Location Identification-harmonic Excitation

8/11/2019 Location Identification-harmonic Excitation

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Mechanical Systems

and

Signal ProcessingMechanical Systems and Signal Processing 23 (2009) 3044

Identification of structural non-linearities using describing

functions and the ShermanMorrison method

Mehmet Bu lent Ozera, H. Nevzat Ozgu ven

b,, Thomas J. Roystonc

aBaxter Healthcare Corporation, 1620 Waukegan Rd., MPGR-D1, McGaw Park, IL 60085, USAbMechanical Engineering Department, Middle East Technical University, 06531 Ankara, Turkey

cThe Department of Mechanical and Industrial Engineering (M/C 251), University of Illinois at Chicago, 2039 ERF,

842 West Taylor Street, Chicago, IL 60607, USA

Received 27 March 2007; received in revised form 5 October 2007; accepted 17 November 2007

Available online 22 November 2007

Abstract

In this study, a new method for type and parametric identification of a non-linear element in an otherwise linear

structure is introduced. This work is an extension of a previous study in which a method was developed to localize non-

linearity in multi-degree of freedom systems and to identify type and parameters of the non-linear element when it is

located at a ground connection of the system. The method uses a describing function approach for representing the non-

linearity in the structure. The describing function contains only the first harmonic terms. The ShermanMorrison matrix

inversion method is used in the present study to put the response expression in a form where the non-linearity term

can be isolated. Using measured responses one can calculate the value of the describing function representation of the non-

linear element and thus perform the identification. This new method can be used for type and parametric identification of a

non-linear element between any two coordinates of the system. Case studies are given to demonstrate the applicability of

the method.

r 2007 Elsevier Ltd. All rights reserved.

Keywords:Non-linear structural dynamics; Non-linear identification; Non-linear structural identification

1. Introduction

Engineering structures generally exhibit some degree of non-linearity. Localization and identification of

non-linearity by using experimentally measured response is important for several reasons. For instance, inmodel updating, a non-linearity in the system will make any linear model incorrect and model updating would

not correct the problem; essentially, different corrections at different forcing levels would be required.

There has been extensive research on localization and identification of non-linearity in structures. Studies in

this field can be classified into two groups: time domain and frequency domain techniques. Time domain

techniques generally focus on time series [1] and force state mapping [24]. Some of the frequency domain

techniques in which first-order or higher-order harmonic responses are used to localize and/or identify

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www.elsevier.com/locate/jnlabr/ymssp

0888-3270/$- see front matterr 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ymssp.2007.11.014

Corresponding author. Tel.: +90 312 2102549; fax: +90 312 2102536.

E-mail address: [email protected] (H.N. Ozgu ven).


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non-linearity in structures include the works of Tanrkulu and Ozgu ven[5], Vakakis and Ewins [6], Lin and

Ewins[7], S-anltu rk and Ewins[8], Richards and Singh[9], and Ozer and Ozgu ven [10].

In the work presented here the method developed in an earlier study [10]is used to localize non-linearity in

multi-degree of freedom (MDOF) systems by using measured harmonic responses. The method is then

extended to identify the type and the parametric values of the non-linear element within the structural system

where there is a single non-linear element between any two coordinates of the system. The identificationmethod presented in the earlier study [10]is restricted to systems with non-linear elements located at ground

connections only. Application of ShermanMorrison matrix inversion method allows the method to be used

to identify a non-linear element located at any place in the system. Some numerical case studies are presented

to demonstrate the application of the method.

2. Theory

The method developed is based on expressing the non-linear forcing vector in a matrix multiplication form

in the differential equation of motion of a MDOF non-linear system with harmonic excitation. This has been

first achieved by Budak and Ozgu ven [11,12] in their studies analyzing harmonic vibrations of non-linear

MDOF systems, and then the method is generalized for any type of non-linearity by Tanrkulu et al. [13]by

using describing functions. This representation has been used in several other studies on harmonic vibrations

of non-linear MDOF systems. Some of the works in this field are: Tanrkulu and Ozgu ven[14]; Co mert and

Ozgu ven [15]; Kuran and Ozgu ven [16]; Ferreira and Ewins [17], Ozer [18]; and Platten et al. [19].

