attached to the wingtip and free to oscillate, the system was excited by a
hammer impact applied to the wing. The excitation and response was recorded at several locations on the wing using
accelerometers. The experimental excitation was then applied to the computational model to verify that was capable of
accurately predicting the nonlinear phenomena. The numerical model can only agree with the experimental results if the
NES model and identified parameters are accurate. We offer one example of many with good agreement between
experimental and numerical results.
The response of the wing with NES attachment was observed for a hammer impact at location “P” in Fig. 15.10a. This
location coincides approximately with the nodal line of the first torsional mode of the wing and the antinode of the second
Fig. 15.9 Photo of the
scale-model wing
0 1 2 3 4 5
0
0.5
1
1.5
Time (ms)
For
ce (
kN)
SimExp
a bFig. 15.10 (a) Locations
of the hammer impact “P” and
the positions of
accelerometers “LT” (leading
tip) and “TT” (trailing tip);
(b) Experimentally measured
force pulse and the
corresponding force pulse
used in simulations
206 S.A. Hubbard et al.
bending mode, thus we expect that the second-bending mode comprises significant component of the response The profile of
the force pulse is shown in Fig. 15.10b, along with the corrected signal that was used in the corresponding simulation.
Figure 15.11a compares the experimentally-measured velocity at “TT” to the simulated velocity, showing very good
agreement.The most notable discrepancy between experiment and simulation is the difference in the fundamental frequency
which is due to error in the finite-element model of the wing. Figure 15.11b shows the frequency content of the simulated
velocity at “TT” as a function of time. It indicates that the energy that was initially in the second bending mode was mostly
dissipated within approximately 5 s. Recall that the excitation was provided near the antinode of the second bending mode
and the wing is lightly damped (aluminum alloy) so that, without an NES, the second bending mode should appear as a more
significant component of the response. Thus, the numerical model, using the identified NES parameters accurately predicted
a strongly nonlinear response.
15.6 Conclusion
A single-degree-of-freedom, rotary nonlinear energy sink has been designed and constructed on the scale of a wind-tunnel
model wing. The essentially nonlinear stiffness and the damping of this device have been identified, showing it to behave as
intended. These properties have been found to be robust, with response curves changing little over successive re-assemblies
of the components. The methods proven here will be used in the development of NESs for aeroelastic stability enhancement.
References
1. Hubbard SA (2009) Targeted energy transfer between a model flexible wing and a nonlinear energy sink: computational and experimental
results. Master’s thesis, University of Illinois at Urbana-Champaign
2. Kerschen G, Golinval JC (2001) Theoretical and experimental identification of a non-linear beam. J Sound Vib 244(4):597–613
3. Kerschen G, Lenaerts V, Golinval JC (2003) VTT benchmark: application of the restoring force surface method. Mech Syst Signal Pr
17(1):189–193
4. Kerschen G, Lenaerts V, Marchesiello S, Fasana A (2001) A frequency versus a time domain identification technique for nonlinear parameters
applied to wire rope isolators. J Dyn Syst Meas Control 123:645–650
5. Masri SF, Caughey TK (1979) Nonparametric identification technique for non-linear dynamic problems. J Appl Mech-T ASME 46(2):433–447
0 1 2 3 4−1
−0.5
0
0.5
1
Time (s)
Vel
.(m
/s)
SimExp
Time (s)
Freq
uenc
y(H
z)
1B
2B
1T
3B
2T4B
0 1 2 3 40
20
40
60
80a bFig. 15.11 (a) Comparison
of experimental and simulated
response at “TT”;
(b) Frequency content
as a function of time in the
simulated response
15 Characterization of a Strongly Nonlinear Vibration Absorber for Aerospace Applications 207
Chapter 16
Identifying and Computing Nonlinear Normal Modes
A. Cammarano, A. Carrella, L. Renson, and G. Kerschen
Abstract Non linear normal modes offer a rigorous framework, both mathematical and physical, for theoretical and
experimental dynamical analysis. Albeit still in its infancy, the concept of non linear normal modes has the potential of
providing to both the academic and the industrial establishment a powerful tool for the analysis of non linear dynamical
systems. However, in order to exploit the full potential of this theory (and its associated simulation capability), there is need
to integrate it with other branches of non linear structural dynamics: namely, in order for the non linear normal modes of a
real—physical—structure to be computed, there is need to identify and quantify its non linearity. In this paper, an
identification method based on the measured Frequency Response Function (FRF) is employed to identify and quantify
the system’s non linearity before computing the system’s non linear normal modes.
16.1 Introduction
The last decades havewitnessed a continuous demand for structures to becomemore light and efficient without loosing in safety
and durability. This approach, which is well known in the aerospace engineering, is slowly influencing other fields of the
automotive engineer as well as new branches of the civil engineer. In practice, the design process which relies on the theoretical
and numerical modelling of a system and on the experimental observation for the identification and validation of these models,
lacks methods which enables to account for the non linearities which occur in operational regime. In the last century the great
advance in computational science and in numerical methods provided indispensable tools for solving the complex system of
equations needed tomodel this type of structures where large displacements might not meet the hypothesis of linearity. This step
is absolutely crucial for their analysis, but nevertheless not sufficient. Themain problem is the definition of themodel itself. How
is it possible to associate a mathematical model with a given structure so that we are able to describe its dynamics? This science,
known as identification, is the real question that we are not able to answer yet. The typology of non-linearity is generally an
unknown of the problem as well as the parameters which characterize the equations. The identification of this information from
experimental data is not easy and requires, in general some assumptions on the non-linearity [1]. Even in that case, it is not
entirely clear which experiments are more useful for a full parameters’ identification and what data are strictly necessary for the
definition of a suitable mathematical model. This work aims to answer some of the questions still open in the world of non-linear
structures.With this purpose inmind, the authors simulated numerically the behaviour of a non-linear systems with three degree
of freedom. The data generated, i.e., the numerical FRFs are analysed with the method presented in [2], (also referred to as
CONCERTO). The main advantage of this approach is that the structure is simulated numerically and both the type of non-
linearity and the equation parameters are known. In this contest a combination of numerical and simulation procedures is used to
enhance the identification and thus the prediction capability. After a short description of the numerical method used to generate
A. Cammarano (*)
Department of Aerospace Engineering, University of Bristol, Unversity Walk, BS8 1TR, Bristol, UK
e-mail: [email protected]
A. Carrella
Faculty of Engineering, Bristol Laboratory for Advanced Dynamic Engineering (BLADE), LMS International,
Queens Building, University Walk, Bristol BS8 1TR, UK
L. Renson • G. Kerschen
Department of Aerospace and Mechanical Engineering, University of Liege, 1, Chemin des Chevreuils, Liege, B-4000, BE
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_16, # The Society for Experimental Mechanics, Inc. 2012
209
the data and themethod behindCONCERTO, some results will be presented. In particular the testwill showhow this can be used
for the identification ofmulti degree of freedom (MDOF) systems , highlighting its advantages and its limitations. Finally, a few
considerations about the method will be presented and some possible directions for future works suggested.
16.2 Continuation of Forced, Periodic Response of Non Linear Systems
The forced response of discrete mechanical systems with n degrees of freedom (DOFs) is considered, assuming that
continuous systems (e.g., beams, shells or plates) have been spatially discretized using the finite element method. The
equations of motion are
M €xðtÞ þ C _xðtÞ þKxðtÞ þ fnl xðtÞ; _xðtÞf g ¼ fðtÞ (16.1)
whereM is the mass matrix; C is the damping matrix;K is the stiffness matrix; x, _x and €x are the displacement, velocity and
acceleration vectors, respectively; fnl is the non linear restoring force vector and f(t) is the external force vector.The numerical method proposed here for the computation of the forced periodic response of non linear systems relies on
the algorithm developed for the computation of non linear normal modes (NNMs), which are periodic responses of the
undamped, unforced system [3]. The NNM algorithm relies on two main techniques, namely a shooting technique and the
pseudo-arclength continuation method and is described below.
16.2.1 Shooting Method
The undamped, unforced equations of motion of system (16.1) can be recast into state space form
_z ¼ gðzÞ (16.2)
where z ¼ x� _x�½ �� is the 2n-dimensional state vector, and star denotes the transpose operation, and
gðzÞ ¼ _x�M�1 Kxþ fnlðx; _xÞ½ �
� �(16.3)
is the vector field. The solution of this dynamical system for initial conditions zð0Þ ¼ z0 ¼ x�0 _x�0� ��
is written as z(t) ¼ z(t, z0)in order to exhibit the dependence on the initial conditions, z(0, z0) ¼ z0. A solution zp(t, zp0) is a periodic solution of the
autonomous system (16.2) if zpðt; zp0Þ ¼ zpðtþ T; zp0Þ, where T is the minimal period.
The computation is carried out by finding the periodic solutions of the governing nonlinear equations of motion (16.2).
In this context, the shooting method is probably the most popular numerical technique. It solves numerically the two-point
boundary-value problem defined by the periodicity condition
Hðzp0; TÞ � zpðT; zp0Þ � zp0 ¼ 0 (16.4)
Hðz0; TÞ ¼ zðT; z0Þ � z0 is called the shooting function and represents the difference between the initial conditions and the
system response at time T. For forced motion, the period T of the response is known a priori.
The shooting method consists in finding, in an iterative way, the initial conditions zp0 and the period T that realize a
periodic motion. To this end, the method relies on direct numerical time integration and on the Newton-Raphson algorithm.
Starting from some assumed initial conditions zp0(0), the motion zp
(0)(t, zp0(0)) at the assumed period T(0) can be obtained by
numerical time integration methods (e.g., Runge-Kutta or Newmark schemes). In general, the initial guess (zp0(0), T(0)) does
not satisfy the periodicity condition (16.4). A Newton-Raphson iteration scheme is therefore to be used to correct an initial
guess and to converge to the actual solution. The corrections Dzp0(k) and DT(k) at iteration k are found by expanding the
nonlinear function
H zðkÞp0 þ DzðkÞp0 ; T
ðkÞ þ DTðkÞ� �
¼ 0 (16.5)
in Taylor series and neglecting higher-order terms (H.O.T.).
210 A. Cammarano et al.
The phase of the periodic solutions is not fixed. If z(t) is a solution of the autonomous system (16.2), then z(t + Dt) isgeometrically the same solution in state space for any Dt. Hence, an additional condition, termed the phase condition, has tobe specified in order to remove the arbitrariness of the initial conditions. This is discussed in detail in [3]. For forced motion,
the phase is fixed by the external forcing.
In summary, an isolated periodic solution is computed by solving the augmented two-point boundary-value problem
defined by
Fðzp0; TÞ �Hðzp0; TÞ ¼ 0
hðzp0Þ ¼ 0
((16.6)
where h(zp0) ¼ 0 is the phase condition.
16.2.2 Continuation of Periodic Solutions
Different methods for numerical continuation have been proposed in the literature. The so-called pseudo-arclength
continuation method is used herein.
Starting from a known solution (zp0, (j), T(j)), the next periodic solution ðzp0;ðjþ1Þ; Tðjþ1ÞÞ on the branch is computed using a
predictor step and a corrector step.
16.2.2.1 Predictor Step
At step j, a prediction ð~zp0;ðjþ1Þ; ~Tðjþ1ÞÞ of the next solution ðzp0;ðjþ1Þ; Tðjþ1ÞÞ is generated along the tangent vector to the
branch at the current point zp0, (j)
~zp0;ðjþ1Þ~Tðjþ1Þ
" #¼
zp0;ðjÞ
TðjÞ
" #þ sðjÞ
pz;ðjÞpT;ðjÞ
" #(16.7)
where s(j) is the predictor stepsize.The tangent vectorp(j) ¼ [pz,(j)∗ pT,(j)]
∗ to the branchdefinedby (16.6) is solutionof the system
@H
@zp0
����ðzp0;ðjÞ;TðjÞÞ
@H
@T
����ðzp0;ðjÞ;TðjÞÞ
@h
@zp0
�����ðzp0;ðjÞÞ
0
266664
377775
pz;ðjÞpT;ðjÞ
" #¼ 0
0
" #(16.8)
with the condition kpðjÞk ¼ 1. The star denotes the transpose operator. This normalization can be taken into account by fixing
one component of the tangent vector and solving the resulting overdetermined system using the Moore-Penrose matrix
inverse; the tangent vector is then normalized to 1.
16.2.2.2 Corrector Step
The prediction is corrected by a shooting procedure in order to solve (16.6) in which the variations of the initial conditions
and the period are forced to be orthogonal to the predictor step. At iteration k, the corrections
zðkþ1Þp0;ðjþ1Þ ¼ z
ðkÞp0;ðjþ1Þ þ DzðkÞp0;ðjþ1Þ
Tðkþ1Þðjþ1Þ ¼ T
ðkÞðjþ1Þ þ DTðkÞ
ðjþ1Þ (16.9)
16 Identifying and Computing Nonlinear Normal Modes 211
are computed by solving the overdetermined linear system using the Moore-Penrose matrix inverse
@H
@zp0
����ðzðkÞ
p0;ðjþ1Þ;TðkÞðjþ1ÞÞ
@H
@T
����ðzðkÞ
p0;ðjþ1Þ;TðkÞðjþ1ÞÞ
@h
@zp0
�����ðzðkÞ
p0;ðjþ1ÞÞ0
p�z;ðjÞ pT;ðjÞ
266666664
377777775
DzðkÞp0;ðjþ1Þ
DTðkÞðjþ1Þ
24
35 ¼
�HðzðkÞp0;ðjþ1Þ; TðkÞðjþ1ÞÞ
�hðzðkÞp0;ðjþ1ÞÞ0
26664
37775 (16.10)
where the prediction is used as initial guess, i.e, zð0Þp0;ðjþ1Þ ¼ ~zp0;ðjþ1Þ and T
ð0Þðjþ1Þ ¼ ~Tðjþ1Þ. The last equation in (16.10)
corresponds to the orthogonality condition for the corrector step.
This iterative process is carried out until convergence is achieved. The convergence test is based on the relative error of
the periodicity condition:
Hðzp0; TÞ
zp0 ¼ zpðT; zp0Þ � zp0
zp0
<e (16.11)
where e is the prescribed relative precision.
The system used for this work consists of three masses (namely M1, M2 and M3) connected between each other and to
the ground through springs and dashpots (as shown in Fig. 16.1). The dashpots and the spring characteristics are linear
excepting for the spring connecting the first mass to the ground, which responds to a movement of M1 with the elastic
force Fe1 given by
Fe1 ¼ Kl x1 þ Knl x31 (16.12)
where x1 is the displacement of the first mass in comparison with its equilibrium position, Kl is the linear stiffness of the
spring and Knl is the coefficient of the non linear term. The forcing term f(t) is a sinusoidal force with amplitude F and is
applied toM1. The system, when the non linear term of Fe1 is set to zero is linear and has three natural frequencies at 0.096,
0.226, and 0.257 Hz. The theory of non linear systems suggests that the position of the peaks in the frequency response is not
influenced by the non-linear term and that for small forcing amplitude the frequency response tends to the linear response. In
the case of non linear behaviour the amplitude of the frequency response function (FRF) has been evaluated considering the
maximum amplitude of time response over a period. It’s phase is computed using the Fourier Transform: the phase of the
applied force is subtracting from the phase of the response. The identification tool uses this information to evaluate the
characteristics of the system.
16.3 Identification Tool: Theory
The method used for the identification procedure is referred tp as CONCERTO and is described in [2], thus only a brief
summary is given here. CONCERTO is based on the assumption that each peak dominates the response in correspondence of
the resonant frequency and therefore the system can be thought as a single-degree-of-freedom (SDOF) system, with
amplitude-dependent damping and/or stiffness—which are the most common classes of non linearity in engineering
F
M1 M2 M3
C1 C2 C3 C4
K1 K2 K3 K4
Fig. 16.1 Schematic of the simulated system: the the arrow on the spring named K1 indicates its non linear stiffness
212 A. Cammarano et al.
structures. The equation of motion for a SDOF is the same as (16.1) with the only difference that the coefficient of the
equation are now scalar terms
M €xðtÞ þ C _xðtÞ þ K xðtÞ þ f nl xðtÞ; _xðtÞf g ¼ f ðtÞ (16.13)
Assuming that the system responds at the same frequency as the excitation, the FRF is measured. The forcing term is a
sinusoidal excitation with constant amplitude and variable frequency. For the point of the FRF at any given response
amplitude, x, the functions fnl(x) in (16.13) are in effect constants. This implies that it is possible to linearise the system at
that specific response amplitude so that the systems FRF is given by
HðX;oÞ ¼ 1oo2ðXÞ � o2 þ joo
2ðXÞ�ðXÞ; (16.14)
where oo are the natural frequency and the modal loss factor at that given amplitude. It is important to note that the
linearisation must refer to a given value of amplitude of displacement. The functions oo(X) and Z(X) can be extracted from
the measured real and imaginary part of (16.14) as follows:
ooðXÞ ¼ ðR2 � R1ÞðR2o22 � R1o1
2Þ þ ðI2 � I1ÞðI2o22 � I1o1
2ÞðR2 � R1Þ2 þ ðI2 � I1Þ2
; (16.15)
�ðXÞ ¼ � ðI2 � I1ÞðR2o22 � R1o1
2Þ þ ðR2 � R1ÞðI2o22 � I1o1
2Þoo ðR2 � R1Þ2 þ ðI2 � I1Þ2
h i������
������; (16.16)
where R1 and R2 are the real parts of the FRF at the amplitude X and I1 and I2 its imaginary parts; o1 and o2 are the
frequencies at which a certain amplitude of displacement occurs: by definition they these frequency values are always
before and after the resonance frequency omegao. If the peak, due to non linear effect, is bent over itself so that a given
value of displacement occurs more than two times, the program consider the extreme values of the interval [omega1,omegan] and the method loose in validity. This effect will be discussed in the following section and will be supported by
graphical examples.
16.4 Results and Discussion
In this section some results will be presented. CONCERTO has been used to identify the backbones of the FRF peaks. In the
procedure the FRF relative to five different levels of excitation have been considered: 0.01, 0.03, 0.04, 0.05, and 0.1 N. For
each level of excitation, and each mass, the displacement values around one peak per time have been considered. The
procedure has been repeated for each level of excitation and than for each peak. The results are presented in Figs. 16.2, 16.3,
and 16.4. For each mass the details of the peak and their backbones are presented. The backbones are evealuated both with
the NNM algorithm described in Sect. 16.2 (red dashed lines) and with FRF based method described in Sect. 16.3 (black
solid lines). In the first case the backbones have been evaluated considering the free oscillations of the undamped system. In
the second case the natural frequency has been computed as function of the response amplitude according to (16.15). From
each case, it can be noticed that the curves evaluated with the FRF method are generally very close to those computed with
the simulation procedure. The error tends to be particularly small around the peak of the considered FRF and tend to diverge
for smaller values of displacement. In previous works the FRF method has been applied only to weakly non linear systems
where no unstable branch exists. In this work all the cases present both stable and unstable branches. An extension to this
identification method to FRF with jumps (i.e., unstable branches) is presented in [4]. It is clear from this study (see for
example Fig. 16.2d), that if the unstable branch of the FRF is provided the method works perfectly. On the other hand, the
strong non linearity highlights a weakness of the method. In presence of strongly non linear effects, e.g., if the peak bends on
itself (see for example Figs. 16.3d and 16.4d) there can be more than two frequencies at which a given amplitude is reached.
In this case CONCERTO is not able to follow the peak because it looks for the minimum and maximum frequency at which
the given displacement occurs. The results is that the found backbone heavily diverge from the real one as shown in the same
figures. However, although of great academic interest, this occurrence is rare in practical engineering structures and
therefore can be object of future studies.
16 Identifying and Computing Nonlinear Normal Modes 213
16.5 Conclusion
The combination of a continuation procedure with an identification method based on the FRF have shown to be a valid
approach to study practical non linear structures. At this stage the data flow has been unidirectional from the simulation
tools to the identification tool. The final goal of this study is the creation of a more complex tool able to extrapolate the
parameter from experimental data and to iterate the passage between the identification program and the simulation tool to
converge toward a model which can predict the behaviour of the analysed system. There are several issues which have
still to be addressed but the good matching between the NNM method and the FRF based procedure is definitely an
interesting result for such an early stage. The next phase will be focussed also on understanding the interaction between
the the modes to see if it is possible to reduce the error in the identification procedure. One issue to tackle is the definition
of a suitable parameter domain for this study. Currently modal (natural frequency) and spacial (displacement) models are
(mis)used. In fact, although it can be considered generally acceptable, the backbones found by the identification tools
diverge if the displacement values are not close enough to the peak. Moreover there are some cases in which poorer
results have been obtained (see Fig. 16.4c). Nevertheless, unless strong non linear behaviours occur, the trend of the
backbone can be easily identified. The possibility to use weighted regression techniques will be strongly considered in
future works to create a complete backbone from a limited set of data. This information is crucial for the estimation of the
stiffness of the system.
0.1 0.15 0.2 0.25 0.30
0.5
1
1.5
2
2.5
Frequency
Displacement
0.09 0.1 0.11 0.120
0.2
0.4
0.6
0.8
1
Frequency
Displacement
0.21 0.22 0.23 0.24
0.2
0.4
0.6
0.8
1
Frequency
Displacement
0.26 0.28 0.3 0.32
0.5
1
1.5
2
Frequency
Displacement
Fig. 16.2 FRF of the displacement of the massM1 (gray solid lines) for different levels of eccitation and backbones line evaluated with the NNMcode (dashed lines) and with CONCERTO (black silid lines)
214 A. Cammarano et al.
0.1 0.15 0.2 0.25 0.3
0.2
0.4
0.6
0.8
1
1.2
1.4
Frequency
Displacement
0.08 0.1 0.12
0.2
0.4
0.6
0.8
1
1.2
1.4
Frequency
Displacement
0.225 0.23 0.2350
0.05
0.1
0.15
0.2
Frequency
Displacement
0.26 0.28 0.3 0.32
0.1
0.2
0.3
0.4
0.5
Frequency
Displacement
Fig. 16.3 FRF of the displacement of the massM2 (gray solid lines) for different levels of eccitation and backbones line evaluated with the NNMcode (dashed lines) and with CONCERTO (black silid lines)
16 Identifying and Computing Nonlinear Normal Modes 215
References
1. Kerschen G, Worden K, Vakakis AF, Golinval JC (2006) Past, present and future of nonlinear system identification in structural dynamics.
Mech Syst Signal Pr 20(3):505–592
2. Carrella A, Ewins DJ (2010) Identifying and quantifying structural nonlinearities in engineering applications from measured frequency response
functions. Mech Syst Signal Pr
3. Peeters M, Viguie R, Serandour G, Kerschen G, Golinval JC (2009) Nonlinear normal modes, part ii: toward a practical computation using
numerical continuation techniques. Mech Syst Signal Pr 23(1):195–216
4. Carrella A (2012) Nonlinear identification using a frequency response functions with the jump. In: IMAC XXX, vol 11, Jacksonville, Feb 2012
0.1 0.15 0.2 0.25 0.30
0.5
1
1.5
Frequency
Displacement
0.08 0.1 0.120
0.2
0.4
0.6
0.8
1
Frequency
Displacement
0.215 0.225 0.235
0.2
0.4
0.6
0.8
1
1.2
1.4
Frequency
Displacement
0.26 0.28 0.3 0.320
0.2
0.4
0.6
Frequency
Displacement
Fig. 16.4 FRF of the displacement of the massM3 (gray solid lines) for different levels of eccitation and backbones line evaluated with the NNMcode (dashed lines) and with CONCERTO (black silid lines)
216 A. Cammarano et al.
Chapter 17
Nonlinear Identification Using a Frequency Response
Function With the Jump
A. Carrella
Abstract Recently, an identification method (referred to as CONCERTO) based on the measured Frequency Response
Function (FRF) has been proposed. Amongst its advantages there are its simplicity and its general applicability using
standard measurement techniques. This makes it particular suitable for use in practical applications and by the wider
engineering community. The method, however, also present some limitations: it applies to weakly-linear structures
(implements the Harmonic Balance method to a first order expansion and assumes that the system is linear at given
amplitude of the response), it is a single-degree-of-freedom (SDOF) method and it fails in the presence of an FRF with the
‘jump’. The aim of this paper is to present the latest findings that show that the nonlinear characteristic can be correctly
identified if the FRF of a system with cubic nonlinearity contains both stable and unstable branches: in order to do so, a
hybrid experimental-analytical approach is proposed.
17.1 Introduction
The majority of dynamical systems in practice behave in a nonlinear fashion. In order to create numerical (e.g. finite
element) models which are faithful of the dynamics of a structure, there is need to measure these nonlinearities. This branch
of engineering, called ‘Identification’, has been very active in producing more and more refined techniques for identification
of dynamic nonlinearities from measurements [1]. Recently, Carrella et al. have proposed a nonlinear identification method
based on the analysis of the measured Frequency Response Function (FRF) [2]. Amongst the advantages of this method there
is its mathematical simplicity, the applicability to standard measurement techniques (it does not require special sensors or
test-set up) and its capability of identifying both stiffness and damping nonlinearities without any a priori knowledge of thestructure. However, this techniques is still in its infancy and presents several shortfalls: is based on a linearisation and
therefore only provides a first degree approximation of the nonlinearity, it is a single-degree-of-freedom (SDOF) method and
hence it assumes that the mode being analysed dominates the response and it fails in the presence of a ‘jump’ in the response.
The aim of this letter is to present some later findings regarding this latter issue. It will be shown that the identification
method proposed in reference [2], also referred to as CONCERTO, is applicable also to the measured FRF of a system with
cubic stiffness excited with an amplitude such as to produce the characteristic ‘jump’. The jump is a well-known
phenomenon which is due to a bifurcation of the system, [3, 4]. At a particular frequency, there are three possible states
that the system can attain: two are stable and one is not. The ‘jump’ marks the passage from one stable solution to the other.
Therefore, in practice the unstable branch cannot be measured. Nonetheless, it is possible to use analytical solutions to
compute the frequencies at which the jump occurs: these expressions for the jump-down and jump-up frequencies have been
proposed in reference [5]. In particular, the jump-down frequency is a function of the damping ratio and the nonlinear
coefficient. It important to point out that, although the hybrid approach proposed in this work is of general validity, it is
limited by the need of having simplified analytical expressions given in [5] and which are based on the two assumptions:
(1) the damping of the system is that of the underlying linear system. In practice, also the energy dissipation mechanism can
be nonlinear. In this case the formulation becomes rather complex and closed form expression become no longer
A. Carrella (*)
Faculty of Engineering, Bristol Laboratory for Advanced Dynamic Engineering (BLADE), LMS International,
Queens Building, University Walk, Bristol BS8 1TR, UK
e-mail: [email protected]
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_17, # The Society for Experimental Mechanics, Inc. 2012
217
advantageous if at all available, [6] and numerical methods are more suitable; (2) the measured frequency at which the
system passes from one stable state to the other is considered to be the jump frequency. In reality this is very sensitive to a
number of parameters, but mainly the step-change of the excitation frequency (during stepped-sine measurements) and its
accurate measurement is a rather challenging task [7]. An experimental validation of the proposed method goes beyond the
scope of this paper but it will be object of future studies.
In this article it is shown that CONCERTO can identify and quantify the system’s stiffness if the FRF analysed contains
both stable and unstable branches. In order to identify the system’s amplitude-dependent natural frequency (or backbone),
a hybrid experimental-analytical approach is proposed:
1. With a low-level excitation, i.e. without exciting the nonlinear behaviour, calculate the system’s damping;
2. Increase the level of excitation until the jump occurs;
3. Using the analytical expression for the jump-down, the nonlinear coefficient can be computed;
4. Regenerate, numerically, the FRF of the system for that level of excitation: this will contain both the stable and unstable
branches;
5. Apply CONCERTO to the numerical curve
17.2 Nonlinear Identification from Measured FRF
A detailed description of the method is presented in reference [2], therefore only its key aspects will be given in here. Briefly,
the identification procedure enables one to extract the amplitude dependent modal parameters (natural frequency and
damping ratio) provided that both FRF and excitation force are recorded (so to calculate the displacement). The algorithm
is based on the assumption that the mode analysed dominates the response (i.e. is a SDOF method). If the system has only
one degree of freedom, then the spatial characteristics (stiffness and damping) can also be computed.
The dynamic equation of a mass m suspended on a parallel combination of an amplitude-dependent damping, c(X), andstiffness, k(X), excited by a harmonic force of amplitude F0 at frequency fe, can be expressed by the following relation:
m€xþ cðXÞ _xþ kðXÞx ¼ F0 sinð2p fe tÞ (17.1)
Assuming that the system’s response is at the same frequency of the excitation force, for a given amplitude of the
response X, the frequency response function (for the mode being considered) can be expressed as:
HðoÞ ¼ XðoÞFðoÞ ¼
Ar þ jBr
o2r ðXÞ � o2 þ jo2
r ðXÞ�rðXÞ(17.2)
In particular, at the same level of response amplitude Xi, there correspond a pair of points on the FRF. Thus
H1ðo1Þ ¼ Xi
Fðo1Þ ¼A1r þ jB1r
o2r ðXiÞ � o2
1 þ jo2r ðXiÞ�rðXiÞ ¼ Re1 þ jIm1
H2ðo2Þ ¼ Xi
Fðo2Þ ¼A2r þ jB2r
o2r ðXiÞ � o2
2 þ jo2r ðXiÞ�rðXiÞ ¼ Re2 þ jIm2
8>><>>: (17.3)
By solving this system (with the unknowns Ar;Br;or; �r) for each value of displacement Xi, the natural frequency and
damping ratio as function of the response amplitude can be found as:
o2r ðXÞ ¼
ðR2 � R1ÞðR2o22 � R1o2
1Þ þ ðI2 � I1ÞðI2o22 � I1 o2
1ÞðR2 � R1Þ2 þ ðI2 � I1Þ2
(17.4)
�rðXÞ ¼ðI2 � I1ÞðR2o2
2 � R1o21Þ þ ðR2 � R1ÞðI2o2
2 � I1o21Þ
o2r ½ðR2 � R1Þ2 þ ðI2 � I1Þ2�
���������� (17.5)
218 A. Carrella
One of the limitations of this approach is that it is necessary to measure the two points which have the same amplitude of
the response (one before, one after the peak). In the occurrence of a jump (e.g. the jump down which may occur for high level
of excitation in a Duffing oscillator as the frequency is increased), there are no measurable FRF points which corresponds to
those on the stable branch. In the next section, it will be shown that these can be numerically generated making the
identification technique applicable.
17.3 The Jump in the Duffing Oscillator
A classical example of a system that presents a bifurcation that yield the characteristic jump is the Duffing oscillator. This is
ubiquitous in the literature [4]. Consider the system with hardening cubic stiffness expressed by the equation of motion:
m€xþ c _xþ kxþ knlx3 ¼ F0 sin ð2pftÞ (17.6)
There are different techniques to solve (17.6). Numerical solutions can be sought by directly integrating the equation of
motion (e.g. using a Runge–Kutta algorithm); there are also approximate analytical expressions which provide a solution to
(17.6). Amongst these, the Harmonic Balance (HB) method offers a good first order approximation. In reference [5]
approximate analytical expressions for the jump-up and jump-down frequencies have been obtained following the HB
approach. It is noticeable that by applying the HB method it is possible to extract an equivalent stiffness function [3]:
keq ¼ k þ 3
4knlx
2 (17.7)
For the purpose of this paper the Duffing oscillator considered has the following value:
The FRF shown in Fig. 17.1. has been obtained by solving the equation of motion (6) with the built-in Matlab ODE45
solver, and then by computing the ratio of the Fourier coefficients of the response and the excitation. As it can be seen, for
very low levels of excitation, the jump does not occur and the system can be considered as linear (F0 ¼ 0.1 N); by increasing
the force level the nonlinearity is excited but the jump not occurs (F0 ¼ 0.6 N). For these low levels of excitation the jump
does not occur and the identification method, also referred to as CONCERTO, yields reliable results, as plotted with the
dotted and solid lines in Fig. 17.2. Furthermore, the validity of the identified nonlinearity is assessed by superimposing the
curve extracted with CONCERTO with that obtained using the analytical expression in (17.7) (dot-diamond line in
Fig. 17.2).
On the contrary, for a higher level of excitation (F0 ¼ 1 N), as shown by the dashed-circle line in Fig. 17.1, the FRF
presents the characteristic jump. When such a FRF is processed with using the identification algorithm described earlier, it is
obtained the dashed curve in Fig. 17.2 which is clearly not representative of the system being analysed. This is due to the lack
9.4 9.6 9.8 10 10.2 10.4 10.6 10.8 110
1
2
3
4
5
6
7
8
9x 10-3
Frequency [Hz]
|H|
F0=0.1 N
F0=0.6 N
F0=1 N
Fig. 17.1 Simulated
frequency response functions
(FRFs) of a dynamic system
with hardening cubic stiffness
(Duffing oscillator). At high
forcing level the jump occurs
(-o line)
17 Nonlinear Identification Using a Frequency Response Function With the Jump 219
of two physical points in the FRF with equal displacement response amplitude. As a result, the algorithm interpolates the
measured curve creating numerical points which have no physical meaning, hence the error of the identified curve.
In order to find a solution to this problem, and considering that the identification technique implements the HB to a first
order expansion, it’s reasonable to assume that the unstable branch would provide the necessary information to apply
CONCERTO algorithm. Because the unstable branch is not measurable, it is proposed a hybrid experimental-analytical
approach. The analytical basis can be found in reference [5] which provides some simple expressions for the jump-up and
jump-down frequencies. The equation of motion of the system, (17.6), can be re-written in a non-dimensional form as [5]:
€xþ 2z _xþ xþ ax3 ¼ cosðOtÞ (17.8)
where x is the non-dimensionalised displacement, z << 1 is the damping ratio and F0
^and Ω are respectively the non-
dimensional magnitude and frequency ratio of the excitation force, the · operator denotes differentiation with respect to non-
dimensional time t and
z ¼ c
2mono2
n ¼k
ma ¼ knlx
20
kO ¼ o
ont ¼ ont €xþ
_x
o2nx0
_x ¼ _x
o2nx0
x ¼ x
x0x0 ¼ F0
k
The application of the Harmonic Balance leads to the cubic equation in X2 (or a quadratic equation in O2) (17.9) which
represents the relation between the response, the amplitude and frequency of excitation:
9
16a2X6 þ 3
2ð1� O2ÞaX4 þ ½ð1� O2Þ2 þ 4z2O2�X2 ¼ 1 (17.9)
and the phase of the frequency response function:
’ ¼ tan�1 � 2zOX
ð1� O2ÞX þ 34aX
3
!(17.10)
Equations (17.9) and (17.10) are plotted in Fig. 17.3.
17.4 Nonlinear Identification Using Hybrid (Experimental-Analytical) Data
In order to build the approximate analytical FRF (using the HB formulation of (17.9)) there is need of the parameters a and z.For what concerns the latter, the damping is taken to be that of the underlying linear system and as such can be retrieved from a
low-level linear identification test. It is noteworthy that from the low-level linear test, also the linear stiffness can be computed.
0 2 4 6 8
x 10-3
10.05
10.1
10.15
10.2
10.25
10.3
10.35
10.4
10.45
10.5
Displacement [m]
Nat
ural
Fre
quen
cy [H
z]
F0=0.1 N
F0=0.6 N
F0=1 N
theoretical trend
Fig. 17.2 Natural frequency
as function of the
displacement amplitude
identified using CONCERTO
algorithm: when the FRF
contains the jump the
extracted natural frequency
does not compare well with
the analytical curve (-* line)
obtained using (17.7)
220 A. Carrella
On the other hand, the nonlinear coefficient of the stiffnessa can be extracted using a hybrid experimental-analytical approach.
From the experimental FRF with the jump the jump-down frequency can be measured. Using the simplified analytical
expression for the jump-down frequency proposed in reference [5],
Od ffi 1ffiffiffi2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 3a
4z2
svuut (17.11)
it is possible to calculate a as:
a ¼ 16
3O4
d � O2d
� �z2 (17.12)
or
knl ¼163
O4d � O2
d
� �z2k
F0
k
� �2 (17.13)
Having retrieved the nonlinear coefficient of the stiffness, it is possible to generate the analytical FRF of the system using
the approximate solution (9). The analytical FRF calculated using (17.9) with the nonlinear coefficient a identified from the
9 9.5 10 10.5 11 11.5 120
1
2
3
4
5
6
7
8
9x 10-3
Frequency [Hz]
|H|
F0=1 N
F0=2 N
F0=3 N
9 9.5 10 10.5 11 11.5 12-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
Frequency [Hz]
φ
F0=1 N
F0=2 N
F0=3 N
a
b
Fig. 17.3 Approximate
analytical FRF, magnitude
(a) and phase (b), of a Duffing
oscillator
17 Nonlinear Identification Using a Frequency Response Function With the Jump 221
hybrid procedure described earlier is shown in Fig. 17.4. The figure also shows that now both stable and unstable branches
are available and approximate fairly accurately the results of the numerical FRF.
