Simulation and Experimental Validation of Modal Analysis for Non-linear Symmetric Systems

Mechanical Systemsand

Signal Processing

www.elsevier.com/locate/jnlabr/ymsspMechanical Systems and Signal Processing 19 (2005) 2141

Simulation and experimental validation of modal analysis fornon-linear symmetric systems

R. Camillaccia,*, N.S. Fergusonb, P.R. Whiteb

aDipartimento de Scienze dellIngegneria Civile, University of Roma TRE, Italyb Institute of Sound and Vibration Research, University of Southampton, UK

Received 14 April 2003; received in revised form 6 May 2003; accepted 14 May 2004

Abstract

A dual approach, direct and inverse, is proposed for the study of a subset of discrete mechanical non-linear systems. Applying the denition of a mode for a non-linear system, the response calculation isrened for a class of mechanical systems possessing elastic restoring forces proportional to the cube of thedisplacement. The relationship between the modal natural frequencies and modal amplitudes of oscillationis analytically computed using the Harmonic Balance Method. An identication method, which operates onthe free response of the system, is presented and it is shown to be capable of recovering these functionalrelationships. Comparisons of the analytical approximation and identication solutions are made in orderto evaluate the methods effectiveness and its range of application. This comparison is achieved throughnumerical simulation.An experiment performed on a physical mechanical system exhibiting non-linear behaviour is then

presented. For such a system, the analytical relationships between modal natural frequencies and modalamplitudes of oscillation are calculated using its mechanical parameters. These relationships are comparedwith those identied experimentally and nally the results and possible extensions to the work arediscussed.r 2004 Elsevier Ltd. All rights reserved.

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*Corresponding author.

E-mail addresses: [email protected] (R. Camillacci), [email protected] (N.S. Ferguson), [email protected]

(P.R. White).

0888-3270/$ - see front matter r 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ymssp.2004.05.002

1. Introduction

The problem of extending the concept of modal analysis to non-linear mechanical systems hasbeen widely studied since the second half of the last century. The rst denition of a Non-linearNormal Mode (NNM) of a conservative system was given by Rosenberg [1], according to whoma NNM is a periodic oscillation where all material points of the system reach their maximumdisplacement at the same instant of time, and they all pass through their equilibrium position atanother instant of time. More recently, this denition was also extended to non-conservativesystems by Shaw and Pierre [2,3]. These authors dene a NNM as a motion which takes place onan invariant manifold of co-dimension two in the phase-space. Several studies have focused on theapproximate analytical computation of the non-linear modal properties both by perturbation [4]and energetic [5] methods that are described in literature.In general, the mode shape of a NNM depends on the amplitude of oscillation, but, as in [1], it

is well known that there are some particular classes of non-linear mechanical systems for whichthe mode shapes are independent of the amplitude; these modes are called similar modes. For suchsystems exact solutions have also been proposed [5]. By analogy to linear systems theory, a similarNNM is associated with a particular set of initial conditions, but since the principle of thesuperposition does not apply in non-linear systems theory, the NNMs, even if they characteriseunequivocally the dynamical system, they cannot be used directly to calculate the response underarbitrary initial conditions. In order to avoid this difculty several approaches have beenproposed in the literature [6]. Among these there is the concept of Coupled Non-linear Modes(CNM) [7]. In this approach, the free vibrations of a non-linear and non-conservative system areapproximated by a linear combination of harmonic terms. Every harmonic term depends on amode shape vector and on its corresponding frequency. In this way it is possible to characterisethe dynamic behaviour of a non-linear system under arbitrary initial conditions.

An important aspect of CNMs is that both the modal frequencies and the mode shapes dependon the amplitudes of all of the modes. It is assumed that if a system is endowed with similarNNMs it has similar CNMs as well, and in that case just the frequencies depend on the modalamplitudes [8]. It is possible to extract the temporal evolution of the CNM from the knowledge ofthe time histories in free vibration (the inverse problem), by use of appropriate identicationtechniques [813].