2.1. Non-linearity matrix

Consider the differential equation of motion of a non-linear MDOF system with harmonic excitation

M f xg C f _xg K fxg fNg ffg, (1)

where [M], [C], and [K] are mass, viscous damping and stiffness matrices of the linear system, respectively.

Here, {f} and {N} represent the external forcing and non-linear internal forcing, respectively. Harmonic

excitation is expressed as

ffg fFg eiot, (2)

where {F} denotes the amplitude vector of forcing and i represents the unit imaginary number. The assumed

response is

fxg fXg eiot, (3)

where {X} is the vector of response amplitudes. Since the response is assumed to be harmonic, the internal

non-linear forces can also be assumed to be harmonic, and then can be written as

fNg fGg eiot. (4)

Substituting Eqs. (2)(4) into Eq. (1) yieldso2 M io C K

fXg fGg fFg. (5)

The amplitude of the internal forcing vector can be expressed [1012]as

fGg D fXg, (6)

where [D] is the response-dependent, so-called non-linearity matrix. The non-linearity matrix can be

formed by using describing function representations of internal non-linear forces. For details of how to

calculate the non-linearity matrix one should refer to the works of Tanrkulu and Ozgu ven [14] and

Tanrkulu et al. [13]. This representation makes it possible to perform harmonic vibration analysis of non-

linear MDOF systems with non-linearities that have describing function representations, provided that

an iterative solution accurately converges to the actual solution. This is basically the application of the

harmonic balance method and describing functions to non-linear MDOF systems. Moreover, by using this

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representation it is also possible to apply almost all the techniques developed for linear MDOF systems to

non-linear MDOF systems, provided that the non-linearity is not so strong so that the harmonic response

assumption holds true.

From Eqs. (5) and (6) the relation between harmonic forcing and response can be obtained as

o2 M io C K D fXg fFg. (7)Then {X} can be written as

fXg h fFg, (8)

where [h] is the pseudo-receptance matrix of the non-linear structure, given by

h o2 M io C K D 1

. (9)

Here, [h] is called pseudo-receptance since it is the receptance matrix of the system only for a specific

response level.

The receptance matrix [a] of the linear part of the same system can be written as

a o2 M io C K 1

. (10)

From Eqs. (9) and (10), [D] can be obtained as

D h1 a1. (11)

Post multiplying both sides of Eq. (11) by [h] yields

D h I Z h, (12)

where [Z] is the dynamic stiffness matrix of the linear part of the system, and is given by

Z a1 K o2 M io C

. (13)

Eq. (12) is the starting point for determining the harmonic response of a non-linear MDOF system by using

the receptances of the corresponding linear system.

2.2. Evaluation of non-linearity matrix using single harmonic response

Tanrkulu et al.[13]proposed the use of describing functions for the evaluation of [D]. The elements of [D]

can be found from urjas follows:

Drr urrXnj1jar

urj, (14)

Drj urj, (15)

whereurjis the harmonic describing function representation of the non-linear force and can be evaluated using

the expression below:

urj i

pYrj

Z 2p0

Nrjeic dc, (16)

whereNrjis the non-linear forcing between the rth and jth coordinates and Yrj is the amplitude of the inter-

coordinate displacement

Yrjifraj; Xr Xj

ifr j; Xr

( ) (17)

and

c ot. (18)

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M.B. Ozer et al. / Mechanical Systems and Signal Processing 23 (2009) 304432


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It should be noted that the harmonic describing functions of many systems have already been derived; a

table of describing functions for different types of non-linearities is given by Gelb and Van der Velde [20].

Common types of structural non-linearities are given in Table 1.

