University of Rome “La Sapienza”, Rome, 00184, ITALY
P. Marzocca,† and A. Behal‡
Clarkson University, Potsdam, 13699, NY, USA
Recently, structural health monitoring and damage detection has reemerged as an areaof interest for the mechanical, automotive, and aerospace industry. Increasingly, there isa need to develop in-service and on-line health monitoring techniques. Such techniquesallow systems and structures to monitor their own structural integrity while in operationand throughout their life and are useful not only to improve reliability, but also to re-duce maintenance and inspection costs. Although, there has been an extensive amount ofwork performed in the area of structural health monitoring and damage sensing technolo-gies, most of the existing methodologies suffer from defects that include low sensitivities,complex FE models that may take long periods of time to calibrate, and modal trunca-tions that may or may not lead to accurate predictions. In this paper, the possibility ofaddressing this problem with different techniques is proposed and applied to both numer-ical and experimental test cases. This work constitutes a first contribution toward thegoal of integrating these techniques for an efficient and robust real-time health monitoringtechnique. The first approach is a novel application of an adaptive control technique fornon-destructive monitoring and evaluation for structural integrity of aerospace structures,while the second method is based on a structural updating technique predicting the eval-uation of the changes of the dynamic characteristics of the structures in the frequencydomain. A numerical and experimental investigation of the above methodologies will becarried out taking into account different problems in system dynamics.
Nomenclature
CR, CP Weighting MatricesIα Mass moment of inertia of the airfoil about the elastic axisL,M Lift and aerodynamicPi i− th Design ParameterS, p Sensitivity matrix, Vector of the Design Parameters Pi
U Freestream velocitya Dimensionless distance from mid-chord to elastic axis positionb Semi-chord of the airfoilclα, cmα Lift and Moment curve slopes per angle-of-attackclβ , cmβ Lift and Moment curve slopes per control surface deflectionch, cα Structural damping coefficients in plunge and pitchh Plunging displacementkh, kα Structural spring stiffness in plunge and pitch
∗Assistant Professor, Dipartimento di Ingegneria Aerospaziale e Astronautica, Via Eudossiana, 16, AIAA Member.†Assistant Professor, Department of Mechanical and Aeronautical Engineering, The Wallace H. Coulter School of Engineering
CAMP 234, AIAA Memeber.‡Associate Professor, Department of Electrical and Computer Engineering, The Wallace H. Coulter School of Engineering
m Mass of airfoilt Timexα Dimensionless distance from the elastic axis to the mid-chord, positive rearwardα Frequency Response Function, Pitch angleβ Flap angleε Vector of the dynamic differencesχs, χa Correlation functions of the “FRFsρ Air densityωk, ωn Frequency point, Eigenvalue
SubscriptA, X Analytical, Experimentala, s Amplitude, Shape
Superscriptk Mode Index
I. Introduction
One of the critical issue in the development of techniques for the identification of a structural damageis related to their capability to warning for failure in a real time monitoring of the structure as well asto detect, identify, and predict the existence and the degree of damage severity from the analysis of lin-ear and nonlinear structural characteristics. The first approach proposed here exploit an adaptive controltechnique,1–3 that shows a considerable potential for detection of small changes in structural properties. Inany structure stabilized by feedback, an introduction of change in a single or a set of structural propertiesleads to deviation in the system states from their equilibrium values. The system is then stabilized throughhigh gain feedback (good dynamic response with a good disturbance rejection, and a poor noise rejection)or through integration of error in the output variable alongside proportional feedback (good steady-state re-sponse, inferior dynamic response, and a good noise rejection). In an attempt to reduce the problems relatedto the low sensitivity to structural modifications of the vibration-based methodologies recently developedby researchers,4 a second approach has been proposed. This approach, based on an updating methodologycalled Predictor-Corrector,5,6 has been further developed, to increase its numerical stability, and appliedto different aerospace structures.7–9 Within this procedure, generally classified as “iterative method”, thehealth of a structure could be monitored through the evaluation of some design parameters, arbitrarily se-lected by the structural analyst, related to the local mass and stiffness, that minimize a non-linear functionrepresenting the differences in the dynamic behavior of the actual and the reference structure.10 Therefore,the developed approach is an iterative non-linear optimization procedure in conjunction with a least squarealgorithm. The solution of this minimization problem is achieved once the unknown changes in the designparameters have been obtained by solving, iteratively, a linear system of equations. The capability of theadaptive control technique has been first investigated through a numerical analysis of a simple aeroelasticsystem, whereas the last technique has been tested on an experimental specimen.