For the specific numerical example presented in this paper, given in Table 17.1, the ‘experimental’ (or measured) FRF
have provided the damping, z, and the natural frequency (i.e. the stiffness k) of the linear system (from the low-level FRF,
F0 ¼ 0.1 N) and the jump-down frequency, Od (from the high level FRF, F0 ¼ 1 N). With these values using (17.13), the
nonlinear coefficient has been calculated to be knl1 ¼ 6:85� 106 N/m (note that this value differs from the exact numerical
value given in Table 17.1 by 2%). With these values is now possible to compute the analytical FRF. Finally, CONCERTO
can be applied to the regenerated analytical curve. The result is shown in Fig. 17.5: it can be noticed that now, unlike the
previous results shown in Fig. 17.2, the method is able to identify the correct natural frequency as function of the
displacement as shown by the good match with the analytical curve.
9.4 9.6 9.8 10 10.2 10.4 10.6 10.8 110
1
2
3
4
5
6
7
8
9x 10-3
Frequency [Hz]
|X|
FRFHB
FRFnumerical
Fig. 17.4 Regenerated FRF:
the solid line is the FRF
reconstructed using the
nonlinear coefficient
calculated using the hybrid
approach and (17.12)
Table 17.1 Value of the system’s parameters used for the numerical simulations
Mass Stiffness Nonlinear stiffness Damping ratio Damping coefficient Force amplitude
m ¼ 1:5 kg k ¼ 6000 Nm knl ¼ 7 � 106 N
m3 z ¼ 0:01 c ¼ 2zffiffiffiffiffiffikm
pF0 ¼ 0:1N
F0 ¼ 0:6N
F0 ¼ 1N
0 2 4 6 8
x 10-3
10
10.05
10.1
10.15
10.2
10.25
10.3
10.35
10.4
Displacement [m]
Fre
quen
cy [H
z]
F0=1 N
Theoretical function
Fig. 17.5 Identified nonlinear
characteristic using
CONCERTO (dotted line):
unlike the plot in Fig. 17.2
this compare very well with
the analytical expression
(solid line)
222 A. Carrella
17.5 Conclusion
A recent nonlinear identification method was shown to be valid for nonlinear systems which do not present the jump
phenomenon. In fact, for the technique (also referred to as CONCERTO) to be applied there is need of measuring two points
at the same displacement amplitude at either side of the resonance. However, if in a system with cubic nonlinearity the jump
occurs there are no measurable points on one part of the FRF.
In this paper it has been shown that it is possible to identify the nonlinear characteristic correctly if CONCERTO is
applied to a FRF which contains also the unstable branch. However, because this cannot be measured it has been proposed to
use numerically generated data. In order to do so, simplified expression for the jump-down frequency has been used and a
hybrid experimental-analytical approach has been proposed to retrieve the nonlinear coefficient and produce a nonlinear
numerical FRF (which contains also the unstable branch). Future works will need to focus on validating this approach using
experimental data.
References
1. Kerschen G et al (2006) Past, present and future of nonlinear system identification in structural dynamics. Mech Syst Signal Process
20(3):505–592
2. Carrella A, Ewins DJ (2011) Identifying and quantifying structural nonlinearities in engineering applications from measured frequency response
functions. Mech Syst Signal Process 25(3):1011–1027
3. Worden K, Tomlison GR (2001) Nonlinearity in structural dynamics. Institute of Physics, UK
4. Jordan DW, Smith P (1999) Nonlinear ordinary differential equations, IIIth edn. Oxford Presss, New York
5. Brennan MJ et al (2008) On the jump-up and jump-down frequencies of the Duffing oscillator. J Sound Vib 23(4–5):1250–1261
6. Peeters M (2011) Theoretical and experimental modal analysis of nonlinear vibrating structures using nonlinear normal modes. Ph. D thesis,
University of Liege
7. Ravindra B, Mallik AK (1994) Stability analysis of a non-linearly damped Duffing oscillator. J Sound Vib 171(5):708–716
17 Nonlinear Identification Using a Frequency Response Function With the Jump 223
Chapter 18
Nonlinear Structural Modification and Nonlinear Coupling
Taner KalaycIoglu and H. Nevzat Ozg€uven
Abstract Structural modification methods were proved to be very useful for large structures, especially when modification
is local. Although there may be inherent nonlinearities in a structural system in various forms such as clearances, friction and
cubic stiffness, almost all of the structural modification methods are for linear systems. The method proposed in this work is
a structural modification/coupling method developed previously, and extended to systems with nonlinear modification and
coupling recently. It is based on expressing nonlinear internal force vector in a nonlinear system as a response level
dependent “equivalent stiffness matrix” (the so-called “nonlinearity matrix”) multiplied by the displacement vector, by quasi
linearizing the nonlinearities using Describing Function Method. Once nonlinear internal force vector is expressed as a
matrix multiplication then several structural modification and/or coupling methods can easily be used for nonlinear systems,
provided that an iterative solution procedure is employed and convergence is obtained. In this paper, formulations for each of
the following cases are given: nonlinear modification of a linear structure with and without adding new degrees of freedom,
and elastic coupling of a nonlinear substructure to a main linear structure with linear or nonlinear elements. Case studies for
three of those cases are given and an application of the method to a real life engineering problem is demonstrated.
Keywords Structural modification • Nonlinear structural modification • Vibration of nonlinear structures • Nonlinear
structural coupling
18.1 Introduction
Over the last five decades or so, the finite element (FE) method has established itself as the major tool for the dynamic
response analysis of engineering structures. However, for the dynamic reanalysis of large engineering structures modified
locally, constituting FE model each time is expensive and time consuming, especially when several alternatives are to be
studied. Therefore, it will be more practical to predict the dynamic behavior of a modified structure by using dynamic
response information of the original structure and dynamic properties of the modifying structure. Various structural
modification methods, focusing on the change of dynamic behavior of a structure due to modifications, have been developed
in order to reduce the effort involved in the dynamic reanalysis of such systems. Although the structural modification
methods based on linearity assumption are available in the literature, a review of which can be found in recent papers of
Hang et al. [1–3], these methods cannot be used when there is nonlinearity in the system.
During the past two decades, several structural modification/coupling methods have been suggested taking the nonlinear
effect into account. Watanabe and Sato [4] used first order describing function in order to linearize the nonlinear stiffness
of a beam structure and developed the so-called “Nonlinear Building Block” approach for coupling nonlinear structures
with local nonlinearity. C€omert and Ozg€uven [5] developed a method for calculating the forced response of linear
T. KalaycIoglu
Department of Mechanical Engineering, Middle East Technical University, 06800 Ankara, Turkey
MGEO Division, ASELSAN Inc., 06011 Ankara, Turkey
e-mail: [email protected]
H.N. Ozg€uven (*)
Department of Mechanical Engineering, Middle East Technical University, 06800 Ankara, Turkey
e-mail: [email protected]
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_18, # The Society for Experimental Mechanics, Inc. 2012
225
substructures coupled with nonlinear connecting elements. Ferreira and Ewins [6] proposed a new Nonlinear Receptance
Coupling Approach for fundamental harmonic analysis based on describing functions. They suggested an approach that is
capable of coupling structures with local nonlinear elements whose describing functions are available considering just the
fundamental frequency. Then, Ferreira [7] extended the approach and introduced Multi-Harmonic Nonlinear Receptance
Coupling Approach. This approach is able to couple linear and nonlinear structures with different types of joints by
specifying the multi-harmonic describing functions for all nonlinear joints. Chong and Imreg€un [8] suggested an iterative
algorithm for coupling nonlinear systems with linear ones. Maliha et al. [9] coupled a nonlinear dynamic model of a spur
gear pair with linear FE models of shafts carrying them, and with discrete models of bearings and disks. Huang [10] worked
on dynamic analysis of assembled structures with nonlinearity.
In a recent work [11], the authors of this paper proposed a new approach for dynamic reanalysis of a large linear
structure modified locally with a nonlinear substructure. The method suggested is an extension of the method developed by
Ozg€uven [12] for structural modifications of linear systems. In the present study, the proposed approach is further extended
for dynamic reanalysis of linear structures coupled with nonlinear substructures by using linear and nonlinear coupling
elements. In this paper, applications of the method to modification analysis problems where a linear system is modified with
a nonlinear system for the following three cases are presented: structural modification with additional degrees of freedom
(DOFs), structural coupling with linear elements and structural coupling with nonlinear elements. Finally, a real life
structure modeled with FE method is used in order to show the applicability of the method to large ordered systems.
18.2 Theory
The structural modification method proposed by Ozg€uven [12] more than two decades ago can be used for modified linear
systems with or without additional DOFs. The method is for dynamic reanalysis of systems where there is a modification in
the mass, stiffness and/or damping of the system. The frequency response functions (FRFs) of a modified system are
calculated from those of the original system and the dynamic stiffness matrix representing the modifications in the system by
using the following equations [12]:
H�11
� � ¼ �½I� þ ½H11�½Z11���1½H11� (18.1)
H�12
� �T ¼ H�21
� � ¼ ½H21� ½I� � ½Z11� H�11
� �� �(18.2)
H�22
� � ¼ ½H22� � ½H21�½Z11� H�12
� �(18.3)
where [H] and [H*] are receptance matrices of the original and modified systems, respectively. [Z] represents the dynamic
stiffness matrix of structural modifications, and superscripts 1 and 2 denote the coordinates on which a modification is
applied and the remaining coordinates, respectively.
The method has been extended in the same work for modifications which require additional DOFs, in other words, for
modifications which also causes coupling of another system to the original one. The resultant equations for such cases are as
given below [12]:
H�ba
� �
H�ca
� �" #
¼ ½I� ½0�½0� ½0�
� �þ ½Hbb� ½0�
½0� ½I�� �
:½Z�� ��1 ½Hba�
½0�� �
(18.4)
H�bb
� �H�
bc
� �
H�cb
� �H�
cc
� �" #
¼ ½I� ½0�½0� ½0�
� �þ ½Hbb� ½0�
½0� ½I�� �
:½Z�� ��1 ½Hbb�
½0�½0�½I�
� �(18.5)
H�aa
� � ¼ ½Haa� ��½Hab� ½0�
�� �½Z� H�ba
� �
H�ca
� �" #
(18.6)
��H�
ab� H�ac
� ��� � ¼ �½Hab� ½0��� � ½I� � ½Z� H�
bb
� �H�
bc
� �
H�cb
� �H�
cc
� �" #" #
(18.7)
226 T. KalaycIoglu and H.N. Ozg€uven
where the subscript a represents the coordinates that belong to the original system only, the subscript b denotes
connection coordinates, and the subscript c represents coordinates that belong to modifying structure only. The
method is most useful when such modifications are on limited number of coordinates, that is, when modification is
local. Then the order of the matrix to be inverted will be reduced considerably, irrespective of the total size of the
original structure.
The method can be extended to modification of a linear system where modifying system has nonlinearity. Nonlinear
internal forces can be included in the analysis by considering an additional equivalent stiffness matrix in the calculations
which is a function of unknown response amplitudes [11]. Then, the dynamic stiffness matrix for the modifying system
showing nonlinear behavior will take the form
ZðXÞ½ � ¼ ½DK� � o2 ½DM� þ jo ½DC� þ j ½DD� þ DðXÞ½ � (18.8)
where [DK], [DM], [DC] and [DD] represent stiffness, mass, viscous and structural damping matrices of the modifying
structure, respectively, and {X} is the amplitude vector of harmonic response of the system. The formulation for [D(X)],named as “nonlinearity matrix”, was first introduced by Budak and Ozg€uven [13, 14] for certain types of nonlinearities, andlater extended by Tanrikulu et al. [15] for any type of nonlinearity by using Describing Function Method. The elements of
nonlinearity matrix are given [15] as
Dkk ¼Xn
m¼1
nkm (18.9)
Dkk ¼ �nkm; ðk 6¼ mÞ (18.10)
where subscripts k andm represent two engagement coordinates of a nonlinear element. Here, due to the nonlinearity matrix,
[Z(X)] will be response level dependent, and therefore solution can only be obtained by employing an iterative solution
procedure. The details of the above formulation can be found in [11].
18.2.1 Formulation for Nonlinear Structural Modification Without Additional DOFs
For nonlinear structural modifications without additional DOFs, (18.1)–(18.3) are used where dynamic stiffness matrix for
the modifying system is expressed as follows:
Z11 ðXÞ½ � ¼ ½DK11� � o2 ½DM11� þ jo ½DC11� þ j ½DD11� þ D11 ðXÞ½ � (18.11)
Here, the subscript 1 refers to the coordinates of the system where there is a modification. To be able to use (18.1)–(18.3),
renumbering of the coordinates of the original system may be necessary. It should be also noted that although there is
a matrix inversion in the formulation, the size of the matrix to be inverted is equal to the total DOFs of the modifying
system (size of the matrix [Z11(X)]). Therefore, the method is most useful when local modifications are applied to large
ordered systems.
18.2.2 Formulation for Nonlinear Structural Modification with Additional DOFs
When the modification is such that modifying nonlinear system does not only change the system properties of the original
system at some coordinates, but also couples another system to the original system, then the total DOF of the modified
system will be increased. Such a structural modification case is referred to as structural modification with additional
DOFs. In such applications, (18.4)–(18.7) are used for the receptance of modified system. [Z(X)] in these equations will
be as shown in (18.8).
18 Nonlinear Structural Modification and Nonlinear Coupling 227
18.2.3 Formulation for Nonlinear Structural Coupling with Linear Elements
The same formulation given in Sect. 18.2.2 can also be used for analyzing coupled systems by treating the problem as an
equivalent structural modification problem as shown in Fig. 18.1. That is, for each connection node on the original system a
massless node is added to the coupled subsystem.
Firstly, the stiffness matrix of the coupled subsystem, [DK], is expanded. For example, if p number of massless nodes are
added to the coupled subsystem where the DOF per node is q, p � q number of rows and columns are added to the stiffness
matrix of the coupled subsystem. Then, the stiffness values of the linear elastic coupling elements are inserted in proper
locations of added rows and columns of [DK]. The mass, nonlinearity, viscous and structural damping matrices of the
coupled subsystem should also be expanded in the same way. However, just zeros will be inserted in these rows and
columns. Lastly, by defining additional massless nodes as new rigid connection nodes of the coupled subsystem, the problem
can be taken as a nonlinear structural modification problem as defined in Sect. 18.2.2.
18.2.4 Formulation for Nonlinear Structural Coupling with Nonlinear Elements
Again, the same formulation given in Sect. 18.2.2 can be used for analyzing coupled systems by treating the problem as an
equivalent structural modification problem as shown in Fig. 18.2. That is, for each connection node on the original system a
massless node is added to the coupled subsystem as in the previous case. The only difference will be the nonlinear character
of the connection elements. Therefore, again; stiffness, mass, nonlinearity, viscous and structural damping matrices of the
coupled subsystem are expanded by adding p � q number of rows and columns, where p is the number of massless nodes
added to the coupled subsystem and q is the DOF per node.
Fig. 18.1 Structural coupling problem with linear elastic elements
Fig. 18.2 Structural coupling problem with nonlinear elements
228 T. KalaycIoglu and H.N. Ozg€uven
However this time, the added rows and columns of the nonlinearity matrix [D(X)] will be filled with proper elements
representing the nonlinear connection elements. If there are linear stiffness counterparts of the connecting elements, these
values will also be properly inserted into the expanded rows and columns of the stiffness matrix of the coupled subsystem.
Lastly, by defining additional massless nodes as new rigid connection nodes of the coupled subsystem, the problem can
be taken as a nonlinear structural modification problem as defined in Sect. 18.2.2.
18.3 Case Studies
In this section applications of the proposedmethod firstly to discrete systems and then to real engineering structuremodeledwith
FEmethod will be given. The first three case studies illustrate modification and coupling analyses of two discrete subsystems in
three main categories, namely, nonlinear structural modification with additional DOFs, nonlinear structural couplingwith linear
elements and nonlinear structural coupling with nonlinear elements. An application for structural modification without
additional DOF is not given in this paper, since it was well investigated in our recent paper [11]. In the last case study, a real
life engineering problem with nonlinear structural modification will be considered.
18.3.1 Nonlinear Structural Modification with Additional DOFs
In this case study, nonlinear modification of a linear discrete system is considered. As can be seen from Fig. 18.3, the
modifying nonlinear system adds a new DOF to the original linear system.
In the same figure, there exists a cubic stiffness element between coordinates 4 and 5 showing hardening behavior.
Parameters of this nonlinear element and the properties of both subsystems are given as follows:
m1 ¼ m2 ¼ m3 ¼ 1 kg and m4 ¼ m5 ¼ 0:5 kg;
k1 ¼ k2 ¼ k3 ¼ k4 ¼ k5 ¼ k6 ¼ 1000 N=m;
nðx; _xÞNL ¼ K0xþ bx3 where K0 ¼ 0 N=m and b ¼ 2x106 N=m3 (18.12)
Structural damping with a loss factor of 0.0015 is assumed in the analysis for all linear elastic elements. Frequency
response of the modified system at the point where a harmonic force of magnitude 4 N is applied is shown in Fig. 18.4.
The results show that the nonlinearity in the modifying system becomes effective in all four modes of the modified system
which reveals the importance of including nonlinearity in this specific case. Although the nonlinearity changes the frequency
responses around resonances considerably by causing a jump, which is a typical response behavior due to cubic stiffness
Fig. 18.3 Structural modification with additional DOFs
18 Nonlinear Structural Modification and Nonlinear Coupling 229
element, no convergence problem is observed in the solution when the proposed method is used. Note that, since the
proposed method is an FRF based method, only the FRFs related with the DOFs we are interested and with the connection
DOFs are to be included in the calculation. Moreover, the size of the matrix to be inverted is 2 by 2 in this example, which
is the order of the modifying system (and therefore it would still be 2, even though the size of the original system were
much higher). These are the important features of the method which makes it more advantageous for large ordered systems
with local modification.
18.3.2 Nonlinear Structural Coupling with Linear Elements
As the second case study, nonlinear structural coupling analysis of the same linear and nonlinear discrete system coupled
with a linear elastic element, as shown in Fig. 18.5, is considered.
Here, the stiffness of the linear elastic element is taken as 200 N/m. Assuming structural damping with a loss factor of
0.0015 again for all linear elastic elements, frequency response of the modified system at the point where a harmonic force
of magnitude 4 N is applied, is obtained as shown in Fig. 18.6.
It can be seen from the figure that nonlinearity is more effective on second, third and fifth modes of the system compared
to the other two modes. Note that, since the proposed method is based on FRFs, in addition to the FRFs related
with the connection DOFs, it is sufficient to include only the FRFs related with the required DOFs in the calculations.
1
0
−1
−2
−3
Log(
Dis
plac
emen
t[m])
−4
−50 2 4 6
Frequency [Hz]
RESPONSE vs FREQUENCY
8 10
Non-linear response – Forward sweep
Non-linear response – Reverse sweepLinear response
12 14
Fig. 18.4 Frequency response of m3 after modification
Fig. 18.5 Structural coupling of linear and nonlinear systems with a linear element
230 T. KalaycIoglu and H.N. Ozg€uven
Moreover, the size of the matrix to be inverted during calculations is again 2 � 2 in this example (which is the size of
the coupled subsystem). This saves considerable computational time especially in large ordered original systems as long as
the modification is of small order. This feature of the method makes it very desirable in parametric studies, for instance,
while investigating the effects of the linear elastic coupling element stiffness on system response. In Fig. 18.7, the effect
of different stiffness values of the linear elastic coupling element on the system response around third resonance
is examined in detail. It can be seen from the figure that increasing values of the linear elastic coupling element stiffness
will not only shift the third natural frequency to higher frequencies, but also will increase the effect of nonlinearity on
this mode.
1
0
−1
−2
−3
Log(
Dis
plac
emen
t[m])
−4
−5
−60 2 4 6
Frequency [Hz]
RESPONSE vs FREQUENCY
8 10
Non-linear response – Forward sweep
Non-linear response – Reverse sweepLinear response
12 14
Fig. 18.6 Frequency response of m3 after coupling
−1
−2
−3
−47.2 7.4 7.6 7.8
Frequency [Hz]
RESPONSE vs FREQUENCYa
c
b
RESPONSE vs FREQUENCY
Frequency [Hz]
RESPONSE vs FREQUENCY
Frequency [Hz]8.2 8.4 8.68
7.2 7.4 7.6 7.8 8.2 8.48
7.2 7.4 7.6 7.8 8.2 8.4 8.68
−1.5
−2.5
Log(
Dis
plac
emen
t[m]}
Log(
Dis
plac
emen
t[m]}
Log(
Dis
plac
emen
t[m]}
−3.5
−4
−1
−2
−3
−0.5
−1.5
−2.5
−3.5
−4
−1
−2
−3
−0.5
−1.5
−2.5
−3.5
Non-linear response – Forward sweep
Non-linear response – Reverse sweepLinear response
Non-linear response – Forward sweepNon-linear response – Reverse sweepLinear response
Non-linear response – Forward sweepNon-linear response – Reverse sweepLinear response
Fig. 18.7 Frequency responses around third resonance for different linear elastic coupling elements (a) kLC ¼ 200 N/m, (b) kLC ¼ 400 N/m,
(c) kLC ¼ 600 N/m
18 Nonlinear Structural Modification and Nonlinear Coupling 231
18.3.3 Nonlinear Structural Coupling with Nonlinear Elements
The third case study discusses nonlinear structural coupling analysis of the same linear and nonlinear discrete systems
with nonlinear elements. The linear coupling element used in the previous case study is kept the same, but an additional
nonlinear coupling element is considered as shown in Fig. 18.8.
As the nonlinear coupling element, a linear spring having a stiffness of magnitude 200 N/m and 0.02 m clearance on each
side is inserted between two coupling coordinates. Assuming structural damping with a loss factor of 0.0015 again for all
elastic elements, frequency response of the modified system at the point where the harmonic force of magnitude 4 N is
applied is obtained as shown in Fig. 18.9.
When Figs. 18.6 and 18.9 are compared with each other, it can be observed that nonlinear coupling element affects
first and third modes of the system more than it does the other modes. Again, the advantage of using only the FRFs related
with the required and connection DOFs, which is the FRF related with the third mass in this example, and inverting
a matrix only in the size equal to the DOF of the modifying system in this method, the method can be used in design
analyses where, for instance, the effects of using different nonlinear coupling elements on the system response are
investigated.
In Fig. 18.10, the effect of different stiffness values of nonlinear coupling element on the system response around third
resonance is examined in detail. It can be seen from Fig. 18.10 that typical response distortion due to clearance type of
nonlinearity is observed as an abrupt change in the frequency response at the point of transition where the response
Fig. 18.8 Structural coupling of linear and nonlinear systems with a linear and a nonlinear element
1
0
−1
−2
−3
Log(
Dis
plac
emen
t[m])
−4
−5
−60 2 4 6
Frequency [Hz]
RESPONSE vs FREQUENCY
8 10
Non-linear response – Forward sweep
Non-linear response – Reverse sweepLinear response
12 14
Fig. 18.9 Frequency response of m3 after coupling with a linear and a nonlinear element
232 T. KalaycIoglu and H.N. Ozg€uven
amplitude reaches to the value of clearance. As an expected result, the displacement value where this abrupt change
occurs differs depending on the value of the clearance (compare Fig. 18.10a and b). On the other hand, for nonlinear
spring elements having different stiffness values but the same clearance, this abrupt change occurs at the same
displacement value but the frequency responses after that point show different behaviors due to different additional
linear spring stiffnesses of the nonlinear coupling elements after the response amplitude reaches to the value of clearance
(compare Fig. 18.10a and c).
18.3.4 Nonlinear Structural Modification: A Real Life Engineering Problem
As the last case study, a real life engineering problem, a shaft and mirror plate assembly usually used in land platforms for
optical purposes, is considered (Fig. 18.11).
The mirror plate is a costly part due to its well machined reflective surface, so once it is designed it is not desired to
be modified further in the design optimization of the assembly depending on the vibration characteristics of the platform it
is mounted. Therefore, when it is used in a platform, it may be necessary to modify the shaft and/or bearings, in order to
minimize the vibration of the mirror plate so that its reflection performance is increased. In order to make a more precis
analysis, nonlinearity introduced by the bearings is included in the dynamic analysis, which can easily be handled by the
nonlinear structural modification analysis method suggested here.
Solid elements are used in the FE model of the mirror plate with three DOFs per node yielding 2,655 total DOFs
(Fig. 18.12). The shaft is also modeled by using solid elements with three DOFs per node resulting in 186 total DOFs. The
FE model of the shaft is also shown in the same figure.
Material properties of the mirror plate made of an aluminum alloy and of the shaft made of a structural steel are given in
Table 18.1.
−1
−2
−3
7 7.5Frequency [Hz]
RESPONSE vs FREQUENCYa
c
b
RESPONSE vs FREQUENCY
Frequency [Hz]
RESPONSE vs FREQUENCY
Frequency [Hz]8.58 7 7.5 8.58
7 7.5 8.58
−1.5
−2.5
Log(
Dis
plac
emen
t[m]}
Log(
Dis
plac
emen
t[m]}
Log(
Dis
plac
emen
t[m]}
−3.5
−1
−2
−3
−1.5
−2.5
−3.5
−1
−2
−3
−1.5
−2.5
−3.5
−4
Non-linear response – Forward sweepNon-linear response – Reverse sweepLinear response
Non-linear response – Forward sweepNon-linear response – Reverse sweepLinear response
Non-linear response – Forward sweepNon-linear response – Reverse sweepLinear response
Fig. 18.10 Frequency responses around third resonance for different nonlinear coupling elements (a) kNLC ¼ 200 N/m, d ¼ 0.02 m;
(b) kNLC ¼ 200 N/m, d ¼ 0.04 m; (c) kNLC ¼ 400 N/m, d ¼ 0.02 m
18 Nonlinear Structural Modification and Nonlinear Coupling 233
Here, mirror plate is taken as the original structure since it is not desired to be changed during the design phase of the
assembly. Shaft and bearing assembly on the other hand is taken as the nonlinear modifying structure where ball bearings at
the two ends of the shaft, shown in Fig. 18.11, are modeled as grounded nonlinear springs in horizontal and vertical
directions. The nonlinear behavior of the ball bearings can be taken to be cubic in nature [16]. The nonlinear parameters of
the ball bearings are taken as follows:
nðx; _xÞNL ¼ K0xþ bx3 where K0 ¼ 2x102 N=m and b ¼ 5x107 N=m3 (18.12)
YZ X
1 Elements
YZ
X
38
8 56
72
541
554
642
45
4624
23
22
21
20
3
48
56
52
54
44
47
49
51
53
43
57
16
15
14
13
1
9
10
17
a bAUG 3 2011
16:55:04
1 Elements
AUG 3 201116:50:18
Fig. 18.12 The FE model of (a) the mirror plate and (b) the shaft
Table 18.1 Material properties
of the mirror plate and the shaftMirror plate Shaft
Young’s modulus 71 GPa 200 GPa
Poisson’s ratio 0.33 0.3
Density 2,770 kg/m3 7,850 kg/m3
Fig. 18.11 The shaft and mirror plate assembly
234 T. KalaycIoglu and H.N. Ozg€uven
In the analysis, firstly the receptances of the mirror plate are calculated for connection points and for any other point we
might be interested in (i.e., points of which response is required or a force is applied to, by using the standard modal
analysis). Then, the structural modification method is employed and the receptances of the required points on the modified
structure are calculated. As the response of corner points on the mirror have the major importance since they are more prone
to effect the reflection performance, the FRF related with a point near one of the corners of the mirror plate is calculated
when a harmonic force of magnitude 2 N is applied to the same point. The calculated frequency response is shown in
Fig. 18.13 with the linear FRF of the assembly without considering bearing nonlinearity. Using structural modification
method, it is very easy and fast to recalculate the response for any design change in the shaft and/or bearings.
Furthermore, the FRF values for the resulting nonlinear system will be a function of the amplitude of the applied
harmonic force, which will require the recalculation of the FRFs for each forcing amplitude level even though nothing
is changed in either of the subsystems. In such analyses the method suggested here, again, provides drastic computational
advantage, since the FRFs of the linear part of the structure (which is usually the major part of the system) are calculated
once and then used to find the FRFs for the nonlinear overall system. In this later phase of the computations, which
requires iterative solution, only the FRFs related with the points we are interested in (in addition to those of the modifying
structure) are used, rather than all DOFs (which would be the case if the coupled nonlinear system were to be analyzed with
standard approaches). In this case study, FRF related with a point near one of the corners of the mirror plate is calculated
at two more different forcing levels. The results are shown in Figs. 18.14 and 18.15. The effect of forcing level on the FRF
related with a point near one of the corners of the mirror plate after modification can easily be observed by comparing
Figs. 18.13, 18.14 and 18.15.
Fig. 18.13 The direct point FRF related with a point near one of the corners of the mirror plate for F ¼ 2 N
Fig. 18.14 The direct point FRF related with a point near one of the corners of the mirror plate for F ¼ 4 N
18 Nonlinear Structural Modification and Nonlinear Coupling 235
18.4 Discussion and Conclusions
The noble structural modification method originally developed for linear systems [12] almost two decades ago was recently
extended for dynamic reanalysis of linear structures modified locally with a nonlinear substructure [11]. In this paper, the
same approach is further applied for dynamic reanalysis of linear structures coupled with nonlinear substructures by using
linear and/or nonlinear coupling elements. In this approach, the frequency responses of the modified structure are calculated
from those of the original structure and the system matrices of the modifying nonlinear structure (which can be in the form of
a coupled nonlinear substructure). The formulation used in this approach is given for each of the four nonlinear modification
or coupling cases investigated. Case studies for three of those cases are presented in this paper. Finally, an application of the
method to a real life engineering problem is demonstrated.
The method is based on the computation of the FRFs of a modified system from those of the original system and the
dynamic stiffness matrix representing themodifications in the system. Due to the nonlinear behavior of the modifying system,
the dynamic stiffness matrix turns out to be response level dependant and therefore the solution requires an iterative approach.
The formulation is for rigid connection of the nodes of the original and modifying systems. For the cases where a
nonlinear subsystem is coupled to a linear system with elastic elements (linear or nonlinear), the problem is treated as an
equivalent structural modification problem where to each free end of a connecting elastic element a massless node is added
and that node is rigidly coupled to the main system. With numerical case studies, applications of the method are
demonstrated. Firstly, a discrete linear system modified with a discrete nonlinear system is considered. The same original
system with different types of modification is used in the first three case studies. The effects of modifications and/or coupling
elements are demonstrated. The iterative numerical solution was found to be successful as far as convergence to a solution is
concerned. It should be noted that since the modified system is a nonlinear system, the calculated FRFs are valid only for the
level of the force applied, and different FRFs are obtained when the amplitude of the external harmonic force is changed.
As the last case study, a real life engineering problem is considered in order to show the applicability of the method to real
structural systems modeled with FE method. In this problem, structural modification analysis of a mirror plate modified with
a shaft-bearing assembly, where bearings at the two ends of the shaft are taken as hardening stiffnesses, is studied.
It should be noted that since the proposed method is an FRF based method, only the FRFs of the original system related
with the DOFs we are interested in, in addition to the ones at the connection DOFs are to be included in the calculations.
Although the formulation includes a matrix inversion, the size of the matrix to be inverted is equal to the DOF of the
modifying system, and therefore the method is most advantages when the modification is local. Especially in the design of
large main structures which may need to be modified locally, the method is very useful and makes it possible for the designer
to try various possible design changes or to make a parametric study with minimum computational cost.
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Fig. 18.15 The direct point FRF related with a point near one of the corners of the mirror plate for F ¼ 6 N
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10. Huang S (2007) Dynamic analysis of assembled structures with nonlinearity. Ph.D. thesis, Imperial College London, University of London
11. Kalaycioglu T, Ozg€uven HN (2011) Harmonic response of large engineering structures with nonlinear modifications. In: Proceedings of the
8th international conference on structural dynamics, EURODYN 2011, Leuven, pp 3623–3629, 4–6 July 2011
12. Ozg€uven HN (1990) Structural modifications using frequency response functions. Mech Syst Signal Process 4(1):53–63
13. Budak E, Ozg€uven HN (1990) A method for harmonic responses of structures with symmetrical nonlinearities. In: Proceedings of the 15th
international seminar on modal analysis and structural dynamics, Leuven, vol 2, pp 901–915, 17–21 Sept 1990
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15. Tanrikulu O, Kuran B, Ozg€uven HN, Imreg€unM (1993) Forced harmonic response analysis of non-linear structures using describing functions.
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16. Tiwari R (2000) On-line identification and estimation of non-linear stiffness parameters of bearings. J Sound Vib 235(5):906–910
18 Nonlinear Structural Modification and Nonlinear Coupling 237
Chapter 19
Nonlinear Dynamic Response of Two Bodies Across
an Intermittent Contact
Christopher Watson and Douglas Adams
Abstract An Improvised Explosive Device models as an outer buffer layer and an inner layer of surrogate material. It is
hypothesized that the area in contact between these layers and pressure across this area determines in large part the
transmission of vibration energy from the outer layer to the inner layer. Even if these layers of material display linear
elastic behavior separately, the combined vibration properties of the coupled bodies will exhibit nonlinear elastic behavior.
To study these interactions, a two-dimensional numerical finite element plate model is developed of the buffer layer and the
interface is represented as a random array of transverse springs simulating the stiffness of the inner layer beneath the buffer
layer. Natural frequencies and corresponding modal deflection shapes are calculated and shown to change depending on the
nodes connected by these springs. Modal experiments are performed with a polycarbonate sheet resting on a polymeric plate.
Experiments for low and high amplitude impulsive forcing functions demonstrate nonlinear vibration behavior in the two
coupled bodies and that this nonlinear behavior is a function of the location of the forcing function as well as the location of
the sensor that is used to measure the outer layer’s response.
19.1 Introduction
Sixty to seventy percent of deaths of coalition forces in Iraq and Afghanistan have been attributed to roadside Improvised
Explosive Devices (IEDs). In particular, as the Afghanistan campaign has intensified in the last few years, total IED
incidents have increased from about 200 per month to over 1,000 per month as of April 2010.1 Current methods of detection,
including infrared imaging of roadside areas beneath which IEDs are placed, are effective to some degree but more
sensitivity in these measurements is desired to reduce the rate of false positives and negative detections. The goal of this
research is to develop a fundamental understanding of how vibrations pass through multi-body solid objects and to explore
the use of acoustic excitations to cause IEDs to vibrate and/or emit energy, and be subsequently detected using other
measurements like infrared imaging. One facet of this research focuses on developing a modeling capability for describing
the dynamic interactions between two or more structural components undergoing vibratory response. Such a vibratory
response may cause intermittent contact between these bodies leading to complex nonlinear dynamic behavior.An improvised explosive device like the one pictured in Fig. 19.1a is conceptualized as consisting of an outer buffer
(inert) layer and an inner target (energetic) material among other components. The material properties and geometry of the
outer layer determine in part the reception of acoustic energy into the outer layer of the device. The amount of area in contact
between the outer layer and inner material along with the pressure between these two materials also determines in part the
transmission of energy in the form of vibrations through the interface. For example, the hypothetical device shown in
Fig. 19.1a has a metallic material only partially in contact with an inner target material. Because of this intermittent contact
between the two bodies, the combined dynamic properties of these bodies will exhibit both linear and nonlinear changes:
(1) changes in the linear modal vibration response of the two bodies will occur as the contact area between the two bodies
changes causing the boundary conditions on each body to fluctuate and (2) changes in the forced vibration response of the
two bodies will occur as the two bodies interact at the interface.
C. Watson (*) • D. Adams
Purdue University, 1500 Kepner Drive, Lafayette, IN 47905, USA
e-mail: [email protected]
1 Center for Strategic and International Studies, IED Metrics for Afghanistan January 2004–May 2010.
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_19, # The Society for Experimental Mechanics, Inc. 2012
239
The nature of the interactions between the outer layer and inner (target) material is important from a standpoint of
detection because these interactions dictate how readily the target material can be stimulated and its response observed based
on surface measurements on the outer layer. Figure 19.1b illustrates a conceptualized view of the transmission path for
vibratory energy. A shield is included for generality in this figure but is not considered in the present analysis.
19.2 Methods and Procedures
In order to study the interactions between the buffer layer and the inner material, a two-dimensional numerical finite element
model is developed for each layer as well as for the interface between these layers. First, a plane stress finite element model
of an upper plate with three degrees of freedom per node was developed. This upper plate represents the buffer layer and was
assumed to be thinner than the target material. The material properties for the buffer layer were chosen to correspond to those
of polycarbonate (Young’s Modulus: 2.6 GPa; Bulk Modulus: 2.3 GPa; Poisson’s Ratio: 0.37). The dimensions of this plate
were set as 254 mm long by 178 mm wide by 4.8 mm thick. In this initial study, the inner target material was modeled using
an elastic foundation that exists under portions of the buffer layer. The target material was designed to mimic the mechanical
and molecular properties of common energetic materials without being energetic itself. A binder material of hydroxyl-
terminated polybutadiene (htpb) was impregnated with inert Ammonium Chloride (NH4Cl) crystals, which replace the
otherwise energetic components typical used (see Fig. 19.2).