If the type of non-linearity is known, using the describing functions one can find the describingfunction representation of the non-linearity and form the [D] matrix using Eqs. (14)(18). It should be noted

that the expressions given above for the elements of [D] are also a function of the response itself. Therefore, in

order to calculate the response one should employ an iterative procedure. The initial guess can start with the

response of the linear system. First, the initial guess is used in the [D] expression (Eqs. (14)(18)) and then one

can solve for the non-linear response (Eq. (9)). In the next iteration step, the calculated responses are used in

obtaining the elements of [D] and non-linear responses are recalculated. This procedure goes on until a

convergence criterion is satisfied for the non-linear responses. Note, although the fundamental harmonic of

the response is considered here, it is always possible to include the higher harmonics, as explained, for

instance, in [16,17].

3. Identification of structural non-linearities using describing functions and ShermanMorrison method

3.1. Localization of structural non-linearity

Ozer and Ozgu ven[10]introduced a non-linearity number (now called non-linearity indexNLI) that can

be obtained from harmonic vibration measurements, in order to localize any non-linear element in the system.

Let us consider the ith column of Eq. (12), which will give

D fhig feig Z fhig, (19)

where {ei} is a vector of which the ith element is unity while all other elements are zero, and {hi} is the ith

column of [y]. The rth row of Eq. (19) gives,

Drfhi

g dir Zr fhi

g, (20)

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Table 1

Describing function representations of common structural non-linear elements (A is the oscillation amplitude)

Type of non-linearity Describing function representation

Cubic stiffness Desc 34

csA2

Coulomb damping Desc i4FfpA

Piecewise stiffnessDesc m1 m2

2p

arcsin dA

d

A

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 d

A

2q m2

Friction-controlled backlashDesc 1

2 1 2

p arcsin 1 b

A

1 b

A

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 1 b

A

2q i 1

p2bA

bA

2



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where [Dr] and [Zr] represent the rth rows of [D] and [Z], respectively, and dir is the Kronecker Delta function.

The termdir Zr fhigis called NLI. If we consider the rth coordinate of the system, NLIris an indication

of non-linear element(s) connected to the rth coordinate. This term is zero if the rth row and column of [D]

have all zeros. Any non-zero element in the rth row or column of [D] will yield a non-zero NLI for the rth

coordinate, since NLIr can be written from Eq. (20) as

NLIr Dr1 y1i Dr2 y2i Drn yni. (21)

Here i can be any coordinate. However, in calculating NLIr from experimental measurements, the right-

hand side of Eq. (20) will be used:

NLIr dir Zr1 y1i Zr2 y2i Zrn yni. (22)

Then, NLIr can be calculated from harmonic vibrations measured at all coordinates connected

to the rth coordinate, while the system is harmonically excited at coordinate i. The measurements

are carried out under high forcing amplitudes to obtain the elements of pseudo-receptance vector {hi}. The

elements of the pseudo-receptance vector can be obtained as in the case of receptances of a linear system, i.e.,

by using the applied harmonic force and the complex displacement responses measured. However, unlike

linear receptances, pseudo-receptances are response amplitude dependant. On the other hand, [ Zr] can beobtained from receptances experimentally determined through response measurements at low forcing levels,

which can be practical only for systems with small number of degrees of freedom (DOFs). For systems with

large number of DOFs, however, [Zr] can also be obtained via analytical and numerical (finite element, etc.)

methods.

Eq. (22) implies that measurements from all DOFs are needed to calculate the NLI of a coordinate r.

However in practical applications, impedance matrices are sparse with several zero elements. Since NLI is a

weighted summation of pseudo-receptances (weights being the elements of impedance matrix), the pseudo-

receptances that are multiplied with zero or small impedance matrix elements do not need measurement.