II. Theoretical Background
A. Damage Identification through Adaptive Control
By exploiting properties of the system dynamic model and using time-domain analysis, a generalized pro-portional integral control scheme emerges that is able to identify linear and nonlinear structural parametricproperties. In the presence of a control input that forces the system output error to be square integrable whena contrived time varying reference trajectory with sufficient “richness” is imposed on the system, changesin parameter estimates can be identified.11 Two types of approaches are possible. The first is responsive tocontinuous changes in structural parameters; however, in the presence of significant noise, the convergenceresponse may not be accurate. The second approach is able to overcome the problem of the first approach viacertain modifications to the overall control and adaptation scheme that retain the identification propertieswhile averaging out the effects of noise. The cost of doing this is a more computationally involved strategy
2 of 13
American Institute of Aeronautics and Astronautics Paper 2005-2227
that might not be very responsive to continuous structural changes. It is worth noting that it seems impor-tant to utilize feed-forward of a sufficient number of derivatives of the reference trajectory to obtain a squareintegrable error. As a consequence, it is implied right away that a bounded control input is achievable only ifthe reference trajectory is chosen to be bounded as well as continuously differentiable - usually, a multi-ratecombination of sinusoids may be shown to suffice. Further implied is the fact that a slow moving referencewould better suit circumstances where the available actuation is limited. For a better understanding of theproblem and of the present procedure, details of the mathematical model are presented together with theconfiguration of the 2-D lifting surface which stiffness changes with time. Based on a representative modelstudied in Ref. 2, several results are presented in this paper.
B. Damage Identification through Sensitivity Analysis
The task of identifying the damage occurring in a vibrating structure could be follow up considering itseffects on the dynamic characteristics of the system. Several techniques have been proposed in literaturemost of them require the evaluation of the modal, or the response model of the actual structure from boththe numerical analysis, and from the experimental survey:4,10 from changes of those global characteristicsthe presence of a damage could be assessed. Here, a sensitivity based structural updating algorithm hasbeen shaped to detect structural failures. Assuming the FRFs of the system, denoted as H, are available,then the difference between the reference, undamaged, and the actual structure could be evaluated usingtwo correlation functions of the FRFs, χs and χa, defined as:5
χs(ωk) =|HH
X(ωk) HA(ωk)|2
[HHX(ωk) HX(ωk)] [HH
A (ωk) HA(ωk)](1)
χa(ωk) =2|HH
X(ωk) HA(ωk)|[HH
X(ωk) HX(ωk)] + [HHA (ωk) HA(ωk)]
(2)
In Eq. (1) and (2) and the following one, the subscripts (s) and (a) identify the experimental and analyticalFRFs. These functions are defined over a discrete number of frequency lines disposable from the test, ωk.The first correlation function is related to the correspondence of the shape of the FRFs, whereas the last isrelative to their amplitudes. Since these two correlation functions are defined in the range [0, 1] (0 if thereis no correlation, whereas 1 if there is a perfect correlation among the structural models), their sensitivityto structural changes could be used to monitor the health of a structure. First, the actual differences in thedynamic behavior could be evaluated arranging in a vector ε so that:
ε =
1− χs(ω1)...
1− χs(ωNf)
1− χa(ω1)...
1− χa(ωNf)
Then, the effects of a generic damage could be included in the analysis introducing a sensitivity matrix,Sp, formed by the derivatives of the correlation functions with respect to a variation of design parameters.Describing the structure through a F.E. model, then a set of P design parameters could be identified andarranged in a vector p. The unknown changes of the i− th design parameters, Pi, arranged in a vector ∆p,could be achieved by iteratively solving the linear system of equations given by:
Sp ∆p = ε (3)
3 of 13
American Institute of Aeronautics and Astronautics Paper 2005-2227
where the sensitivity matrix could be expressed as:
Sp =
∂χs(ω1)∂p1
∂χs(ω1)∂p2
· · · ∂χs(ω1)∂pP
· · · · · · · · · · · ·∂χs(ωNf
)
∂p1
∂χs(ωNf)
∂p2· · · ∂χs(ωNf
)
∂pP∂χa(ω1)
∂p1
∂χa(ω1)∂p2
· · · ∂χa(ω1)∂pP
· · · · · · · · · · · ·∂χa(ωNf
)
∂p1
∂χa(ωNf)
∂p2· · · ∂χa(ωNf
)
∂pP
(4)
in which the derivatives of the correlation functions of the FRFs with respect to the unit variation of thedesign parameters are numerically evaluated considering the derivatives of the dynamic stiffness. Moreover,this system of equations is heavily rectangular since the number of rows are related to the number of frequencylines disposable from the experimental analysis. Therefore, the solution of the equation could be achievedby minimizing the functional:
F = εT CRε + ∆pT CP ∆p (5)
where those frequency lines which correspond to the maximum correlation between the FRFs could beweighted through the weighting matrices CR and CP . An estimate of the damaged areas of the structurecould be finally obtained evaluating, using a Baeysian-based least square approach, the change in designparameters given by:
∆p =(ST
p CRSpCP
)−1ST
p CR ε (6)
Therefore, the detection of a damaged region of a structure could be achieved identifying the design para-meters of the finite element model that reduce the actual difference between the dynamic characteristics ofsystem. This is done by iteratively solving Eq. 6 until the convergence of the values associated to the designparameters is reached.