The interface between the outer plate and the inner target was observed to contain gaps and voids due to the nature of the
surrogate material (see Fig. 19.3), so this interface is represented in the model as a random array of transverse springs, which
restrain the transverse vibrations of the plate (see Fig. 19.4). These springs were used to simulate the stiffness supplied by the
inner (target) material that is beneath the buffer layer. The natural frequencies and corresponding modal deflection shapes
were calculated and were observed to change depending on the nodes that were connected by these springs. The possible
modal properties for each mode of vibration of the body were calculated by randomly connecting nodes and then solving the
corresponding eigenvalue problems. The resulting set of modal properties defined the bounds of the natural frequencies and
modal deflection shapes for the coupled vibrating system. Modal experiments were performed with a polycarbonate sheet
resting on a surrogate plate to motivate the model approach.Modal testing with a modal impact hammer was performed in order to identify the modal frequencies and modal
deflection shapes of the polycarbonate outer plate and the combination of the polycarbonate plate sitting on the inner target
material. To perform these tests, the plates were subdivided to contain 70 nodes. Two accelerometers were attached to the
plate, and each node was excited with the modal impact hammer in order to gather data (Fig. 19.5).
The effect of dynamic coupling on the two bodies was investigated by placing the polycarbonate plate on top of the target
material. In order to investigate potential nonlinearities involved when coupling the outer plate and the inner target material,
modal tests were performed with two different magnitudes of force input. If the system was linear, it was expected that no
change in the magnitude or phase of the measured frequency response functions or modal properties would be observed
(Fig. 19.6).
The numerical model consisted of a finite element model of the polycarbonate outer plate. A random array of transverse
springs was then attached to the nodes of this model and then the other ends of the springs were attached to ground. The
number of randomly arrayed springs was adjusted to represent differing levels of contact between the outer plate and the
inner target material. The stiffness of the springs can also be adjusted in the model. The intent is to also introduce Rayleigh
damping, at some point, and be able to use experimental results to help tune the model.
Fig. 19.1 (a) Inert improvised explosive device. (b) Conceptual layering of materials for use in developing intermittent contact model to study
transmission of acoustic energy from one body into the other
240 C. Watson and D. Adams
Fig. 19.3 (a) Polycarbonate outer layer resting on surrogate target layer and (b) close up of edge indicating some gaps and voids between these
two layers of material
Fig. 19.4 Outer (e.g., buffer) layer ofmaterial couples to inner (e.g., target) layer ofmaterial thatwill be simulatedusing a randomized elastic foundation
Fig. 19.2 (a) Polycarbonate (buffer) layer and (b) surrogate (target) layer of materials
19 Nonlinear Dynamic Response of Two Bodies Across an Intermittent Contact 241
19.3 Results
Numerical modeling is still underway. Figure 19.7 shows the simulated natural frequencies for an undamped polycarbonate
plate and for an undamped plate with transverse springs running to ground. Figure 19.7 also shows the first four mode shapes
for the simulated polycarbonate plate.
Modal testing of the polycarbonate outer layer resting on the surrogate target layer at two different levels of excitation
yielded the frequency response functions shown below in Fig. 19.8.
19.4 Conclusions
At this point we have determined, in a qualitative sense, that the combined dynamic properties achieved when coupling the
two bodies together do have a nonlinear component to them. As seen in Fig. 19.8, the amplitude of the frequency response
functions does change with a change in the excitation force.
Fig. 19.6 Polycarbonate
outer layer resting on
surrogate target layer
undergoing modal impact
testing
Fig. 19.5 Polycarbonate outer layer undergoing modal impact testing
242 C. Watson and D. Adams
It should be noted that nodes 1 and 5 are at locations where there are gaps and voids, whereas nodes 6 and 11 are areas of
more substantial contact. Regardless of the level of excitation, the response seems to be quite different between these
two groups.
Continued development of the numerical model is proceeding to predict a range of modal frequencies that can account for
the random nature of the contact interface between the outer and inner layers.
Fig. 19.7 Frequency vs. mode number plots for polycarbonate only layer model and polycarbonate on random elastic foundation indicating a
change in the resulting natural frequencies as a result of the foundation provided the surrogate layer of material
19 Nonlinear Dynamic Response of Two Bodies Across an Intermittent Contact 243
101
101
100100
H(jw) vs FrequencyNode 1
H(jw) vs FrequencyNode 6
Low ImpactHigh Impact
Low ImpactHigh Impact
Low ImpactHigh Impact
Frequency (Hz) Frequency (Hz)
H(jw) vs FrequencyNode 11
H(jw) vs FrequencyNode 5
101
100
100
h(jw
)h(
jw)
h(jw
)H
(jw)
10−1
10−1
10−1
10−2
10−2
100 200 300 400 500Frequency (Hz) Frequency (Hz)
600 700 800 900 1000
100 200 300 400 500 600 700 800 900 1000 100 200 300 400 500 600 700 800 900 1000
100 200 300 400 500 600 700 800 900 1000
Fig. 19.8 Frequency response functions for polycarbonate outer layer resting on surrogate target layer at two different levels of excitation at four
different locations
244 C. Watson and D. Adams
Chapter 20
Application of Continuation Methods to Nonlinear
Post-buckled Structures
T.C. Lyman, L.N. Virgin, and R.B. Davis
Abstract Continuation and path following methods have been applied to many nonlinear problems in mathematics and
physics. There is less widespread application of these methods, however, to structural systems. Since structural buckling and
stability problems are primarily concerned with system behavior as a control parameter (most often the load) varies, they are
particularly well suited for continuation methods and bifurcation analysis. In this work, the continuation package AUTO is
utilized to calculate post-buckled configurations, natural frequencies, and mode shapes of flat plates. Additionally, the
continuation analysis identifies bifurcation points and is also adapted to plate configurations that include slight initial
imperfections. Finally, the path following methods are also applied to track the unstable snap-through solution and natural
frequencies of post-buckled plates subject to a transverse load.
Keywords Buckling • Post-buckling • Continuation • Nonlinear • Snap through
20.1 Introduction
In modern engineering, there is a considerable interest in predicting the behavior of post-buckled structures. With current
lightweight aerospace and high performance applications, structural elements frequently operate beyond their buckled load.
This is especially true of plates, which are capable ofmaintaining stability at loads several times their critical buckling load [1].
Previous studies [2, 3] have investigated the post-buckled natural frequencies of plates by first linearizing the von Karman
plate equations about equilibrium points, and then solving the linear perturbation vibration problem. This method is
effective, although requires analytically determining the linearized equations of motion describing the behavior of small
perturbations around an equilibrium point.
Alternatively, given the equations of motion, natural frequencies and stability can be extracted from a numerical
linearization, taken at any given equilibrium point for the system. Continuation methods, specifically, take linearizations
at each step to aid in finding equilibrium solutions and stability information; however, this linearization can also be used to
find natural frequencies and mode shapes for dynamic mechanical systems. Continuation methods have been the basis of
study for a variety of dynamical systems in mathematics and physics [4], but to the extent of the authors’ knowledge have
been applied to problems in mechanics in a very limited sense [5].
The work presented herein introduces the basic steps and procedure behind continuation methods. Then a continuation
package, AUTO, is used to solve a Galerkin approximation applied to the von Karman plate equations. Both pre- and post-
buckled static and modal behavior is extracted. Finally, the methods are also applied to briefly look into the snap-through of
a transversely loaded, buckled, square plate.
T.C. Lyman (*)
Department of Mechanical Engineering and Materials Science, Duke University, Box 90300, Durham, NC 27708, USA
e-mail: [email protected]
L.N. Virgin
Department of Mechanical Engineering and Materials Science, Duke University, Box 90300, Durham, NC 27708, USA
R.B. Davis
Aerospace Engineer, NASA Marshall Space Flight Center, Mail Code ER41, Huntsville, AL 35812, USA
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_20, # The Society for Experimental Mechanics, Inc. 2012
245
20.2 Continuation Methods
Numerical continuation is a tool most often used to study nonlinear sets of equations, and is normally based on predictor-
corrector type methods. The main advantage of continuation methods is they handle solutions of nonlinear systems better
than conventional Newton methods, especially when turning points are encountered. Seydel [6, 7] lists that the predictor-
corrector continuation methods consist of four main components:
1. Predictor step,
2. Choice of parametrization,
3. Corrector iteration,
4. Step length control.
Figure 20.1 graphically shows the procedure of the predictor step. The predictor utilizes a tangent of the system at the
current solution point, x1, by numerically calculating a Jacobian of the system, and then takes a step towards the new
solution. The ‘predicted’ new point is illustrated in Fig. 20.1 by point x2P.
The next step in the continuation procedure is to choose a parameterization. The parameterization decides which set of
variables the corrector iteration will operate on to converge to a new solution. The parameterization can be done in many
different manners, several of which are shown in Fig. 20.2. It is implemented by adding a constraint equation to the system
of equations that will hold the increment of a certain parameter, or set of parameters, equal to a constant value. Traditionally
parameterizations based on holding the control parameter increment, Dl, constant are chosen. This choice encounters
problems if turning points are reached, in which case the system will have a vertical tangent and the corrector iteration will
not be able to successfully iterate to a solution. Another option is to parameterize the system via some combination of system
parameters, Dx, called a local parameterization. Although more robust, turning points can still arise and cause difficulty for
the corrector iterations. Finally, the most commonly used approach is to use an arc-length parameterization, where the
incremental arc-length of the overall solution from step to step is set and controlled. This method alleviates the problem of
navigating around turning points.
After the parameterization is chosen, a corrector iteration, usually Newton’s method, is applied to the system starting at
the predicted state, x2P, to find the solution at the next step, x2. Given the specific parameterization, it is likely that the Newton
iteration steps will correct in both the solution parameters, x, and the control parameter, l. Figure 20.1 shows corrector
iterations solely in x from point x2P to point x2, and another parameterization that corrects in both x and l parameters from
point x2P to point x2
0.The final step includes adjusting the step size that is taken in the predictor step. Using some type of metric, usually a check
on the number of Newton iterations needed for convergence, the step size of the next predictor is adjusted accordingly.
Fig. 20.1 Predictor step of
continuation method
246 T.C. Lyman et al.
Continuation methods are frequently researched in the mathematics and physics communities and several software
packages are freely available and maintained for solving problems in this manner. For the work presented here, the AUTO
continuation package, maintained by Eusebius Doedel of Concordia University and his collaborators, is used [8].
20.3 Application to Dynamic Mechanical Systems
Continuation methods are particularly well suited to solving dynamic systems with a changing control parameter, such as a
structure with an applied load. Second order mechanical systems can be written as a set of first order state space equations,
_xðtÞ ¼ fðx; lÞ; (20.1)
where x is a vector of state variables, f is a vector of linear or nonlinear equations and l is a control parameter. Continuation
methods not only solve the set of equations given by (20.1), but also calculate a Jacobian matrix for the system at each
equilibrium point in order to assist in finding the next incremented point. For dynamic systems, the eigenvalues of the
Jacobian will indicate the stability of the system [9]. Additionally, for dynamic mechanical systems, the eigenvalues will
actually take on a physical meaning and relate to the natural frequencies of the system while the eigenvectors will represent
the small perturbation vibration modes [10].
20.4 Plate Results
For finite deflections of a plate, the von Karman plate equations are used [11]. The von Karman plate equations are given in
dimensional form as [5]
Dr4ðw� w0Þ þ m@2w
@t2þ C
@w
@tþ Dp ¼ Fyywxx þ Fxxwyy � 2Fxywxy; (20.2)
and
r4F ¼ EhððwxyÞ2 � ðw0xyÞ2 � wxxwyy þ w0xxw0yyÞ: (20.3)
a b c
Fig. 20.2 Different types of variable parameterizations used in continuation methods, after [7]. (a) Parameterization of control parameter.
(b) Local parameterization. (c) Arc length parameterization
20 Application of Continuation Methods to Nonlinear Post-buckled Structures 247
The following non-dimensionalizations, also from [5], are used:
x ¼ a~x y ¼ a~y
w ¼ffiffiffiffiffiffi
D
Eh
r
~w F ¼ D ~F
Px ¼ D
a2~Px Py ¼ D
a2~Py
Pxy ¼ D
a2~Pxy m ¼ D
a4~m
C ¼ffiffiffiffiffiffiffi
Dmp
a2~C Dp ¼ D
a4
ffiffiffiffiffiffi
D
Eh
r
dp
t ¼ffiffiffiffiffiffiffiffi
a4m
D
r
t r ¼ b
a; (20.4)
where D ¼ Eh4=12ð1� n2Þ is the plate stiffness, m is the density per unit area, C is a damping coefficient, Dp is a pressure
applied to the plate, w is the out-of-plane deflection, F is the Airy stress function, and w0 is the initial out of plane
imperfection. Finally, the biharmonic operator, ∇4(�) is defined as
r4ð�Þ ¼ @4ð�Þ@x4
þ 2@4ð�Þ@x2@y2
þ @4ð�Þ@y4
: (20.5)
The Airy stress function, F, is defined as follows:
Fxx ¼ Py; (20.6)
Fyy ¼ Px; (20.7)
and
Fxy ¼ Pxy: (20.8)
Substituting the expressions from (20.4) into (20.2) and (20.3), results in the following set of non-dimensional von Karman
equations,
~r4ð~w� ~w0Þ þ ~w00 þ ~C~w0 þ dp ¼ ~F~y~y~w~x~x þ ~F~x~x~w~y~y � 2~F~x~y~w~x~y; (20.9)
and
~r4 ~F ¼ ~w2~x~y � ~w2
0~x~y � ~w~x~x~w~y~y þ ~w0~x~x~w0~y~y: (20.10)
For simply supported boundary conditions, Fourier solutions were assumed for both the out of plane displacement and the
Airy stress function,
~wðxÞ ¼X
m
X
namnðtÞfmðxÞcnðyÞ; (20.11)
and
~FðxÞ ¼X
m
X
nbmnðtÞfmðxÞcnðyÞ; (20.12)
where
fm ¼ sin mpxð Þ; (20.13)
and
cn ¼ sinnpyr
� �
: (20.14)
248 T.C. Lyman et al.
Equations 20.11 and 20.12 were substituted into (20.9) and (20.10). The series was truncated to a finite number of terms,
and then a Galerkin approximation was applied in order to minimize the errors associated with truncating the series. The
result is a set of nonlinear differential equations in terms of the Fourier coefficients, amn and bmn, that describe the deflectionof a plate with an applied axial load.
To solve the set of equations, a nine-term approximation was used with m, n ¼ {1, 2, 3} for a square plate, r ¼ 1.
A simple representation of the axially-loaded plate is shown in Fig. 20.3. The equations were solved using the AUTO
continuation package installed on Mac OS X. Using a uniaxially applied load, the bifurcation diagram showing non-
dimensional load versus the L2-norm of the Fourier coefficients is shown in Fig. 20.4. The solid lines in the figures indicate
stable branches, while the dotted lines indicate unstable branches.
The square markers along the abscissa axis indicate bifurcation points. The first square marker, at Px ¼ 1, indicates the
first linear elastic buckling load of the uniaxially compressed plate in the (m, n) ¼ (1, 1) mode. The following squares along
the axis are the linear buckling loads of the higher order modes. The buckling levels of the higher order modes correspond
with those predicted by energy methods [12].
As the axial load is applied to the plate, the natural frequencies will change with load and at least one frequency will drop
to zero as the plate approaches the buckling limit. As the plate follows a stable post-buckled equilibria, it will regain its
stiffness and, in turn, the natural frequencies will all be nonzero. Figure 20.5b shows the variation in natural frequency,
calculated by AUTO, of the plate, about the trivial, flat, solution from Px ¼ [0, 1] and then about the (m, n) ¼ (1, 1)
dominated equilibrium solution, shown in Fig. 20.5a, after buckling. The four lowest mode shapes, calculated from the
eigenvectors of the Jacobian matrix produced by AUTO, for the unloaded case are shown in Fig. 20.6.
Fig. 20.4 Bifurcation
diagram for a uniaxially
loaded square plate
Fig. 20.3 Schematic
representation of uniaxially
compressed square plate
20 Application of Continuation Methods to Nonlinear Post-buckled Structures 249
0
00
1
2
3
4
5
6
1 2 3 4 5 6 7
10.8
0.60.4
0.20 0 0.2 0.4 0.6 0.8 1
0.5
1
1.5
2
2.5
3
a b
Fig. 20.5 Static deflection and modal behavior of a uniaxially compressed square plate. (a) (m,n) ¼ (1,1) mode post-buckled static deflection,
Px / Pcritical ¼ 4.0, L2 norm ¼ 11.1933. (b) Variation of four lowest, linearized, natural frequencies for the main bifurcation branch of a square
plate
0.35
a
c d
b
0.4
0.3
0.2
0.1
−0.1
−0.2
−0.3
−0.4
0
0.4
0.3
0.2
0.1
−0.1
−0.2
−0.3
−0.410.9
0.80.70.6
0.50.40.3
0.20.1 0.1 0.2 0.3
0.40.5 0.6
0.7 0.8 0.91
0 0
0
10.90.8
0.70.6
0.50.4
0.30.2
0.10 0 0.1 0.2 0.3
0.4 0.5 0.6 0.7 0.8 0.9 1
0.4
0.3
0.2
0.1
−0.1
−0.2
−0.3
−0.4
0
10.90.8
0.70.6
0.50.40.3 0.2
0.1 0.1 0.20.3 0.4 0.5 0.6 0.7 0.8
0.9 1
0 0
0.25
0.15
0.05
0.1
00.9
0.80.7
0.60.50.4
0.30.2
0.10 0.1 0.2 0.3 0.4 0.5 0.6
0.7 0.8 0.9 1
0
1
0.3
0.2w
w
w
w
y
y
y
y
x
x
x
x
Fig. 20.6 Vibration mode shapes corresponding to the four lowest natural frequencies
250 T.C. Lyman et al.
Additionally, AUTO predicts that the secondary post-buckled equilibrium branch, which has the buckled mode shape
(m, n) ¼ (2, 1) shown in Fig. 20.7a, will stabilize after approximately Px/Pcritical ¼ 2.0 as shown in Fig. 20.4. Figure 20.7b
shows the natural frequencies of the secondary buckling branch after it regains stability.
20.5 Initial Imperfections
In order to model more physical systems, initial imperfections in the plate are taken into account since they are always
present in axially-loaded physical specimens. The continuation problem can be reformulated to start with a perfect plate,
x0 ¼ 0, and use the prediction correction steps until a desired imperfection level is reached. The plate was given an initial
imperfection in the (m, n) ¼ (1, 1) mode with a maximum central displacement equivalent to one plate thickness. The
imperfect plate bifurcation diagram and natural frequencies are shown in Fig. 20.8. Since there is no distinctive buckling
point (when compared with the perfect case) none of the frequencies for the main bifurcation branch drop to zero as before.
Additionally, with the initial imperfection present, the secondary branch still maintains stability through a portion of the
post-buckled regime as with the perfect case.
20.6 Snap Through
The final aspect that the continuation methods were applied to, and are particularly well suited to investigate, was the
snap through behavior of the post-buckled plate. Starting with a perfect plate, in the post-buckled region with an axial load of
Px/Pcritical ¼ 1.63, a transverse point load was applied to the center point of the plate. The standard bifurcation diagram, with
the pressure loading included is shown in Fig. 20.9a. The axial load is held constant while the continuation method varies
and follows the path of the transverse load needed for equilibrium. Figure 20.9b shows the equilibrium path as the pressure
load is applied. As the load is applied, the plate reaches a limit point where it becomes unstable and then eventually moves
dynamically, regains stability, and reaches the symmetric equilibrium condition on the opposite side. The points labeled 1–4
on Fig. 20.9a–c indicate the loading pattern. Figure 20.9c shows the lowest natural frequency as the central point load is
applied to the plate. The frequency drops as the load is applied, and eventually goes to zero as the equilibrium becomes
unstable. After the plate regains stability the lowest frequency becomes positive again and eventually reaches its original
starting point again as the plate reaches the symmetric equilibrium state.
1.5
0.5
−0.5
−1.5
−21
0.80.6
0.40.2
0 000
1
2
3
4
5
6ba
1 2 3 4 5 6 70.20.4
0.60.8
1
−1
0
1
2
Fig. 20.7 Static deflection and modal behavior of secondary buckling branch for a uniaxially compressed square plate. (a) (m,n) ¼ (2,1) mode
post-buckled static deflection, Px / Pcritical ¼ 4.0, L2 norm ¼ 6.6834. (b) Variation of four lowest, linearized, natural frequencies for the secondary
bifurcation branch of a square plate
20 Application of Continuation Methods to Nonlinear Post-buckled Structures 251
a b
c
Fig. 20.8 Buckling and post-buckling behavior of an imperfect, uniaxially compressed square plate. (a) Bifurcation diagram of an initially
imperfect, uniaxially loaded square plate. (b) Variation of natural frequencies for main branch of an imperfect, uniaxially loaded square plate.
(c) Variation of natural frequencies for secondary branch of an imperfect, uniaxially loaded square plate
252 T.C. Lyman et al.
20.7 Conclusions and Future Work
Continuation methods were applied to a problem of plate buckling to determine both pre and post-buckled behavior. The
continuation method produced results that correspond well with previously published data. Additionally, the continuation
code, AUTO, produced information corresponding to the post-buckled stability, natural frequency, and mode shapes. AUTO
was also utilized to trace an initially unstable secondary buckling branch that was found to stabilize after a certain increment
in the axial load. Finally the behavior of an imperfect, post-buckled plate, and snap through behavior of a post-buckled plate
were investigated using the continuation methods. Experiments will need to be conducted as future work to validate several
of the results found using the continuation analytical methods. Most importantly, verifying that the post-buckled stable
region of the secondary buckling path for the square, simply supported plate is physically realizable.
r
a
c
Fig. 20.9 Snap through behavior of post-buckled plate. (a) Bifurcation diagram showing both main buckling branch and snap through branch.
(b) Bifurcation diagram showing the snap through equilibrium path. (c) Variation of lowest natural frequency of square plate for snap through
equilibrium path
20 Application of Continuation Methods to Nonlinear Post-buckled Structures 253
Acknowledgements The authors would like to acknowledge the NASAGraduate Student Researchers Program (GSRP) grant NNX09AJ17H and
the Air Force Office of Scientific Research grant FA9550-09-1-0204 for support.
References
1. Murphy KD (1994) Theoretical and experimental studies in nonlinear dynamics and stability of elastic structures. PhD thesis, Duke University
2. Hui D, Leissa AW (1983) Effects of geometric imperfections on vibrations of biaxially compressed rectangular flat plates. J Appl Mech
50:750–756
3. Ilanko S (2002) Vibration and post-buckling of in-plane loaded rectangular plates using a multiterm Galerkin’s methods. J Appl Mech
69:589–592
4. Doedel E, Keller HB, Kernevez JP (1991) Numerical analysis and control of bifurcation problems (II): bifurcation in ininite dimensions.
Int J Bifurcat Chaos 1:745–772
5. Chen H, Virgin LN (2004) Dynamic analysis of modal shifting and mode jumping in thermally buckled plates. J Sound Vib 278:233–256
6. Seydel R (1991) Tutorial on continuation. Int J Bifurcat Chaos 1:3–11
7. Seydel R (2010) Practical bifurcation and stability analysis. Springer, New York
8. Doedel EJ, Oldeman BE (2009) AUTO-07P: continuation and bifurcation software for ordinary differential equations. Concordia University,
Oct 2009
9. Strogatz SH (2001) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Westview Press,
Cambridge,
10. Meirovitch L (2001) Fundamentals of vibrations. McGraw-Hill, Boston
11. Dowell EH (1975) Aeroelasticity of plates and shells. Noordhoff International, Leyden
12. Bulson PS (1970) The stability of flat plates. Chatto and Windus, London
254 T.C. Lyman et al.
Chapter 21
Comparing Measured and Computed Nonlinear Frequency
Responses to Calibrate Nonlinear System Models
Michael W. Sracic, Shifei Yang, and Matthew S. Allen
Abstract Many systems of interest contain nonlinearities that are difficult to accurately model from first principles, so it
would be preferable to characterize the system experimentally. For many nonlinear systems, it is now possible to measure
frequency response curves with stepped sine testing and to compute frequency response curves with numerical continuation.
Nonlinear frequency response curves are very sensitive to the system model and the nonlinearities and they provide a lot of
insight into the response of the system to a variety of inputs. This paper explores the feasibility of a nonlinear model updating
approach based on nonlinear frequency response and the experimental and analytical tools that are needed. For the
experiment, a cantilever beam with an unknown nonlinearity is driven with a harmonic force at various frequencies.
The steady-state response is measured and processed with the fast Fourier transform to obtain the frequency response
curve. Some subtle yet important details regarding how this is implemented are discussed. An analytical model is also
constructed and its frequency response computed using a recently developed technique. The measured and simulated
frequencies are then compared and used to tune the analytical model.
Keywords Nonlinear model updating • Frequency response • Stepped sine testing
21.1 Introduction
The frequency response function has long been used to characterize the dynamic response of linear systems. For nonlinear
systems, the nonlinear frequency response has also proven very useful, even with the additional complexities that are
introduced. Several different approaches have been taken to calculate frequency response from a known mathematical
model. A few studies combine the Harmonic Balance Technique with a numerical continuation algorithm to compute the
frequency response curves of torsional sub-systems with clearance nonlinearities [1] or nonlinear mesh phase interactions in
multi-gearbox drive systems [2]. This method is a semi-analytical approach since it uses the HBM formulation and closed-
form equations of motion. In another paper [3], Padmanabhan and Singh used a purely numerical approach based on
shooting methods to compute nonlinear frequency response curves and to characterize jump phenomena, subharmonic
responses, and chaos. However, as they discuss in [3], the algorithm seems to break down for higher-order nonlinear
systems. Ribeiro also used a numerical shooting approach to study frequency responses of geometrically nonlinear beams
and plates [4], but the algorithm used in that study doesn’t appear to be equipped to follow turning points [5] on branches that
lead to unstable periodic solutions. Finally, Gibert [6] used a numerical continuation approach based on nonlinear normal
modes (NNMs) to compute the nonlinear frequency responses of a beam model, and then used model updating method to
match those frequency responses to an actual beam. However, his method requires one to choose a specific number of NNMs
to use in the calculation, and this approximation may break down for certain nonlinearities or if an insufficient number of
nonlinear normal modes is used.
In this paper, the numerical algorithm presented by the authors in [7] is used to calculate nonlinear frequency responses.
The algorithm is based on shooting and continuation techniques that are very similar to those used in [8] to find the nonlinear
M.W. Sracic (*) • S. Yang • M.S. Allen
Department of Engineering Physics, University of Wisconsin-Madison, 535 Engineering Research Building,
1500 Engineering Drive, Madison, WI 53706, USA
e-mail: [email protected]; [email protected]; [email protected]
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_21, # The Society for Experimental Mechanics, Inc. 2012
255
normal modes of unforced systems. Here, the system is excited with a sinusoidal input force and the response is found over
one period of the excitation. The initial conditions are then adjusted until a steady state response is obtained and numerical
continuation techniques are used to calculate a branch of periodic responses for different forcing frequencies (i.e. essentially
a frequency response). The algorithm used in this work is identical to the one presented in [7] except that in this work the
Jacobians that are required were computed numerically using finite differences, so the equations of motion do not need to be
known in closed form.
Once a model has been formed and its frequency response has been calculated, one can use experiments to validate and
update the model. In the paper by Ribeiro [4] this was done by calculating one nonlinear frequency response and comparing
it to a measured frequency response near one of the beam’s resonances. In [9], Carella and Ewins measure nonlinear
frequency response a system using a stepped-sine testing approach and extract a first order approximation (linear) model for
specific forcing amplitudes. This approach though is applied to a single mode of the system and doesn’t seem to consider any
higher harmonic information associated with the nonlinearity. In [6], Gibert compares frequency responses between an
analytical model of a beam and an actual beam near three resonances, and in this paper a similar approach will be employed.
A cantilever beam was constructed in the laboratory, and a geometric stiffness nonlinearity was created by attaching a small
strip of nylon between the beam’s free end and a fixture. The nonlinear frequency response was then estimated using
stepped-sine excitation. A numerical model of the beam was constructed using a Ritz-Galerkin approach and its frequency
responses were computed using a numerical continuation approach and compared to the measured frequency responses of
the actual beam. The information from the frequency responses and from the harmonic information contained in the periodic
time history responses are used to evaluate the accuracy of the model, and some initial is made to update the model to
correctly capture the beam’s dynamics.
Themethodsused in this paper shouldbe applicable to a broad range of systems.The nonlinear frequency responseof a system
can be computed over a wide range of frequencies using efficient shooting and continuation techniques [3, 7, 8] and they are not
reliant on any approximation, such as an assumed number of polynomial terms approximating the nonlinearity. The resulting
nonlinear frequency responses can be readily compared to measurements, paralleling the way in which linear finite element
models are sometimes validated by comparing their linear frequency response functions with measured ones [10].
21.2 Theoretical Development
Generally, a forced nonlinear system can be represented in state space with the following equation
_x ¼ f x; uð Þ (21.1)
where f is a function that describes how the time-dependent state of the system, x(t), and the time-dependent inputs applied to
the system, u(t), influence the dynamics of the system. The output response of the system yðtÞ is often simply a subset of the
states, but in general it can be a general nonlinear function of the state and input as follows.
y ¼ h x; uð Þ (21.2)
When the nonlinear system is subjected to a periodic input with period T (i.e. uðtþ TÞ ¼ uðtÞ), the state and output will
often be periodic with xðtþ TÞ ¼ xðtÞ and yðtþ TÞ ¼ yðtÞ. (In the most general case, the nonlinear system may respond with
a longer or shorter period or may respond chaotically even in the presence of periodic input [5, 11], but these issues will not
be addressed in this paper.) Thus, it shall be assumed that the period of the forcing will always equal the period of the
response. It is more convenient to work with frequencies rather than periods, oT¼ ð2pÞ T= , so they will be used in place of
the period in the rest of this paper.
21.2.1 Frequency Response Attributes
The nonlinear frequency response curves describe how the periodic response(s) of a nonlinear system change as the input
frequency is varied. For linear systems this is captured by the frequency response function, which relates the magnitude and
phase of the input to that of the output. Each point on the frequency response function curve describes a periodic orbit g,which is a trajectory in the state space that contains the state �x for every time t. A linear FRF typically has a large magnitude
256 M.W. Sracic et al.
peak near each natural frequency corresponding to high amplitude periodic orbits in the state space. Similarly, the nonlinear
frequency response has resonance peaks near its nonlinear normal mode frequencies [12], but there are a few other important
differences. Resonance peaks in nonlinear systems can occur away from the nonlinear normal modes (e.g. superharmonic
resonances [5]), resonance peaks can contain multi-valued regions where several periodic orbits are possible for single
forcing frequencies, certain branches on the nonlinear frequency response curve may contain unstable periodic responses,
and finally the law of superposition does not hold for nonlinear systems so frequency responses cannot be linearly scaled
when the input amplitude is scaled. Based on the latter point, a nonlinear frequency response is defined by the response alone
(i.e. there is no scaling by the input such as in the linear frequency response), and it is only valid for a specific forcing
function and amplitude.
Using some of the previous facts, nonlinear frequency response curves can be built from time domain periodic responses,
and it is advantageous to have time domain signals because they contain more information about the nonlinearity in the
system. For example, consider the Duffing oscillator equation of motion with o1 ¼ 1, o3 ¼ 0:5, and sinusoidal forcing at a
frequency oT ¼ 1.
€xþ 0:02 _xþ o21xþ o2
3x3 ¼ sinðoTtÞ (21.3)
The response of this system to initial conditions x; _x½ � ¼ �0:056; 1:594½ � is periodic. The displacement is plotted over
five cycles of the response in Fig. 21.1a, and the signal appears to be sinusoidal. The magnitude of the frequency spectrum of
this response is plotted in Fig. 21.1b revealing that this is not a pure sinusoid but that there are several frequencies present at
1, 3, 5, etc. rad/s (labeled with open circles). The magnitude of these frequency components diminishes with increasing
frequency. There are also sharp peaks that rise above the noise floor at 2, 4, 6, etc. rad/s.
The system is clearly nonlinear since the input was a single frequency sinusoid, yet the system responded at several
different frequencies. In particular, if the forcing frequency is designated as the m ¼ 1 harmonic then the odd harmonics of
the system (i.e. the m ¼ 3, 5, . . .) designate peaks at 3 rad/s, 5 rad/s, etc. The even harmonics also appear in the spectrum at
2 rad/s, 4 rad/s, etc., however, those peaks are more than ten orders of magnitude smaller than the dominant peak and may be
the result of numerical integration error. The harmonic information can be used to build a single point on the frequency
response curve. For example, one could record the amplitude of a certain harmonic term in the response for a number of
different forcing frequencies and then plot the magnitude of that harmonic term versus the forcing frequency. In some cases a
harmonic other than the m ¼ 1 (i.e. the driving frequency) harmonic might dominate the responses, so it might be beneficial
to choose a harmonic other than the driving frequency. One can also sum the complex amplitudes of all the harmonics at
each driving frequency and then plot the magnitude of the sum of the harmonics versus the forcing frequency. No matter
which method is chosen, one needs to repeat this process for each stable periodic response on the frequency response curve.
21.2.2 Simulated Frequency Responses
Numerical continuation techniques can be used to efficiently calculate all of the periodic responses of a system over a wide
forcing frequency range. In this paper, a Newton–Raphson correction technique is used to solve the following two-point
boundary value problem.
0 5 10 15 20 25 30-2
-1
0
1
2
time, s
Dis
p x
Periodic Response
0 5 10 15
10-10
100
Frequency, rad/s
|FF
T(x
) |
FFT of Periodic Responsea b
Fig. 21.1 Periodic time domain response and frequency domain response of the Duffing system
21 Comparing Measured and Computed Nonlinear Frequency Responses to Calibrate Nonlinear System Models 257
Hðx; TÞ ¼ xðTÞ � xð0Þ ¼ 0 (21.4)
In particular for a system that is forced at the frequency oT¼ ð2pÞ T= , if the response vector xðTÞ is equal to the initial
condition vector xð0Þ, then the response is periodic. This solution provides a single point on the frequency response curve.
Once a solution has been established, the following system of equations can be solved to make a prediction for the next
solution on the curve.
@H@x
��xðTÞ;Tð Þ
@H@T
��xðTÞ;Tð Þ
h iPf g ¼ 0f g
Pf g ¼ PTx PT
� �T (21.5)
The Jacobian matrices @H @x=½ � and @H @T=½ � can be calculated using closed form equations of motion and the methods
described in [7, 13] or by using numerical integration and finite difference equations. In this paper, the latter method is used.
The vector P is in the null space of the matrix defined by the Jacobian matrices, so it is tangent to the frequency response
curve by construction [7, 8]. Once P has been calculated, it can be used to calculate the initial conditions for the prediction
of the next periodic response: xjþ1ð0Þ ¼ xjð0Þ þ sPx and Tjþ1 ¼ Tj þ sPT , where j ¼ 0 defines the first periodic solution
(i.e. j ¼ 0, 1, 2, 3, . . .) and s is a step-size parameter which can be automatically changed to increase the efficiency of the
calculations [7, 8]. The new prediction may need to be corrected so that (21.4) is satisfied. Therefore, corrections can be
calculated from the following system of equations.
@H@x
��xðkÞðjþ1Þ; T
ðkÞðjþ1Þ
� � @H@T
��xðkÞðjþ1Þ; T
ðkÞðjþ1Þ
� �Pzf gT PT
24
35 DxðkÞðjþ1Þ
DTðkÞðjþ1Þ
( )¼ �H x
ðkÞðjþ1Þ; T
ðkÞðjþ1Þ
� �0
( )(21.6)
The solutions of the previous system have been constrained to be orthogonal to the tangent prediction vector. This was
enforced to increase the convergence rate of the calculations. The variable k is used to track the number of corrections that
are required until the (21.4) has converged. In this paper, the solution has converged when H x; Tð Þk k< xð0Þk k � 10�6.
The prediction-correction process is repeated for each periodic response within a desired forcing frequency range.
These equationswere incorporated into an algorithm that calculates the entire frequency response curve over a large frequency
range for one forcing amplitude. The algorithm is automated and efficient, and can be used for relatively high order systems.
21.3 Measuring Frequency Response Curves
Swept sine tests are often used to measure the frequency response of linear systems. A similar approach can be used for
nonlinear systems; the frequency of the forcing signal is changed step by step in the tests, while the forcing amplitude is the
same for all the forcing frequencies. Then, the system’s response at each forcing frequency is measured and the nonlinear
frequency response curves for this specific forcing amplitude can be constructed within the tested frequency range.