Similarly, the NLI of each coordinate is usually not required. Most of the time structural systems consist of

mainly known linear materials, but have non-linearities due to joints, ground connections, etc. Therefore, for

any coordinate that is in the linear region of the system there is no need to calculate a NLI. The NLI should be

calculated only for coordinates around potential non-linearities, and this will considerably decrease the

number of coordinates that need measurement.

3.2. Analysis of non-linear structures using Sherman Morrison method

In this section, it will be shown that if there is a localized non-linear element in a vibrational system, the type

and parametric identification is possible if the ShermanMorrison matrix inversion formula is used in the

expression for response amplitudes. The ShermanMorrison matrix inversion formula was first introduced by

Sherman and Morrison[21]and it has been applied to various vibrations and acoustics problems by Ozer and

Royston[2225]and Hong and Kim [26].

The system under consideration is shown inFig. 1. The non-linearity can be between any of the DOFs of the

system. Let us assume that the non-linearity exists between rth andjth coordinates. The non-linearity matrix

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Fsin(t)Non-linear elements

k1

m1

k2 k* kn

cn

mnm3m2

c1 c2 c*

Fig. 1. N-degree of freedom non-linear system with a localized non-linearity.



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[D] can be represented as

D

0

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

0u u

u u

0

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

0

2

6666666666666666664

3

7777777777777777775

. (23)

Since the non-linearity exists between the rth and jth coordinates the number of non-zero

elements of the [D] matrix is four. One can write the [D] matrix as a multiplication of two matrices as

shown below:

D fd1gfd2gT, (24)

where

fd1g

0

.

.

.

u

u

.

.

.

0

266666666664

377777777775; fd2g

0

.

.

.

1

1

.

.

.

0

266666666664

377777777775

. (25,26)

It can be seen that only therth andjth elements of the above vectors are non-zero. Using the above equation

in Eq. (8), which gives the non-linear response, one can obtain the following equation:

fXg Z fd1gfd2gT

1fFg, (27)

where the definition of [Z] is given in Eq. (13).

The ShermanMorrison matrix inversion formula is given as

A fugfvgT 1

A1 A1fugfvgTA1

1 fvgTA1fug, (28)

where [A] is anN Nmatrix with full rank and {u} and {v} areN 1 vectors. Then, Eq. (27) can be written as

fXg a afd1gfd2g

Ta

1 fd2gTafd1g

fFg. (29)

Recall that [a] is the receptance of the linear system and is given as

a Z1. (30)

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Using Eq. (29) one can write the response of each coordinate as follows:

X1

.

.

.

Xr

Xj

.

.

.

Xn

2666666666664

3777777777775

Xlin1

.

.

.

Xlinr

Xlinj

.

.

.

Xlinn

2666666666664

3777777777775

afd1gfd2g

T

1 fd2gTafd1g

Xlin1

.

.

.

Xlinr

Xlinj

.

.

.

Xlinn

2666666666664

3777777777775

. (31)

The non-linear responses are on the left-hand side of the equation and they are experimentally measured

quantities. The vectors on the right-hand side of the equation are the linear response of the system. They can

be obtained from measured receptances of the system under low-amplitude loading conditions, where non-

linearities would not be excited. The only unknown in this equation is u, which is the describing function

representation of the non-linearity. Eq. (31) can be solved for the value of u, and the values of u at

different amplitudes of response can be plotted, from which the type and the parametric value of the non-

linearity can be identified.

4. Case studies

4.1. Localization of non-linearity in the structure

It was explained in Section 3.1 that the describing function representation of non-linearities and the non-

linearity matrix concept can be used to identify the location of non-linear elements. Unlike the type and

parametric identification methods developed for non-linearity, in this method there is no limitation to the

number of non-linear elements; so the non-linearity can be distributed to many DOFs. The localization

procedure can be used to locate the non-linearity infected coordinates. As a case study, the system shown in

Fig. 2is investigated. This is a massspringdamper system with three DOFs. There is a non-linear macro-slipelement (Coulomb friction) between the first and the second DOFs. The numerical values of the system are

given as follows:

M1 1 kg; M2 2 kg; M3 1 kg; k1 1000 N=m; k2 1000 N=m; k3 1000 N=m,

k4 1000 N=m; c1 8 N s=m; c2 2 N s=m; c3 2 N s=m; c4 2 N s=m; F1 N,

cf 0:25 NCoulomb friction force.