This methodology has been further optimized for the damage identification problems. Since it requiresthe search for both local, and small changes in the structural properties, a “by-step” technique has beenintroduced,12 i.e., the damages are identified by successive applications of the presented approach. In orderto increase the stability of the numerical algorithm, and to reduce the computational resources, for each step,only the design parameters, most sensitive to the type of the actual damage are considered as active variables.Finally, the proposed methodology iteratively solve Eq. 6 until the most sensitivity design parameters arehighlighted.
III. Results and Discussion
In the next subsection, the effectiveness of the adaptive control based approach for damage identificationis demonstrated via the use of an aeroelastic system,13 controlled by a single trailing edge flap, representedby a 2-D plunging-pitching-flapping lifting surface, depicted in Fig. 1. Moreover, the capability to identifystructural damages using the Predictor-Corrector approach represents the item of the last subsection, wherean aluminum plate is considered.
LE
TE
Kh
Kα
b b
α
db
h
VMC
cbβ
Kβ
xy
EAAC
L
MT
xEA
Figure 1. 2-D wing section aeroelastic model
4 of 13
American Institute of Aeronautics and Astronautics Paper 2005-2227
A. Numerical simulation: damage identification in aeroelastic system
Figure 1 shows a schematic for a plunging-pitching typical wing-flap section that is considered in the presentanalysis. This model has been widely used in aeroelastic analysis.13,14 The plunging (h) and pitching (α)displacements are restrained by a pair of springs attached to the elastic axis of the airfoil (EA) with springconstants kh and kα (α), respectively. Here, kα (α) denotes a continuous, linear parameterizable nonlinearity,i.e., the aeroelastic system has a continuous nonlinear restoring moment in the pitch degree of freedom. Suchcontinuous nonlinear models for stiffness result from a thin wing or propeller being subjected to large torsionalamplitudes.15,16 Similar models1,3, 13,17 have been examined and provide a basis for comparison.
1. Configuration of the 2-D Wing Structural Model
The aeroelastic system permits two degree-of-freedom motion, whereas an aerodynamically unbalanced con-trol surface is attached to the trailing edge. In addition to controlling the aerodynamic flow and providingincreased maneuverability, the flap is used here to suppress instabilities. Conventionally, the plunge dis-placement is positive downward, the pitch angle is measured from the horizontal at the elastic axis of theairfoil, positive nose-up and the aileron deflection (beta) is measured from the axis created by the airfoilat the control flap hinge, positive flap-down. The governing equations of motion for the aeroelastic systemunder consideration are given by:13,17[
m mxαb
mxαb Iα
] [h
α
]+
[ch 00 cα
] [h
α
]+
[kh 00 kα (α)
] [h
α
]=
[−L
M
](7)
As an example, the aerodynamic lift (L) and moment (M) can be modeled in a form as18
L = ρU2bclα
[α + h
U +(
12 − a
)b α
U
]+ ρU2bclββ
M = ρU2b2cmα
[α + h
U +(
12 − a
)b α
U
]+ ρU2b2cmββ
(8)
One can transform the governing equations of motion, EOM, of (7) into the following equivalent state spaceform as follows17
z = f(z) + g(z)βy = z2
(9)
where z (t) =[
z1 (t) z2 (t) z3 (t) z4 (t)]T
∈ <4 is a vector of system states that is defined as follows
z =[
h α h α]T
(10)
β (t) ∈ <1 is a flap deflection control input, y (t) ∈ <1 denotes the designated output, while f(z), g (z) ∈ <1
assume the following form
f(z) =
z3
z4
−k1z1 −(k2U
2 + p (z2))z2 − c1z3 − c2z4
−k3z1 −(k4U
2 + q (z2))z2 − c3z3 − c4z4
g(z) =
00g3U
2
g4U2
, g4 6= 0 (11)
where p (z2) , q (z2) ∈ <1 are continuous, linear parameterizable nonlinearities in the output variable resultingfrom the nonlinear pitch spring constant kα(α).