However, there are some important differences between swept sine tests for linear and nonlinear systems. A nonlinear
system generally has different frequency response curves for different forcing amplitudes. Furthermore, even with a fixed
forcing amplitude, as implied earlier, the system may have several periodic orbits in some frequency ranges. The higher
energy periodic orbits are often less stable, so when increasing/decreasing the forcing frequency the system may jump to a
lower energy orbit when a disturbance occurs, making the high energy branch more difficult to capture. The high energy
branch often contains some of the most valuable information, so it is desirable to capture as much of that branch as possible.
Therefore, it is necessary to keep the forcing amplitude constant during the whole test. Sometimes the force can be
monitored and controlled to reject disturbances or fluxuations by using a feedback control algorithm [9].
In this work, a National Instrument PXI system was used to generate the step-sine forcing signal and to acquire the
responses. A LabView program was designed to generate a continuous forcing signal that has a fixed number of samples per
period, ensuring that zero amplitude is obtained at the beginning and the end of each period. Because there are a fixed
number of samples per period, the generation rate of the forcing signal varies with the forcing frequency (or the period).
When the forcing frequency is stepped, the program waits until the end of a period (i.e. when the amplitude is zero) to change
the frequency so that the transient response (disturbance) incurred to the system is minimized. The input force and the
response are monitored and the responses are recorded with a fixed sampling rate after the transient responses disappear.
258 M.W. Sracic et al.
At each forcing frequency, the data acquisition program waits prescribed amount of time (typically a few seconds) for the
response to reach steady state prior to recording the response and exporting the time series to Matlab for analysis. The
magnitudes of the response versus frequency can then be computed using the procedure described in Sect. 21.2.1.
21.4 Updating Models Using Nonlinear Frequency Response Curves
Although nonlinear frequency response curves can be substantially more complicated than linear FRFs, general dynamics
principles can still be used to interpret the curves. Figure 21.2 shows several frequency response curves of the Duffing
system in (21.3), which were calculated using the numerical continuation technique that was summarized previously. The
results are plotted in terms of the magnitude of the periodic displacement. For each curve, a different set of values of linear
and nonlinear stiffness terms (i.e. o1 and o3, respectively) was used. All the curves have well known bent resonance shape
with a region where multiple solutions are possible for a single forcing amplitude and frequency. For some sets of the
parameters o1 and o3 this multi-valued solution region extends over a larger range of frequency than for others. When the
linear stiffness is increased, the whole resonance peak shifts to higher frequencies. For example, the base of the curve shown
with a solid line (i.e. o1 ¼ 1, o3 ¼ 0:5) seems to originate near 1 rad/s. When the linear stiffness is increased to o1 ¼ 2 the
resonance shifts upward, as shown by the curve with open circles. When the linear stiffness is held constant and the nonlinear
stiffness is varied, the shape of the peak changes. For example, the curve with a dashed line is for o3 ¼ 0:25, and is seen to
bend less severely than the curve for o3 ¼ 0:5.Figure 21.2 also shows that the linear and nonlinear stiffness terms have coupled effects on the shape of the shape of the
nonlinear FRF. For example, the shape of the frequency response curves can be seen to change when the linear stiffness is
varied, even though the nonlinear stiffness term is held constant at o3 ¼ 0:5. Furthermore, none of the curves in the figure
has the same peak magnitude, even thought the forcing amplitude was the same in all cases (i.e. it was equal to 1 in all cases).
Nevertheless, the nonlinear frequency response curves do change in a fairly straightforward way so it would appear that,
given a reasonable set of measurements, one could adjust the system model until it has the same frequency response as the
measurement. The dynamics of higher order nonlinear systems can be more complicated, but they often show this same
characteristic shape and hence these same principles could be used to tune a model until it reproduces each resonance
accurately.
0 2 4 6 8 1010-2
10-1
100
101
102
Frequency, rad/s
Mag
Dis
p
Duffing Frequency Responses
ω1=1, ω3=0.5
ω1=2, ω3=0.5
ω1=4, ω3=0.5
ω1=1, ω3=0.25
ω1=1, ω3=1
Fig. 21.2 Frequency
response curves of the Duffing
system for different linear
(o1) and nonlinear (o3)
stiffness values
21 Comparing Measured and Computed Nonlinear Frequency Responses to Calibrate Nonlinear System Models 259
21.5 Nonlinear Cantilever Beam System
In a few recent papers, the Sracic and Allen have worked with a cantilever beam that has a geometric nonlinearity at its free
end [14, 15]. Figure 21.3 below shows a top view photograph of the actual experimental setup. An aluminum 6061 alloy
beam is bolted to a fixture that approximates a fixed base. A small strip of nylon is bolted to the free end of the cantilever and
clamped to the fixture. The beam is oriented such that the bending axis is parallel to the plane of the table top. Figure 21.4
shows a close top and front view of the nylon strip that is clamped between the tip of the beam and the right hand side
support. Table 21.1 below provides the physical dimensions of the beam and the nylon strip in millimeters.
The nylon strip at the tip of the beam adds stiffness at that point that depends nonlinearly on the tip displacement. This
setup was originally proposed in [16], although their beam had a strip of spring steel instead of nylon, and other researchers
have studied similar beam setups [17–19].
Fig. 21.3 Top view of the experimental nonlinear beam setup
Fig. 21.4 Top view (a) and front view (b) of the spring steel connected to free end of the cantilever beam
Table 21.1 Dimensions, in millimeters, of the 6061 aluminum beam and the nylon strip, and the location of measurement sensors from
the approximated fixed end
Dimension Al 6061 beam Nylon 6/6 strip Sensor location
Length 1016 53.2 a1 ¼ 45
Width 25.4 25.4 a2 ¼ 508
Thickness 9.5 0.254 a3 ¼ 984
260 M.W. Sracic et al.
An analytical model of this system was created, as described in the schematic in Fig. 21.5. The beam is modeled as a
uniform, prismatic cantilever beam with material density r, elastic modulus Eb, cross sectional area Ab, bending area
moment of inertia I, and length L. The position along the length of the beam is given by the variable ‘x’. The deflection of thebeam is designated with the variable y. The beam has a nonlinear spring at its tip with stiffness knl. The equations of motion
of the analytical beam are derived in Appendix A.
In order to mimic the experimental system, the following parameters were used for the original model, which are based on
the nominal properties of the experimental hardware: r ¼ 2,700 kg/m3, Eb ¼ 68 GPa, Ab ¼ 3.23 � 10�4 m2, I ¼ 4.34
� 10�9 m4, L ¼ 1.016 m. Using these properties with the Ritz-Galerkin method, the two linear natural frequencies of the
system are o1/(2p) ¼ 9.97 Hz and o2/(2p) ¼ 62.51 Hz. Modal damping was added to the model assuming a coefficient of
critical damping of z ¼ 0.01 for all modes. The transverse stiffness of the nylon strip is approximated in the model as
k3 ¼ EnAn ð2l3nÞ�
, where En, An, and ln are the elastic modulus, cross sectional area, and length of the nylon strip. For the
original model, the nylon was assumed to have an elastic modulus of En ¼ 3.9 GPa, which makes the nonlinear stiffness
k3 ¼ 8.75 � 107 N/m3. A derivation of this approximation can be found in the appendix in [14], although in that work spring
steel was used at the beam’s free end instead of nylon.
The experiment was designed to produce a cubic nonlinearity. However, several practical design aspects could lead to
other forms of nonlinearity in the response. For example, Fig. 21.4a shows that the nylon strip is offset from the neutral
bending axis of the beam, which has the potential to produce a quadratic nonlinear contribution to the response. Addition-
ally, the fixed end of the cantilever is realized using bolts, and the effective length of the beam may change depending on the
direction of the deflection. The finite order model of the beam was developed assuming a perfect cantilever and a perfectly
cubic spring at the beam’s free end. Therefore, the goal is to use the frequency responses to improve the model so that it more
closely represents the actual beam.
21.5.1 Frequency Responses of the Nonlinear Beam
The procedure described in Sect. 21.3 was used to measure the periodic responses of the nonlinear beam in order to calculate
the beam’s frequency responses. Harmonic excitation was applied to the beam with a model 2100E11-100 lb Modal Shaker
from The Modal Shop, Inc. The beam was approximated as a having a fixed support, so the shaker was freely hung from a
lateral excitation stand, as recommended in [20]. A thin steel stinger was used to transmit the excitation from the shaker to
the beam. One end of the stinger was clamped inside the shaker armature and the other end was fixed to a force transducer,
model 208 C04 from PCB Piezotronics, Inc. (PCB), which was bolted to the beam at a location x ¼ 45 mm from the fixed
end of the beam. Harmonic forcing was provided by the National Instruments PXI system described in Sect. 21.3. The peak
force amplitude, which was measured by the force transducer, was 70 N. The response was measured with two Endevco
model 66A12 triaxial accelerometers (only the z-channels were used) located at x ¼ 45 mm (DOF 1; shaker location) and
x ¼ 508 mm (DOF 2; beam center) and with an Edevco model 256-100 isotron accelerometer located near the free end of the
beam at x ¼ 984 mm (DOF 3; beam tip). The degree-of-freedom locations are labeled with arrows in Fig. 21.3. All of the
accelerometers were secured to the beam with wax.
Initially, a fast sweep was performed using a large step size (frequency increment) in order to estimate the resonance
frequencies. The first three resonances were found to be near 14, 45, and 120 Hz. The program was then set to automatically
perform forward and backward frequency sweeps using a variable step size. Specifically, a step size of 0.1 Hz was used near
the resonances and 0.5 Hz in the regions away from the resonances. This approach decreased the testing time significantly
while still capturing all of the features of the nonlinear frequency response accurately.
After the measurements were obtained, the method described in Sect. 21.2.1 was used to calculate each point on the
frequency response curve from the measured periodic responses. In particular, the spectra of the measured accelerations
were calculated with the Fast Fourier Transform, and the amplitudes of all the harmonic peaks were collected (since the
knl = k3 y(L)3
xy
DOF 2 DOF 3 DOF 1
Fext = Aext sin(w Tt )
r, Eb, Ab, I, L
Fig. 21.5 Schematic of the
nonlinear beam
21 Comparing Measured and Computed Nonlinear Frequency Responses to Calibrate Nonlinear System Models 261
steady-state response is periodic, these are the coefficients of the Fourier Series description of the response). In order to
estimate displacement, each harmonic peak was integrated twice by dividing frequency squared. Then, the displacement
amplitudes of all the harmonics were summed to give the estimate for the magnitude of displacement.
Next, the frequency response of the model beam was calculated with the numerical continuation technique. The algorithm
was set to find the nonlinear frequency response between 2 and 150 Hz. For each periodic response, the instant at which
the displacement of DOF 3 (i.e. the tip displacement) was maximum was found and recorded, and the displacements of the
other DOF were recorded at the same instant. These amplitudes were used to plot the magnitude of the frequency response.
Note that this assumes that all the degrees-of-freedom reach their extreme displacement at the same instant, which may not
be the case for very nonlinear systems or for when there are complicated modal interactions. However, it will be shown that
the accuracy of the model can still be inferred using this approach.
Figure 21.6 shows the frequency response functions that were calculated from the measurements of the beam as well from
the numerical continuation simulations. The dashed curves with markers show the results from the experiment (open blue
squares-accelerometer a1, open green circles-accelerometer a2, red dots-accelerometer a3). The solid lines are the frequencyresponse curves that were calculated with the numerical continuation technique (blue-DOF 1, green-DOF 2, red-DOF 3).
There are three dominant resonance peaks in most of the curves except for the DOF 2 curves (green). The first two resonance
peaks in the model curves occur near 10 and 47 Hz and are bent to higher frequencies, while the third peak near 130 Hz
appears to be predominantly linear. In the curves from the experiment, the first two peaks occur near 14 and 44 Hz. They
seem to have a bent shape, but when they reach a certain frequency the curve sharply drops in magnitude, which many would
recognize as the well know jump-phenomenon for the multi-valued region (i.e. the response jumps from a large amplitude
periodic response to a small one). Additionally, the first resonance of the model seems to bend more significantly than the
curves from the experiment and the curves from the model contain several superharmonic resonances below 5 Hz. This
range was not tested in the experiment because the freely suspended shaker moved quite a bit in that frequency range causing
the measurements to be unreliable. Lastly, the amplitude of the resonance peaks in the model’s responses for DOF 1 (i.e. the
blue curve, especially for the peaks near 47 and 130 Hz) appear to be much smaller than those of the actual system.
The frequency response curves show that the model accurately characterizes the number of modes in this frequency band
of interest, and the fact that the nonlinearity is most apparent in the first and second modes. However, there are several
significant discrepancies. The amplitude of the peaks in the model’s curve for DOF 1 are too small, indicating that this
DOF’s amplitude is smaller than the actual hardware. This discrepancy may arise because the model has a perfect fixed base,
while the fixture in the experiment has finite compliance. Another important issue is that the frequencies where the
resonances occur differ between the model and the actual beam. The frequencies of the second two modes are too high in
the model. As in the single degree-of-freedom case, this property is influenced by the linear stiffness of the system.
Therefore, the elastic modulus of the beam can be reduced to decrease the linear stiffness and hence the frequency of all
the resonance peaks. Figure 21.7 shows the second comparison between the measured frequency responses and those from
0 50 100 15010-7
10-6
10-5
10-4
10-3
10-2
Frequency, Hz
Mag
nitu
de D
isp
Nonlinear Beam Frequency Response
Exp DOF 1
Exp DOF 2
Exp DOF 3
Mod DOF 1
Mod DOF 2
Mod DOF 3
Fig. 21.6 Comparison #1 of frequency response curves for the experiment (Exp) and the model (Mod) for model parameters: Eb ¼ 68 GPa,
En ¼ 3.9 GPa, k3 ¼ 8.75 � 107 N/m3
262 M.W. Sracic et al.
the model after reducing the elastic modulus of the beam to Eb ¼ 55.5 GPa (note that this value is still in the nominal range
for the aluminum). No other parameters in the model were changed.
With a lower elastic modulus, the location of the resonance peaks has changed to be more in line with the measurements,
but the shapes of the peaks has changed as well. The resonance of the third mode near 119 Hz seems to align very well with
the measurements, and the peaks seem to have the same magnitude except for DOF 1. The model’s second resonance peak
also seems to align more closely with the measurements. However the peaks in the simulated frequency responses bend over
more than the measurements and more than they did in Fig. 21.6. This is evidence that the linear and the nonlinear stiffness
of the model are coupled, since only the linear stiffness was changed but a nonlinear feature of the second resonance also
changed. The same seems to be true for the first resonance. However, one must use caution when comparing the simulated
frequency responses with the measurements, since the actual beam may have jumped to a lower amplitude response before
reaching the absolute peak of the frequency response curve. The phase of the response can be used to investigate this, since
the phase should be 90� at the extreme of the resonance curve. The phase of the response was calculated at the jump
frequency for the resonance near 14 Hz by computing the difference between the phase of the fundamental harmonic of the
response and the fundamental harmonic of the force. For DOF 3 at 14.3 Hz the phase difference was found to be �83.26�,suggesting that the experiment captured nearly the full nonlinear resonance curve for DOF 3 before the jump happened. In
contrast, the phase of DOF 3 at the 45.11 Hz jump frequency was 74.86�, revealing that portion of the second resonance
curve was probably missed in the experiment. However, the peaks in the model’s responses still seem to bend over larger
frequency ranges than those of the actual beam suggesting that the nonlinear stiffness of the model is too large. Hence, the
model was updated again by reducing the modulus of the nylon spring to En ¼ 0.24 GPa, which yields a nonlinear stiffness
coefficient of k3 ¼ 5.38 � 106 N/m3. The frequency responses of the model were recomputed and are shown in Fig. 21.8.
This has caused the shape of the model’s resonances to change so that the first two resonances do not bend as much towards
higher frequencies. Unfortunately, the peaks near 44 Hz now do not seem to bend as much as those in the measurements. The
peaks near 12 Hz have a larger maximummagnitude than the previous case and yet they still seem to bend more than those in
the measurements. It also seems apparent now that the model’s resonance curves are centered at a significantly lower
frequency than those in the measurements.
21.5.1.1 Discussion
By simply adjusting the modulus of the beam (i.e. in Fig. 21.7), the nonlinear frequency responses of the model were made to
agree much more closely with the actual measurements, at least for the second and third resonances. The shape of the
resonance curve for the second mode was also improved as well. However, the linear and nonlinear stiffness terms have a
coupled effect on the nonlinear FRFs so it was not trivial to bring the first mode into agreement nor to match the shape of that
resonance. It is likely that the current beam model cannot be updated to agree more closely with the measurements without
0 50 100 15010-7
10-6
10-5
10-4
10-3
10-2
Frequency, Hz
Mag
nitu
de D
isp
Nonlinear Beam Frequency Response
Exp DOF 1
Exp DOF 2
Exp DOF 3
Mod DOF 1
Mod DOF 2
Mod DOF 3
Fig. 21.7 Comparison #2 of frequency response curves for the experiment (Exp) and the model (Mod) for model parameters: Eb ¼ 55.5 GPa,
En ¼ 3.9 GPa, k3 ¼ 8.75 � 107 N/m3
21 Comparing Measured and Computed Nonlinear Frequency Responses to Calibrate Nonlinear System Models 263
relaxing certain assumptions in the model. For example, it may be important to consider the finite stiffness of the support at
the base of the cantilever, rather than treating it as a fixed support. Likewise, the nylon strip may actually impart a linear
stiffness to the beam tip and the mass of the bolts, accelerometers and/or load cell may be significant.
By construction, the methods in this paper can also be used to validate the terms that are used to describe the nonlinearity.
For example, Fig. 21.9 shows the Fourier coefficients of the estimated displacement response of the actual beam for the first
resonance. The coefficients of the first three harmonics (i.e.m ¼ 1, 2, and 3) are shown in subplots (a), (b), and (c), respectively.
12 12.5 13 13.5 14 14.5 15 15.5 1610-6
10-4
10-2 Fourier Coefficients: m=1
Frequency, Hz
|Dis
p|, m
DOF 1
DOF 2
DOF 3
12 12.5 13 13.5 14 14.5 15 15.5 16
10-5
Fourier Coefficients: m=2
Frequency, Hz
|Dis
p|, m
12 12.5 13 13.5 14 14.5 15 15.5 16
10-5
Fourier Coefficients: m=3
Frequency, Hz
|Dis
p|, m
a
b
c
Fig. 21.9 Fourier coefficients of the actual beam’s steady state forced response, extracted during the nonlinear frequency response calculation.
The first three harmonics are shown in (a), (b), and (c)
0 50 100 15010-7
10-6
10-5
10-4
10-3
10-2
10-1
Frequency, Hz
Mag
nitu
de D
isp
Nonlinear Beam Frequency Response
Exp DOF 1
Exp DOF 2
Exp DOF 3
Mod DOF 1
Mod DOF 2
Mod DOF 3
Fig. 21.8 Comparison #3 of frequency response curves for the experiment (Exp) and the model (Mod) for model parameters: Eb ¼ 55.5 GPa,
En ¼ 0.24 GPa, k3 ¼ 5.38 � 106 N/m3
264 M.W. Sracic et al.
The degrees-of-freedom are shown with open squares (DOF 1), open diamonds (DOF 2), and dots (DOF 3). Recall that for
a linear system only the coefficients in 9a would be nonzero. Hence, by viewing the responses in this way one can see that the
first resonance is dominated by the m ¼ 1 and m ¼ 3 responses, while the m ¼ 2 harmonic is significantly weaker. Other
harmonics are present but are not shown.
The harmonics in the response provide information on the type of nonlinearity that is present in the system. In particular,
the m ¼ 3 harmonic is associated with a cubic nonlinearity. The presence of a m ¼ 2 harmonic could be due to the existence
of a quadratic type nonlinearity in the system. So, one could continue to interrogate the harmonics of the responses to see
what terms provide a contribution, and ideally the responses of the model should also contain these harmonic terms. The
model in this paper was not designed with any quadratic nonlinear terms, but these observations suggest that a term of the
form k2yðLÞ2 � C1ðLÞ � � � CNðLÞ½ �T should be inserted into the equations of motion in Appendix A.
21.6 Conclusions
This paper has explored model updating for nonlinear systems by measuring the nonlinear frequency response of the system
and comparing those with the simulated frequency responses of a representative model. The experiment was performed with
a controlled stepped-sine test that measures the steady state, periodic response of the nonlinear system at each forcing
frequency. Then, the measured time histories can be processed to display the nonlinear frequency response over a range of
forcing frequencies. The nonlinear frequency response of the system model was found using a numerical continuation
technique. The algorithm employed a Newton–Raphson shooting and updating technique and finite differences gradients so
that the closed form equations of motion were not needed. One advantage of this approach is that the time histories used by
this algorithm could be supplied by a finite element analysis package, simplifying the modeling process. The frequency
responses contain many characteristics that are similar to linear frequency response functions (e.g. resonance peaks that are
related to the modes of the system), so many of the same principles from linear modal analysis can be used to evaluate and
update the system model. In addition, the nonlinear characteristics of these curves (e.g. the shape of bent resonance peaks),
can be used to validate the nonlinear parameters of the model.
This approach was evaluated using an actual cantilever beamwith a geometric nonlinearity at its tip. The beam’s nonlinear
frequency responses were measured (up to the limits of the modal shaker) and nonlinearity was clearly observed. A simple
Ritz-Galerkin model of the beam was created and its parameters were adjusted to more closely reproduce the three measured
resonance peaks. The locations of the resonance peaks were used to update the linear stiffness of the model, which brought the
model and measurements into much closer agreement. On the other hand, the model had difficulty reproducing the shapes of
the first two resonance curves for any value of the model parameters. The results suggest that other features of the model must
be adjusted to obtain better agreement. For example, the response of the beam was interrogated, revealing that a weak
quadratic term was present and should be added to the beam model in order to more faithfully reproduce the response.
Appendix A: Ritz-Galerkin Discrete Model
A Galerkin approach was used to create a finite-order model of the experimental structure. Assuming that the beam behaves
linear-elastically, mode shapes corresponding to transverse bending motion were used as shape functions to construct the
Ritz-Galerkin representation [21]. The displacement of the beam at a position x was approximated as
y x; tð Þ ¼XNm
r¼1
CrðxÞqrðtÞ (A.1)
where cr(x) is the rth Euler-Bernoulli beam mode shape for a cantilever, qr(t) is the rth generalized coordinate, and Nm is the
number of modes used. The system’s undamped equations of motion are provided in the following equation, where the
coordinates are the amplitudes of the basis functions.
rAbL M½ �€qþ EbI
L3K½ �q ¼ Q ¼
XfextCrðxf Þ (A.2)
21 Comparing Measured and Computed Nonlinear Frequency Responses to Calibrate Nonlinear System Models 265
For the generalized coordinates, a time derivative is denotedwith an over-dot (e.g. the generalized acceleration vector is €q).Modal damping was added to the system by performing an eigenvector analysis on the linear system and then using,
C½ � ¼ ðrAbLÞ2 M½ � fb½ � diag 2zrorð Þ½ � fb½ �T M½ � (A.3)
where [fb] is a matrix containing the eigenvectors in the columns, or is the rth circular natural frequency, and zr is the rthdesired damping ratio. The generalized force vectorQ is a sum of the product between all external forces and the value of the
shape functions at the point where the force is applied, xf. Therefore, Q includes the applied or external forces, Fext in
Fig. 21.5, as well as the nonlinear restoring force due to the spring [21]. The beam provides linear stiffness at the tip due to its
flexural rigidity, so the discrete spring’s stiffness was chosen to be purely nonlinear
knl ¼ k3yðLÞ2 (A.4)
where k3 is a stiffness constant associated with the nonlinear spring. The physical restoring force due to the spring is then
equal to fsp ¼ k3yðLÞ3. The generalized force vector then has components corresponding to the nonlinear spring located at
x ¼ L and the externally applied force located at x ¼ xf.
Qf g ¼ k3yðLÞ3c1ðLÞ
..
.
cNðLÞ
8><>:
9>=>;þ Aext sin oTtð Þ
c1ðxf Þ...
cNðxf Þ
8><>:
9>=>; (A.5)
Aext is the amplitude and oT the frequency of the external forcing term that produces the limit cycle.
After using the Ritz-Galerkin method to form the discrete beam model and to account for the nonlinear applied force of
the spring, the equations of motion were transformed back into physical coordinates using the relationship in (A.1). The
differential equations of motion can then be arranged in state space format.
_y
€y
� ¼
_y
� Mp
� ��1Cp
� �_yþ Kp
� �yþ Ff g �
( )
Mp
� � ¼ rAbL C½ ��T M½ � C½ ��1; Cp
� � ¼ C½ ��T C½ � C½ ��1;
Kp
� � ¼ EbI
L3C½ ��T K½ � C½ ��1; Ff g ¼ C½ ��T Qf g
(A.6)
The matrix C½ � has the numerical values of the mode vectors for specific position coordinates on the beam. Then, C½ � cancontain shape functions evaluated at the nodal degrees of freedom on the beam. In this study the number of mode shapes used
in the expansion and the number of degrees of measurement points (shown in Fig. 21.5) was N ¼ Nm ¼ 3. The nodes were
located at the center and tip of the beam as shown in Fig. 21.5.
References
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international modal analysis conference (IMAC XXIX), Jacksonville
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11. Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, vol 42. Springer, New York
12. Kerschen G, Peeters M, Golinval JC, Vakakis AF (2009) Nonlinear normal modes, part I: A useful framework for the structural dynamicist.
Mech Syst Signal Process 23:170–194
13. Sracic MW (2011) A new experimental method for nonlinear system identification based on linear time periodic approximations. Ph.D.
Engineering Mechanics, Department of Engineering Physics, University of Wisconsin-Madison, Madison
14. Sracic MW, Allen MS (2011) Identifying parameters of nonlinear structural dynamic systems using linear time-periodic approximations.
Presented at the 29th international modal analysis conference (IMAC XXIX), Jacksonville, 2011
15. Sracic MW, Allen MS. Identifying parameters of multi-degree-of-freedom nonlinear structural dynamic systems using linear time periodic
approximations. Mech Syst Signal Process, submitted July 2011.
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17. Kerschen G, Lenaerts V, Golinval JC (2003) Identification of a continuous structure with a geometrical non-linearity. Part I: Conditioned
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20:505–592
19. Worden K, Tomlinson GR (2007) A review of nonlinear dynamics applications to structural health monitoring. Struct Control Health Monitor
15:540–567
20. Mayes RL, Gomez AJ (2006) What’s Shakin’, Dude? Effective use of modal shakers. Exp Tech 30:51–61
21. Ginsberg JH (2001) Mechanical and structural vibrations theory and applications, 1st edn. Wiley, New York
21 Comparing Measured and Computed Nonlinear Frequency Responses to Calibrate Nonlinear System Models 267
Chapter 22
Identifying the Modal Properties of Nonlinear Structures
Using Measured Free Response Time Histories
from a Scanning Laser Doppler Vibrometer
Michael W. Sracic, Matthew S. Allen, and Hartono Sumali
Abstract This paper explores methods that can be used to characterize weakly nonlinear systems, whose natural
frequencies and damping ratios change with response amplitude. The focus is on high order systems that may have several
modes although each with a distinct natural frequency. Interactions between modes are not addressed. This type of analysis
may be appropriate, for example, for structural dynamic systems that exhibit damping that depends on the response
amplitude due to friction in bolted joints. This causes the free-response of the system to seem to have damping ratios
(and to a lesser extent natural frequencies) that change slowly with time. Several techniques have been proposed to
characterize such systems. This work compares a few available methods, focusing on their applicability to real
measurements from multi-degree-of-freedom systems. A beam with several small links connected by simple bolted joints
was used to evaluate the available methods. The system was excited by impulse and the velocity response was measured with
a scanning laser Doppler vibrometer. Several state of the art procedures were then used to process the nonlinear free
responses and their features were compared. First the Zeroed Early Time FFT technique was used to qualitatively evaluate
the responses. Then, the Empirical Mode Decomposition method and a simple approach based on band pass filtering were
both employed to obtain mono-component signals from the measured responses. Once mono-component signals had been
obtained, they were processed with the Hilbert transform approach, with several enhancements made to minimize the effects
of noise.
Keywords Iwan joint • System identification • Damping • Bolted joint • Nonlinear joint
22.1 Introduction
Many built-up systems consist of substructures that are assembled with bolted joints. Although some significant strides have
been made in recent years, it is still exceedingly difficult to predict the nonlinear damping behavior of bolted joints, caused
by micro- and macro-slip in the bolted joint interfaces. New experimental methods are needed to allow one to characterize
the nonlinear damping in real structures so better models can be created. In the recent literature, researchers have applied
several approaches in order to identify nonlinear damping from structures. The most common approach involves using some
form of time-frequency analysis [1]. For example, the Hilbert transform [2, 3] has been widely used to estimate the
instantaneous frequency and phase of a signal. This method is quite satisfactory for single frequency component signals
of single degree-of-freedom systems. Furthermore, the method is extended to multi-frequency component signals with the
Hilbert-Huang transform, which uses Empirical Mode Decomposition [4] to decompose the original response into several
single frequency component signals. In a more recent paper [5], the authors relate the Empirical Mode Decomposition
approach to the analytical slow flow analysis, and this approach provides a theoretical basis that promises to extend these
M.W. Sracic (*) • M.S. Allen
Department of Engineering Physics, University of Wisconsin-Madison, 535 Engineering Research Building,
1500 Engineering Drive, Madison, WI 53706, USA
e-mail: [email protected]; [email protected]
H. Sumali
Component Science and Mechanics, Sandia National Laboratories, 5800, Albuquerque, NM 87185-1070, USA
e-mail: [email protected]
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_22, # The Society for Experimental Mechanics, Inc. 2012
269
concepts to multi-degree of freedom nonlinear systems. The wavelet transform is an alternative to the Hilbert-Huang
approach. In [6], the authors used the wavelet transform to analyze free-decay time responses of a built-up beam system,
but this type of analysis becomes challenging if the damping is not very light or if the nonlinearity is strong causing the
spectra to become difficult to interpret visually. Peeters et al. [7] have proposed an important extension to this approach
where the system is excited at a specific nonlinear normal mode and allowed to freely decay along that mode. A controlled
input (e.g. a sinusoidal input from a shaker) is typically required. However, attaching a shaker adds mass and damping to the
structure and inhibits its free response.
Several methods have been suggested to identify joint properties using measured frequency response functions,
for example [8–11]. However, these methods may be sensitive to measurement noise, may only provide valid models for
certain frequency ranges, or may require one to assume some information regarding the model for the joints a priori. In any
event, these approaches rely on linear theory, so they don’t seem to be able to predict the amplitude dependent damping that
is characteristic of many systems with bolted joints.
While several methods are available to identify nonlinear models of systems with bolted joints, all of the available
methods have limitations and none has proved to be the best method in all situations. Furthermore, few of the methods have
been applied to real measurements from high order systems. This work will compare several of the most promising methods
in order to evaluate their relative merits. In order to ground the comparison in a real, yet relatively simple system, a test
structure was created that consists of a free-free steel beam (i.e. suspended with elastic strings) with several steel links
attached. The links are bolted to the beam in various combinations using various torque values, and the beam is excited by an
impulsive force and allowed to freely vibrate while its velocity response is measured with a scanning laser Doppler
vibrometer (SLDV). The SLDV is non-contact, so the ring-down responses are not affected by the sensor. The time response
is recorded as the beam freely vibrates. Additionally, the linear frequency response function is also estimated using the
impulsive force, which was also measured. Both the time histories and linear frequency response functions are used to
characterize the damping of this high order nonlinear system. First, standard experimental modal analysis is performed and
the best-fit linear damping is extracted at different bolt torques in order to get a baseline linear approximation of the damping
trends for different torques. Then, the time histories are interrogated using both the Hilbert-Huang Transform with Empirical
Mode Decomposition and single-mode band-pass filtering, in order to isolate individual frequency component signals
(Intrinsic Mode Functions). In the end, a curve-fitting procedure that was presented in [12] is used to fit the nonlinear time
dependent properties of the Intrinsic Mode Function, and the nonlinear time dependent frequency and damping is extracted.
The rest of this paper will review the theory for the methods that will be used, introduce the linked-beam experiment,
show and discuss the results from when the proposed methods are applied to the responses from the experiment, and discuss
and present some conclusions based on the applied techniques.
22.2 Theory
The free response of a general nonlinear system can be represented by the following state space equation
yðtÞ ¼ f ðx; tÞ (22.1)
where xðtÞ is the time dependent state vector and f is a nonlinear function that describes how the state and input combine to
define the response. The function f is assumed to be sufficiently smooth so that all partial derivatives are well defined.
When this is the case, the system can be linearized about specific points in the state space or about entire trajectories (e.g.
periodic orbits [13]). In general the linearized modes of a nonlinear system interact and exchange energy, so one must
consider all of the linearized modes and their nonlinear couplings to construct the free response. Indeed, this characteristic
has even been exploited to create a very effective nonlinear vibration absorber [14]. On the other hand, systems with weak
nonlinearities are frequently observed to have insignificant modal interactions, in which case the free response can be
expressed as follows. Let the response y(t) define the free-decay velocity of the system such that yj(t) ¼ vj(t) is the jthmeasured velocity, which conforms to the property just described. From here forward, the response v(t) will be assumed to
be from the jth degree of freedom and the subscript will be dropped. Then, for a quasi-linear system (i.e. a nonlinear
system with smooth nonlinearities that vary slowly with time), the free decay velocity can be represented with the
following equation
yjðtÞ ¼ vðtÞ ¼Xmr¼1
ArðtÞ cos od;rtþ ’0;r
� �(22.2)
270 M.W. Sracic et al.
where m is the number of frequencies present in the response, ArðtÞ is the time varying amplitude for the rth frequency
component, od;r is the rth damped natural frequency, and ’0;ris the rth phase variable. If the response is linear, then the
frequency will be constant and the amplitude given by ArðtÞ ¼ A0e�zron;r t, where on;r is the natural frequency and is related
to the damped natural frequency by od;r ¼ on;r
ffiffiffiffiffiffiffiffiffiffiffiffiffi1� z2
p. If the velocity response is nonlinear, then the damped natural
frequency and hence the argument of the cosine function may vary with time and the amplitude may not be a simple
exponential. The experimental methods of interest in this work seek to characterize the time dependent frequency and
damping in order to obtain the amplitude-frequency and amplitude-damping relationships for the system.
Although it will not be pursued in this work due to space limitations, the complexification approach discussed in [5]
can be used to compute the time varying amplitude and frequency of a nonlinear system from its equation of motion,
establishing a more solid theoretical foundation for (22.2).
22.2.1 Zeroed-Early Time Fast Fourier Transform
The first method considered is the Zeroed early-time fast Fourier transform (ZEFFT) that was presented recently by Allen
and Mayes [15]. This frequency domain technique was shown to be quite effective at detecting nonlinearity even in
relatively high order systems with quite severe nonlinearities. This method is briefly reviewed below.
For many systems, nonlinearities are only present when the system responds with large amplitude, so the response
becomes more linear as the response amplitude diminishes. At very low amplitude one may reach a point where the response
is linear so that the free response can be written as
vðtÞ ¼Xmr¼1
Rr expðlrtÞ þ R�r expðl�r tÞ (22.3)
where Rr and lr are the rth residue and eigenvalue, respectively, and the complex conjugate is denoted with ()*. If the
system has under-damped modes, then the complex conjugate eigenvalue pairs are defined by lr ¼ �zron;r þ iod;r where
the coefficient of critical damping, zr, has been introduced and i ¼ ffiffiffiffiffiffiffi�1p
. The Fourier transform can be used to compute
the frequency domain counterpart to (22.3) (assuming that the response is zero for t < 0).
VðoÞ ¼Xmr¼1
Rr
io� lrþ R�
r
io� l�r(22.4)
As described in [15], this is identical to the expression for the frequency response of a linear system in terms of its modes,
except that the residues in (22.4) have a different definition than the residues of a linear frequency response function. Even
then, the spectrum has the same shape as the frequency response of a linear mode near each natural frequency.
Now if the signal is artificially set to zero up to a certain time denoted tz (i.e. vz(t) ¼ 0 for t < tz and vz(t) ¼ v(t)otherwise), then the Fourier transform of vz(t) is
VzðoÞ ¼Xmr¼1
Rrelr tz
io� lrþ ðRre
lr tz�io� l�r
� �eiotz (22.5)
and the residues change to reflect the initial value of the response at time tz but otherwise the spectrum has approximately the
same shape, especially near the peaks. If the spectra are compared for various zero times (various amounts of the initial
response erased) one would see peaks that have essentially the same shape but with decreasing amplitude.