In this case study, as well as in all others, the experimental results are simulated with the time domain

numerical solution of the differential equation of motion. A fifth-order RungeKutta numerical integration

method is used to solve the equations of motion. The steady-state response (amplitude and phase) information

is obtained from the numerical solution. The response after 40 cycles is taken as the steady-state response, and

the time step is determined by dividing each cycle into 400 equal step sizes (i.e. a fixed step size is used). Linear

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Fig. 2. The three-degree of freedom system with a macro-slip (Coulomb damping element).



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responses are calculated by transforming the equations into the frequency domain. Using Eq. (22) one can find

the NLIs for each coordinate. The NLIs are plotted out for frequencies between 40 and 80 rad/s.Figs. 35

show the plots of the NLIs for each DOF.

FromFigs. 3 to 5it can be observed that the NLI has finite values at coordinates 1 and 2, while the value of

the NLI for coordinate 3 is very close to zero almost at all frequencies. Figs. 35show that the coordinates to

which a non-linear element is connected can be accurately identified using the method proposed in this study.The primary reason for having a very small, but non-zero NLI at a coordinate that is not connected to a non-

linear element is the numerical inaccuracies in the simulated experiments. Similar results are expected in real

experimental cases due to experimental measurement errors.

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Fig. 3. The value of the non-linearity index at coordinate 1 evaluated at different frequencies.





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4.2. Localization of non-linearity in the structure: multiple non-linear elements

Eq. (22) can be used also for localizing multiple non-linear elements. The system in this example is a 15

DOF system where the forcing is applied at the eighth DOF. The system is shown inFig. 6and the numerical

data for the system are as follows:

M1...15 1 kg; k1...15 20; 000N=m; c1...15 30 N s=m; F 3 Nlocated on the 8th coordinate,

k6 k6 3107 x2 N=m; k11 k11 310

7 x2 N=m.

The response of the system is calculated by a fifth-order RungeKutta numerical integration method. The

simulation was run for 250 cycles at each frequency to ensure that transients die out. The time step used in the

simulation is set such that each cycle is divided into 650 time steps. The frequency range used during

the simulations is between 2 and 4 Hz with frequency increments of 0.025 Hz. The NLI is calculated for each

coordinate using Eq. (22).Fig. 7shows the non-linearity infected coordinates correctly. Since the two cubic

stiffness elements are between coordinates 5 and 6, and 10 and 11, these coordinates have higher NLI values as

compared to the rest of the coordinates.

4.3. Type and parametric identification of non-linearity: single degree of freedom system

A single degree of freedom (SDOF) system with cubic type of stiffness non-linearity will be used in this case

study to demonstrate that the type and parametric identification is possible applying the method suggested in

Section 3.2. An experimentally measured response will be simulated with a numerical solution of the

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Fig. 6. The 15 degree of freedom system with two cubic stiffness elements.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0

2

4

6

8

10

12

14

16

18

Coordinate Number

Sumo

fNon-linearityIndices

Fig. 7. The summed values of NLI at all frequencies over the whole frequency range for all coordinates.



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differential equation of motion. The numerical data for the system in Fig. 8are

M1 kg; k 100 N=m; k 104 N=m3; c 1 N s=m.

For a SDOF system the [D] matrix is a scalar, which is equal to u. Therefore one can take {d1} to be u and

{d2} to be unity in Eq. (29). The nonlinear responseXis equal toytimesF(the amplitude of the forcing on the

system). One can substitute these scalar values into the right- and left-hand side of Eq. (29) and obtain thefollowing expression for the describing function of the non-linearity:

ua y

ya , (32)

where y is the pseudo-receptance of the non-linear SDOF system.