The auxiliary constants ki, ci ∀ i = 1...4 as well as g3, g4 that were introduced in the model descriptionof (11) are explicitly defined as follows
k1 = Iαkhd−1 k2 = d−1[Iαρbclα + mxαb3ρcmα
]k3 = −mxαbkhd−1 k4 = d−1
[−mxαb2ρclα −mρb2cmα
]c1 = d−1
[Iα (ch + ρUbclα) + mxαρUb3cmα
]c2 = d−1
[IαρUb2clαa−mxαbcα + mxαρUb4cmαa
]c3 = d−1
[−mxαbch −mxαρUb2clα −mρUb2cmα
]c4 = d−1
[mcα −mxαρUb3clαa−mρUb3cmαa
]g3 = d−1
[−Iαρbclβ −mxαb3ρcmβ
]g4 = d−1
[mxαb2ρclβ + mρb2cmβ
]d = m
(Iα −mx2
αb2)
a = (0.5− a)
5 of 13
American Institute of Aeronautics and Astronautics Paper 2005-2227
Motivated by the desire to rewrite (9) in a form that is amenable to output feedback control design, Kalman’sobservability test is applied to the pair
(A, C
)where A ∈ <4×4, C ∈ <1×4 are explicitly defined as
A =
0 1 0 0−k4U
2 −c4 −k3 −c3
0 0 0 1−k2U
2 −c2 −k1 −c1
, C =
1000
. (12)
A sufficient condition for the system of (9) to be observable from the pitch angle output α (t) is given as
dobs = k23 − k3c3c1 + c2
3k1 6= 0
which is easily satisfied by the nominal model parameters of a 2-DOF aeroelastic system presented in17
and adopted in this paper. After a series of coordinate transformations, the governing equations for theaeroelastic model can be expressed in the following convenient state-space form
x = Ax + Φ(y) + Bβ
y = CT x =[
1 0 0 0]x
(13)
Herein, β = U2β is an auxiliary control input, x (t) =[
x1 (t) x2 (t) x3 (t) x4 (t)]T
∈ <4 is a new
vector of system states, A ∈ <4×4, B ∈ <4 are explicitly defined as
A =
0 1 0 00 0 1 00 0 0 10 0 0 0
, B =
0θ2
θ3
θ4
(14)
where θi ∀ i = 2, 3, 4 are constants that are explicitly defined as follows
θ2 = g4
θ3 = −g3c3 + c1g4
θ4 = k1g4 + k1c23g3k
−13 − c1g3c3 − dobsg3k
−13
In Eq. (13), Φ(y) ∈ <4 is a smooth vector field that can be linearly parameterized as follows
Φ(y) = W (y)γ =p∑
j=1
γjWj(y) (15)
where γ ∈ <p is a vector of constant unknowns, W (y) ∈ <4×p is a measurable, nonlinear regression matrixwhile the notation Wj (·) ∈ <4 denotes the jth column of the regression matrix W (·) ∀ j = 1...p. Thenonlinear pitch spring stiffness kα(α) was represented by explicit expressions for α in terms of a quinticpolynomial as follows
kα(α) =5∑
i=1
τiαi−1 (16)
where the unknown τi extracted from experimental data17 are given as
{τi} =[
2.8 −62.3 3709.7 −24195.6 48756.9]T
A regulation control objective can be further modified in order to design a tracking controller with a desiredoutput trajectory, αd, with a broad spectrum that allows for satisfaction of a mild persistence of excitation(PE) condition.19 This would allow one to exactly identify the model and the nonlinearity associated withthe aeroelastic model. We can define a tracking error as follows
e = αd − α
6 of 13
American Institute of Aeronautics and Astronautics Paper 2005-2227
where the desired value αd (t) is chosen as
αd = αd0 +n∑
i=1
sin (αdit) ; i = 1, 2, 3, ... (17)
Furthermore, we define a filter tracking error as follows
r =.e +λe (18)
After taking the time derivative of (18) and substituting the dynamics of (13), the control input β (t) canbe obtained as
β =(krr − Fφ + αd − λ
.e −Rθ
)U−2g−1
4 (19)
where the auxiliary signal Fφ (t) is defined as follows
and θ (t) denotes a dynamic parameter estimate that is generated as follows.