On the other hand, if the system has the nonlinearity that was described previously then the influence of the nonlinearity
will diminish as tz increases, and when one compares VðoÞ and VzðoÞ the spectrum will show that the effective frequency
(and perhaps damping) of the system has changed. Plots of the spectra versus the truncation time, tz, give a qualitative
description of the nonlinearity in the system. The method can be extended to give quantitative measures of nonlinearity
(i.e. using the backwards extrapolation for nonlinearity detection (BEND) technique described in [15]), although in this work
it will be used primarily to detect nonlinearity and evaluate it qualitatively.
22 Identifying the Modal Properties of Nonlinear Structures Using Measured. . . 271
22.2.2 Hilbert-Transform
Slow-flow analysis is often realized through the Hilbert transform, which can be applied as follows. First, the discrete-time
analytic signal is formed by augmenting the real response signal v(t) with its Hilbert transform ~vðtÞ as follows.
VðtÞ ¼ vðtÞ þ i~vðtÞ (22.6)
Note that in this case, the signal v(t) is assumed to be mono-component (contain only one frequency component), so the
‘r’ subscript is dropped from these equations. The magnitude of the analytic signal, VðtÞj j, is the envelope of the response.If the signal is mono-component then the envelope can be readily related to the damping in the system.
AðtÞ ¼ VðtÞj j (22.7)
Assuming that the damping and frequency are slowly varying functions of time, the amplitude or decay envelope can be
estimated as the following
AðtÞ ¼ A0 exp �zðtÞonðtÞð Þ (22.8)
where A0 is the initial amplitude, and the natural frequency onðtÞ and coefficient of critical damping zðtÞ are both functions
of time for a general nonlinear system. The phase can be obtained from the analytic form of the measured signal (i.e. (22.6))
using the following equation.
’ðtÞ ¼ tan�1 ~vðtÞ=vðtÞð Þ (22.9)
In order to obtain the damped natural frequency, some authors have time-differentiated the phase signal [2]. However,
most measured signals contain a certain amount of noise which can corrupt the time-differentiated signal. Following [12], an
alternative approach is to use the measured response at time instants t ¼ t0; t1; . . . ; tN�1, (N ¼ number of data points)
and fit the phase signal ’ðtÞ with a polynomial of degree, p.
’ðt0Þ’ðt1Þ...
’ðtN�1Þ
8>>><>>>:
9>>>=>>>;
¼t0p � � � t0 1
t1p � � � t1 1
..
. � � � ...
1
tN�1p � � � tN�1 1
26664
37775
bp
..
.
b1b0
8>>><>>>:
9>>>=>>>;
(22.10)
The polynomial coefficients b0; b1; . . . ; bp can be obtained by a least squares solution of the above system of equations.
Then, since the instantaneous frequency is the time derivative of the phase, the time-varying damped oscillation frequency
can be estimated as the time-derivative of the previous equation.
odðtÞ ¼ d’ðtÞdt
¼pt0
p�1 � � � 1 0
pt1p�1 � � � 1 0
..
. � � � ... ..
.
ptN�1p�1 � � � 1 0
26664
37775
bp
..
.
b1b0
8>>><>>>:
9>>>=>>>;
(22.11)
The next step is to estimate the decay envelope. Again, assuming that the signal is nonlinear, the decay envelope will be
time varying and can be well approximated with a polynomial. The coefficients of the polynomial can be calculated from the
following equation (assuming a third order polynomial).
ln V t0ð Þj jln V t1ð Þj j
..
.
ln V tN�1ð Þj j
8>>><>>>:
9>>>=>>>;
¼t03 t0
2 t0 1
t13 t1
2 t1 1
..
. ... ..
. ...
tN�13 tN�1
2 tN�1 1
26664
37775
c3c2c1
ln A0ð Þ
8>><>>:
9>>=>>;
(22.12)
272 M.W. Sracic et al.
Now the above cubic regression analysis gives a nonlinearly decaying envelope,
AðtÞ ¼ A0 exp �c1t� c2t2 � c3t
3� �
(22.13)
which implies that the following relationship holds.
zðtÞonðtÞ � CðtÞ ¼ c1 þ c2tþ c3t2
� �(22.14)
Then, using the relationship between the damped natural frequency, the natural frequency, and the coefficient of critical
damping, ðodðtÞÞ2 ¼ ðonðtÞÞ2 1� ðzðtÞÞ2� �
, the time-varying natural frequency onðtÞ can be computed.
onðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiodðtÞð Þ2 þ CðtÞð Þ2
q(22.15)
Finally, the time-varying damping ratio z(t) can be computed.
zðtÞ ¼ �CðtÞonðtÞ (22.16)
If the measured response is linear, then only the linear terms will be significant in the polynomial regressions in
(22.10)–(22.12), and the procedure identifies the constant linear natural frequency and coefficient of critical damping.
22.2.3 Empirical Mode Decomposition
In general the response of a nonlinear system is composed of oscillations of multiple different frequencies, so the method in
the previous section cannot be directly applied. One must first isolate individual oscillatory components (often called
intrinsic mode functions or IMFs) so that their time varying frequency and damping can be characterized. The Empirical
Mode Decomposition method was developed to isolate the intrinsic mode functions, which are constrained to obey two
properties. For each intrinsic mode function, the number of local extrema and the number of zero crossings must be equal or
differ by no more than one. Additionally, the envelopes defined by the local maxima and the local minima must have a mean
of zero. Using these two properties, the IMFs can be successively removed from the full, multi-component signal with
an iterative process, which is well described in a few references [1, 4]. To start the process, the measured response (i.e. vðtÞ)is interrogated for its local maxima and minima. A cubic spline is fit to the local maxima and then to the local minima to form
the upper and lower envelops, respectively. Then, the mean of these curves is calculated and designated as m1. The first
estimate for the IMF is formed by subtracting the mean signal from the original signal.
h1 ¼ vðtÞ � m1 (22.17)
The estimate may need to be refined in order to satisfy the intrinsic mode function criteria, so one iterates on the estimate
by repeating the previous process until the kth IMF estimate
h1k ¼ h1ðk�1Þ � m1k (22.18)
satisfies the criteria for an intrinsic mode function. Then, the first intrinsic mode function c1 ¼ h1k can be subtracted from the
original signal
r1 ¼ vðtÞ � c1 (22.19)
In order to start sifting for the remaining IMFs, the procedure is repeated now using the residual signals (i.e. r2 ¼ r1 � c2,. . ., rm ¼ rm�1 � cm). Once the mth residual rm is monotonic, or has only one local extremum [4], then the decomposition is
complete. The procedure is ad hoc, in general, but it often works quite well and, as was shown in [5], it can be directly linked
to the theoretical slow-flow analysis.
22 Identifying the Modal Properties of Nonlinear Structures Using Measured. . . 273
22.3 Experimental Application to Linked-Beam System
As mentioned previously, a structure was designed in order to evaluate these methods. The structure was designed to have
the following features:
• Numerous modes in the 0–2,000 Hz range.
• The modes are well separated in frequency.
• Modular attachments can be bolted to the main structure in various combinations to vary the nonlinear damping.
• The attachments do not cause modes to switch order (e.g., the third lowest mode in frequency should remain the third no
matter which links are attached). This feature would allow modal damping to be more easily related to the combination of
attachments.
• The structure can be assembled and disassembled with good repeatability.
• The joints can be readily modeled with sufficient detail using the finite element method (FEM).
• The structure can be modeled analytically with reasonable accuracy.
• The structure can be accurately measured with out-of-plane scanning LDV (so data from many measurement points could
be obtained without modifying the structure).
Figure 22.1 shows a schematic of the beam and three links that were bolted to the beam. The beam was 508 mm long,
50.8 mm wide, and 6.35 mm thick. Six through-holes were drilled on the beam’s midline for ¼-in. fine-thread bolts, which
were used to attach the three links. The bolt hole pattern started at 63.5 mm from one end, and the spacing between holes was
76.2 mm. The three links were 12.7 mm wide and 3.175 mm thick. Through-holes were also drilled near the ends of the links
for the ¼-in. bolts. The links were fastened to the beam with the bolts and a washer between each pair of facing surfaces. All
parts of the structure were made of AISI 304 stainless steel. All mating surfaces on the beam and on the links were polished
to a roughness of 0.1 mm or smoother.
The experiment was designed to minimize the effects of the boundary conditions on the measured damping. A clamped
boundary condition, for example, would cause significant damping at the clamp which could dominate the measured
damping and make it difficult to calculate the damping caused by the bolted joints. Thus, a free-free boundary condition
was chosen. The beam was excited by impacts in order to avoid using excitation hardware with surfaces that rub such as
those of attached force transducers and stingers. The cables required to attach conventional sensors (e.g. accelerometers) can
also introduce damping so the vibration responses were measured with a laser Doppler vibrometer.
The finite element method (FEM) was used to predict the natural frequencies and mode shapes of the structure. Because
FEM modal analysis cannot account for nonlinear rubbing interfaces, the interfaces were fused together in the model so that
the whole built-up structure was modelled as monolithic. The FEMmodel was used to iterate the design towards the features
mentioned above.
While the structure was designed to allow various combinations of link attachments, this paper discusses only a case
where all three links are bolted to the beam. To characterize the effects of bolt torques on modal damping, the torque on all
the six bolts was varied using 9.04 Nm (80 Lbf.in.), 10.2 Nm (90 Lbf.in.), and 12.4 Nm (110 Lbf.in.).
22.3.1 Experiment
The beam was suspended in a manner that emulated the free-free boundary condition as shown in Fig. 22.1a. Two strings
suspended the beam from overhead points. Four elastic strings kept the beam from swinging too much out-of-plane since that
would cause the laser spot to depart from the measurement point of interest. A small patch of retro-reflective tape was
Beam
Link C
Link B
Link A
Washers
View C-C: Nut
BeamLink
Bolt head
C-C
Fig. 22.1 Finite element
model of the linked-beam
274 M.W. Sracic et al.
adhered to the beam at each measurement point (see Fig. 22.3). These patches ensured that the LDV sensor received
adequate light even if the test article rotated relative to the laser.
The structure was excited with an impulsive force from a shaker-impactor as follows. An APS 400 long-stroke shaker
carried a force transducer at the end of its armature. A special waveform command was designed and applied to
the amplifier, which caused the shaker to push the force transducer into the beam then to retract quickly after impact. The
force transducer was then held in the retracted position so that the swinging of the beam did not cause a second impact. After
the rigid-body deflections of the structure dissipated, the shaker brought the force transducer close to the structure again and
the process was repeated.
The peak force of the impact was around 150 N. Upon each impact, the laser Doppler vibrometer (LDV) computed the
mobility (frequency response function) between the input and response. The sampling rate of 12,800 samples/s gave a
bandwidth of 5,000Hz. The scanning LDV recorded the mobility at 63 points shown in Fig. 22.3 using four averages at each
point. The impact location was behind point 28. Point 22 was on link B, and point 53 was on the main beam.
22.3.2 Nonlinearity Detection with Zeroed-Early Time Fast Fourier Transforms
The measurements were first interrogated using the Zeroed Early-time FFT (ZEFFT) method [15], in order to obtain a
qualitative understanding of the degree to which the structure is behaving nonlinearly. The measurements with the bolts at
a torque of 10.2 Nm were first investigated. There were 62 measurement points, so the ZEFFTs of the entire dataset were
computed and the average of the magnitude of the ZEFFTs was computed over all of the measurement points for each zero
time. Figure 22.4 shows the resulting average spectrum for various values of tz. The system is predominantly linear, so the
spectrum near each peak must be closely examined to see any sign of nonlinearity. Figures 22.5 and 22.6 show expanded
views near the first and third bending modes respectively. The former shows that the first natural frequency appears to be
constant, within the resolution of the measurement at least. The third natural frequency seems to show a very small shift from
Fig. 22.2 (a) Test structure
suspended by two vertical in-
plane strings and four out-of-
plane elastic strings and
(b) close up showing the
beam, one link with its bolted
joints, and the force transducer
on the shaker armature
Fig. 22.3 Laser Doppler
vibrometer scan points
on the structure
22 Identifying the Modal Properties of Nonlinear Structures Using Measured. . . 275
801.1 to 801.5 Hz (at tz ¼ 1.2 s. or later). All of the other peaks were similarly inspected and small, 0.5 Hz frequency shifts
were also noticeable in the modes at 1,192 and 2,871 Hz. All of the other peaks seemed to have constant frequencies.
While this analysis suggests that the frequencies of these modes remain essentially constant, visual inspection does not
reveal whether the damping is changing nonlinearly. The BEND procedure described by Allen and Mayes [15] could be used
to assess this to some extent, although this will not be pursued in this work. Since the nonlinearity seems to be quite weak,
it might be reasonable to approximate this system as linear. This is explored in the following section.
22.3.3 Linear Modal Analysis Using Low-Excitation Frequency Response Functions
Since the ZEFFT method showed weak evidence of nonlinearity, the standard approach using linear modal analysis was first
attempted to see whether a linear model might be adequate to characterize the system at each torque value. This section
discusses results from a separate modal test where the peak impact force was 50 N instead of 150 N. The purpose of the test
0 500 1000 1500 2000 2500 300010-2
10-1
100
101
102
103
Frequency - Hz
Mag
nit
ud
e
NLDetect: Composite of FFTs of Time Response
0.025977
0.32199
0.61801
0.91402
1.21
1.5061
1.8021
Fig. 22.4 Composite ZEFFT
of the response of beam at a
torque level of 10.2 Nm
135 136 137 138 139 140
101
102
103
Frequency - Hz
Mag
nit
ud
e
NLDetect: Composite of FFTs of Time Response
0.025977 0.32199
0.61801
0.91402
1.21
1.5061 1.8021
Fig. 22.5 Expanded view
of Fig. 22.4 near the natural
frequency of the first bending
mode
276 M.W. Sracic et al.
was to investigate the effect of bolt torques on linearized damping. The LDV software computes an ‘average spectrum’,
which is an average of the mobilities at all the scanned points. The resonant frequencies of the structure correspond to the
peaks of the average spectrum. In this paper, those peaks will be called ‘peaks of average mobility’ (PAM). The PAM gives a
good estimate of the natural frequencies of the structure. These estimates were useful in determining poles in the linear
experimental modal analysis (EMA). The LDV software also plotted the mobility values at all scanned points at a chosen
frequency. At each PAM, the spatial distribution of those mobility values (i.e. operating deflection shape) gives a good
estimate of the mode shape since none of the modes have close natural frequencies.
Linear curve fitting was performed using accelerances (computed from the mobilities) from the measurements at each
torque level. ATA Engineering’s AFPoly software was used to perform the curve fitting. The low end of the analysis
frequency band was set to 30 Hz to exclude all rigid-body modes caused by the soft free-free suspension cords. The mode
indicator function (MIF) and ‘stability’ diagram showed estimates of where the natural frequencies are likely to be. In almost
all cases, the peaks of the MIF were collocated with the trains of poles on the stability diagram. These frequencies were very
close to the PAM from the LDV software. Thus, for the most part the EMA was straightforward. Appendix A shows the MIF
and stability diagram, along with sketches of the mode shapes. Near 2,500 and 4,000 Hz there is strong indication of a mode.
However, observation of the shape from the PAM led to the conclusion that the modes around those two frequencies involve
primarily in-plane motion. Because the links were attached to only one side of the beam, the in-plane motion resulted in a
little out-of plane motion. Those modes are probably not well measured by the laser and hence will not be analyzed here.
EMA was difficult near certain frequencies because the three nominally identical links resonate at similar frequencies.
The combinations of one, two and three links resonating together created a high modal density; five modes were found
between 2,870 and 3,015 Hz. In that range, the PAM and stability diagram had to be used together to identify the modes.
Despite the high modal density, the five modes in that range were identified. For example, Fig. 22.7 shows good agreement
between accelerances from measurement (solid curves) and from the synthesis (dashed curves) of the identified modes for
the bolt torque case of 9.04 Nm. A careful comparison does show differences between the reconstruction and the
measurements that are on the order of 10% of the peak, but differences such as this can arise for many reasons and they
are not large enough that one would typically call the linear model into question.
Table 22.1 shows the modal damping ratios for 21 modes with three bolt torques: ‘low’ torque of 9.04 Nm, ‘medium
torque’ of 10.2 Nm, and ‘high’ torque of 12.4 Nm. The plots of most of the mode shapes are from a finite element analysis
(FEA). The rest of the plots (modes 4c, A1, etc.) came from the spatial distribution of mobility amplitudes at the PAM,
because the monolithic FEA did not predict those modes. (The modes in question seemed to depend strongly on
the characteristics of the interfaces in the joints.) The modes are shown in Table 22.1 from top to bottom in the order
of increasing natural frequencies. The bars in the chart show the identified linear modal damping of each mode as percentage
of critical damping. Three bars are shown for each mode: the bar on the top represents damping for the low bolt torque,
the middle bar for the medium torque, and the bottom bar for the high torque.
799 800 801 802 803 804 805 806 807
10-1
100
101
Frequency - Hz
Mag
nit
ud
e
NLDetect: Composite of FFTs of Time Response
0.025977
0.32199
0.61801
0.91402
1.21
1.5061
1.8021
Fig. 22.6 Expanded view of
Fig. 22.4 near the natural
frequency of the third bending
mode
22 Identifying the Modal Properties of Nonlinear Structures Using Measured. . . 277
Mode 4c had much higher damping than the rest of the modes, so its damping was multiplied by 0.1 prior to displaying it
on the plot. This mode is of questionable accuracy; its shape is very similar to mode 4, but it has a strong local motion of link
C. This mode could be an artifact caused by processing nonlinear measurements with linear frequency response estimation
and modal analysis procedures.
Using Table 22.1, one can assess the effect of bolt torques on modal damping. The damping for these 13 modes clearly
decreases as the bolt torques increase. In five other modes, the low torque still gives higher damping than the high torque, but
the medium torque gives the lowest or highest damping among the three torques. Three modes (1, 4c and 6) exhibit damping
that increases with higher torque, which is the opposite trend from other modes.
22.3.4 Nonlinear Modal Analysis Using Free Ring-Down History
The linear modal analysis in the preceding section gave an estimate of the modal damping of the structure. The following
section discusses three techniques to study the nonlinear damping. The free response velocity due to 150 N peak impacts was
measured at each of the points in Fig. 22.3 for 2.56 s at all three of the torque levels. Figure 22.8 shows the response at the
torque level of 10.2 Nm for point 28, which is at the same location as the impactor. The dashed blue curve is the measured
response. There is a considerable amount of low frequency vibration in the response (likely due to rigid body motion of the
beam on the suspension system). The response was high-pass filtered with an eighth order Butterworth filter and a cut off
frequency of 30 Hz. The filtered response is plotted with the red curve, which seems to remove the low frequency motion.
This signal will be used as a benchmark in this paper, although there were many other measurement points available. The
bolted joints in this beam provide very complicated nonlinear damping relationships, and before the spatial information
provided by the numerous measurement points can be taken into account we seek first to characterize a single response.
Figure 22.9 shows the FFT magnitude of the velocity response of point 28 (after filtering and drop out bridging). There are
many sharp peaks in the response out to 12 kHz, so the signal certainly has multiple frequency components.
The response of point 28 is characteristic of many of the responses measured on the beam. There are many modes
involved in each response, but most of the modes are well separated. The goal is to determine which modes contain
nonlinearity and to try to identify the time dependent properties of those nonlinearities. Since the free velocity response
contains multiple significant modes, it is necessary to first isolate mono-component signals which contain the time
dependent properties of a single mode.
22.3.4.1 Application of Empirical Mode Decomposition
In order to isolate mono-component signals, the Empirical Mode Decomposition (EMD) method was applied to the free
velocity response. For this paper, the EMD implementation from Ortigueira that is available on MATLAB Centrals File
exchange was used, and this algorithm was based on [16]. The EMD procedure extracted 12 intrinsic mode functions (IMFs)
2850 2900 2950 3000 305010-1
100
101
102
103
104
Frequency, Hz
|Acc
eler
atio
n/F
orce
|, m
/s2 /
N
22 Meas22 Synth53 Meas53 Synth
Fig. 22.7 Accelerances at
points 22 (higher magnitudes)
and 53 (lower magnitudes) in
the 2,850–3,050 Hz range for
bolt torque case 9.04 Nm;
solid curves are frommeasurements; dashed curvesare AFPoly synthesis
278 M.W. Sracic et al.
from the free velocity response. Figure 22.10 shows plots of the first three IMFs. The time domain signals are plotted
in Fig. 22.10a, and the frequency domain signals are plotted in Fig. 22.10b from 0 to 7,000 Hz. The spectra of the first IMF
(top plot of (b)) still contains several significant peaks near 4,000, 5,600, and 6,600 Hz as well as several less coherent peaks
at lower frequencies. The spectrum of the second IMF contains one dominant peak near 400 Hz, and the third IMF contains
two dominant peaks near 130 and 360 Hz.
The EMD algorithm sifted through the components of the free velocity response and extracted several signals that
contained significantly less frequency content. The second IMF, for example, seems to contain only one major component.
However, as seen in the first and third IMF, the signals still contain the effects of several modes. Moreover, the EMD process
adds a certain amount of broad band noise to the responses, which can be seen in the frequency spectra of the IMF signals.
Table 22.1 Beam with three links ABC, various bolt torques
# Shape 1
2
T1
3
4
4c *0.1 T2
5
6
T3
A1
T4
A+ B-C- B+ C- B1
7
T5
8
T6
7b
9
Bar chart on the right depicts the critical damping ratio for each mode for low torque ¼ 9.04 Nm (blue bar), medium torque ¼ 10.2 Nm
(green bar), and high torque ¼ 12.4 Nm (red bar)Mode 4c was much more heavily damped than the rest, so its damping ratios were multiplied by 0.1 in order to fit on the bar chart
The table on the left depicts the corresponding mode shapes
22 Identifying the Modal Properties of Nonlinear Structures Using Measured. . . 279
0 0.5 1 1.5 2 2.5-0.2
-0.1
0
0.1
0.2
0.3
time, s
Poi
nt 2
8 ve
loci
ty, m
/s
Drive Point (28) Velocity Ringdown
MeasuredHP Filt, 30Hz
Fig. 22.8 Free velocity response of point 28: measured (dashed blue), high-pass filtered with a 30 Hz cut off frequency (red)
10−1
10−2
10−3
10−4
10−5
IFF
T(V
eloc
ity)
I, m
/s
10−6
10−7
0 2000 4000 6000Frequency, Hz
FFT of Response 28
8000 10000 12000
Fig. 22.9 FFT magnitude of the free velocity response of point 28
0 0.5 1 1.5 2 2.5 3-0.05
0
0.05
Vel
ocity
, m/s
IMF 1
0 0.5 1 1.5 2 2.5 3-0.05
0
0.05
Vel
ocity
, m/s IMF 2
0 0.5 1 1.5 2 2.5 3-0.05
0
0.05
time, s
Vel
ocity
, m/s IMF 3
0 1000 2000 3000 4000 5000 6000 7000
10-5
|Vel
ocity
|, m
/s
IMF 1
0 1000 2000 3000 4000 5000 6000 7000
100
|Vel
ocity
|, m
/s IMF 2
0 1000 2000 3000 4000 5000 6000 7000
10-5
Frequency, Hz
|Vel
ocity
|, m
/s IMF 3
a b
Fig. 22.10 Application of Empirical Mode Decomposition where the first three intrinsic mode functions are shown: (a) time domain,
(b) frequency domain
Since there are numerous modes in the free velocity response, it is possible that the EMD procedure has
difficulty isolating all of the individual components. The algorithm may perform better if the response was first low-pass
or high-pass filtered to reduce the number of modes that need to be separated. The authors are exploring these ideas, but
the Hilbert transform method used in this paper performs best when the signal contains only one frequency component,
so the EMD signals were not processed further. Instead, a different sifting procedure was implemented as described in the
next section.
22.3.4.2 Band-Pass Filtering of Individual Modes
Next, the authors applied a single mode filtering procedure where different Butterworth filters were applied to the time
domain free velocity response in order to isolate individual frequency components in the signal. The ZEFFT analysis did not
show much frequency shift in any of the modes, so the stiffness of the system is predominantly linear. Hence, it is unlikely
that any of the modes contain harmonic peaks at higher frequencies, but the spectra can always be visually scanned to see
if this type of the nonlinear phenomenon is occurring. The free velocity response did not show signs of harmonics of any of
the modes occurring at higher frequencies, so it was assumed that each mode could be isolated with a single bandpass filter.
Figure 22.11 shows the results when the first two modes were isolated using this filtering procedure. First, the free velocity
response was filtered with a tenth order low-pass Butterworth filter with a cut-off frequency of 240 Hz. The frequency
spectra of the measured signal and the filtered signal are shown in Fig. 22.11a. The filtered time domain signal is shown
10−10
10−10
10−8
10−8
10−6
10−6
10−4
10−4
10−2
10−2
100
100
0.08
0.05
−0.05
0
0.06
0.04
0.02
−0.02
−0.04
−0.06
−0.08
0
0 500 1000
|Vel
ocity
|, m
/s|V
eloc
ity|,
m/s
Vel
ocity
, m
/sV
eloc
ity,
m/s
1500
Filtered Response
Filtered Response
Measured Response
Measured Response
2000 0 1 20.5 1.5 2.52500
0 500 1000 1500 2000 0 1 20.5 1.5 2.52500
Filtered Mode 1a
c d
b Filtered Mode 1
Filtered Mode 2 Filtered Mode 2
Frequency, Hz
Frequency, Hz
time, s
time, s
Fig. 22.11 Single mode filtering of the free velocity response of point 28: (a) tenth order low-pass Butterworth filter for Mode 1 with cutoff
frequency of 240 Hz, (b) filtered time domain response of Mode 1, (c) fifth order band-pass Butterworth filter for Mode 2 with cutoff frequencies
300 and 450 Hz, (d) filtered time domain response of Mode 2
22 Identifying the Modal Properties of Nonlinear Structures Using Measured. . . 281
in Fig. 22.11b. In order to isolate the second mode, the measured signal was filtered with a fifth order band-pass Butterworth
filter with cut-off frequencies of 300 and 450 Hz. The frequency spectra and the time domain response are shown
in Fig. 22.11b,c, respectively.
The single mode filter approach effectively isolates a single mode and produced a mono-component time domain signal
for both the two modes that are shown. This could be applied to most of the modes in the free velocity response, especially
since the modes are well separated. For a structure with close modes, a different approach may be needed. For example a
band of close modes could first be isolated using a band-pass filter similar to the one used for Mode 2 above. Then, one might
try Empirical Mode Decomposition on the resulting signal, and it may be more effective since the overall number of modes
in the band is likely to be much fewer than the measured signal. These ideas are being explored for future work.
22.3.4.3 Hilbert Transform with Curve Fitting
Once one has isolated mono-component signals from the measured response, the Hilbert transform curve fitting approach
described in Sect. 22.2.2 can be applied to extract the nonlinear time dependent properties of the system. First, the phase
signal from (22.6) is fit with a 5th degree polynomial. The time derivative of the polynomial fit is equal to the damped natural
frequency, and this can be formed with (22.11). Then, the amplitude of the Hilbert transformed signal can be fit with a cubic
polynomial in order to extract the time dependent damping. Figure 22.12 shows the Hilbert transformed velocity Amplitude
(a) and phase (b) as well as the curve fits of those signals. Figure 22.13 shows the time dependent natural frequency
and damping that were extracted from the curve fits. The natural frequency changes by less than 1% over the course of the
free response, however, this is likely spurious since the modes were found to be predominantly linear in frequency.
The damping changes a significant amount over time, and as shown in Fig. 22.14, the damping is clearly nonlinear since
it is a function of the amplitude of the response. The same procedure can be followed for the filtered signal of Mode 2.
The damping-amplitude relationship of Mode 2 is plotted in Fig. 22.15.
0 1 2 3-500
0
500
1000
1500
2000
2500
Ang
le (
rad)
time, s
Curve Fit of Phase
Hilbert-Phase5th Degree Polynomial
0 1 2 310-8
10-7
10-6
10-5
Mag
nitu
de
time, s
Curve Fit of Amplitude
Hilbert-AmplitudeCubic Polynomial
a bFig. 22.12 Curve fit of the
amplitude (a) and phase
(b) of the Hilbert transform
of the filtered Mode 1
0 1 2 3135.8
135.9
136
136.1
136.2
Nat
ural
Fre
quen
cy, H
z
Time, s
Natural Frequency vs Time
0 1 2 30
0.05
0.1
0.15
0.2
0.25
zeta
, % o
f Crit
ical
Time, s
Damping vs Timea bFig. 22.13 Time varying
natural frequency and
damping that were extracted
from the curve fits in
Fig. 22.12
282 M.W. Sracic et al.
22.3.4.4 Discussion
The Empirical Mode Decomposition method and the single mode filtering method were both used to isolate
mono-component signals from the measured response. The single mode filtering method was found to produce very clean
signals with a single, distinct frequency component. The identified nonlinear damping functions had clear amplitude
dependence, and those functions were different due to the differences in filtered modes. Since the beam has well spaced
modes, the single mode filtering approach could be used to identify the nonlinear damping functions on the remainder of the
modes. For other systems with closely space modes, a combination of band-pass filtering and Empirical Mode Decomposi-
tion may produce the mono-component signals needed to apply the Hilbert transform curve fitting method. In any event,
it seems that these tools can be readily applied to characterize the damping nonlinearity in this structure.
22.4 Conclusions
This paper reviewed a few methods for system identification of nonlinear systems with slowly time-varying nonlinear
properties, especially damping. In particular, the Empirical Mode Decomposition method was compared with a method
based on band-pass filtering around each resonance, in order to obtain single mode responses that could be processed using
the Hilbert transform. The Zeroed Early-Time FFT (ZEFFT) was also discussed and applied to the measurements, and was
found to allow one to quickly and quite robustly identify which modes had natural frequencies that were amplitude
dependent. That method was not explored further in this work, although it might be helpful in obtaining more quantitative
results in the future. The Hilbert transform method seemed to provide the most convenient approach for quantifying the
10-7 10-6 10-5-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
Amplitude
zeta
, % o
f crit
ical
Damping vs. AmplitudeFig. 22.14 Nonlinear
damping plotted versus the
amplitude of the Hilbert
transformed signal of Mode 1
10-8 10-7 10-60
0.02
0.04
0.06
0.08
0.1
0.12
Amplitude
zeta
, % o
f crit
ical
Damping vs. AmplitudeFig. 22.15 Nonlinear
damping plotted versus the
amplitude of the Hilbert
transformed signal of Mode 2
22 Identifying the Modal Properties of Nonlinear Structures Using Measured. . . 283
dependence of the frequency and damping on time (and hence amplitude), however the signal of interest must be first
decomposed into mono-component signals. The Empirical Mode Decomposition method and a band-pass filtering method
were both used, but the former gave quite unsatisfactory results and was not pursued further. This may not be a fault of the
algorithm; the authors are certainly not experts and were at the mercy of a particular implementation of the EMDmethod. On
the other hand, simple band-pass filtering was quite effective for the system considered here, although it is difficult to be sure
that the filter has not distorted the signal of interest. In any event, once a mono-component response was available the Hilbert
transform method was quite effective.
The proposed methods were applied to free velocity response measurements of a linked-beam system that contains bolted
joints. The ZEFFTs of the measured responses showed little sign of variation in the natural frequencies of the modes, and this
seems to imply that the stiffness of the system is predominantly linear. Therefore, linear modal analysis was performed at
different torque values to assess any trends in damping versus torque. Some trends were clearly observed but it is difficult to
assess the accuracy of the damping measurement in each configuration and to determine whether the trends observed, where
damping both increased and decreased with increasing torque, were meaningful. The measurements were then processed
with Empirical Mode Decomposition and with band-pass filtering in order to isolate individual modes. The Empirical Mode
Decomposition method was not successful in extracting mono-component signals from the measurements, perhaps because
so many modes were excited in the free response. The band-pass filtering approach worked very well to isolate mono-
component signals, since this system has well spaced modes. Once the modes were isolated, the Hilbert transform curve fit
approach identified significantly nonlinear damping from the measurements, and the damping-amplitude relationships were
displayed. Despite the initial success of the band-pass filtering and Hilbert transform approach, all of the methods are being
investigated further to understand which methods work best in a variety of situations.
Acknowledgment Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned
subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-
AC04-94AL85000.
284 M.W. Sracic et al.
Appendix A
Linear Curve Fitting of the 9 Nm Data
with ATA Engineering’s AFPoly
Mode shapes and “Stability Diagram” are shown below.
# Shape Modal Indicator Function (curve) and pole orders (squares or triangles)
1
2
T1
3
4
4c
T2
5
6
T3
A1
T4
A+ B-C-
B+ C- B1
7
T5
8
T6
7b
9
22 Identifying the Modal Properties of Nonlinear Structures Using Measured. . . 285
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9. KimW-J, Lee B-Y, Park Y-S (2004) Non-linear joint parameter identification using the frequency response function of the linear substructure.
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11. Tsai J-S, Chou Y-F (1988) The identification of damping characteristics of a single bolt joint. J Sound Vib 125:487–502
12. Sumali H, Kellogg RA (2011) Calculating damping from ring-down using Hilbert transform and curve fitting. In: Presented at the 4th
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Process 25:2705–2721
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essentially nonlinear attachment. Commun Nonlinear Sci Numer Simulat 15:2617–2633
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286 M.W. Sracic et al.
Chapter 23
Nonlinear System Identification of the Dynamics
of a Vibro-Impact Beam
H. Chen, M. Kurt, Y.S. Lee, D.M. McFarland, L.A. Bergman, and A.F. Vakakis
Abstract We study the dynamics of a cantilever beam with two rigid stops of certain clearances by performing nonlinear
system identification (NSI) based on the correspondence between analytical and empirical slow-flow dynamics. First, we
perform empirical mode decomposition (EMD) on the acceleration responses measured at ten, almost evenly-spaced,
spanwise positions along the beam leading to sets of intrinsic modal oscillators governing the vibroimpact dynamics at
different time scales. In particular, the EMD analysis can separate any nonsmooth effects caused by vibro-impacts of
the beam and the rigid stops from the smooth (elastodynamic) response, so that nonlinear modal interactions caused by
vibro-impacts can be explored only with the remaining smooth components. Then, we establish nonlinear interaction models
(NIMs) for the respective intrinsic modal oscillators, where the NIMs invoke slowly-varying forcing amplitudes that can be
computed from empirical slow-flows. By comparing the spatio-temporal variations of the nonlinear modal interactions for
the vibro-impact beam and those of the underlying linear model (i.e., the beam with no rigid constraints), we demonstrate
that vibro-impacts significantly influence the lower frequency modes introducing spatial modal distortions, whereas the
higher frequency modes tend to retain their linear dynamics in between impacts.
Keywords Nonlinear system identification • Empirical mode decomposition • Vibro-impact beam • Intrinsic mode
oscillation • Nonlinear interaction model
23.1 Introduction
Experimental modal analysis based on Fourier transforms (FTs) has beenwell established based on the assumption of linearity
and stationarity of the measured signals (see, for example [1]). In many practical situations, however, the measured data is
likely to exhibit strong nonlinearity and nonstationarity, particularly when the tested systems involve nonlinearities due to
complexity caused by multi-physics nonlinear interactions [2]. In addition, FT-based methods are not able to properly isolate
and extract nonlinearity and nonstationarity from themeasured data, frequently leading to wrong conclusions (for example, to
misinterpretations of internal and combination resonances as natural frequencies). As a result, there has been the need for an
effective, straightforward, system identification and reduced-ordermodelingmethod for characterizing strongly nonlinear and
nonstationary, complex, multi-component systems in multi-physics applications. Reviews of nonlinear system identification
(NSI) and reduced-order modeling (ROM)methods are provided in [3, 4]. Typical nonparametric NSImethods include proper
orthogonal decomposition (POD, also known as Karhunen-Loeve decomposition [5–8]), smooth orthogonal decomposition
[9], Volterra theory [10, 11], Kalman filter [12], and so on. As for the methods of nonlinear parameter estimation, we mention
H. Chen • Y.S. Lee (*)
Department of Mechanical and Aerospace Engineering, New Mexico State University, 1040 S. Horseshoe St., Las Cruces, NM 88003, USA
e-mail: [email protected]; [email protected]
M. Kurt • A.F. Vakakis
Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, 104 S. Wright St., Urbana, IL 61801, USA
e-mail: [email protected]; [email protected]
D.M. McFarland • L.A. Bergman
Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206 W. Green St., Urbana, IL 61801, USA
e-mail: [email protected]; [email protected]
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_23, # The Society for Experimental Mechanics, Inc. 2012
287
the restoring force surface method [13], NARMAX (Nonlinear Auto-Regressive Moving Average models with eXogenous
inputs) methods [14], harmonic balance method [15], methods based on Hilbert transform [16, 17], and others.