The receptance of the linear and non-linear system is given in Fig. 9. A typical jump phenomenon can be

observed in the non-linear response. The non-linearity value (D) (which is a scalar in this case) is calculated

using the linear and non-linear receptance data. Delta values for different response amplitudes can be

calculated using Eq. (32). The change of delta values with harmonic vibration amplitude is plotted in Fig. 10.

It can be seen fromFig. 10that the non-linearity is a polynomial type of non-linearity. The scalar value of

delta is the describing function of the cubic type non-linearity. The describing function of a cubic type of

stiffness non-linearity is given as

ucubic 34csX2, (33)

where cs is the cubic stiffness coefficient and Xis the response amplitude of the non-linear system. The solid

line in Fig. 10 is the data fit (obtained through non-linear regression) to the delta values calculated from

Eq. (32). As depicted in Fig. 10, the curve fit is closely tracing the numerically obtained delta values. Using

Eq. (33) and the quadratic equation of the fit as given in Fig. 11, one can identify the cubic stiffness coefficient

to be 9765 N/m3, which is very close to the cubic stiffness coefficient of the system: 10,000 N/m3.Fig. 11shows

the response of the original non-linear system and the response of the identified system. It can be seen that the

difference in the cubic stiffness coefficient result does not translate into an appreciable difference in the

frequency domain response.

4.4. Type and parametric identification of non-linearity: multi-degree of freedom system-I

In this case study, type and parametric identification of a MDOF system will be performed. The system that

will be identified is the same system as the one in Section 4.1. Using Eq. (31) one can solve for the unknown

describing function representation of the non-linearity.

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Fig. 8. A single degree of freedom system with cubic stiffness.



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Using Eq. (31) one can write the response of the kth coordinate when the non-linearity is located betweenthe rth and jth coordinates as follows:

XkXlinkakr akj Xlinr Xlinj

u

1uarr 2arj ajj . (34)

From Eq. (34) one can solve for the describing function representation of the non-linearity, u, as shown

below:

uXlinkXk

Xk

Xlink

arr

2arj

ajj

akr

akj

Xlinr

Xlinj

. (35)

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Fig. 10. The delta values, ( ) calculated values, and (_____) obtained by performing a curve fit on calculated data.

Fig. 9. The receptance of the SDOF system case study: (UUUUUUUUUUUUUUUU

) linear receptance, (_____) non-linear pseudo-receptance

(increasing frequency), and ( ) non-linear pseudo-receptance (decreasing frequency).



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In this case study, the kvalue (measured coordinate) is taken to be the first coordinate, the r and jvalues

(the coordinates to which the non-linear element is attached) are 1 and 2, respectively.

Using the measured linear and non-linear responses of the kth coordinate (which are all simulated) along

with the linear receptance matrix, one can find the values of the describing function. Note that the linear

response can be obtained as the response of the non-linear system at a low forcing level. If the describing

function of the non-linearity and the amplitude of the response are plotted, behavior of the curve allows us to

identify the type of the non-linearity. If a simple curve fitting algorithm is applied, the parametric

identification is also possible.

It can be observed fromFig. 12that the describing function representation of the non-linearity decreases as

the amplitude of the response increases. Table 1shows that the only non-linearity that would cause such a

behavior is the Coulomb damping type of non-linearity. Using the describing function representation for

Coulomb damping one can fit a curve to the data given in Fig. 10. The curve fit (using non-linear regression

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Fig. 11. The receptance of (______) original system, ( ) original system (decreasing frequency), and (UUUUUUUUUUUUUUUU) identified

system.

Fig. 12. The plot of the calculated describing function values at different measured response amplitudes.



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analysis) gives a Coulomb friction force of 0.251 N, which means less than 1% error in estimating the

frictional force.