θ= RT r (21)
where R (t) is the following measurable regression vector
R =[−α −α2 −α3 −α4 −α5
](22)
2. Damage identification in the 2-D wing structural model
0 1010 2020 3030 4040 5050 6060
Persistent Random Disturbance (ts = 15s)
Time, sec
Esti
mat
e K
α
Persistent Steady Disturbance at t = 15sTriangular Blast Loading at tin = 20s; tfin = 30s
No Disturbance
10% reduction in Kα
4545
4040
3535
42.542.5
37.537.5
Figure 2. Effect of selected disturbances on stiffness estimation in response to step change in torsional stiffnessat t = 10 [s]
The model described in (9) was simulated using the output feedback observation, control, and estimationalgorithm. The values for the model parameters are listed in Table 1, see Ref. 13.
Results illustrating the technique used for the structural health monitoring are presented next and aredemonstrated via the use of a 2-D plunging-pitching-flapping lifting surface that is controlled by a singletrailing edge flap, see Fig. 1. In Fig. 2, selected types of unmodeled disturbances (step, blast, and randomloading) are applied to the aeroelastic system while also introducing a step (-10%) in the pitching stiffness
7 of 13
American Institute of Aeronautics and Astronautics Paper 2005-2227
0 1010 2020 3030 4040 5050 6060
4545
4141
3737
Persistent Steady Disturbance at t = 15sTriangular Blast Loading at tin = 20s; tfin = 30s
No Disturbance
Time, sec
Esti
mat
e K
α
20%
15%
NominalValue
10%
5% reduction in Kα4343
3939
Figure 3. Estimate of the pitching stiffness for varying steps in system stiffness.
that occurs during flight at t = 10 [s]. The change in stiffness has been identified in about 20 [s] after theinitiation of the change. A random gust or a transitory loading does not affect the prediction of the stiffness.The only case in which there is a small error (< 3%) in the estimation is when an unmodeled persistentunit load is applied to the system. We should emphasize here that our technique can certainly account foran unknown step disturbance in the system model when it is modeled; then it is easily recognizable as soand the estimation of the structural parameters in the model can be very accurate. Figure 3 shows that
0 0.0.5 1 1.1.5 2 2.2.5 3 3.3.5 4 4.4.5 50
2020
4040
6060
8080
Change in Linear Stiffness KChange in Linear Stiffness Kα
Figure 4. Stiffness estimate vs amplitude of persistent disturbance and change in linear pitching stiffness.
a decrease in the linear portion of the pitching stiffness by 5%, 10%, 15% and 20% from its nominal valuecan be quickly estimated via the adaptive control technique. Moreover, the same figure shows also that thepresence of disturbances, such as a persistent steady gust load (occurring at t = 20 [s]) or a triangular blastloading (with a positive phase starting at t = 15 [s]) affects the estimation only minimally - in the sensethat a minor change in the final estimated value occurs only in the presence of a persistent disturbance asexplained above. Some conclusions can be extracted from Fig. 4, where the changes in the linear stiffness due
Figure 5. Estimate of the linear pitching stiffness in response to concurrent step change in multiple parametersat t = 10 [s].
to the damage, in the presence of disturbance, is highlighted. It can be seen that the slope of the estimatestiffness with respect to the changes in the stiffness due to the structural degradation is much larger thanthe one due to the persistent disturbance. Notice that the product of the slope with τi of Eq. 16 will give
8 of 13
American Institute of Aeronautics and Astronautics Paper 2005-2227
the scaled values of the structural pitching stiffness. This ensures the robustness of the method in estimating
Figure 6. Estimate of the nonlinear pitching stiffness in response to concurrent step change in multipleparameters at t = 10 [s].
changes in structural parameters. We note here that the technique can be utilized to predict changes inother structural parameters, such as mass, damping, etc. Figures 5 and 6 depict as a possible scenario achange in linear and nonlinear structural stiffness coefficients that occur due to deterioration of structuralstiffness due to, say, the degradation of the material properties or a failure of a structural component. It
0 5 1010 1515 2020 2525 3030 3535 4040 4545 5050
1515
3030
4545
6060Persistent Disturbance at t = 0s; amplitude = - 5;
Time, sec
Esti
mat
e K
α
Kr = 1.104
Kr = 1.103Kr = 1.102
Kr = 10Kr = 1
Nominal Value
10% Kα linear reduction at t = 0sFeedback gains:
Figure 7. Estimate of the linear stiffness component in response to step change in torsional stiffness at t = 0[s] and with a persistent disturbance at t = 0 [s] for selected values of the feedback gain Kr.
clearly appears from these figures that this methodology can predict the changes in both linear and nonlinearstiffness components. Assuming that there is a change in linear and nonlinear stiffness characteristics (5%,10% and 15% reduction) the stiffness characteristics are estimated in a short time. As can be seen in Fig.7, the time of estimation is a function of the feedback gains that have to be optimized for disturbance andnoise rejection as well as speed of convergence. This methodology can be successfully applied toward theestimation of other physical characteristics, such as a change in the mass of the system that can occur whenthe payload is delivered, or also a change in material properties, due to a thermal degradation or relaxationof the structure.