Use of POD has been rather popular in studying system identification and nonlinear normal modes of coupled beams [18]
and rods [19], and in structural damage detection [20]. For example, the method of POD has been utilized for studying
chaotic vibrations of a 10-degree-of-freedom (DOF) impact oscillator and a flexible-beam impact oscillator [21, 22]. In these
studies, the spatial structure of impacting responses under a harmonic excitation of the boundary was demonstrated to be
close to what can be obtained by averaging over many impulse-response tests on the linear system (even if the system is
strongly nonlinear). Moreover, POD was applied for model reduction of a vibro-impact (VI) rod [23], and also for extracting
dominant coherent structures of a VI beam from experimental time-series data [24] with the goal to eventually derive low-
dimensional ROMs through a Galerkin reconstruction process based on the extracted mode shape functions.
We note, however, that these techniques are only applicable to specific classes of dynamical systems; in addition, some
functional form is assumed for modeling the system nonlinearity. Recently, a nonlinear system identification (NSI) method
with the promise of broad applicability was proposed by Lee et al. [25]. This method was based on empirical mode
decomposition (EMD) [26], under the key assumption that the measured time series can be decomposed in terms of a finite
number of oscillating components. These are in the form of fast (nearly) monochromatic oscillations modulated by slowly
varying amplitudes. The empirical slow-flow model of the dynamics is obtained from EMD, and its correspondence with the
analytical slow-flow model has been established [27], paving the way for constructing physics-based local nonlinear
interaction models (NIMs) [28]. A NIM consists of a set of intrinsic modal oscillators (IMOs) that can reproduce the
measured time series over different time scales and account for (even strongly) nonlinear modal interactions across scales.
Hence, it represents a local model of the dynamics, identifying specific nonlinear transitions. By collecting energy-
dependent frequency behaviors from all identified IMOs, a frequency-energy plot can be constructed, which depicts global
features of the dynamical system. The method requires no a priori system information but only measured (or simulated) time
series; i.e., it is purely an output-based approach. Applications of the proposed NSI methodology have been provided with
studies of targeted energy transfers in a 2-DOF dynamical system [28], instability generation and suppression in a 2-DOF
rigid aeroelastic wing model [29], and the dynamics of a rod coupled to an essentially nonlinear end attachment [30].
In this paper, we explore the nonlinear dynamics of a VI beam (whose setup is similar to that used in [24]) by performing
the aforementioned NSI method [25] to reveal coherent structures (e.g., Dawes [31]) in terms of IMOs of strongly nonlinear
dynamics due to vibro-impacts. Study of such systems will provide essential dynamical features of structures with defects
with applications to structural health monitoring and damage detection (e.g. [32, 33]). For this purpose, this paper has the
following structure. Section 2 provides a discussion of the VI beam model including geometry, measurement locations,
method of excitation, (linearized) natural frequencies and mode shapes; in Sect. 3 the proposed NSI method is applied to the
numerically-obtained acceleration data of the VI beam and those of the underlying linearized beam for comparison
purposes; then, concluding remarks are provided in Sect. 4.
23.2 System Descriptions
We consider the uniform, homogeneous cantilever beam (made of steel with the density r ¼ 7850 kg/m3 and Young’s modulus
E ¼ 200 GPa) depicted in Fig. 23.1, with dimensions Lxhxt ¼ 1.311 � 0.0446 � 0.008 m so that the cross-sectional area and
the second moment of area with respect to the z axis are A ¼ 3.57 � 10�4 m2 and Izz ¼ 1.9 � 10�9 m4, respectively (we refer
to Fig. 23.1 for a definition of the system of axes). Table 23.1 summarizes the positions of the accelerometers x1–x10 along thebeam span, the position of the laser displacement sensor xLDS, and the placement of the two symmetric rigid stops xSTP causingvibro-impacts. The leading ten natural frequencies (theoretical and experimental) on in Hz are listed in Table 23.2, with
the corresponding normalized mode shape functions fn(x/L), n ¼ 1,. . ., 10, being presented in Fig. 23.2 [34, 35].
1x 2x 3x 4x 5x 6x 7x 8x 9x 10xLDSx
x
y
y
zh
t1x 2x
3x 4x
5x 6x 7x 8x9x 10x
LDSx
STPx( )p t
( , )v x t
L
z
x
Fig. 23.1 Experimental setup for the VI beam: x1–x10, xLDS, and xSTP respectively denote the spanwise locations of the accelerometers, of the laser
displacement sensors, and of the rigid stops
288 H. Chen et al.
Two clearance levels between the cantilever beam and the rigid stops are considered, namely, infinite clearance
corresponding to the case of the linear cantilever beam, and 4 mm clearance corresponding to the case of the strongly
nonlinear VI beam. Experimental procedures for measuring time series data involve (1) applying impulsive excitation p(t)by varying magnitude at position x3 by means of an impact hammer, selecting the excitation frequency band by using several
types of tips on the impact hammer (e.g., plastic, rubber and metal), and (2) measuring the resulting accelerations at x1�x10and the displacement at xLDS.
In thisworkwe utilize numerically generated acceleration signals froma reduced-ordermodel based on the assumed-modes
method, and such numerical solutions are updated and validated by the experimental measurements. That is, the beam
was excited at each node with an impact hammer, and averages of four measurements were taken at each node; from
the resulting 100 transfer functions, the leading ten mode shapes, modal damping factors, and natural frequencies were
obtained and used to update the assumed-modes model. In the assumed-modes method the analytical natural frequencies
were replaced with the measured ones, and numerically simulated time series were obtained by solving the reduced
system of differential equations. Details of this computation can be found in [34]. We remark that the 5th mode, whose
linearized natural frequency is equal to 209 Hz, has a node at x9, which is located very close to the point of vibro-
impacts xSTP. Furthermore, the impulsive excitation is applied at location x3, which is also close to another node of the
5th mode. As shown below, this will affect the results of EMD analysis used for reconstructing the 5th mode at those
particular points (i.e., there will arise issues of observability) in the sense that the flexible dynamics of the beam at
these locations is small and consequently is dominated by the vibro-impacts (non-smooth effects). Similar observations
apply for the 8th mode, which possesses a node near the excitation point (x3).
23.3 Nonlinear System Identification of the VI Beam
In this section we apply the NSI methodology to two typical cases: (1) The linear beam (i.e., a cantilever beam with infinite
clearances at the impact boundaries); and (2) the vibro-impact (VI) beam with 4 mm symmetric clearances. In particular, by
comparing the system identification results for the VI beam to those of the linear beam, we study the effects of the strongly
Table 23.1 Positions of the accelerometers, rigid stops and laser displacement sensors of the VI beam
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 xSTP xLDS
Positions (mm) 131 263 395 527 657 787 917 1,052 1,215 1,311 1,185 1,230
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
−0.5
0
0.5
1
fn(x
/L
)
x/L
f1 f2 f3 f4 f5 f6 f7 f8 f9 f10
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
Fig. 23.2 Experimental setup for the VI beam: x1–x10, xLDS, and xSTP respectively denote the spanwise locations of the accelerometers, of the laser
displacement sensors, and of the rigid stops
Table 23.2 The leading ten linear natural frequencies in Hz for the beam in Fig. 23.1
o1 o2 o3 o4 o5 o6 o7 o8 o9 o10
Theoretical 3.8 23.8 66.6 130.5 215.7 322.2 450.0 599.1 769.5 961.2
Experimental 3.7 23.2 64.9 126.9 209.4 314.7 433.9 580.7 751.3 926.7
23 Nonlinear System Identification of the Dynamics of a Vibro-Impact Beam 289
nonlinear dynamics induced by the vibro-impacts. Typically, we will consider the acceleration signals at position x9 for thiscomparison because the VI effects are expected to be most significant there due to its proximity to the impact position. We
also illustrate overall spatio-temporal variations of the beam dynamics caused by vibro-impacts. The basic elements of the
NSI methodology are referred to [25].
23.3.1 Linear Beam
By linear beam we mean the cantilever beam in Fig. 23.1 without any rigid stops (or with impacting boundaries of infinite
clearances). Then, since the beam is homogeneous and uniform, we can assume that its transverse vibrations can be
approximately governed by the Bernoulli-Euler beam model with the following equation of motion,
rA€vðx; tÞ þ EIzzv0000ðx; tÞ ¼ pðtÞdðx� x3Þ (23.1)
where v x; tð Þ denotes the displacement of the beam in the transverse (y) direction at ðx; tÞ (cf. Fig. 23.1); pðtÞ ¼ P0dðtÞ is theimpulsive excitation at t ¼ 0, where dðtÞ and dðxÞ denote Dirac delta functions; and primes and dots are partial differentia-
tion with respect to x and t, respectively. Then, the general solution for (23.1) can be written as
vðx; tÞ ¼X1
m¼1AmfmðxÞe�zmomt cosðomdt� ymÞ (23.2)
where omd ¼ omð1� z2mÞ1=2; om is the natural frequency of the m-th linear bending mode; zm is the modal damping factor
(when a certain viscous damping is assumed in the system); and fmðxÞ is the normalized mode shape function for the m-th
mode (cf. Fig. 23.2). The corresponding acceleration can be written as
aðx; tÞ€vðx; tÞ ¼X1
m¼1�AmfmðxÞe�zmomt cosðomdt� �ymÞ (23.3)
where �Am ¼ Amo2m and �ym ¼ ym þ 2tan�1½ð1� z2mÞ1=2=zm�.
Consider now the acceleration response of the linear beam at position x9 depicted in Fig. 23.3. The wavelet and Fourier
transforms clearly depict the ten dominant fast frequencies identified from experimental modal analysis (see Table 23.2).
−1000
−500
0
500
1000
a(x
9,t)
Fre
quen
cy (
Hz)
Time (s)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
500
1000
0 100 200 300 400 500 600 700 800 900 1000
100
0 10 20 30
10−5
Frequency (Hz)
Frequency (Hz)Frequency (Hz)200 210 220
10−8
10−7
10−6
(A) (B)
Close−up (A)
Close−up (B)
Fig. 23.3 Wavelet and Fourier transforms of the acceleration for the linear beam at position x9
290 H. Chen et al.
As discussed in Sect. 2, the harmonic at 209 Hz appears to be negligible because the position x9 is close to one of the nodesfor the 5th mode. We write
a9ðtÞ � aðx9; tÞ �X10
m¼1�Amfmðx9Þe�zmomt cosðomt� �ymÞ (23.4)
for small damping zm. By means of EMD analysis, we wish to obtain the relation between the acceleration time series and the
intrinsic mode functions (IMFs) such that
a9ðtÞ �X10
m¼1cmða9; tÞ �
X10
m¼1�Amfmðx9Þe�zmomt cosðomt� �ymÞ (23.5)
where cmða9; tÞ denotes the m-th IMF of the acceleration at position x9 (and is usually associated with the m-th normal mode
vibration that can be observed at the same position of the beam). Figure 23.4 depicts the ten dominant IMFs from the
advanced EMD analysis algorithm introduced in [27], demonstrating that the relation (23.5) is valid except for the 5th mode
of the beam due to observability issues. We establish the reduced-order model (ROM) for the acceleration in Fig. 23.3 for the
linear beam dynamics at position x9 in terms of intrinsic modal oscillators (IMOs). That is, we write the IMO corresponding
to each IMF [27], and the instantaneous slowly-varying envelope and phase of the m-th IMF are computed respectively as
AmðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficmða9; tÞ2 þ H½cmða9; tÞ�2
q; ymðtÞ ¼ tan�1f H½cmða9; tÞ�=cmða9; tÞg � omt (23.6)
where m ¼ 1,. . .,10. Since the slowly-varying complex forcing amplitude LmðtÞ is computed from the time series (or IMF)
in an effort to match the solution xmðtÞ of the IMO with the corresponding IMF, we can write xmðtÞ � cmða9; tÞ. During this
validation process the damping factor 0<zm<1 is chosen such as to minimize the error between xmðtÞ and cmða9; tÞ. Then, theoriginal response can be reconstructed as the sum of all IMO solutions; that is, the following expression holds:
a9ðtÞ �X10
m¼1xmðtÞ �
X10
m¼1cmða9; tÞ (23.7)
Figure 23.5 compares the 10th and 5th IMFs with the corresponding IMO solutions, exhibiting good agreement. Finally,
we consider the physical meaning of the complex-valued forcing function LmðtÞ for the m-th IMO of the linear problem,
since such a term is known to be associated with nonlinear modal interactions in nonlinear dynamical systems [28, 29]. In
our linear beam problem, the slowly-varying envelope AmðtÞ and phase ymðtÞ can be identified from (23.5) as
AmðtÞ ¼ �Amfmðx9Þe�zmomt; ymðtÞ ¼ ��ym ¼ constant (23.8)
Then, the slow-flow variable can be expressed as
’mðtÞ � jom�Amfmðx9Þe�j�yme�zmomt ) _’mðtÞ � �jzmo
2m�Amfmðx9Þe�j�yme�zmomt (23.9)
If zm ¼ zm (i.e., the damping factor in the IMO is the same as the modal damping factor identified from experimental
modal analysis, and carries a direct physical meaning), then we can easily show thatLmðtÞ � 0. This idea may sound feasible
and reasonable, because the resulting reduced-order model will be the same as that obtained from the typical linear modal
analysis with the coordinates, xm;m ¼ 1; � � � ; 10; being the modal coordinates. Furthermore, the solution for the IMO will
appear as a free damped response, which may naturally satisfy the relation in (23.5). However, as is the case for many other
nonlinear system identification methods where it is of more interest to check whether the proposed parametric model is able
to reproduce the measured (or simulated) dynamics, the damping factor in the IMO is not necessarily the same as the
physical one (i.e., zm 6¼ zm, in general). In this case, the complex forcing amplitude LmðtÞ can be expressed as
LmðtÞ � jðzm � zmÞo2m�Amfmðx9Þe�j�yme�zmomt (23.10)
The absolute value of the complex number (23.10) is a monotonically and exponentially decaying function; such forcing
function will not generate any modal interactions (as is supposed to be the case for a linear system). Nonetheless, the solution
for the IMO, which is strongly driven by the forcing LmðtÞejomt because zm � zm, can approximately reproduce the IMF in
(23.5). Similar discussions can be made not only for the response at position x9, but also for those at all other positions alongthe linear beam.
23 Nonlinear System Identification of the Dynamics of a Vibro-Impact Beam 291
−50
0
50
c 8(a
9,t
)F
requ
ency
(Hz)
Time (s)0 0.2 0.4 0.6 0.8 1
0
500
1000
≈ 580Hz
Time (s)
−500
0
500
c 9(a
9,t
)F
requ
ency
(Hz)
0 0.2 0.4 0.6 0.8 10
500
1000
≈ 750Hz
−500
0
500
c 10(a
9,t
)F
requ
ency
(Hz)
Time (s)0 0.2 0.4 0.6 0.8 1
0
500
1000 ≈ 926Hz
−100
0
100
c 7(a
9,t
)F
requ
ency
(Hz)
Time (s)0 0.2 0.4 0.6 0.8 1
0
500
1000
≈ 436Hz
−50
0
50
c 6(a
9,t
)F
requ
ency
(Hz)
Time (s)0 0.2 0.4 0.6 0.8 1
0
500
1000
≈ 314Hz
−5
0
5c 5
(a9,t
)F
requ
ency
(Hz)
Time (s)0 0.2 0.4 0.6 0.8 1
0
200
400
600
≈ 209Hz
−20
−10
0
10
20
c 4(a
9,t
)F
requ
ency
(Hz)
Time (s)0 0.2 0.4 0.6 0.8 1
0
50
100
150
200
250
≈ 125Hz
−50
0
50
c 3(a
9,t
)F
requ
ency
(Hz)
Time (s)0 0.2 0.4 0.6 0.8 1
0
50
100
150
200
250
≈ 64Hz
−10
0
10
c 2(a
9,t
)F
requ
ency
(Hz)
Time (s)0 0.2 0.4 0.6 0.8 1
0
50
100
≈ 23Hz
−4
−2
0
2
4
c 1(a
9,t
)F
requ
ency
(Hz)
Time (s)0 0.2 0.4 0.6 0.8 1
0
10
20
30
40
≈ 4Hz
a b
c d
e f
g h
i j
Fig. 23.4 The ten dominant IMFs extracted from the acceleration response in Fig. 23.3: (a) through (j) sequentially depict the tenth to first
IMFs, respectively
292 H. Chen et al.
23.3.2 Vibro-Impact Beam
We now consider the cantilever beam in Fig. 23.1 but with the two symmetric rigid stops of 4 mm clearances at position xSTP.If the displacement jvðxSTP; tÞj<4mm, then the dynamics of the beam is linear and can be described by (23.1). Whenever the
beam displacement jvðxSTP; tÞj ¼ 4mm, a vibro-impact occurs resulting in a new impact load �pðtÞdðx� xSTPÞ applied to the
beam as well as causing energy dissipation due to inelastic impact. Mathematically speaking, the nonsmoothness due to the
vibro-impacts means that the displacement response is of class C0 (i.e., continuous but not continuously differentiable). For
this strongly nonlinear nonsmooth dynamical system, there is no closed-form solution available, in general. Furthermore,
such a VI dynamical system may possess a very complicated topological structure of periodic orbits (e.g., see [36]). This is
mainly because nonsmooth dynamical systems may involve complicated dynamics such as grazing bifurcations [37] and
chaos [21]. We wish to model and understand the nonlinear dynamics of the VI beam by applying the proposed NSI method.
As for the case of the linear beam problem of the previous section, we consider the acceleration signal at position x9(depicted in Fig. 23.6), where the effects of vibro-impacts generate multiple broadband perturbations in the wavelet
transforms. In particular, comparing the Fourier transform of the linear beam response (dashed line) with that of the VI
beam, this broadband excitation of the beam due to vibro-impacts is significant. Figure 23.7a depicts the numerically
computed displacement and the corresponding impact load on the beam at xSTP in order to identify the instants of vibro-
impacts (i.e., the time instants when the beam displacement at xSTP reaches the thresholds4 mm). It was shown in [38] that
the nonlinear modal interactions due to vibro-impacts are purely due to the smooth parts of the VI dynamics, whereas the
nonsmooth parts tend to create frequency-energy relations involving numerical artifacts. Such numerical artifacts could lead
to wrong conclusions regarding the nonlinear resonances involved in the nonlinear modal interactions between the measured
IMFs. Furthermore, it was demonstrated that the smooth parts of the VI dynamics can be obtained by separating the
nonsmooth effects by means of EMD analysis [38]. Typically, the nonsmooth part is computed as the first IMF with the help
of masking and mirror-image signals [27]. The characteristics of the nonsmooth IMF were explored in previous works by
relating them to Fourier series expansions of saw-tooth wave signals [39], and also by a partial-differential-equation-based
sifting process [38] noting that EMD acts, in essence, as a dyadic filter bank.
Figure 23.7b depicts such a nonsmooth IMF for the acceleration signal in Fig. 23.6. Superposition of the impact instants
identified from Fig. 23.7a illustrates that the isolated nonsmoothness agrees reasonably well with the time instants of vibro-
impacts. We note that the numerical displacement was calculated from the reduced-order model through the assumed-modes
method, which means that some other modes higher than 10th may need to be included to get a better match between the
numerical simulations and experimental measurements. Some quantitative discrepancies prevail after 0.2 s with the current
reduced-order model.
Now EMD is applied to the remaining smooth part of the acceleration signal after subtracting the nonsmooth IMF in
Fig. 23.7b from the original acceleration in Fig. 23.6. The ten dominant IMFs are depicted in Fig. 23.8. By superimposing the
vertical dashed line at each impact instant identified from Fig. 23.7a, one can at least qualitatively observe the effects of
vibro-impacts on each IMF at position x9; for example, the vibro-impacts seem to directly influence higher IMFs (above the
5th). Indeed, considering these higher frequency IMFs we note linear dynamical behavior between consecutive vibro-
impacts, in the form of exponentially decaying damped responses. On the other hand, lower IMFs do not seem to exhibit
such straightforward patterns, implying that these IMFs may undergo more strongly nonlinear modal interactions and may
be more significantly influenced by the strong nonlinearities due the vibro-impacts.
0 0.2 0.4 0.6 0.8 1−600
−400
−200
0
200
400
600
Time (s)
c10(a9; t) IMOClose−up
Close−up
0 0.2 0.4 0.6 0.8 1−4
−2
0
2
4
Time (s)
c5(a9; t) IMO
Close−up
Close−up
a b
Fig. 23.5 Comparison of the IMFs in Fig. 23.4 with their corresponding IMO solutions: (a) tenth IMF; (b) fifth IMF
23 Nonlinear System Identification of the Dynamics of a Vibro-Impact Beam 293
As in the linear beam, we can also establish a nonlinear interaction model (NIM) for the IMFs obtained in Fig. 23.8 in
terms of a set of IMOs. Computing the nonlinear modal interaction forcing LmðtÞ from each IMF by means of the slow-flow
correspondence, we solve the ten IMOs respectively. Figure 23.9 compares the IMFs with the corresponding IMO solutions
for 10th and 5th IMFs, which show good agreement. We sum all IMO solutions to reconstruct the original signal, and this
exhibits a perfect match as depicted in Fig. 23.10. That is, the NIM we established has been validated so that it can be used to
study the nonlinear dynamics of the VI beam (at position x9) as an alternative reduced-order model.
Now, the physical meaning of Lmðak; tÞ;m; k ¼ 1; � � � ; 10; in the nonlinear dynamics of the VI beam can be explored by
comparing it with that for the linear beam. We first note that the magnitude of Lmða9; tÞ for all IMOs of the linear beam
appears as almost a straight line on a logarithmic scale (cf. Fig. 23.11), which makes sense due to the form of (23.10).
−4000
−2000
0
2000
4000
a(x
9,t
)
0 500 1000 1500
100
Frequency (Hz)
Fre
quen
cy(H
z)
Time (s)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
500
1000
1500
Fig. 23.6 Wavelet and Fourier transforms of the acceleration for the VI beam at x9 (the Fourier transform in Fig. 23.3 is superimposed as a dashed
line to illustrate the effects of vibro-impacts in frequency domain)
−5
0
5
v(x
ST
P,t
) (m
m)
t1 t2,3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13
0 0.2 0.4 0.6 0.8 1−2
−1
0
1
2−2
Time (s)
p(t
) (k
N)
t1t2
t3t4
t5t6
t7t8
t9t10
t11t12
t13
−2000
−1000
0
1000
2000
c NS(a
9,t
)
t1 t2,3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13
Fre
quen
cy (
Hz)
Time (s)0 0.2 0.4 0.6 0.8 1
0
500
1000
1500
a b
Fig. 23.7 Depiction of the vibro-impacts: (a) the displacement response of the VI beam simulated at position xSTP and its corresponding impact
loads on the beam from the rigid stops; (b) the nonsmooth component of the acceleration in Fig. 23.6 is decomposed via EMD analysis (note that
the dashed lines at t ¼ tk; k ¼ 1; � � � ; 13 imply the impact instants identified from the impact force �pðtÞ)
294 H. Chen et al.
−2000
0
2000
c 10(
a9,t
)F
requ
ency
(Hz)
Time (s)0 0.2 0.4 0.6 0.8 1
0
500
1000
1500
≈ 926Hz
−1000
−500
0
500
1000
c 9(a
9,t
)F
requ
ency
(Hz)
Time (s)0 0.2 0.4 0.6 0.8 1
0
500
1000
1500
≈ 750Hz
−500
0
500
c 8(a
9,t
)F
requ
ency
(Hz)
Time (s)0 0.2 0.4 0.6 0.8 1
0
500
1000
1500
≈ 580Hz
−200
−100
0
100
200
c 7(a
9,t
)F
requ
ency
(Hz)
Time (s)0 0.2 0.4 0.6 0.8 1
0
500
1000
1500
≈ 436Hz
−100
−50
0
50
100
c 6(a
9,t
)F
requ
ency
(Hz)
Time (s)0 0.2 0.4 0.6 0.8 1
0
100
200
300
400
500
≈ 314Hz
−50
0
50c 5
(a9,t
)F
requ
ency
(Hz)
Time (s)0 0.2 0.4 0.6 0.8 1
0
100
200
300
400
500
≈ 209Hz
−100
0
100
c 4(a
9,t
)F
requ
ency
(Hz)
Time (s)0 0.2 0.4 0.6 0.8 1
0
100
200
300
400
500
≈ 125Hz
−100
0
100
c 3(a
9,t
)F
requ
ency
(Hz)
Time (s)0 0.2 0.4 0.6 0.8 1
0
50
100
150
200
≈ 64Hz
−50
0
50
c 2(a
9,t
)F
requ
ency
(Hz)
Time (s)0 0.2 0.4 0.6 0.8 1
0
50
100
≈ 23Hz
−2
0
2
c 1(a
9,t
)F
requ
ency
(Hz)
Time (s)0 0.2 0.4 0.6 0.8 1
0
10
20
30
40
50
≈ 3.5Hz
a b
c d
e f
g h
i j
Fig. 23.8 The ten dominant IMFs extracted from the acceleration response in Fig. 23.6: (a) through (j) sequentially depict the 10th to 1st IMFs,
respectively (note that the dashed lines imply the impact instants identified in Fig. 23.7)
23 Nonlinear System Identification of the Dynamics of a Vibro-Impact Beam 295
Similarly, jLmða9; tÞj for the VI beam can also exhibit linearity with the same slope on average on a logarithmic scale as in
the case of the linear beam, but such a linear pattern appears only in between impact instants and, in particular, when m 6
(cf. Fig. 23.11a for the 10th IMO). The trajectory of L10ða9; tÞ in the complex plane for the linear beam appears as a single,
monotonic, decaying pattern (i.e., time-like behavior on a logarithmic scale), which implies no modal coupling or
interactions in the ROM. The trajectory of L10ða9; tÞ in the complex plane for the VI beam also exhibits such monotonic
behavior but only in between vibro-impacts (denoted by the intervals In; n ¼ 1; 2; � � � ); the role of the vibro-impacts is to
cause phase shifts of the slowly-varying forcing L10ða9; tÞ at the instants of vibro-impacts. On the other hand, the slowly-
varying complex forcing function for the 4th IMO of the VI beam does not exhibit any linear behavior but only a slowly-
varying wavy envelope regardless of vibro-impacts (cf. Fig. 23.11b). Such wavy patterns in the plot of jL4ða9; tÞj indicatethat certain modal interactions occur through nonlinear resonant conditions such as internal resonances or resonance
captures [29]. Also, nonlinear modal interactions are evidenced by the spiral (or non-time-like) patterns of the trajectory
in the complex plane.
From these two typical examples, we may conjecture the following: Whereas the higher IMOs (i.e., the IMOs associated
with higher frequency components) tend to maintain their linear dynamics in between impacts (although the overall
dynamics is strongly nonlinear), the lower IMOs exhibit strongly nonlinear modal interactions independent of vibro-impact
patterns. The role of vibro-impacts is just to exert broadband impulsive excitations on the linear beam causing instantaneous
phase shifts in the higher IMOs. To verify this conjecture we first compute Pearson’s linear correlation coefficient [40] for
the slowly-varying complex forcing amplitudes Lmðak; tÞ;m; k ¼ 1; � � � ; 10; of all IMOs for the linear and VI beams at all
the positions along the beam. This correlation coefficient is widely utilized in statistics as a measure of the linear dependence
between two variables, and a Matlab command, ‘corr.m’, was used in this work.
Figure 23.12 depicts the interpolated contour map of the absolute value of the linear correlation coefficient for each mode
number along the beam span. Note that by ‘mode number’ m in Fig. 23.12 we mean the IMO which is associated with the
0 0.2 0.4 0.6 0.8 1−3000
−2000
−1000
0
1000
2000
3000
Time (s)
c10(a9; t) IMO
0 0.2 0.4 0.6 0.8 1−50
0
50
Time (s)
c5(a9; t) IMO
a b
Fig. 23.9 Comparison of the IMFs in Fig. 23.8 with their corresponding IMO solutions: (a) tenth IMF; (b) fifth IMF
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−5000
0
5000
Time (s)
a(x
9;t
)
Original Reconstructed
Fig. 23.10 Comparison of the reconstructed acceleration from the ten IMO solutions plus the nonsmooth IMF with the original response
in Fig. 23.6
296 H. Chen et al.
m-th linear mode; hence, there is no such continuous distribution with respect to the vertical axis. From these simple
calculations, we find that the IMOs higher than the 4th possess strong linear dependence (linear correlation coefficient above
90%) between the linear and VI responses of the beam, regardless of the position along the beam. Again, it is noted that the
low correlation for the 3rd, 5th, 7th and 9th IMOs at the midspan of the beam is due to the fact that the position is very close
to one of the nodes for the respective linear modes. Similar explanations can be given for the 5th and 7th IMOs at position x9,and for the 8th IMO at position x8. Therefore, the aforementioned conjecture is confirmed by means of the linear correlation
coefficients between jLmðak; tÞj (and hence the corresponding IMO responses) for the linear and VI beams. That is, vibro-
impacts do not significantly alter the linear dynamics for the higher modes (typically, higher than 4th), but they significantly
affect the lower modes through strongly nonlinear modal interactions. This result agrees with Cusumano’s previous work
[22], where the topological characterization of the spatial structure of the VI beam vibrations was studied by means of the
two-point spatial correlation (i.e., correlation dimension) and POD. In particular, the estimate for the correlation dimension
of the VI dynamics obtained was lower than but near four, which dictates that a low-dimensional model can capture the
overall complicated, chaotic-like VI beam dynamics. Furthermore, if such complicated dynamics can be captured by a low-
dimensional model with several lower IMOs, then energy transfers (or cascades) from the higher to the lower modes through
certain nonlinear modal interactions, such as internal resonances, may be responsible [22].
While the linear correlation coefficient provides excellent physical insights into the VI beam dynamics, we note that it can
be regarded as a static global measure; that is, it does not contain any information regarding the temporal variations of the
vibro-impacts throughout the beam. A high linear correlation coefficient for certain IMOs at some position may indicate a
strong linear dependence between the VI beam and the underlying linear beam, and imply that the corresponding IMO of the
VI beam behaves linearly for that position. Nonetheless, this will not be apparent in the local dynamics (e.g., propagation
and/or localization of the effects of nonlinear modal interactions caused by vibro-impacts, such as the temporal localization
of the nonlinear dynamics for the 7th IMO). In particular, the linear correlation becomes a poor measure when the issue of
observability is involved (e.g., all the odd-number IMOs higher than 3rd at position x5 in Fig. 23.12).
b
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
106
107
108
Time; t (s)
|Λ4(a
9;t
)|
−1 −0.5 0 0.5 1
x 108
−4
−2
0
2
4
6
8
x 107
Re Λ4(a9;t )Im
Λ4(a
9;t
)
Linear Beam
VI BeamLinear Beam
VI Beam
I1 I2
I3I4
I1
I2
I3
I4
I5I5
a
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110
6
107
108
109
1010
1011
Time; t (s)
|Λ10
(a9;t
)|
−2 −1 0 1 2 3 4
x 1010
−2
−1.5
−1
−0.5
0
0.5
1x 10
10
Re Λ10(a9;t )
ImΛ
10(a
9;t
)
Linear Beam
VI Beam
I1 I2
I3
I4
I5
I6
Linear Beam
I6
I5
I4
I1
I2
I3
VI Beam
Fig. 23.11 Comparison of the slowly-varying forcing functionsL10ða9; tÞ: (a)m ¼ 10 (10th IMO) and (b)m ¼ 4 (4th IMO) (Note that the dashedlines imply the impact instants identified in Fig. 23.7)
23 Nonlinear System Identification of the Dynamics of a Vibro-Impact Beam 297
23.4 Conclusions
We presented the dynamics of a cantilever beam with two symmetric rigid stops with prescribed clearances by performing
nonlinear system identification (NSI) based on the correspondence between analytical and empirical slow-flow dynamics.
Performing empirical mode decomposition (EMD) analysis of the numerically-computed acceleration responses at ten,
almost evenly-spaced, spanwise positions along the beam, we constructed sets of intrinsic modal oscillators at different time
scales of the dynamics. In particular, the EMD analysis can separate nonsmooth effects due to vibro-impacts between the
beam and the rigid stops from the underlying smooth dynamics of the flexible beam, so that nonlinear modal interactions can
be explored only based on the remaining smooth components. Then, we established nonlinear interaction models (NIMs) for
the respective intrinsic mode oscillations, where the NIMs invoke slowly-varying forcing amplitudes (or nonlinear modal
interaction terms) that can be computed from empirical slow-flows and directly dictate nonlinear modal interactions between
different-scale dynamics. By comparing the spatio-temporal variations of the nonlinear modal interactions for the vibro-
impact beam and the corresponding linear beam model, we demonstrated that vibro-impacts significantly influence the lower
intrinsic mode functions through strongly nonlinear modal interactions, whereas the higher modes tend to retain their linear
dynamics between impacts. Also, computation of linear correlation coefficients as measures for linear dependence between
the dynamics of the linear and VI beams manifested the same results but only with spatial information about this correlation.
Acknowledgments This material is based upon work supported by the National Science Foundation under Grants Number CMMI-0927995 and
CMMI-0928062.
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23 Nonlinear System Identification of the Dynamics of a Vibro-Impact Beam 299
Chapter 24
Modeling of Subsurface Damage in Sandwich Composites
Using Measured Localized Nonlinearities
Sara S. Underwood and Douglas E. Adams
Abstract Composite materials are being used more frequently in commercial and military aircraft structures. Many
nondestructive techniques have been developed to inspect composite materials for subsurface damage; however, many of
these existing inspection techniques aim to detect either linear changes in the material properties of the composite or
geometrical changes in the material to determine the presence of damage. Subsurface damage in a sandwich composite panel
is tested using a scanning laser vibrometer, and nonlinear vibration response properties are identified in the forced frequency
response of the composite panel. The nonlinear behavior identified in the composite panel is applied to a homogeneous,
isotropic beam model such that the forced frequency response of localized damage in the beam resembles the behavior
measured in the sandwich composite panel. Stiffness and damping nonlinearities induced locally in the analytical model are
used to show that multi-amplitude frequency response functions may be used as a means of detecting nonlinear behavior
attributed to composite damage in a composite material. The results of the analytical model show that the nonlinear behavior
due to damage displays a global behavior in an analysis of the frequency response of the system and is able to be identified
locally when consideration of the linear system dynamics are taken into account.
Keywords Composite damage • Multi-amplitude frequency response functions • Nonlinear behavior • Scanning laser
vibrometry
24.1 Introduction
Composite materials are being used more frequently in commercial and military aircraft structures. For example, honeycomb
core sandwich composites are being used in fixed wing structures, floor panels, rotor blades, and other parts of air vehicles in
which high strength and low weight material properties are needed. A drawback to using sandwich composite materials for
aircraft applications is that damage in composite materials often occurs beneath the surface, making it difficult to inspect these
aircraft using visual or line-of-sight techniques. Many existing methods for inspecting composite materials for subsurface
damage involve using linear or geometrical changes in the structural properties of the composite, typically measured between
undamaged and damaged states, as an indicator of damage. A nonlinear approach has the potential to remove the dependence
on historical baseline data or a reference specimen with manufactured defects to detect damage. Several researchers, for
example, [1–3], have applied nonlinear methods to detect damage in composite structures, which rely on the assumption that
damage displays localized nonlinear properties.
Nonlinearmethods for detecting damage in compositematerials have significant advantages over linearmethods in that they
are less susceptible to variations in environmental conditions and may be used to detect damage without the need for historical
baseline measurements or reference standards. Thesemethods address the nonlinear nature of damage; however, there is not an
agreement in the literature on the nature of the nonlinear behavior identified in the vicinity of subsurface composite damage.
An understanding of the type of nonlinear behavior seen in the vicinity of composite damage, such as one which can be applied
through measurements of forced harmonic excitation, may help guide inspection methods for composite materials.
S.S. Underwood (*) • D.E. Adams
Purdue University, School of Mechanical Engineering, Center for Systems Integrity, 1500 Kepner Drive, Lafayette, IN 47905, USA
e-mail: [email protected]; [email protected]
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_24, # The Society for Experimental Mechanics, Inc. 2012
301
In this paper, nonlinear behavior in the vicinity of core crack damage is investigated on a fiberglass sandwich panel. A
scanning laser vibrometer is used to collect frequency response measurements from the surface of the panel as it is excited at
multiple amplitudes of excitation. High and low amplitude frequency response functions measured with the laser vibrometer
are used to investigate and identify nonlinear behavior at the damage location. A single degree-of-freedom model is used to
confirm the nonlinear behavior identified in the frequency response function measurements and then the identified
nonlinearities are used to simulate damage in an analytical finite element model of a Bernoulli-Euler beam. The analytical
model is used to study the effects of the localized nonlinearities on the vibration response of the structure and the ability to
discern the damage location by considering the localized nature of composite damage identified through multi-amplitude
frequency response functions.
24.2 Experimental Investigation
In order to investigate nonlinear behavior due to subsurface damage in a composite material, a fiberglass, honeycomb core
sandwich panel, which resembles the material commonly used in rotor blade trailing edge structures, was obtained and
damaged. The panel was made of 0.060 in. fiberglass facesheets and a 0.50 in. polypropylene honeycomb core with a 10 mm
cell size. The panel measured 84 in. long by 6.5 in. wide and was clamped at the top and bottom to a rigid frame. Core crack
damage, such as that shown in Fig. 24.1a was introduced to the panel by placing a cut in the honeycomb core at the location
indicated by the small, solid rectangle in Fig. 24.1b.