In this paper a specic class of non-linear mechanical systems is considered, that is systems witha linear and cubic forcedisplacement relationships that possess similar amplitude independentmodes. Initially the relationships between the modal frequencies and the modal amplitudes arecalculated analytically using the method of Harmonic Balance (HB). Then an identicationtechnique that is able to identify experimentally these relationships from the response of thesystem in free vibration is presented. Existing methods [1013] for this identication determine therelationship between the modal frequencies and time. Consequently they are unable tocharacterise the dynamical behaviour of the system, since the results depend upon on the initialconditions. In this work, the proposed procedure is able to identify the dynamic behaviour ofsystems by relating the frequencies of the CNM to the modal amplitudes and not to time.A validation is undertaken based on numerical simulations with the goal of determining the

limits of application of the procedure. Finally, the results from applying this technique tomeasurements from an experimental system containing geometric non-linearities are presented.

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R. Camillacci et al. / Mechanical Systems and Signal Processing 19 (2005) 214122

2. Analytical CNM for non-linear systems

Consider a n degree of freedom non-conservative mechanical system, described by the followingequation of motion:

M .x xC x Fx 0; 1

where M and xC are the n n mass matrix and n n damping matrix, respectively, x is a realpositive parameter such that 0ox51; x is the n element displacement vector and F(x) is the vectorof restoring forces. For such a system, the displacement response of the ith degree of freedom canbe approximated by the following relationship under the assumption of small amplitudes:

xi Xnj1

Yijaaj cos Fja; 2

where a a1; a2;y; an is the vector of amplitudes of the n harmonic components of theresponse. Since the system is damped a depends on time. Yij and Fj; are, respectively, the modalcoefcients and the total phase; both of them depend on all of the modal amplitudes, i.e. on thevector a. Fj is the sum of the instantaneous frequencies Oj and the initial phase jj:

Fj Z t0

Ojas ds jj: 3

If one collects all of the j components Yij for every degree of freedom, the modal shape vectorYj is obtained:

Yj

Y1ja1 a2?an

Y2ja1 a2?an

^

Ynja1 a2?an

0BBB@

1CCCA: 4

Every pair Yja;Oja denes a CNM. Thus a CNM depends on all of the modal amplitudesa1; a2;y; an:By application of the HB Method it is possible to calculate the CNM [7] and so by (2) the

approximate response of the non-linear system. It is necessary that (2) represents a goodapproximation of the response when the amplitudes of oscillation are small.If the modal shape Yj does not depend on the amplitude the mode is called similar [1]. Amongst

all mechanical systems possessing similar modes, one class of non-linear systems have beenanalysed here, where the restoring force is expressed by

Fi kliixi Xiah

klihxi xh knlihx

3i

Xiah

knlih xi xh3; 5

where kl and knl are the linear and non-linear stiffnesses, respectively.One can normalise the mode shapes with respect to the pth degree of freedom Ypj 1; j

1;y; n such that

xp Xnj1

aj cos Fja; xh Xnj1

Yhjaj cos Fja: 6

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R. Camillacci et al. / Mechanical Systems and Signal Processing 19 (2005) 2141 23

The restoring force for the pth degree of freedom is

Fp Fpl Fpnl klppxp X

h;pah

klphxp xh

" # knlppx

3p

Xh;pah

knlphxp xh3

" #; 7

where Fpl and Fpnl are the linear and non-linear restoring force, respectively. Because of thelinearity of the integration operator one can calculate separately the coefcients of the Fourierexpansion:1

1

2Jpj

ZFp cos Fj dU

ZFpl cos Fj dU

ZFpnl cos Fj dU Ijl Ijnl : 8

Substituting (6) into (7) and (7) into (8), see the appendix for denition of the integrals, it can beshown that

Ijl g0jaj; Ijnl Xnh1

ghjaja2h; 9

where ghj are constant coefcients resulting from the evaluation of the integrals in Eq. (8). HBrequires that [8]

1

ajJpj mijO

2j : 10

For the class of systems under consideration a general relationship between the jth modalfrequency Oj and the modal amplitudes is obtained:

O2j c0j Xnh1

chja2h; 11

where chj are constants that depend on the mechanical parameters of the system. Eq. (11)represents the relationships between the main frequency of each CNM and all of the modalamplitudes. In other words, the coefcients chj dene, unequivocally, the dynamical properties ofthe non-linear system.Eq. (11) thus represents the frequency modulation law of the jth modal component for a n-dof

lightly damped system possessing similar modes and cubic restoring force terms.It is well known [4] that non-linear mechanical systems can possess internal resonances. In such

a condition bifurcation of modes can arise, which means that the number of modes can be greaterthan the number of degrees of freedom. Yet, it is also known that even if the ratio of two (ormore) modal frequencies is an integer, but appropriate orthogonal conditions are satised [14,15],the modes do not have internal resonances, and thus bifurcation cannot occur. Since the class ofsystems under consideration possess similar modes, that is to say that the non-linear mode shapes

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1where Jij are the coefcients of the rst-order Fourier expansion:

Jij 2Z

Fi cos Fj dU;

whereR: dU

1

2pnR 2p0 ?