B. Experimental investigation: damage identification in a structural component
The second approach, has been applied on an aluminum plate (with dimensions 0.3×0.2×0.003[m3
]) under
free-free boundary conditions, experimentally achieved by elastic suspensions. The modal survey has beenperformed in a frequency band of [0, 640] [Hz] using 4096 spectral lines with 3 averages for each signal, andexciting the structure, with an impulse force hammer. The transversal accelerations of 81 points of the plate,equally distributed in the plane of the plate, have been measured with the aid of 3 roving accelerometers.Moreover, the leakage effects have been reduced introducing the well known exponential windows for boththe input, and the output signals. As a results, the first 4 natural frequencies are collected in Tab. 2: Thesame aluminum plate experienced a cut, approximately near the center of the structure, in order to obtaina slight material removal from its skin, producing a local thickness reduction of about 20%, as reported inFig. 8. The corresponding changes in the modal parameters are summarized in Tab. 3. As one can seefrom the previous tables, the damage introduced in the structural component produces an almost immaterialshift of the natural frequencies without any changes in the mode shapes as one would expect. Indeed, theminimum change in the natural frequency is about 0.6%, corresponding to the 1st mode, but higher than the
9 of 13
American Institute of Aeronautics and Astronautics Paper 2005-2227
Table 2. Natural Frequency Estimate for the Undamaged Plate Structure
overall uncertainty of the experimental set up since this last is about 10−3 [Hz]. The reference numericalfinite element model was formed by 64 plate elements equally distributed. Since the effects of the damageintroduced in the actual structure are mostly related to changes in the local stiffness distribution, i.e., themass changes could be considered as a second order effect, only design parameters linked to elemental stiffnessare considered. Therefore, the damage identification procedure considered a total of 64 design parameters.The sensitivity matrix was built using the frequency response functions associated to 12 points, as depictedin Fig. 9. From the numerical point of view, the linear system to be iteratively solved for the structuralchanges ∆p, Eq. 6, required a large amount of computer memory, and long computational time. To reduceboth the items, and to investigate the numerical capabilities of the developed procedure, a frequency bandof 0 − 300 [Hz], that includes the first two modes only, have been considered. Thus, the dimensions of thesensitivity matrix reduced to 2450× 64.
In the first of the the damage identification procedure, only 12 design parameters, over 64 available,were found to be the most sensitivity to the experimental identification. Only these parameters have beenconsidered in the subsequent second step, keeping the others constant. From the convergence history reportedin Fig. 10, where the value of the selected design parameter for each iteration step has been reported, it waspossible to further reduce the number of the design parameters interested in the damage identification. Infact, the design parameters that exhibited the major changes were only 4, i.e., design parameters number28, 36, 54, and 58. Next, a third step of the identification procedure has been performed. From theanalysis of the corresponding convergence history, reported in Fig. 11, it seemed that design parametersnumber 28, 36, and 58 were the most representative of the damage acting on the structure. Besides, theparameter number 58 was found to be not only almost insensitive to the iteration procedure, but also the
Table 3. Natural Frequency Estimate for the Damaged Plate Structure
Figure 10. Convergence History of the Design Parameters - Step #2
suggested changes were relative low compared with the others, during the second step. Therefore, it waspossible to discard such design parameter in a further damage identification step since its numerical nature:this was probably the effect of the reduction of the frequency band considered. Finally, the behavior of
0 5 10 150.9985
0.999
0.9995
1
1.0005
1.001
1.0015
1.002
Iteration Step
Element 28Element 36Element 54Element 58
Figure 11. Convergence History of the Design Parameters - Step #3
design parameters number 28, and 36 were reported in Fig. 12. As one can see, after few iterations, thenumerical investigation suggested that both the elemental stiffness, related to such design parameters, werea representation of the unknown damage condition. Indeed, the cut produced in the structure would hadled to the identification of the 28th elemental plate only, without any suggestion regarding element 36.Nevertheless, this results could be considered completely acceptable since element 36 is adjacent to the finiteelement 28. Although the localization of the damaged region is correctly identified, the identification of itsentity still needs further improvement since the proposed correction was not representative of the imposeddamage. For a further validation of the results achieved, a comparison among the initial and final correlationof the dynamic behavior of the free-free plate has been performed, and reported in Fig. 13. Specifically, acomparison between the frequency response functions for the damaged, initial, and final numerical model