A piezoelectric actuator (PCB model 712A02) with an attached 50 g mass was attached to an impedance head (PCB
model 288D01). The actuator stack was mounted to an aluminum block which was attached to the panel at a skewed angle in
order to excite the structure in multiple directions simultaneously. A three-dimensional scanning laser vibrometer (PCB
PSV-400-3D) was used to measure the surface velocity of the panel in three orthogonal directions as an excitation was
applied through the actuator. Frequency response functions relating the input force to the panel, measured through the
impedance head, to the output surface velocity of the panel in the transverse, lateral, and longitudinal directions were
obtained from the laser vibrometer measurements.
Three measurement points on the surface of the fiberglass panel were considered for analysis of the frequency response
behavior of the panel in the vicinity of composite damage. The locations of these points are indicated by the dark, medium,
and light colored circles in the panel diagram shown in Fig. 24.1b. The actuator was set to a sine sweep excitation from 100
to 5,000 Hz and laser vibrometer measurements were taken at high and low amplitudes of excitation. For each measurement
Fig. 24.1 Fiberglass panel (a)
core crack damage and (b)
diagram showing the damage
location and measurement
points (dark, medium, andlight colored circles)considered in the experimental
investigation
302 S.S. Underwood and D.E. Adams
point and excitation amplitude, five measurement averages were taken where the sample time for each measurement was
1.28 s and the frequency resolution was 781.25 mHz. The frequency response functions measured in the transverse direction
were considered for analysis of nonlinear behavior in the vicinity of damage.
24.3 Nonlinear Behavior of Damage
The frequency response functions obtained from high and low amplitude excitation levels were compared in order to
determine if nonlinear behavior is discernable in the vibration response of the panel in the vicinity of the core crack damage.
By using multi-amplitude frequency response functions, nonlinear behavior is identified by determining measurement points
where the measured frequency response functions change due to a change in excitation amplitude. The frequency response
function is an input–output relationship, and in a nonlinear dynamic system, the output response does not change
proportional to changes in input excitation as it does in a linear dynamic system. This non-proportional change in the
nonlinear case leads to a change in the measured frequency response function. Therefore, nonlinear behavior is able to be
identified when frequency response functions collected from multiple amplitudes of excitation are compared.
The high and low amplitude frequency response function measurements considered in the analysis of the frequency
response behavior of the panel in the vicinity of composite damagewere scaled and plotted simultaneously, alongwith a scaled
difference between the two values. This result is shown in Fig. 24.2 for the three measurement points depicted by the
corresponding colors in Fig. 24.1b. In Fig. 24.2, the bold line of each the dark, medium, and light colored lines representsthemeasured high amplitude response while the thin line represents the low amplitude response and the dashed line representsthe difference between the two responses. As seen in Fig. 24.2, nonlinear behavior, which is identified by large differences
between the high and low amplitude frequency response functions, occurs for each of the measurement points considered. In
addition, the point directly on the damage location, depicted by the dark colored lines, shows a significantly larger change infrequency response behavior than the points on the edge and away from the damage location, depicted by themedium and lightcolored lines, respectively.
The nonlinear behavior identified in the high and low amplitude frequency response comparison shown in Fig. 24.2
shows trends that resemble behavior which is expected in the presence of stiffness and damping nonlinearities. A single
degree-of-freedommodel was used to identify the form of the identified nonlinear stiffness and damping behavior seen in the
measured frequency response functions. A simple spring-mass-damper system, with mass M, stiffness K, and damping C,
was used for the single-degree-of freedom model. Newmark’s method was used to evaluate the response when an impulse
input was used to excite the mass. The parameters used in the model are shown in Table 24.1.
Frequency ranges from the frequency response comparison shown in Fig. 24.2, where the nonlinear behavior was most
apparent, were selected for further analysis. Figure 24.3a shows a frequency range where nonlinear stiffness behavior is
400 450 500 550 600 650 700 750 8000
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Frequency (Hz)
Mag
nitu
de (
mm
/s/N
)
Low Amplitude Response
High Amplitude Response
Difference (Scaled)
Low Amplitude Response
High Amplitude Response
Difference (Scaled)
Fig. 24.2 Frequency response function comparison for three measurement points on the fiberglass panel
24 Modeling of Subsurface Damage in Sandwich Composites Using Measured Localized Nonlinearities 303
apparent. Here, the frequency response functions are diverging, and the largest difference between them is seen to occur
before the peak in the response. Similar behavior was simulated in the single-degree of freedom model using a hardening
cubic stiffness nonlinearity. This result is shown in Fig. 24.3b. The nonlinearity causes an upward shift in frequency of the
resonance frequency between the high and low amplitude simulations. The largest difference between the simulated
frequency response functions occurs prior to the resonance peaks as is seen in the measured frequency response data.
In addition to the nonlinear stiffness behavior identified in the frequency response comparison, nonlinear damping
behavior was also identified. Figure 24.4a shows an example of a frequency range where nonlinear damping behavior is
apparent. Here, the amplitude of the frequency response functions differs with the high amplitude response significantly
lower in magnitude than the low amplitude response. The largest difference between the responses is seen to occur over the
peak area in the response. Similar behavior was simulated in the single-degree of freedom model using a cubic damping
nonlinearity. This result is shown Fig. 24.4b. The nonlinearity causes a decrease in the magnitude of the response with no
affect on frequency, and the largest difference between the frequency response functions is seen at the resonance frequency.
This behavior is similar to what is seen in the measured frequency response data.
The analysis of the measured frequency response functions for the core cracking damage mechanism using a single
degree-of-freedom model for comparison showed that the data displays behavior similar to behavior seen for cubic stiffness
and cubic damping nonlinearities. With these nonlinearities, the internal forces in the structure can be described by:
f u; _uð Þ ¼ C _uþ Kuþ m1u3 and f u; _uð Þ ¼ C _uþ Kuþ m2 _u
3 (24.1)
for the cubic stiffness nonlinearity and the cubic damping nonlinearity, respectively. In (24.1), C and K represent the
damping and stiffness matrices of the structure, respectively, u and _u represent the displacement and velocity of the structure,
respectively, and m1 and m2 represent the respective coefficients for the nonlinearities.
a
400 410 420 430 440 4500
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Frequency (Hz)
Mag
nitu
de
(m
m/s
/N)
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
1.2x 10-3
Frequency Ratio, f/fn
Ma
gnitu
de
(m/s
/N)
Low AmplitudeHigh AmplitudeDifference (Scaled)
b
Fig. 24.3 (a) Frequency response comparison displaying a stiffness nonlinearity and (b) the single degree-of-freedom model result with a cubic
stiffness nonlinearity
Table 24.1 Parameters for the
single degree-of-freedom modelParameter Value
Mass, M (kg) 5
Stiffness, K (N/m) 1.0E + 07
Damping, C (N-s/m) 1840
g (Newmark parameter) 1/2
b (Newmark parameter) 1/6
Nonlinear stiffness constant, m1 (N/m3) 2.0E + 20
Nonlinear damping constant, m2 (N-(s/m)3) 4.0E + 10
304 S.S. Underwood and D.E. Adams
24.4 Analytical Model
A finite element model was developed in Matlab in order to qualitatively understand the frequency response behavior of a
beam-like structural component with localized nonlinearities as the excitation amplitude is varied. An isotropic, homoge-
nous beam, such as that shown in Fig. 24.5, was used to develop the analytical model. The properties of the beam were
determined through micromechanical methods such that the first few bending modes of the beam matched the first few
bending modes of the panel used in the experimental investigations. The properties used in the model are given in
Table 24.2.
Subsurface damage
xa
d
n(u,u)• u(x,t)f(a,t)
Subsurface damageSubsurface damage
L
Fig. 24.5 Beam model with a localized nonlinearity n u; _uð Þ to represent damage
Table 24.2 Model properties
selected to simulate the fiberglass
sandwich panel
Property Value
Density, r (kg/m3) 460
Young’s modulus, E (Pa) 2.3e9
Beam length, L (m) 2.032
Beam width, b (m) 0.1
Beam height, h (m) 0.017
Cross-sectional area, A (m2) 0.0017
Area moment of inertia, I (m4) 4.1e-8
Mass proportional damping constant, a (s�1) 0.005
Stiffness proportional damping constant, b (s) 0.00005
a
560 570 580 590 600 610 6200
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Frequency (Hz)
Mag
nitu
de (
mm
/s/N
)
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
1.2x 10-3
Frequency Ratio, f/fn
Mag
nitu
de (
m/s
/N)
Low AmplitudeHigh AmplitudeDifference (Scaled)
b
Fig. 24.4 (a) Frequency response comparison displaying a damping nonlinearity and (b) the single degree-of-freedom model result with a cubic
damping nonlinearity
24 Modeling of Subsurface Damage in Sandwich Composites Using Measured Localized Nonlinearities 305
The beam was modeled using Ne Bernoulli-Euler beam elements with two nodes per element. This divided the beam into
Ne finite length segments of length Le ¼ L Ne= . For a beam response u x; tð Þ given as a function of the position x along the
beam and the time t, the discretized equation of motion for the finite element model of the beam is given by:
M½ � €uf g þ C½ � _uf g þ K½ � uf g ¼ ff g � n uf g; _uf gð Þf g (24.2)
In (24.2), the mass and stiffness matrices, [M] and [K], respectively, are determined by the Bernoulli-Euler beam
formulation, and a proportional damping matrix [C] is given by:
C½ � ¼ a M½ � þ b K½ � (24.3)
In (24.3), a and b are mass proportional and stiffness proportional damping constants, respectively. Since each
Bernoulli-Euler beam element has two nodes per element, there are four degrees of freedom per element, leading to a
total of 2Ne þ 2 degrees of freedom for the assembled finite element model. A fixed boundary condition was selected to
resemble the boundary condition used in the experimental investigation, giving 2Ne � 2 degrees of freedom of interest.
Two applied forces were used, including an applied forcing function represented by the vector ff g, and a nonlinear
forcing function represented by the vector n uf g; _uf gð Þf g. The applied forcing function models the excitation source as a
point excitation, fa(t), on the nodes of one or more elements, where the excitation is applied to the transverse degrees of
freedom at a location a along the beam. The nonlinear forcing function simulates damage and was applied across a single
element at a location d along the beam such that Newton’s third law of equal and opposite forces is enforced across the nodes
of the element on which the nonlinearities are applied. The cubic stiffness and damping nonlinearities identified using the
single degree-of-freedom model in the previous section were used for the nonlinear forces to represent composite damage.
The applied and nonlinear force vectors are given by:
ff g ¼
0
..
.
faðtÞ...
0
8>>>>>>><>>>>>>>:
9>>>>>>>=>>>>>>>;
2Ne�2x1
and n uf g; _uf gð Þf g ¼
0
..
.
�m1 uj � uk� �30
þm1 uj � uk� �3
..
.
0
8>>>>>>>>>>>>><>>>>>>>>>>>>>:
9>>>>>>>>>>>>>=>>>>>>>>>>>>>;
2Ne�2x1
or n uf g; _uf gð Þf g ¼
0
..
.
�m2 _uj � _uk� �30
þm2 _uj � _uk� �3
..
.
0
8>>>>>>>>>>>>><>>>>>>>>>>>>>:
9>>>>>>>>>>>>>=>>>>>>>>>>>>>;
2Ne�2x1
(24.4)
In (24.4), m1 and m2 are coefficients for the stiffness or damping nonlinearity, respectively, and the subscripts j and krepresent the nodes which the damaged element connects.
The number of elements selected for the finite element model simulation was Ne ¼ 13. For the fixed boundary condition,
this gave a total of 24 degrees of freedom. Two excitation types were studied, including an impulse input and a step sine
input. In both cases, the excitation was applied at a single node to a transverse degree of freedom. In each simulation, a single
element was damaged, and the excitation force was applied at a transverse degree of freedom away from the applied
nonlinear force simulating the damage. Newmark’s method, using Newmark parameters g ¼ ½ and b ¼ 1/6, was used to
evaluate the time response of the beam, and frequency response functions relating the displacement at each degree of
freedom to the applied excitation force were determined for high and low amplitudes of excitation.
24.5 Model Results
High and low amplitude excitations were applied to the analytical model, and frequency response functions were determined
from the simulated response. Figure 24.6a shows the high and low amplitude frequency response functions obtained at a
single node for the cubic stiffness nonlinearity case simulated with an impulse excitation. The linear and nonlinear restoring
force curves for this case are depicted in Fig. 24.6b. Comparison of the high and low amplitude frequency response functions
306 S.S. Underwood and D.E. Adams
for this case shows that the cubic stiffness nonlinearity applied locally to a single element, had a global effect on the system.
Each transverse degree of freedom experienced a change in frequency response between the high and low amplitude
simulations.
A damage index computed by taking difference between the high and low frequency response functions across a selected
frequency range did not clearly indicate the damage location. Further investigations led to the hypothesis that the linear
response of the beam may mask the nonlinear response, making the localized nonlinearities difficult to detect. In the
frequency domain, the simulated output, {X}, in the model can be represented by:
Xf g ¼ H½ � Ff g þ FNf g½ � (24.5)
In (24.5), [H] is the frequency response function relating the simulated output to the simulated input, {F} is the frequencydomain spectrum of the linear applied input force, and {FN} is the spectrum of the nonlinear applied force used to
simulate damage. For an input location q, an output location p, and a nonlinear location n, the relationship used to compare
high and low amplitude frequency response functions, Hpqhigh and Hpqlow, respectively, is given by:
Hpqhigh � Hpqlow ¼ Xphigh
Fqhigh� Xplow
Fqlow¼ HpqFqhigh
Fqhigh� HpqFqlow
Fqlow
� �linear
þ HpnFnhigh
Fqhigh� HpnFnlow
Fqlow
� �nonlinear
¼ HpnFnhigh
Fqhigh� Fnlow
Fqlow
� �nonlinear
¼ 0 for all p except damaged p (24.6)
In (24.6), Hpq represents the linear frequency response function of the system, and Hpn represents the nonlinear frequency
response function of the system. The resulting relationship from the frequency response comparison evaluated in (24.6)
suggests that the global dynamics of the system affect the ability to discern the location of the applied nonlinearities.
However, if the frequency response function of the linear system is known, the output of the system can be pre-filtered by the
frequency response function, resulting in an ability to separate the effects of the linear and nonlinear applied forces by
considering a pre-filtered output variable Xpf:
Xf g ¼ H½ � Ff g þ FNf g½ �H½ ��1 Xf g ¼ Ff g þ FNf g
Xpf
� � ¼ Ff g þ FNf g (24.7)
-3 -2 -1 0 1 2 3
x 10-9
-1
-0.5
0
0.5
1
x 10-3
Deflection (m)
For
ce (
N)
Linear Restoring ForceNonlinear Restoring ForceLinear Restoring ForceNonlinear Restoring Force
0 100 200 300 400 500 600
10-13
10-12
10-11
10-10
Frequency (Hz)
Mag
nitu
de (
m/N
)
Low AmplitudeHigh Amplitude
a b
Fig. 24.6 For the cubic stiffness nonlinearity case with an impulse excitation: (a) high and low amplitude frequency response functions and
(b) restoring force curve for the linear and nonlinear restoring forces
24 Modeling of Subsurface Damage in Sandwich Composites Using Measured Localized Nonlinearities 307
In this case, a similar comparison between high and low amplitude outputs yields a result in which the nonlinear force is
able to be distinguished separate from the linear dynamics of the system:
Xpfhigh
Fqhigh� Xpflow
Fqlow¼ Fqhigh
Fqhigh� Fqlow
Fqlow
� �linear
þ Fnhigh
Fqhigh� Fnlow
Fqlow
� �nonlinear
¼ Fnhigh
Fqhigh� Fnlow
Fqlow
� �nonlinear
¼ 0 for all p except damaged p (24.8)
The result in (24.8) was applied to the finite element simulation for both cubic stiffness and cubic damping nonlinearity
cases when a step sine excitation at selected frequencies was used for the applied force. Figure 24.7a shows damage index
results for the cubic stiffness case where the damage was applied between the seventh and eighth transverse degrees-of-
freedom. Similarly, Fig. 24.7b shows damage index results for the cubic damping case where the damage was applied
between the third and fourth transverse degrees-of-freedom. In both cases, the damage index was computed using the
pre-filtered displacement output.
24.6 Discussion
The result of frequency response comparison for the finite element model of the beam using a pre-filtered output variable
showed that the locally applied nonlinearities were able to be detected by comparing the high and low amplitude responses.
Prior to filtering the output by the frequency response function of the linear system, the location of the nonlinearities was not
able to be distinguished due to being masked by the linear dynamics of the system. By pre-filtering the output by the linear
frequency response function of the system, the damage location in each case was able to be determined.
Prior to pre-filtering the response by the frequency response function of the system, the high and low amplitude frequency
response function comparison created a damage index which indicated that the regions with the largest nonlinear response
occurred near antinodes of the mode shape of the system which was most easily excited by the applied excitation. After
removing the influence of the linear system dynamics, the largest nonlinear response was located at the nodes where the
localized nonlinearities were applied. The step sine excitation provided a clearer indication of the damage location than the
impulse excitation, due to more energy being put into the system at frequencies where the nonlinearities are excited. In
addition, the step sine excitation results showed that the nonlinear behavior due to the simulated damage becomes more
pronounced at higher frequencies.
60
70
1234567891011120
0.5
1
0
10
20
30
40
50
60
70
1234567891011120
0.5
1
0
10
20
30
40
50
60
70
1234567891011120
0.5
1
0
10
20
30
40
50
60
70
1234567891011120
0.5
1
a bX
pfhi
gh
Transverse degree of freedom
Freq
uenc
y (H
z)
Fqh
igh
Xpf
low
Fql
ow
Xpf
high
Transverse degree of freedom
Freq
uenc
y (H
z)
Fqh
igh
Xpf
low
Fql
ow
Fig. 24.7 Damage index results obtained from a step sine excitation for (a) a cubic stiffness nonlinearity at transverse degrees-of-freedom seven
and eight and (b) a cubic damping nonlinearity at transverse degrees-of-freedom 3 and 4
308 S.S. Underwood and D.E. Adams
24.7 Conclusion
The nonlinear nature of subsurface damage in composite materials was investigated through experimental measurements
and an analytical model of an isotropic, homogenous beam. A fiberglass sandwich panel was obtained and damaged in
order to investigate the nonlinear behavior measured using a scanning laser vibrometer. The panel was excited at multiple
amplitudes of excitation to allow for nonlinear behavior to be identified in a comparison of the high and low amplitude
frequency response measurements. Stiffness and damping nonlinearities were identified in the high and low
amplitude frequency response comparison, and a single degree-of-freedom model was used to show that the nonlinear
behavior identified resembles cubic stiffness and cubic damping behavior. The cubic stiffness and damping nonlinearities
were then investigated in an analytical model to study effects of the localized nonlinearities on the vibration response of the
system. It was found that, in order to locate the locally applied nonlinearities, it was necessary to pre-filter the simulation
output by the frequency response function of the linear system prior to taking into account the applied excitation force. This
pre-filtering was necessary to separate the linear system behavior from the nonlinear behavior of the damaged element. In
application to subsurface damage detection in composite materials, comparison of multi-amplitude frequency response
functions provides a method for identifying the presence of nonlinear behavior in the system. In order to locate the damage,
it is important to consider the linear dynamics of the system, where the challenge arises in separating the linear system
dynamics from the overall response of a structure.
References
1. Wong LA, Chen JC (2000) Damage identification of nonlinear structural systems. AIAA J 38:1444–1452
2. Zwink B, Koester D, Evans R, Adams DE (2008) Damage identification in composite sandwich helicopter blades using point laser velocity
measurements. In: proceedings of the sensor, signal and information processing workshop, Sedona, in print
3. Vanlanduit S, Guillaume P, Schoukens T, Parloo E (2000) Linear and nonlinear damage detection using a scanning laser vibrometer. Proc SPIE
4072:453–466
24 Modeling of Subsurface Damage in Sandwich Composites Using Measured Localized Nonlinearities 309
Chapter 25
Parametric Identification of Nonlinearity from Incomplete
FRF Data Using Describing Function Inversion
Murat Aykan and H. Nevzat Ozg€uven
Abstract Most engineering structures include nonlinearity to some degree. Depending on the dynamic conditions and level
of external forcing, sometimes a linear structure assumption may be justified. However, design requirements of sophisticated
structures such as satellites require even the smallest nonlinear behavior to be considered for better performance. Therefore,
it is very important to successfully detect, localize and parametrically identify nonlinearity in such cases. In engineering
applications, the location of nonlinearity and its type may not be always known in advance. Furthermore, in most of the
cases, test data will be incomplete. These handicaps make most of the methods given in the literature difficult to apply to
engineering structures. The aim of this study is to improve a previously developed method considering these practical
limitations. The approach proposed can be used for detection, localization, characterization and parametric identification of
nonlinear elements by using incomplete FRF data. In order to reduce the effort and avoid the limitations in using footprint
graphs for identification of nonlinearity, describing function inversion is used. Thus, it is made possible to identify the
restoring force of more than one type of nonlinearity which may co-exist at the same location. The verification of the method
is demonstrated with case studies.
Keywords Nonlinear identification • Nonlinear model testing • Experimental verification • Nonlinear parametric
identification
25.1 Introduction
System identification in structural dynamics has been thoroughly investigated over 30 years [1]. However, most of the
studies were limited to the linear identification theories. This short literature review does not cover linear identification
theories which are well documented [2, 3].
In the last decade, with the increasing need to understand nonlinear characteristics of complicated structures, there were
several studies published on nonlinear system identification [4–16]. Nonlinearities can be localized at joints or boundaries or
else the structure itself can be nonlinear. There are various types of nonlinearities, such as hardening stiffness, clearance,
coulomb friction, etc. [5].
Nonlinear system identification methods can be divided into two groups, either as time and frequency domain methods [4],
or as discrete and continuous time methods [6]. Most of the methods available require some foreknown data for the system.
Some methods require all or part of mass, stiffness and damping values [8–10] whereas some methods [4, 11–16] require
linear frequency response function (FRF) of the analyzed structure. In these methods nonlinearity type is usually determined
by inspecting the describing function footprints (DFF) visually. However, although the user interpretationmay be possible for
a single type of nonlinearity, it may not be so easy when there is more than one type of nonlinearity present [5]. Furthermore,
obtaining the linear FRF, which is usually presumed to be an easy task, may be difficult when nonlinearity is dominant at low
M. Aykan
Department of Mechanical Engineering, Middle East Technical University, Ankara 06800, Turkey
Defense Systems Technologies Division, ASELSAN Inc., Ankara 06172, Turkey
H.N. Ozg€uven (*)
Department of Mechanical Engineering, Middle East Technical University, Ankara 06800, Turkey
e-mail: [email protected]
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_25, # The Society for Experimental Mechanics, Inc. 2012
311
level excitations. There are also methods using neural networks and optimization for system identification [6]. Application of
optimization methods in nonlinear system identification is rather a new and promising approach. The major disadvantage of
these methods is generally the computational time required.
Nonlinearity identification method presented in this study consists of four main stages. Firstly, existence of nonlinearity
in the system is detected by performing step sine tests with different loads. Secondly, the location of the nonlinearity is
determined by using incomplete FRF data. The next step is the determination of the type of nonlinearity which is achieved by
investigating the restoring force function. Finally, in the parametric identification stage the coefficients of the nonlinear
elements are obtained by curve fitting techniques. The method proposed in this study is mainly an improved version of the
method developed earlier by Ozer et al. [12]. The improvement includes using incomplete FRF data which makes the
method applicable to large systems, and employing describing function inversion in order to reduce the effort in identifica-
tion of nonlinearity. Furthermore, using describing function inversion rather than footprint graphs makes it possible to
identify the total restoring force of more than one type of nonlinearity that co-exist at the same location.
25.2 Theory
Representation of nonlinear forces in matrix multiplication form using describing functions has been employed in identifi-
cation of structural nonlinearities by Ozer et al. [12]. They developed a method starting from the formulation given in their
earlier work [4] to detect, localize and parametrically identify nonlinearity in structures. As the basic theory of the method is
given in detail in reference [12], here it is briefly reviewed just for the completeness.
The equation of motion for a nonlinear MDOF system under harmonic excitation can be written as
½M�f€xg þ ½C�f _xg þ ½K�fxg þ j½D�fxg þ fNðx; _xÞg ¼ ffg (25.1)
where [M], [C], [K] and [D] stand for the mass, viscous damping, stiffness and structural damping matrices of the system,
respectively. The response of the system and the external force applied on it are shown by vectors {x} and {f}, respectively.{N} represents the nonlinear internal force in the system, and it is a function of the displacement and/or velocity response of
the system, depending on the type of nonlinearity present in the system. When there is a harmonic excitation on the system in
the form of
ff g ¼ Ff gejot (25.2)
the nonlinear internal force can be expressed as [17]
N x; _xð Þf g ¼ D x; _xð Þ½ � Xf gejot (25.3)
where [D(x, _x)] is the response dependent “nonlinearity matrix” and its elements are given in terms of describing functions
v as follows:
Dpp ¼ vpp þXnq¼1
q 6¼p
vpq p ¼ 1; 2; :::; n(25.4)
Dpq ¼ �vpq p 6¼ q p ¼ 1; 2; :::; n (25.5)
From the above equations it is possible to write the pseudo-receptance matrix for the nonlinear system, [HNL], as
HNL� � ¼ �o2 M½ � þ jo C½ � þ j D½ � þ K½ � þ D½ �� ��1
(25.6)
The receptance matrix of the linear counterpart of the nonlinear system can also be written as
H½ � ¼ �o2 M½ � þ jo C½ � þ j D½ � þ K½ �� ��1(25.7)
312 M. Aykan and H.N. Ozg€uven
From (25.6) and (25.7), the nonlinearity matrix can be obtained as
D½ � ¼ HNL� ��1 � H½ ��1
(25.8)
Post multiplying both sides of (25.8) by [HNL] gives
D½ � HNL� � ¼ I½ � � Z½ � HNL
� �(25.9)
where [Z] is the dynamic stiffness matrix of the linear part:
Z½ � ¼ H½ ��1 ¼ �o2 M½ � þ jo C½ � þ j D½ � þ K½ �� �(25.10)
In order to localize nonlinearity in a system, a parameter called “nonlinearity index” is used. The nonlinearity index (NLI)for an pth coordinate is defined by taking any ith column of [HNL] and the pth row of [D] from (25.9) as follows:
NLIp ¼ Dp1 � HNL1i þ Dp2 � HNL
2i þ . . .þ Dpn � HNLni (25.11)
Here, theoretically, i can be any coordinate; however, in practical applications it should be chosen as an appropriate
coordinate at which measurement can be made and also be close to suspected nonlinear element. Equation (25.11) shows that
any nonlinear element connected to the pth coordinate will yield a nonzero NLIp. On the other hand, NLIp can be
experimentally obtained by using the right hand side of (25.9), which requires the measurement of the receptances of the
system at high and low forcing levels, presuming that low level forcing will yield FRFs of the linear part:
NLIp ¼ dip � Zp1 � HNL1i � Zp2 � HNL
2i � . . .� Zpn � HNLni (25.12)
25.2.1 Nonlinearity Localization from Spatially Incomplete FRF Data
The main disadvantage of the method discussed in [12] is that in order to calculate the NLIp the whole linear FRF matrix may
be required (if instead of theoretically calculated dynamic stiffness matrix, inverse of experimentally measured receptance
matrix is used). When this is the case, it may not be feasible to apply the method. In this study it is proposed to use
theoretically predicted values for unmeasured receptances calculated from the measured ones, and it is shown with case
studies that this approach yields acceptable results.
In modal testing of complicated structures usually a shaker is attached to a specific location on the test structure and
measurements are made at several locations. Usually test engineer excites the structure from 1 or 2 locations and measures
the responses from many points using accelerometers. This yields 1 or 2 columns of the FRF matrix. The number of
unknown elements can be reduced if reciprocity is used, which is one of the main assumptions of linearity. However, there
will be still unknown terms in the FRF matrix, especially the ones related with rotational degrees of freedommay be missing.
Although there are various methods to obtain FRFs at rotational degrees of freedom [18], measuring FRFs for rotational
degrees of freedom is usually found very difficult and it is avoided.
Nonlinearity localization by using the right hand side of (25.9) requires either the system matrices (that can be obtained
from the FE model) or the complete receptance matrix of the linear part so that it can be inverted to find [Z]. In order to
obtain the missing elements of the experimentally obtained receptance matrix, the application of a well known method is
proposed. Theoretically, if the modal parameters (natural frequency, damping ratio, modal constant, lower and upper
residues) of a structure are obtained by linear modal identification then missing elements of the receptance matrix can be
synthesized. In this study, the linear modal identification is performed by using LMS Test Lab software.
Once the modal parameters are identified, the unmeasured elements of the receptance matrix are calculated by using [2]
HpqðoÞ ¼XNr¼1
1
j2Or
ffiffiffiffiffiffiffiffiffiffiffiffi1�ðzrÞ2
p fprfqr
Orzr þ jðo� Or
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðzrÞ2
qÞþ
ð 1
j2Or
ffiffiffiffiffiffiffiffiffiffiffiffi1�ðzrÞ2
p fprfqr�
Orzr þ jðoþ Or
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðzrÞ2
qÞþ UApq � LApq
o2(25.13)
25 Parametric Identification of Nonlinearity from Incomplete FRF Data Using Describing Function Inversion 313
where,
Or: undamped natural frequency of mode r
zr: damping ratio of mode r
fpr;fqr: mass normalized eigenvectors for mode r
UApq: upper residual
LApq: lower residual
N: number of modes considered
25.2.2 Nonlinearity Type Determination
After determining the locations of nonlinear elements in a structural system from nonzero NLI values, (25.8) is used to
evaluate the numerical values of describing functions for each nonlinear element at various response levels. The value of the
describing function, when there is single nonlinearity present in the system can be obtained from experimental data at
different response amplitudes by using Sherman-Morrison formulation to avoid inversion (see [12] for details). However,
when there are multiple nonlinearities present in the system, Sherman-Morrison formulation cannot be employed. Yet,
simultaneous solution of all describing function values is possible as long as the number of nonlinear elements do not exceed
the total DOF of the system, which would be rather unusual in practical applications. Then, the value of each describing
function can be plotted at different response amplitudes for obtaining Describing Function Footprints (DFF) which can be
used for determining the type of nonlinearity, as well as for parametric identification of nonlinear element(s). Another
common approach used for the same purpose is to obtain Restoring Force (RF) plots. Figure 25.1 presents RF and DFF plots
for some common nonlinear elements. It is clear that RF plots contain more physical information compared to DFF plots.
In this study, DFF calculated as described above is inverted to obtain RF function, which is graphically investigated to
evaluate the type of nonlinearity.
Nonlinearities in a structural system are usually due to nonlinear stiffness (piecewise stiffness, hardening cubic stiffness,
etc.) and/or nonlinear damping (coulomb friction, quadratic damping, etc.). Describing function formulation makes it
possible to handle stiffness and damping nonlinearities separately [19]. The real part of the describing function corresponds
to stiffness nonlinearities whereas the imaginary part corresponds to damping nonlinearities.
The DFF inversion has to be performed using different approaches for stiffness and damping nonlinearities when using
experimental data with no knowledge on the type of the nonlinearity.
The inverse of the describing function can be obtained approximately or analytically. Gibson [20] derived inverses for
real, imaginary and mean parts of a describing function. However, in this formulation the inversion of the real part and the
mean of the describing function requires the information about the type of nonlinearity, but the inversion for the imaginary
part works for any describing function and it does not require information about the type of nonlinearity. The only limitation
for the imaginary part is that the damping nonlinearity, which yields the imaginary part of DF, should not be frequency
dependent. The inverse of the imaginary part of the describing function is given as follows:
NðXÞ � p2
d
dXX2vðXÞ� �
(25.14)
In order to obtain the describing function inversion for the real part, the approximate inversion equations suggested by
Gelb and Vander Velde [19] are used:
NðXÞ � 3XX1
i¼0�2ð Þiv 2iþ1X
� �for n Xð Þ increasing with X (25.15)
NðXÞ � 3X
2
X1i¼0
� 1
2
� �i
vX
2i
� �for n Xð Þ decreasing with X (25.16)
where {N} represents the nonlinear internal force in the system.
314 M. Aykan and H.N. Ozg€uven
The major drawback of these formulations is that when the describing function is inversely proportional to X, for instance
due to Coulomb friction, the summation gives alternating series and a correct result cannot be obtained. However for
damping the imaginary part of the describing function is to be inverted and this is achieved analytically as explained above.
Consequently, in this study it is proposed to use (25.15) or (25.16) for the real part of DF, which is due to stiffness type of
nonlinearity, and to employ (25.14) for the imaginary part of DF, which is due to damping type of nonlinearity.
25.2.3 Parametric Identification of Nonlinearity
There are numerous ways to calculate parametric values for DFF and RF functions. Optimization and black box methods
such as neural networks provide promising results if they are well guided. More direct approaches like graphical methods
require the engineer to be experienced.
In this study the parametric values of the nonlinearity are obtained from RF plots by curve fitting. It is also possible to
obtain the coefficients from DFF when the type of nonlinearity is known. However, for most of the nonlinearity types, DF
representation is far more complicated than the corresponding RF function. It should be noted that when the RF representa-
tion of nonlinearity is already obtained, it is of little importance what the coefficients of RF function are. All the required
information about nonlinear element is stored in the RF function itself which can be further employed in dynamic analysis
for different inputs. Determining RF function, rather than DF may be more important when there is more than one type of
nonlinearity at the same location, in which case it will be very difficult if not impossible to make parametric identification for
each nonlinearity by using DFF.
3000Cubic stiffnesscs.x3
x
x
m2m1
x•
f
f
f
RF
Ff
−Ff
RF
RF
δ
Coulomb damping
Piecewise stiffness
2000
1000
15
4
3
2
1
0
10
5
00 0.1 0.20.150.05
0
� 104
0 0.1
DFF
DFF
DFF
Displacement Amplitude
Displacement Amplitude
0 0.1 0.20.150.05Displacement Amplitude
Del
ta V
alu
eD
elta
Val
ue
Del
ta V
alu
e
0.20.150.05
Fig. 25.1 RF and corresponding DFF plots
25 Parametric Identification of Nonlinearity from Incomplete FRF Data Using Describing Function Inversion 315
25.3 Case Study
The nonlinear identification approach proposed in this study is applied to a 4 DOFs discrete system with a nonlinear elastic
element represented by k1* (a linear stiffness of 100 N/m with a backlash of 0.005 m) between ground and coordinate 1, and
a nonlinear hardening cubic spring k4* (¼ 106*x2 N/m) between coordinates 3 and 4, as shown in Fig. 25.2.
The numerical values of the linear system elements are given as follows:
k1 ¼ k2 ¼ k3 ¼ k4 ¼ k5 ¼ 500N=m
c1 ¼ c2 ¼ c3 ¼ c4 ¼ c5 ¼ 5Ns=m
m1 ¼ 1kg;m2 ¼ 2kg;m3 ¼ 3kg;m4 ¼ 5kg
(25.17)
The time response of the system is first calculated with MATLAB by using the ordinary differential equation solver
ODE45. The simulation was run for 32 s at each frequency to ensure that transients die out. The frequency range used during
the simulations is between 0.0625 and 16 Hz with frequency increments of 0.0625 Hz. The linear FRFs are obtained by
applying a very low forcing (0.1 N) from first coordinate as presented in Fig. 25.2. The nonlinear FRFs are obtained
by applying high forcing (10 N) to the system from the first coordinate as shown in Fig. 25.2. Before using the calculated
FRFs as simulated experimental data, they are polluted by using the “rand” function of MATLAB with zero mean, normal
distribution and standard deviation of 5% of the maximum amplitude of the FRF value. A sample comparison for the
nonlinear and linear FRFs (H11) is given in Fig. 25.3.
It is assumed in this case study that we have only the first columns of the linear and nonlinear receptance matrices. Then,
firstly the missing elements of the linear FRF matrix are calculated by using the approach discussed in Sect. 2.1, and the NLIvalues are calculated for each coordinate by using (25.12). The calculated values are shown in Fig. 25.4a. From Fig. 25.4a it
can easily be concluded that there are nonlinear elements between ground and coordinate 1, and between coordinates 3 and
4. Furthermore, since the nonlinearity can be stiffness and/or damping type, it is possible to make this distinction at this stage
by investigating the real and imaginary parts of the describing function. The real and imaginary parts of the describing
function can be summed over the frequency range and compared with each other. Figure 25.4b reveals that system has
stiffness type of nonlinearity since DF has much higher real part compared to imaginary part.