R 2p0 : dU1 dU2ydU3:


are the same as those of the linearised system, the orthogonality conditions between mode shapesare always satised for these systems [14], and hence bifurcation of modes can never occur.

3. Identication procedure

A two-level identication procedure has been developed in order to experimentally identify thefrequency modulation laws, as predicted by (11), from time histories of a system undergoing freevibration. The rst level of identication computes the evolutionary modal parameters, i.e. themodal parameters are computed as a function of time via a timefrequency representation. In thesecond identication level, the evolutionary modal parameters are used to estimate the coefcientschj in Eq. (11), allowing the frequency modulation laws to be expressed as a linear function of thesquare of the modal amplitudes.

3.1. First level of identification

In the literature [8,10,11,13] several timefrequency transforms have been developed in order toprovide information about the temporal evolution of modal parameters. The objectives here areto extract both amplitude and phase information from the data, so we naturally seek to employ alinear (rather than quadratic) timefrequency transform. The representation adopted here is theGabor transform [11].The Gabor transform Gxt;o of a signal xt is a complex function of two variables, dened by

Gxt;o Z NN

wt txt expjot dt; wt 1p

pDt

exp t

Dt

2 ; 12

where t is time, o is the angular frequency and wt is a Gaussian windowing function with widthparameter Dt.To conduct the analysis we require access to n time histories corresponding to each of the

degrees of freedom in the system. For each of the time histories, xit; the corresponding Gxi t;ois calculated, from which, the modal frequencies fijtk are identied for time instants tk (withk 1yN) by following the ridges in the modulus of Gxi t;o:Several techniques are proposed in the literature in order to identify the location of the peaks in

the transform Gxit;o [11,13]. The most important problem affecting this operation is thepresence of close modes, i.e. a high modal density, and the possible signicant overlap in thefrequency domain. This problem can be overcome if a curve-tting procedure is used [11] and isessentially the same problem as that encountered in standard modal analysis when seeking toextract parameters of multi-modal system. The majority of these forms of procedure are iterative,and hence the main drawback of the implementation of a curve-tting procedure is thecomputational cost. In order to reduce the computational costs, in this work, the modalcomponents are simply identied following the ridges of the Gxi t;o modulus, and, exactly, fromthe modulus of every Gxit;o; the amplitudes of the peaks Aijtk are extracted along with thecorresponding phases Pijtk: One degree of freedom is selected as the reference, this is designatedthe pth degree of freedom. The modal components of the remaining modes are normalised withrespect to the reference degree of freedom in the following manner. The modulus Yij and the sign

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sij of the normalised modal components are dened as

jYijtkj AijtkApjtk

; sijtk signcosPij Ppj: 13

Of course the identication error of Yij for every instant of time is greater than that by applyinga curve-tting estimation, but it is noted that for the class of systems under consideration themode shapes do not depend on time. Thus temporal averaging can be used to enhance theestimates of modal components by computing

Yij PN

k1 sijtkjYijtkjN

; 14

where N is the total number of time samples analysed.In this way, a very good estimation of the modal components is performed. In the presence of a

reasonably high modal density this has been checked with numerical validation of the procedure[10] and found to be reasonable.At this point it is possible to calculate the time histories in the modal domain by the modal

matrix containing the mode shapes

Y

Y11 y Y1n^ & ^

Yn1 ? Ynn

0B@

1CA

using

q Y1x; 15

where the dimensions of q and x are n N and they represent the time histories in the physicaland modal space, respectively. From the time histories in the modal space the modal amplitudesajtk are extracted using the Hilbert transform [12]. An example of this is illustrated in Fig. 1,where the Gabor transform is given for a simulated time history of a 2 dof system, thecorresponding time histories are shown in Fig. 2.

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Fig. 1. Modulus of the Gabor transform for a simulated two degree of freedom coupled non-linear system.


It is also observed, using the Hilbert Transform, that the modal frequencies fijtk; alreadyidentied by following the ridges of the Gabor Transform could also be again identied fromevery modal time history qjtk: The comparison of the two modal frequency identications wasused as a check during the application of this rst level of identication.