11 of 13
American Institute of Aeronautics and Astronautics Paper 2005-2227
0 2 4 6 8 10 12 14 16 18 200.999
0.9995
1
1.0005
1.001
1.0015
1.002
Iteration Step
Element 28Element 36
Figure 12. Convergence History of the Design Parameters - Step #4
- evaluated at the driving point, and denoted as Hd, Hi, and Hf respectively -, is reported in the upperpart of such figure, where the amplitudes and phase are depicted, whereas the corresponding effects on theeigenfrequency shifts, for the two modes considered in the analysis, are reported in the lower bar plot of thesame Fig. 13. As expected, the differences in the dynamic behavior are practically immaterial, nevertheless,
120 140 160 180 200 220 240 260 280 300
2
4
6
8
10
12x 10
−5
f [Hz]
Am
plitu
de
Hd
Hi
Hf
120 140 160 180 200 220 240 260 280 300
−2
0
2
f [Hz]
Pha
se
Hd
Hi
Hf
1 20
0.5
1
f #n
Shi
ft [%
]
Initial ShiftFinal Shift
Figure 13. Comparison Between the Driving point FRFs, and Natural Frequency Shifts
an overall reduction in the natural frequency shifts has been achieved when considering the suggestions ofdamage identification procedure in the numerical model. When the initial and final frequency correlationis considered, although the first mode has been identified with an increasing error in the natural frequency,i.e., from 0.9267% to 0.9396%, the second eigenfrequency showed a much higher reduction of the relativeshift, i.e., from 1.1777% to 1.1729% as reported in Tab. 4, confirming then the correctness of the damageidentification procedure.
Table 4. Comparison Between Initial and Final Eigenfrequency Shifts
Non-destructive health monitoring techniques for structural integrity of aerospace components are pre-sented. The first technique is based on an adaptive control methodology, which allows for real-time mon-itoring of the structure and earlier warning of failure. The detection, identification, and prediction of theexistence and the degree of damage severity are done from the analysis of linear and nonlinear structuralcharacteristics. This structural health monitoring technique can be applied to any generic structural sys-tem and herein the technique has been demonstrated on a simple aeroelastic model for which a premature
12 of 13
American Institute of Aeronautics and Astronautics Paper 2005-2227
aeroelastic instability (flutter) due to a structural damage has been identified. Such approach allows theevaluation of modified flight and/or operative boundary of a damaged structural system. Numerical resultshave demonstrated that this SHM technique can be suitable for real-time identifications of structural para-meters and therefore structural failures. The second method of evaluating structural integrity is based onthe Predictor-Corrector technique, an extended version of the so called “updating” technique which predictchanges of the dynamic characteristics of the structures in the frequency domain. This method has itsfoundation on an iterative non-linear optimization procedure in conjunction with a least squares algorithm.Numerical and experimental results on a flat aluminium plate have been provided to demonstrate the efficacyof the method.
Acknowledgments
P. Marzocca would like to thank Prof. L. Librescu of Virginia Tech for his support, guidance andcollaboration.
References
1Zhang, R. and Singh, S.N., Adaptive output feedback control of an aeroelastic system with unstructured uncertainties, J.Guidance, Control, and Dynamics, Vol. No. 3, pp. 502-509, 2001.
2Behal, A., Marzocca, P., Dawson, D.M.,Lonkar, A., Nonlinear Adaptive Model Free Control of an Aeroelastic 2-D LiftingSurface, AIAA Guidance, Navigation, and Control Conference and Exhibit, Providence, Rhode Island, 16-19 Aug 2004.
3Xing, W. and, Singh, S.N., Adaptive output feedback control of a nonlinear aeroelastic structure, J. Guidance, Controland Dynamics, Vol. 23, No. 6, pp. 1109-1116, 2000.
4Doebling, S. W., Farrar, C. R., and Prime, M. B., A Summary Review of Vibration-Based Damage Identification Methods,The Shock and Vibration Digest, Vol. 30, No. 2, pp. 91-105, 1998.
5Grafe, H., Model Updating of Large Structural Dynamics Models Using Measured Response Functions, Ph. D. Thesis,Imperial College of Science, Technology and Medicine, University of London, UK, 1998.
6Friswell, M., I., and Mottershead J., Finite Element Model Updating in Structural Dynamics, Vol. 38 of Solid Mechanicsand Its Application, Kluwer Academic Publisher, Dordrecht / Boston / London, 1995.
7Balis Crema, L., Coppotelli, G., Effect of Limited DOFs and Noise in Structural Updating, in Proceedings of XXInternational Modal Analysis Conference, p. 1091-1097, Los Angeles, CA, USA, 2002.