Using the method proposed, the describing functions representing these nonlinear elements are calculated at different
response amplitudes and are plotted in Fig. 25.5. From the general pattern of the curves it may be possible to identify the
types of nonlinearity. Fitting a curve to the calculated values makes the parametric identification easier. Although
identification of backlash may not be so easy from DFF, it is quite straightforward to identify the type of cubic stiffness
from Fig. 25.5b.
Alternatively, the types of nonlinear elements can be identified more easily if DF inversion method proposed in this study
is used. The inversion of DF is calculated for this case study by using the formulation given in Sect. 2.2, and RF plots
obtained are presented in Fig. 25.6. Figure 25.6a gives the RF plot for the nonlinearity between the first coordinate and
ground, whereas Fig. 25.6b shows the RF plot for the nonlinearity between coordinates 3 and 4. By first fitting curves to the
calculated RF plots, parametric identification can easily be made. The parametric identification results for the nonlinear
elements are tabulated in Table 25.1. As can be seen from the table, the identified values do not deviate from the actual
values more than 12%.
Although the DF inversion formulations are based on polynomial type describing functions, it is shown in this case study
that they work, at an acceptable level, for discontinuous describing functions such as backlash as well.
Fig. 25.2 Four DOFs discrete system with two nonlinear elements
316 M. Aykan and H.N. Ozg€uven
25.4 Experimental Study
The proposed approach is also tested on the experimental setup used in a recent study [21]. The experimental setup and FRF
plots obtained with constant amplitude harmonic forces are given in Figs. 25.7 and 25.8, respectively.
The test rig consists of a linear cantilever beam with its free end held between two thin identical beams which generate
cubic spring effect. The cantilever beam and the thin nonlinear beams were manufactured from St37 steel. The beam can be
taken as a single DOF system with a nonlinear cubic stiffness located between the ground and the equivalent mass
representing the cantilever beam. This test rig is preferred for its simplicity in modeling the dynamic system since the thin
beams yield only hardening stiffness nonlinearity and the structure itself can bemodeled as a single degree of freedom system.
14a b
9000
Sum
of D
escr
ibin
g F
unct
ion
Val
ues
8000
7000
6000
5000
4000
3000
2000
1000
0
12
10
8
NLI
Val
ue
6
4
2
01 2 213
Coordinate Number
NLI Chart Sum of DF Values
Sum of Real PartsSum of Imaginary Parts
Coordinate 0-1 Coordinate 3-44
Fig. 25.4 (a) Nonlinearity index chart, (b) sums of real and imaginary parts of DF values at high forcing excitation
6�10−3
Linear
Nonlinear
Driving Point Linear and Nonlinear FRF Plots
5
4
3
Rec
epta
nce
(m/N
)
2
1
00 2 4 6 8
Frequency (Hz)10 12 14 16
Fig. 25.3 Driving point linear and nonlinear FRF plots
25 Parametric Identification of Nonlinearity from Incomplete FRF Data Using Describing Function Inversion 317
For a single degree of freedom system, the nonlinearitymatrix reduces to the describing function defining the nonlinearity [4]:
v ¼ H � HNL
HNLH(25.18)
150
100
50
00.005 0.01 0.015 0.02
Displacement (m)
Des
crib
ing
Fun
ctio
n V
alue
(N
/m)
Des
crib
ing
Fun
ctio
n V
alue
(N
/m)
Describing Function Values for Backlash Describing Function Values for Cubic Stiffness
Displacement (m)0.025 0.03 0.035 0.002
0
50
100
150
200
250
300
350
400
450
500a b
Calculated DFExact DF
Calculated DFExact DF
0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022
Fig. 25.5 Identified and exact DFs. (a) For nonlinear element between coordinate 1 and ground, (b) For nonlinear element between coordinates
3 and 4
00 0.005 0.01 0.015
Displacement (m) Displacement (m)
DF InvertedExact RF
DF InvertedExact RF
RF Plot for Backlash RF Plot for Cubic Stiffness
0.02 0.025
a b
0.03 0.035
12
10
8
6
4
2
00 0.005 0.01 0.015 0.02 0.025
0.5
1
1.5
RF
(N
)
RF
(N
)2
2.5
3
3.5
Fig. 25.6 Identified and exact RF plots. (a) For nonlinear element between coordinate 1 and ground, (b) for nonlinear element between
coordinates 3 and 4
Table 25.1 Parametric identification results for the nonlinear elements
Actual Identified Error %
Backlash (m) 0.0050 0.0044 12
Linear stiffness part of k1* (N/m) 100 95 5
k2* (cubic stiffness constant) N/m3 1,000,000 956,800 4
318 M. Aykan and H.N. Ozg€uven
The describing function representation of the nonlinearity (n) can be graphically shown as a function of response amplitude,
which makes it possible to identify the type of nonlinearity and to make parametric identification by using curve fitting.
The nonlinear coefficient for the hardening cubic stiffness is first obtained by a static test. In the static test a load cell is
used to measure force and a linear variable differential transformer is used to measure displacement for stepped loadings
with 5 N increments. The force is applied at the point where the cantilever beam is attached to thin beams. The deflection is
also measured at the same point. The results of this test are presented as a force versus deflection curve in Fig. 25.9.
Then, by using the DFF and DF inversion approaches for nonlinear identification, both DF and RF plots are obtained for
the nonlinear element between the tip point of the cantilever beam and the ground (Figs. 25.10, 25.11). The cubic stiffness
constants identified by using DF and RF curves are 2.667 � 108 N/m3 and 2.656 � 108 N/m3, respectively. The cubic
stiffness constant obtained from static test, on the other hand is 2.437 � 108 N/m3. For visual comparison, force deflection
curves obtained from static test and DF inversion approaches are compared with the force deflection characteristics obtained
from DFF approach in Fig. 25.11. As can be seen, DFF and DF inversion approaches yield very close results.
Thus, it can be concluded that the accuracy in parametric identification of nonlinearity by DF inversion is comparable to
that of DFF method. However, the main advantage of DF inversion is that it gives better insight into the type of the
nonlinearity. Furthermore, when the RF function is obtained by DF inversion, it may be directly used in nonlinear model of
the system when time domain analysis is to be used. Then, it will be possible to identify the restoring force of more than one
type of nonlinearity which may co-exist at the same location.
Fig. 25.7 Setup used in the experimental study
Fig. 25.8 Constant force driving point FRF curves
25 Parametric Identification of Nonlinearity from Incomplete FRF Data Using Describing Function Inversion 319
Fig. 25.10 Measured describing function values and the curve fitted
Fig. 25.11 RF plots of nonlinearity for experimental study
Fig. 25.9 Static force–deflection curve for the cubic stiffness
320 M. Aykan and H.N. Ozg€uven
25.5 Conclusions
It was recently shown [21] with an experimental case study that the method developed by Ozer et al. [12] for detecting,localizing and parametrically identifying nonlinearity in MDOF systems is a promising method that can be used in industrial
applications. In the study presented here some improvements are suggested to eliminate some of the practical limitations of
the previously developed method. The verification of the approach proposed is demonstrated with two case studies. The
main improvements are using incomplete FRF data which makes the method applicable to large systems, and employing
describing function inversion which makes the identification of nonlinearity easier.
The method requires dynamic stiffness matrix of the linear part of the system which can be obtained by constructing a
numerical model for the system and updating it using experimental measurements. In this study, however, it is proposed to
make linear modal identification by using one column of the receptance matrix of the system experimentally measured at
low forcing level, and then to calculate the missing elements of the complete FRF matrix so that the dynamic stiffness matrix
required for the identification can be obtained. Note that low forcing testing will not give the linear receptances if
nonlinearity is due to dry friction, since its effect will be dominant at low level vibrations. For this type of nonlinearity
high forcing testing will yield the linear receptace values. The approach suggested is first applied to a lumped parameter
system and it is shown that detection, localization and identification of nonlinear elements can successfully be achieved by
using only one column of the linear FRF matrix.
Secondly, it is proposed in this study to use RF plots obtained from DF inversion for parametric identification, instead of
DFF plots, in order to avoid the limitations in using footprint graphs. It is found easier to determine the type of nonlinearity
by using RF plots, rather than DFF plots, especially for discontinuous nonlinear functions such as backlash.
The application of the approach proposed is also demonstrated on a real structural test system, and it is concluded that the
accuracy in parametric determination of nonlinearity by DF inversion is comparable to that of DFF method, and since RF
plots give better insight into the type of nonlinearity this approach may be preferred in several applications to identify the
type of nonlinearity. Furthermore, when the RF function is obtained, it may be directly used in nonlinear model of the system
if time domain analysis is to be made. Using describing function inversion rather than footprint graphs makes it possible to
identify total restoring force of more than one type of nonlinearity that may co-exist at the same location. Thus, DF inversion
yields an equivalent RF function that can be used in further calculations without any need to identify each nonlinearity
separately. Consequently, it can be said that the approach proposed in this study is very promising to be used in practical
systems, especially when there are multiple nonlinear elements at the same location.
References
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5. G€oge D, Sinapius M, F€ullekrug U, Link M (2005) Detection and description of non-linear phenomena in experimental modal analysis via
linearity plots. Int J Non-linear Mech 40:27–48
6. Worden K, Tomlinson GR (2001) Nonlinearity in structural dynamics. Institute of Physics Publishing, Bristol
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London/University of London
8. Narayanan S, Sekar P (1998) A frequency domain based numeric–analytical method for non-linear dynamical systems. J Sound Vib
211:409–424
9. Muravyov AA, Rizzi SA (2003) Determination of nonlinear stiffness with application to random vibration of geometrically nonlinear
structures. Comp Struct 81:1513–1523
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46:433–445
11. Elizalde H, Imregun M (2006) An explicit frequency response function formulation for multi-degree-of-freedom non-linear systems. Mech
Syst Signal Process 20:1867–1882
12. Ozer MB, Ozg€uven HN, Royston TJ (2009) Identification of structural non-linearities using describing functions and the Sherman–Morrison
method. Mech Syst Signal Process 23:30–44
13. Thothadri M, Casas RA, Moon FC, D’andrea R, Johnson CR Jr (2003) Nonlinear system identification of multi-degree-of-freedom systems.
Nonlinear Dyn 32:307–322
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14. Cermelj P, Boltezar M (2006) Modeling localized nonlinearities using the harmonic nonlinear super model. J Sound Vib 298:1099–1112
15. Nuij PWJM, Bosgra OH, Steinbuch M (2006) Higher-order sinusoidal input describing functions for the analysis of non-linear systems with
harmonic responses. Mech Syst Signal Process 20:1883–1904
16. Adams DE, Allemang RJ (1999) A New derivation of the frequency response function matrix for vibrating non-linear systems. J Sound Vib
227:1083–1108
17. Tanrikulu O, Kuran B, Ozg€uven HN, Imregun M (1993) Forced harmonic response analysis of non-linear structures. AIAA J 31:1313–1320
18. Duarte MLM (1996) Experimentally-derived structural models for use in further dynamic analysis. PhD thesis in mechanical engineering.
Imperial College London
19. Gelb A, Vander Velde WE (1968) Multiple-input describing functions and nonlinear system design. McGraw Hill, New York
20. Gibson JE (1963) Nonlinear automatic control. McGraw-Hill, New York
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322 M. Aykan and H.N. Ozg€uven
Chapter 26
Finding Local Non-linearities Using Error Localization
from Model Updating Theory
Andreas Linderholt and Thomas Abrahamsson
Abstract Within the aerospace industry, linear finite element models are traditionally used to describe the global structural
dynamics of an aircraft. Ground vibration test data serve to facilitate the validation of models which are then used to
characterize the aeroelastic behavior of the aircraft and to predict the responses due to dynamic loads. Thus, it is vital that the
models contain the essential dynamics of the aircraft. Observed nonlinearities are judged to be local in nature whereas the
main part of the structure behaves linearly under normal loading. In this work we focus on the identification of nonlinear
effects and do that based on model updating theory. That includes methods for error localization with proper selection of
candidate error parameters. The nonlinearities are treated as local modeling errors not considered in the linear system model.
The error localization behavior is studied using synthetic test data from a simple system, known as the ECL Benchmark, with
known nonlinear properties.
Keywords Non linear •Model updating • Error localization • Parameter identifiability • Data informativeness • Optimization
26.1 Introduction
The coupling between an overall linear structure and local nonlinearities is a classical problem in industry. That is also the
case in the aircraft industry where the major part of a structure may be satisfactorily described by a linear model although
local nonlinearities do exist. Typically, joints between stores and the aircraft introduce nonlinear characteristics due to e.g.
gap and dry friction. Traditionally, linear finite element models are used to describe the global behavior of an aircraft since
they are computationally inexpensive and at the same time good from a global perspective. However, linear models may not
be capable of reproducing all the characteristics found in test data. In aircraft industry flight test data and ground vibration
test (GVT) data complement each other in characterizing the structural dynamics. The loading condition in the two situations
are different and during flight the effects of some types of nonlinearities may be suppressed while others are engaged/
increased compared to the loading conditions during a GVT. However, test data stemming from ground vibration tests form
the primary source for detailed model validation and updating. Thus, such data are concerned here.
The observed deviations between test data and the corresponding analytical data stemming from a linear model are here
thought of as being caused by errors within the nominal, linear, model. The task then becomes a model updating problem.
The parameters needed to substantially increase the model’s capability of representing the real structure are most likely not
included in the initial model. Therefore, a set of candidate parameters controlling nonlinear effects, opposite to what is used
within the vast majority of model updating exercises, have to be added. The candidate parameters have to be chosen using
engineering insight into the structure at hand.
A. Linderholt (*)
Department of Mechanical Engineering, Linnaeus University, SE-35195 V€axj€o, Swedene-mail: [email protected]
T. Abrahamsson
Department of Applied Mechanics, Chalmers University of Technology, SE-41296 G€oteborg, Swedene-mail: [email protected]
D. Adams et al. (eds.), Topics in Nonlinear Dynamics, Volume 3,Conference Proceedings of the Society for Experimental Mechanics Series 28,
DOI 10.1007/978-1-4614-2416-1_26, # The Society for Experimental Mechanics, Inc. 2012
323
Before any updating exercises take place, the selection of the data to use in combination with the selection of model
parameters to be involved should be examined. Test data to be used for computational model updating have to fulfil at least
two requirements. Firstly, they should be informative with respect to the parameters used for updating meaning that the
test data should be sensitive to changes of the parameter values. Secondly, they have to change differently for changes
of different parameters or group of parameters which is to say that the parameters should be identifiable. Should these
two requirements not be fulfilled, it is simply impossible to get a reliable parameter value estimation made using that test
data [1].
The Cramer-Rao lower bound quantifies a limit to the accuracy of parameter estimates from the information in the test
data. The inverse of the Cramer-Rao lower bound is known as the Fisher information. The Cramer-Rao lower bound and the
FIM are useful quantities to assess test data informativeness and parameter identifiability and they can be estimated a priori
using a calculation model. The test data should be chosen such that the expected variances of the estimated parameters are
small. For multi-parameter problems, the FIM and its inverse are matrices. Hence, the quantification of an, in some aspect,
large FIM, or a small inverse of it, is not clear [2]. This is described further in Sect. 2.3.
The FIM and its inverse, and thereby the data informativeness and the parameter identifiability, varies with the response
data used. It is therefore important to include all available data that carries information that differentiate parameter settings.
This is partly controlled by the selection of excitation and response measurement during test; that is the actuator and
sensor placement together with the excitation time history. Another part is the choice of the perspective on which the test
data are looked upon. Sometimes, processing the data may hide or destroy information that are in the original data. Some
test data simply destroys or decreases the goodness of the overall test data; such data should be excluded before the
updating is made.
Another issue regarding measurements on nonlinear systems is which sampling frequency to use [3]. The well known
Nyquist’s sampling theorem, or Shannon’s criterion, states that the sampling frequency should be at least twice the highest
frequency of interest which for a linear system is equal to twice the largest excitation frequency. Ljung also shows that the
sampling frequency can be high which leads to increase of the variance of estimated parameters [4]. This phenomena is not
further considered here.
Here, the focus is on the selection of data to use for the optimization process, known as model updating, and the parameter
selection; these are coupled. The paper consists of theoretical part and a numerical example illustrating the theory. For this, a
model setting out from the ECL (Ecole Centrale de Lyon) nonlinear benchmark is used [5]. To the ECL-structure nonlinear
springs; one linear, one quadratic and one cubic, are added. Although, Georg Duffing treated only a few nonlinear systems,
the accepted de notion for any differential equation in which a cubic nonlinearity is included is nowadays a Duffing equation
[6]. Hence, the numerical example consists mathematically of the Duffing equation type.
26.2 Theoretical Background
In general, the forced vibration response of a nonlinear deterministic system can be expressed by the state-space system of
ordinary differential equations as
_x ¼ f ðx; uÞ; y ¼ gðx; uÞ (26.1a,b)
Here u is the excitation vector, x the state vector and y the response vector. For steady-state periodic excitation it is well-known that the linear system reacts to such stimulus by a periodic steady-state response with same periodicity after initial
transients have settle. However, for nonlinear systems it is also well-known that this is not generally the case. The steady state-
response of a nonlinear systemmight be periodicwith stimulus periodicity orwith other periodicity, or therewill be no periodic
response at all [7]. For stationary harmonic excitation with excitation frequency O, corresponding to period T ¼ 2p/O,the non-linear system may respond in a periodic steady-state with the periodicity nT; n 2 Zþ which we will refer to as being
n-cycle periodic.We confine our study to n-cycle periodic steady-state conditions and thereby rejects response solutions thatare chaotic, non-periodic or periodic with any other periodicity.
For a linear system with periodic excitation
uðtÞ ¼ < u1 exp ðiO tÞ þ u2 exp ð2iO tÞ þ . . .þ un exp ðn i O tÞ þ . . .þ u1=2 exp ðiO t=2Þ þ . . .þ u1=n exp ðiO t=nÞ� �(26.2)
324 A. Linderholt and T. Abrahamsson
where the harmonic order is v 2 Zþ, the response may be written as
yðtÞ ¼ < y1 exp ðiO tÞ þ y2 exp ð2iO tÞ þ yn exp ðn i O tÞ þ . . .þ y1=2 exp ðiO t=2Þ þ . . .þ y1=n exp ðiO t=nÞ�(26.3)
or
yðtÞ ¼ < HðOÞu1 exp ðiO tÞ þ H ð2OÞu2 exp ð2iO tÞ þ . . .þ H ðO=nÞ u1=n exp ðiO t=nÞ� �(26.4)
where HðoÞ is the system’s complex-valued frequency dependent transfer function at frequency o. For the mono-frequency
excitation
uðtÞ ¼ < u1 exp ðiO tÞð Þ (26.5)
the linear system’s stationary response is thus
yðtÞ ¼ < HðOÞ u1 exp ðiO tÞð Þ (26.6)
However, for the nonlinear system in n-cycle periodic steady-state, exposed to a mono-frequency loading, the response is
generally not mono-frequency and may be written as
yðtÞ ¼ < �y1 exp ðiO tÞ þXn max
n¼2
�yn exp ðiO tÞ þ �yn exp ðiO t=nÞð Þ( )
(26.7)
where �yn and
y�n are the superharmonic and subharmonic distortion amplitudes of order v respectively. The magnitude of
these generally depend on the stimulus magnitude u1. It is convenient for the sequel to denote the stimulus and response
magnitudes by load, order and frequency indices such that ulvk denotes the excitation level l of harmonic order v at the kthdiscrete frequency Ok and ylvk is then the vth order response due to that excitation.
26.2.1 d-Level Multi-Harmonics Frequency Response Functions
In lab testing, the mono-frequency excitation is an anomality and harmonic distortion is always present due to imperfections
in the test setup. Instead of trying to achieve a pure sinusoidal excitation in lab, which requires some closed loop control [8],
one might have a better chance to enforce a multi-sine stimulus with a distortion level that overshadows the intrinsic test
setup distortion. Let the d-level multi-harmonic excitation of load level l at the fundamental frequency Ok be defined as
uklðtÞ ¼ < ul exp ðiOk tÞ þ dulXn max
n¼2
exp ði n Ok tÞ þ exp ðiOk t=nÞð( )
(26.8)
We note that by letting d � 1 we create a harmonic function with small distortion to the fundamental harmonics atOk. We
may use that as the stimulus signal in lab testing or in simulation. After initial transients have settle, under n-periodic steady-stateconditions, the system response will eventually become periodic and before the full periodic condition arise we have
yklðtÞ ¼ < �yl1 exp ðiO tÞ þXn max
n¼2
�yln exp ði nOktÞ þ y�ln exp ðiOk t=nÞð( )
þ residual (26.9)
In testing or numerical simulation, we can find the complex-valued amplitudes of �y and y� by regression. We define the
d-level multi-harmonic frequency response function that relates the input at the ith dof to the response at the jth dof to be
the complex-valued quantities
�HlnijðOkÞ ¼ �ylniðOkÞ=ulj and HlnijðOkÞ ¼ y�lniðOkÞ=ulj (26.10a,b)
26 Finding Local Non-linearities Using Error Localization from Model Updating Theory 325
for the zeroth order and higher order harmonics and subharmonic orders respectively. We note in particular that, by
construction, for a linear system the following relations hold
�HlnijðOkÞ ¼ d �Hl1ijðnOkÞ and HlnijðOkÞ ¼ d �Hl1ijðOk=nÞn 6¼ 1 (26.11a,b)
i.e. the superharmonic and subharmonic frequency response function can by obtained from the frequency response function
of the fundamental harmonics by a scaling with the level parameter d.For a nonlinear system, relations as those of (26.11a,b) do not hold and the d-level frequency response functions
will depend on the excitation level l. A study of a quantity that relates to the difference between these functions of the
nonlinear system and the corresponding functions of the linearized systems might the give an insight into the nonlinear
behavior of the system. Before we proceed, we make a slight adjustment of our notation. By the d-level frequency responsefunction Hlvkij we mean the function value that pertain to the excitation level of index l (l ¼ 1,2,. . .,lmax), of indexed order
n (n ¼ 1,2,. . .,nmax) that includes both superharmonic and subharmonic orders, and at indexed fundamental frequency
Ok (k ¼ 1,2,. . .,kmax). The frequency response function value Hlvkij relates the output at dof i with the input at dof j. We also
use the Matlab colon notation in the respect we define the vector H:vkij as the vector with elements as function values Hlvkij
for indices l of all available loads as
H:nkij ¼ 8lvect
ðHlnkijÞ (26.12)
with fixed order index, frequency index and dof index. Similarly we mean by the vectors Hl: kij and Hlv: ij the vectors
Hl:kij ¼ 8nvect
ðHlnkijÞ and Hln:ij ¼ 8kvect
ðHlnkijÞ (26.13)
Also we define the vector operator for defined subsets of loads ls as
Hlsnkij ¼ l 2 lsvect
ðHlnkijÞ (26.14)
with similar notation for subsets of order indices and frequency indices. We also generalize this vectorization concept,
such that e.g.
H::kij ¼ 8ðl; nÞvect
ðHlnkijÞ (26.15)
is the vectors of data for all available load levels and orders and
H:nskij ¼ 8l; n 2 nsvect
ðHlnkijÞ (26.16)
and is the vector of data for all available load levels and for orders from the subset vs. The use of any other combination
of indices should now be obvious.
26.2.2 A Deviation from Linearity Criterion Function
For the linear system, the frequency response function are independent of the load level l and we denote it with Hlinvkij. In the
sequel we focus on the difference between the nonlinear system’s frequency response behavior and the linearized systems
counterpart. We define the deviation from linearity as
elnkij ¼ Hlnkij � Hlinnkij (26.17)
326 A. Linderholt and T. Abrahamsson
and use the same vectorizing notation for this quantity as before for H. In short we call the vector of elements elnkijthe vector e, with proper indexing when needed for clarity. The difference from linearity is a complex-valued
number and consequently e is a complex-valued vector. We define the real-valued scalar deviation from linearity criterion
function E as
E ¼ �eT e (26.18)
where overbar denotes the complex conjugate. This function is obviously zero for the linear system. In the following,
for identifiability purpose, we study the gradient, rE, of this function with respect to model parameters, pm, m ¼ 1,2,. . ..
rE ¼ @E=@p1 @E=@p2 . . .½ � (26.19)
The m:th component can be expressed as
@E=@pm ¼ 2< �eT @e=@pm� � ¼
Xk
< 2�e::k:: @e::k::=@pmf g ¼defXk
ek; m (26.20)
These gradient component functions ek;m are functions of discrete frequency Ok. We use these to evaluate the parameter
identifiability.
26.2.3 Ranking Identifiability of Parameters
The requirements for a test design is that the resulting test data should be informative with respect to the parameters and that
the parameters should be identifiable. The first means that a change of the value of any parameter should change the observed
response in a noticeable way. The latter is to say that test data should differentiate changes of different parameter values. In
computational model updating, these requirements are coupled to the usefulness of the test data in estimating model
parameter values accurately. The two requirements can jointly be stated as; regardless of which parameters that are
estimated, precise estimates require a small parameter covariance matrix. This should, if possible, be examined prior to
the test using a model.
The Fisher information, which is a scalar, is a measure of the information an observable variable, i.e. test data, Xcarries with respect to a single unknown parameter that the probability of X depends on. The probability function of X is
also the likelihood function for X given a certain parameter value p. That function is denoted ƒ(X: p). The partial
derivative of the natural logarithm of f with respect to p is denoted the score. An unbiased estimator is an estimator
having a mean that converges to the correct parameter setting as the number of test data tends towards infinity. That
means that the first moment of the score vanish. Further, the second moment of the score is known as the Fisher
information. The corresponding measure for a multi parameter problem is the Fisher information matrix (FIM) having
size p x p. When it comes to estimating parameter values, pest, the Fisher information matrix, here denoted JðpÞ, plays animportant role, see Walter and Pronzato [9] and Spall [10]. The reason for this is the Cramer-Rao theoretical lower bound
which establishes a limit on the expectation of the covariance matrix of the estimate of the parameter values. This limit is
coupled to the FIM according to;
X½pe s t � p�½pe s t � p�T � J�1 (26.21)
in which X denotes the expectation. The Cramer-Rao bound implies that, irrespective of the method used to quantify the
parameters from the data, there is a bound on the estimation precision that can not be overcome. Similar to the single
parameter situation, the Fisher information matrix is determined by the joint probability density function, f;
JðpÞ ¼ X@
@plog f
� �T @
@plog f
� �(26.22)
26 Finding Local Non-linearities Using Error Localization from Model Updating Theory 327
Here, the measurement noise is assumed to be of the simplest kind, from an analytical point of view. It consist of
independent Gaussian sequences having zero mean values. Further, it is assumed that the noise is uncorrelated and
statistically equivalent for all measured quantities; the noise variance equals s2. Then, the FIM expression simplifies into:
JðpÞ ¼ 1
s2XNk¼1
rET rE (26.23)
The gradient rE is a 1� m vector. Each of its columns is formed as a summation, see (26.20). Let all, N, possiblecomponents contributing in the summation be stacked on top of each other to form an N � mmatrix that is here denotedr ~E.The FIM can now be calculated as
JðpÞ ¼ 1
s2XNk¼1
r ~ET r ~E (26.24)
with unchanged result. However, the formation of r ~E is useful in the study of data informativeness and parameter
identifiability which are complementary. When the value of a model parameter is changed, the measurement data should
change noticeable in order to the data to be informative with respect to the parameters involved in the model updating.
Complementary, changing the values of two parameters should affect the measurement data differently. A combination of a
parameter selection and a test design rendering in that all pairs of columns of r ~E are orthogonal is the ultimate. The angle
between two columns is easily calculated and using the definition of the modal assurance criterion gives a good, normalized,
measure of the linear dependencies. When two columns associated with parameters pi and pj are orthogonal, the element of
the ith row and the jth column of the Fisher information matrix is zero.
If the ideal situation for which columns are orthogonal is not fulfilled, the linear relationship between groups of columns has
to be examined since changing one parameter (or a group of parameters) may change the dynamics of themodel in the sameway
as a change of another parameter (or another group of parameters). When this happens, the parameters are not identifiable from
test data, which means that the updating is unreliable. Proper measures have to be taken, possibly involving re-parameterization
or a change of the test design, see Linderholt and Abrahamsson [11]. The conclusion is that it is not enough to examine only
relations between pairs of parameters and this is where the FIM and its inverse is a more useful measure.
A number of criteria ranking the goodness of the FIM or its inverse have been proposed. The A-optimality focus on
minimizing the trace of the inverse of the Fisher information matrix. The D-optimality aims at maximizing the determinant
of the FIM. E-optimality, maximizes the minimum eigenvalue of the FIM. Further, the T-optimality maximizes the trace of
the FIM. Finally, the G-optimality seeks to minimize the maximum expected variance estimated parameter values. These are
just examples of frequently used matrix criteria. In this study, yet another criterion is chosen; that is the condition number of
the FIM. A low condition number indicated a well conditioned matrix which in turn is necessary for small parameter
estimate variances given by the inverse of the FIM.
Since each column in r ~E is partitioned into contributions from different sources such as different sensors or different
excitation frequencies etc., also the FIM is built up by a summation of FIM:s stemming from each combination of sensor,
load level, excitation frequency etc. Such a combination is here denoted with the index s.
JðpÞ ¼XSs¼1
sJðpÞ (26.25)
It is not known a priori how many and which of the contributing data that should be taken into account to get a low
condition number of the FIM. Examining all possible combinations of reductions is time consuming since the number of
combinations grows rapidly as a function of the number of possible sensors etc. Using a sub-set selection technique is an
inexpensive choice to achieve a good, although not guarantied to be optimal, solution, see Miller [12].
26.3 Numerical Example
We illustrate the identifiability properties of the parameters with respect to the deviation from linearity criterion function by
considering a simulation model of the ECL Benchmark setup [5]. The ECL Benchmark was designed with the parameter
identification of nonlinear systems in mind. It consists of a cantilever beam of simple cross section, supported in the free
328 A. Linderholt and T. Abrahamsson
end by a thin structural element that is subjected to tension and mainly adds membrane stiffness to the system. Since the
membrane is short it gives a restoring force to the beam end that is rather strongly cubic in the beam tip deflection. The beam
is loaded by a concentrated transversal force not far from its fixed support end. The system can be seen in Fig. 26.1.
In our simulation model we use three planar linear Euler-Bernoulli finite elements of equal length to model the beam.
That introduces three nodal translational and three nodal rotational degrees-of-freedom. The membrane member is modelled
as a short taut string which introduces nonlinear effects because of its nonlinear kinematics. The elasticity of the supporting
structure is represented by the elasticity of a fictitious rod element. The taut force of the membrane in its neutral position and
the total stiffness of the membrane and support rod are two introduced free parameters of the identifiability problem. To
increase the identifiability complexity, three more free parameters are introduces as the linear, quadratic and cubic stiffness
coefficients of a discrete spring that supports the free end of the beam. The discrete spring is not part of the ECL Benchmark.
Numerical data is given by the caption of Fig. 26.1.
In the simulation we apply a periodic force u(t) with three different magnitudes (ul ¼ 2; 5 and 10N for l ¼ 1,2,3) and
evaluate the response during each full period. We do a stationarity check by comparing the response from period to period
and assume that stationarity is achieved when the responses between two consecutive periods are the same, i.e. when
the norm of the difference of responses between periods is small. All cases here studied settled to periodic solutions.
We evaluate the multi-level FRF:s of the last simulated period for the given condition.
Figure 26.2 shows the d-level multi-harmonics frequency response functions with d ¼ 0.01. It can be noted that
the frequency response functions deviates from the linearized system’s frequency response functions mainly in the vicinity
of structural resonances where the deviation is significant.
Figures 26.3 and 26.4 show the gradient components ek;m, see (26.20), for two load levels ul ¼ 1 ¼ 2N and ul ¼ 2 ¼ 5N.
The gradients were evaluated by finite differentiation of the parameters pi with Dpi ¼ 0:01. We use a vectorizing operation
such that
e; m ¼ 8kvect
ek; m (26.26)
and study the correlation of these vector functions for the five parameters, m ¼ 1,2,. . .,5. We use as correlation index Cmn
the cosine-square of the angle between the vectors (similar to the well-known MAC index for eigenvectors) as
Cmn ¼ cos2 ffðe; m; e; nÞ� �
(26.27)
Graphical illustrations of these indices are shown as inserts in Figs. 26.3 and 26.4.
EI,m,Lb
q2q5q3
q1 q7q6q4
(1+p2)EA
1 32
u(t)(1+p2)EA
kNL(1+p1)R
kNL= (1+p3)k0(1+p4 / +p5( )2)q5 q52 q5 q53/
(1+p1)R Lm Lr
2´N(q5,q7)
N(q5,q7)N=EA(Lm(q5,q7)-Lm0)/Lm0
Nsin φ
φ
Fig. 26.1 Seven-degree-of-freedom model of ECL Benchmark setup. Photo of testbed from [Thouverez]. Beam 1 bending stiffness, mass per unit
length and total length are EI, m and Lb respectively. Taut membrane 2 has tensional stiffness EA and free length Lm0. Elastic supporting rod 3 hastensional stiffness EA and free length Lr. System pretension is R and the periodic loading is u(t). The loading of displaced membrane 20 is shown ininsert. The restoring force from the membraneNsinf acting at the beam end is a nonlinear function of the nodal displacements q5 and q7 because themembrane rotationf is not small. Spring stiffness kNL is nonlinear in q5. The freemodel parameters of the study are p1, p2, . . ., p5which in the nominal
setting are all zero. Numerical data: EI ¼ 672 Nm2, m ¼ 3.36 kg/m, EA ¼ 3150kN, Lb ¼ 593 mm, Lr ¼ 57 mm, R ¼ 10 N, k0 ¼ 175 N/m,
q52 ¼ 30 mm and q53 ¼ 33.5 mm
26 Finding Local Non-linearities Using Error Localization from Model Updating Theory 329
The seven degrees of freedom together with the three load levels (2, 5 and 10N) and the five orders (1, 2, 3, 1/2 and 1/3),form 105combinations each contributing to the FIM. By using a sub-set selection technique, one contribution at the time is
removed. The selection is made such that the condition number of the remaining FIM is kept as low as possible within each
step. The result is shown in Fig. 26.5. The result shows that by reducing the set of combinations by five, the condition number
of the FIM is lowered considerably.
Fig. 26.2 d-Level multi-
harmonics frequency response
functions at load amplitudes
Fl ¼ 2; 5 and 10 N for
l ¼ 1; 2; 3
330 A. Linderholt and T. Abrahamsson
Fig. 26.3 Functions ek;m evaluated for m1 ¼ 2N together with correlation indices Cmn or corresponding gradient vectors. We note that the vectors
are highly correlated for harmonic orders 1 and 3 at this load level
26 Finding Local Non-linearities Using Error Localization from Model Updating Theory 331
Fig. 26.4 Functions ek;mevaluated for load levels ml ¼2; 5; and 10 N together with
correlation indices Cmn for
corresponding gradient
vectors of harmonic order 1/3.
We note that the vectors are
highly uncorrelated for load
level 2N
332 A. Linderholt and T. Abrahamsson
The data stemming from degree-of-freedom no.6, a load level equal to 10N and order 1 is the first to be removed
according to the reduction criterion. The Auto MAC of the part of r ~E gradient stemming from that combination is shown
in Table 26.1. It is obvious that the column vectors are highly correlated. Thus, it is understandable that excluding such
data strengthens the information which results in a decrease of the condition number of the Fisher information matrix,
see Fig. 26.5.
26.4 Conclusions
It is shown that test data should be looked upon from different perspectives; processing data may hide or destroy
information. Here, information from sub and superharmonic component found from steady state oscillation of nonlinear
systems assist the information within the fundamental harmonic response. Instead of trying to get a pure mono-harmonic
excitation, which is hard in practise, it is here proposed to include a few sub and superharmonics having amplitudes that are
low compared to the amplitude of the fundamental harmonic but still large enough to dominate over the excitation noise.
Furthermore, the condition number of the Fisher information matrix, associated with the physical parameters selected, is
chosen as the optimization objective. Finally it is shown that disregarding data that have low information value can actually
increase the parameter identifiability and data informativity.
Acknowledgement We gratefully acknowledge the Swedish National Aviation Engineering Programme (NFFP) for their kind support of
this work.
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107
108
Five data sets removed from the setbuilding up the FIM
No. of remainingcombinations
Condition no.
Fig. 26.5 The condition number of the Fisher’s information matrix as a function of the remaining of contributing data sets i.e. combinations of
degree’s of freedom, load levels and orders
Table 26.1 The AutoMAC matrix associated with the data that should be excluded first among the 105 candidates.
The data stem from degree of freedom no. 6, loadlevel equal to 10N and order 1
1.00 0.85 1.00 0.98 0.89
0.85 1.00 0.85 0.84 0.76
1.00 0.85 1.00 0.98 0.89
0.98 0.84 0.98 1.00 0.96
0.89 0.76 0.89 0.96 1.00
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