3.2. Second level of identification

The frequency modulation laws are identied by a linear regression analysis, tting the data to alinear relationship in the squared modal amplitudes a2j tk:Dening *Oijtk 2pfijtk2 and *ahtk a2htk; the error is dened as the difference between

the squared angular frequency calculated using Eq. (11) O2j tk and that identied experimentally*Oijtk

eijtk O2j tk *Oijtk *ci0j

Xnh1

*cihj *ahtk *Oijtk: 16

Minimization of the norm of the temporal error vector jjeij jj2 where eij feijtkgk; leads to

identication of the coefcients *ci0j; *cihj:

These coefcients do not depend on i; i.e. they are the same for all degrees of freedom. Thisfeature can be exploited to average results across the degrees of freedom, in order to furtherreduce the identication errors.

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Fig. 2. Time histories for simulated data in the physical space (x) and modal space (q).


3.3. Numerical validation

Various numerical simulations have been performed in order to validate the developedprocedure. In the numerical validation, the results obtained by analytical analysis (HB) arecompared with those obtained from the simulated time histories. The aim of the numericalvalidation is to investigate both the characteristics and the limits of the whole proposedprocedure. The differences between the analytical and identied results are due to two mainsources of error: the rst one is related to the analytical computation of the modal parameters bythe approximate HB method; the second one is related to the approximate nature of theparameter estimates obtained from the timefrequency analysis. Unfortunately, no exactanalytical methods exist for the computation of CNM parameters of a non-linear system, andthus the two sources of error cannot be distinguished.Briey presented are results in terms of the percentage difference E, which represents the

difference between the coefcients of the modal relationships calculated analytically and thecoefcients of the modal relationships identied by analysis of the time series

Ehij 100cihj *c

ihj

cihj: 17

In this way it has been possible to use the numerical validation in order to assess the sensitivity ofthe procedure to several factors, specically, the amplitude of oscillation, the damping, the signal-to-noise ratio and the modal density. Just the main conclusions concerning this sensitivity analysisare reported herein for the sake of brevity [10].The amplitude of oscillation are required to be small, in order to neglect the effects of sub and

super harmonics. Numerical simulations have been performed on a 2-dof symmetrical system(with kl11 k

l12 k

l22 k

l and knl11 knl12 k

nl22 k

nl) varying just the initial displacements. Theresults are presented as a function of the parameter K, which corresponds to the ratio between thenon-linear and linear restoring forces

K knlx301klx01

knlx201

kl; 18

where x01 is the initial displacement given to one degree of freedom, the remaining initialdisplacements are 0. Results of the error investigation for these simulations are depicted in Fig. 3for the two modes of oscillation.Inspecting the results in Fig. 3, one can see that the largest errors occur for both high and small

levels of excitation. The larger errors at high amplitudes arise because of the approximate natureof the HB solution, whilst the increased error at low amplitude relates to the fact that at theseamplitudes the changes in frequency that occur during the response are small, so are necessarilymeasured with greater uncertainty. To illustrate the latter effect, plots of two Gabor transformsare presented in Fig. 4 for small and large amplitudes. In Fig. 4 the frequency variations areclearly evident on the high amplitude trace, but are almost imperceptible in the low-amplitudeexample. One can also see that the terms c01 and c02; which relate to the linear component ofstiffness, are always accurately identied.The damping of the system is assumed to be light in order that the damped natural frequency

can be approximated by the undamped natural frequency. To assess the errors introduced by this

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approximation the percentage difference E is shown as a function of the linear modal damping inFig. 5. It is observed that an increase in the errors occurs with an increase in modal damping. Fig.6 relates to the identication with the greatest error and compares the results of analyticalcomputation by HB with the results identied from the procedure developed here. From this itcan be observed that one can achieve a good identication even if there is a large error (3040%)on the individual coefcients. This leads one to suggest that, for reasonable values of modaldamping, relationship (11) is a good representation of the modal frequency behaviour.Finally, a random signal was added to the simulation outputs in order to mimic the presence of

Gaussian noise, with zero mean. The results are given in Fig. 7 as a function of the parameter R,which is the ratio between the standard deviation of noise and the amplitude of overall oscillation,effectively a measure of noise-to-signal ratio. As can be observed, the procedure is able to givegood results for R up to 40%. Further investigations [10] have demonstrated that the procedure isreasonably independent of the modal separation.To summarise, the numerical validation has shown that the proposed procedure works well

within a certain range of excitation amplitudes and when the damping is light. Moreover themethod is robust to modest levels of noise and independent of modal density.