8Balis Crema, L., Coppotelli, G., Experimental Investigation into Sensitivity-Based Updating Methods, in Proceedings ofXXI International Modal Analysis Conference, Kissimmee, p. 125-133, FL, USA, 2003.
9Balis Crema, L., Coppotelli, G., La Scaleia, B., Identification and Updating of AB204 Helicopter Blade F.E. Modelby Means of Static and Dynamic Tests, 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and MaterialConference, Palm Spring, CA, USA, 2004.
10Stubbs, N., Global Non–Destructive Damage Detection in Solids – Experimental Verification, Int’l Journal of Analyticaland Experimental Modal Analisys, 5(2), pp. 81-97, Apr. 1990.
11Slotine J., and Li, Applied Nonlinear Control, Prentice Hall Co., Englewood Cliffs, NJ, 1991.12Balis Crema, L., Coppotelli, G., Structural Damage Identification from Frequency Response Function Estimates to be
published in Proc. of IFASD 2005, June 28 - July 01, Munich, D, 2005.13Ko, J., Kurdila, A.J., Strganac, T.W., Nonlinear Control of a Prototypical Wing Section with Torsional Nonlinearity,
Journal of Guidance, Control and Dynamics, Vol.20, No.6, 1997, pp. 1181-1189.14Block, J.J., Strganac, T.W. , Applied Active Control for a Nonlinear Aeroelastic Structure, Journal of Guidance, Control,
and Dynamics, Vol.21, No.6, 1998, pp.838-845.15Dowell, E.H., ”A Modern Course in Aeroelasticity”, Sijthoff and Noordhoff, 1978.16Jones, D.J., Lee, B.H.K., Time Marching Numerical Solution of the Dynamic Response of Nonlinear Systems, National
Aeronautical Establishment, Aeronautical Note - 25, National Research Council (Canada) No. 24131, Ottawa, Quebec, Canada,1985.
17Ko, J., Strganac, T.W., Stability and Control of a Structurally Nonlinear Aeroelastic System, Journal of Guidance,Control, and Dynamics, Vol.21, No.5, 1998, pp.718-725.
18Fung, Y.C., An Introduction to the Theory of Aeroelasticity, Dover, New York, 1955.19Khalil, H. K., ”Nonlinear Systems”, 3rd ed., Upper Saddle River, New Jersey, Prentice Hall, 2002.
13 of 13
American Institute of Aeronautics and Astronautics Paper 2005-2227
![Page 1: [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC](https://reader043.documents.pub/reader043/viewer/2022020615/5750952a1a28abbf6bbf7255/html5/page/1.jpg)
![Page 2: [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC](https://reader043.documents.pub/reader043/viewer/2022020615/5750952a1a28abbf6bbf7255/html5/page/2.jpg)
![Page 3: [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC](https://reader043.documents.pub/reader043/viewer/2022020615/5750952a1a28abbf6bbf7255/html5/page/3.jpg)
![Page 4: [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC](https://reader043.documents.pub/reader043/viewer/2022020615/5750952a1a28abbf6bbf7255/html5/page/4.jpg)
![Page 5: [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC](https://reader043.documents.pub/reader043/viewer/2022020615/5750952a1a28abbf6bbf7255/html5/page/5.jpg)
![Page 6: [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC](https://reader043.documents.pub/reader043/viewer/2022020615/5750952a1a28abbf6bbf7255/html5/page/6.jpg)
![Page 7: [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC](https://reader043.documents.pub/reader043/viewer/2022020615/5750952a1a28abbf6bbf7255/html5/page/7.jpg)
![Page 8: [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC](https://reader043.documents.pub/reader043/viewer/2022020615/5750952a1a28abbf6bbf7255/html5/page/8.jpg)
![Page 9: [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC](https://reader043.documents.pub/reader043/viewer/2022020615/5750952a1a28abbf6bbf7255/html5/page/9.jpg)
![Page 10: [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC](https://reader043.documents.pub/reader043/viewer/2022020615/5750952a1a28abbf6bbf7255/html5/page/10.jpg)
![Page 11: [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC](https://reader043.documents.pub/reader043/viewer/2022020615/5750952a1a28abbf6bbf7255/html5/page/11.jpg)
![Page 12: [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC](https://reader043.documents.pub/reader043/viewer/2022020615/5750952a1a28abbf6bbf7255/html5/page/12.jpg)
![Page 13: [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC](https://reader043.documents.pub/reader043/viewer/2022020615/5750952a1a28abbf6bbf7255/html5/page/13.jpg)