4.5. Type and parametric identification of non-linearity: multi-degree of freedom system-II

The previous two case studies have small numbers of DOFs. In this case study, a system with more DOFs

will be used (15 DOF). The type of non-linearity is coulomb friction as in the previous example. In this case

study, the coulomb friction element is not located in between the system coordinates. It is located between the

eighth DOF and ground. The system is depicted in Fig. 13

M1...15 1 kg; k1...16 20; 000N=m; c1...16 30 N s=m; F 3 N; Ff 0:12N.

Eq. (35) shows the describing function representation when the non-linear element is between the

coordinates r and j. Eq. (35) can be modified for a non-linearity between ground and a certain DOF by

omitting all the terms that contain coordinate j:

uXlinkXk

Xk Xlink

arr ak Xlinr . (36)

Since an increased number of DOFs is used in this example, the number of oscillations to reach the steady

state is increased to 250 in the numerical solution, and the responses at 500 time instants are determined for

each cycle. This is done to ensure that all of the coordinates reach steady state. The frequency range for the

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Fig. 13. 15 DOF system with macro-slipMDOF system-II.

Fig. 14. The summed values of NLI at all frequencies over the whole frequency range for all coordinatesMDOF system-II.



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solution is taken to be 3.85 Hz with a frequency increment of 0.025 Hz. This frequency range covers the first

natural frequency of the 15 DOF system.

In this case study the NLI is calculated at each frequency simulation location. The amplitude of NLI at each

frequency is summed over the whole frequency range for each DOF. The plot of the sum of the amplitudes of

NLI for the simulation frequency range for all DOFs is shown in Fig. 14.

As shown inFig. 14, the non-linearity infected coordinate is successfully identified as coordinate 8. The

change of the value of describing function for different vibration amplitudes around the first resonance is

shown inFig. 15. Using a least-squares curve fitting algorithm the frictional force is determined as 0.133 N.

It should be noted that due to increased number of DOFs, the computational time required for

solving equations of motion (for simulating experiments) has increased drastically. Therefore, in thenumerical solutions the response may not have reached steady state and/or it may contain slight contributions

from higher harmonics which are not considered in identification. Using the method described in [16]the first-

order harmonic response has also been calculated for the same system, and about a 2% difference was

observed in the mean value between first-order harmonic solution and the steady-state time domain solution.

This error in numerical solution gives some idea on how the method would respond to measurement error.

The effect of this difference can be seen inFig. 15. This difference leads to a deviation of 10% in the estimated

friction force.

5. Conclusions

A new method is introduced for type and parametric identification of a non-linear element in an otherwise

linear structure. This method is an extension of the method developed by the first two authors in an earlier

work [10] to localize the non-linearity in a MDOF system and to identify type and parameter values of the

non-linear element when it is located between the system and ground. The non-linearity matrix concept is

used also in this study, and the ShermanMorrison matrix inversion method is applied to put the response

expression in a form where the non-linearity term can be isolated. This made it possible to identify type and

parameters of a non-linear element between any two coordinates of the system. Case studies are given to show

that the method can be used to find the location of the non-linear element, and also to identify types of non-

linearities successfully. In the case studies given, cubic stiffness and Coulomb damping non-linearities are

considered, and both localization and parametric identification of the non-linearities are demonstrated. In the

last case study it is shown that the method can successfully be applied to large-scale systems. However, it is

observed in this case study that larger measurement errors may yield inaccurate values in parametric

identification. As it is aimed in this paper to introduce an index, which can be used to locate non-linearities in

ARTICLE IN PRESS

Fig. 15. The plot of the calculated describing function values at different measured response amplitudes.



15/15

structural dynamics by using FRF measurements, and give the theory of a new method to parametrically

identify such structural non-linearity, the case studies are given just to verify the theory developed. In order to

investigate further the applicability of the method suggested to real structures, it will be necessary to study the

accuracy of the method when real experimental measurements are used.

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