4. Description and mathematical model of an experimental system

On the basis of the knowledge obtained from the numerical validation, the procedure has beenimplemented on data collected from experimental tests.

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Fig. 3. Relative difference on the frequency modulation law coefcients between the identied and analytically results

Eij vs. K where () c0j is the constant term and (y) c1j and (- - -) c2j are the factors multiplying the squared amplitudesof the rst and second mode, respectively.


A physical system was designed to be exible enough to allow variation of the physicalparameters controlling the dynamics; the system is shown schematically in Fig. 8 andphotographed in Fig. 9. The system can be modelled as a two degree of freedom non-linearmechanical system. It consists of 2 masses connected with 2 parallel tinned wires with diameter0.0254 cm, the total length of each wire is 122.1 cm. The masses comprise of aluminum plates(80.8 g for each mass). The wires have a mass of 1.94 g (1.2% of the total mass of the system) so

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Fig. 4. Comparison of the Gabor transforms of the reference system at low (a) and at high (b) amplitudes of oscillation.


can be neglected. The wires are arranged vertically, as can be seen in Fig. 9. Clamps are used toallow the tension in the wire to be adjusted. Changing the type or number of wires connecting themasses can be used to affect the axial stiffness in the system.The non-linearity of the system is due to the high amplitudes of oscillation and depends on the

elastic restoring force of the wire as is evident in Eq. (21). The system is symmetric, since themasses are equal and they were set in symmetric positions, thus the NNMCNM of the systemcan be considered similar, i.e. the mode shapes do not change with amplitude of oscillation.

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Fig. 6. Comparison of analytically calculated and identied modal frequency modulation laws for a simulation with

modal damping 1.06% for the rst mode and 1.84% for the second mode. ai is the amplitude of ith mode.


Eij versus modal damping v, where () c0j is the constant term and (y) c1j and (- - -) c2j are the factors multiplying thesquared amplitudes of the rst and second mode, respectively.


The system is excited into free oscillation orthogonal to the plates (masses), thus a 2D model isassumed and, taking into account the large displacements, the restoring forces on the two massesare, respectively,

F1 S R1 sin y1 S R2 sin y2;

F2 S R3 sin y3 S R2 sin y2; 19

where yi represents the angles depicted in Fig. 8, S is the static tension applied to the wires (whichremains constant during all the tests duration) and Ri is the elastic restoring force due to thedynamic axial strain

Ri AE

lidi kaidi; 20

where A is a cross section, E is the Young modulus, li is the length of each section of wire, kai isthe axial stiffness and di is the axial dynamic extension.For the model libxi; hence a third order expansion is used both for sin yi and di in terms of xi

for which it leads

sin yi xi

li1

3

x3ili ox5i ; di

l2i x

2i

q li

1

2lix2i ox

4i : 21

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Eij vs. signal noise, where () c0j is the constant term and (y) c1j and (- - -) c2j are the factors multiplying the squaredamplitudes of the rst and second mode, respectively.


Substituting (21) into (19) and neglecting terms of order greater than three, leads to

F1 KSL1x1 KSNL1x31 KNL1x31 KSL2x1 x2 KSNL2x1 x2

3 KNL2x1 x23;

F2 KSL3x2 KSNL3x32 KNL3x32 KSL2x2 x1 KSNL2x2 x1

3 KNL2x2 x13;

22

where KSLi S=li; KSNLi S=3l3i and KNLi kai=2l2i :

The maximum displacements in all the tests performed were of the order of 102m (a fewcentimeters) and because the ratio KSNLi=KNLi 2S=3AE is also of the order 10

3 this means thatthe contributions which are in terms of KSNLi are small compared to the terms arising from thedynamic strain KNLi and have been neglected.Hence, only KNLi and KSLi terms are considered in Eq. (22). The ratio KNLi=KSLi AE=2l2i S is

of order 100 and since they are the restoring force coefcients for the cube of the actual

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S

l

1. M

2.

S

1

3

2

1

l2

x2

l3

Fig. 8. Mathematical mode of the two degree of freedom system under investigation.


displacements (order 106m) and the linear (order 102m) terms in the displacement, respectively,the ratio between the non-linear and linear part of the restoring force is of order 104, that is tosay that the system is weakly non-linear.Moreover as l1 l3 l and dening l2 a1l; allows the equations of motion (22) to be

expressed as

F1 S

l1 ax1

S

lax2

ka

2l2x31

ka

2l2a3x1 x2

3;

F2 S

l1 ax2

S

lax1

ka

2l2x32

ka

2l2a3x2 x1

3; 23

with ka AE=l:The analytical CNM calculated by HB can then be obtained. For the rst mode

c1 1

1

; O21 cl cnll1 24

and for the second mode

c21

1

; O22 1 2acl cnll2 25

in which l1 0:375a21 0:75a22; l2 0:75a

21 0:375 3a

3a22; cl S=lM and cnl ka=Ml2; with

M the value of each mass. Eqs. (24) and (25) represent the modal frequency modulation laws of

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Fig. 9. Physical model of the two degree of freedom system under investigation.


the two modes of the system. One can observe that the rst modal frequency law does not dependon the stiffness between the two mass, since it is independent of a.The physical system was instrumented by placing two accelerometers on the edge of each mass

parallel to the direction of the motion. The signals recorded by the accelerometers were integratedtwice in order to construct displacement records, because the identication procedure developedneeds displacement time histories as inputs.

5. Results

The system was tested in various congurations, these are summarised in Table 1. Fig. 10 showsan example of the time histories recorded on the two masses, with the system in conguration 1,from this data it is evident that the system has light damping, so satises the hypothesis discussedin Section 2. In Fig. 11 the Gabor Transforms of these time histories are given. From the GaborTransforms, both the modal components normalised respect to one degree of freedom and themodal frequencies are identied, as illustrated in Fig. 12. The symmetry in the system ensures thatthe modal components do not change with the amplitude of oscillation, although, of course, themodal frequencies do change as a function of amplitude. The modal components are identiedwith an average error of 3% on the rst mode and 1% on the second mode.The methodology presented in Section 3 is used to construct estimates *cl and *cnl of the

coefcients cl and cnl in (24) and (25). The results are presented in Table 2 based on two errorparameters el and enl ; dened as

el 100cl *cl

cl; enl 100

cnl *cnlcnl

: 26

For each conguration 3 tests were conducted using nominally the same initial conditions.Experimentally ensuring repeatability of the exact initial conditions is difcult and in thisapplication unnecessary, so that each realisation was initiated by release from approximately thesame location. This uncertainty introduced some variation in the measured time series, howeverthe measured coefcients are insensitive to such modest variations as demonstrated in Section 3.3.Each test allows one to construct 2 estimates of the coefcients cl and cnl (one estimate from eachof the measured time series). The values in Table 2 are computed by averaging the absolute valuesof the errors el and enl over these 6 estimates.

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Table 1

Summary of system congurations

System conguration a S (N) ka (kN/m) cl (s2) cnl (s

2m2)

1 1 38.5 50 1170 3.72 106

2 1 40.6 50 1234 3.72 106

3 1 44.2 100 1344 7.44 106

4 2.67 38.0 50 921 1.88 106

5 0.72 40.0 50 1373 5.35 106


Inspecting the results in Table 2, one can see that the linear parameter is always well identied(with an average error less than 2.5%) and that the non-linear parameters are always subject togreater uncertainty. It is informative to examine more closely the worse case in Table 2,corresponding to conguration 3. Fig. 13 depicts the percentage error in the estimates of modalfrequency for one test in this conguration (the realisation chosen is the one that gives rise to thelargest individual error). As observed in the numerical simulations, the observed differencesbetween the identied modal frequency and the frequency calculated by the analytical model aresignicantly smaller than that observed between the coefcients. Also the error decreases (inmagnitude), with time or, equivalently, that the error increases with the amplitude of oscillation.This aspect is in agreement with the theory and the numerical validation presented in Section 3.The effect of repositioning the masses along the wires, altering a; is to shift the frequencies;

more precisely, if the reciprocal distance between the two masses is increased ao1 the stiffnessbetween them decreases and the stiffness between the mass and the clamp increases. Hence therst modal frequency, which is independent of the central stiffness, is invariant whilst the secondmodal frequency decreases. As the distance between the two masses is increased so the separationin frequency between the two modes is decreased, as is the case in conguration 5, whilst inconguration 4 the modal separation is increased. From Table 2 it is evident that the performanceof the estimation method is not aversely affected by such changes.

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0 5 10 15 20 25 30 35-0.04

-0.02

0

0.02

0.04

time [sec]

time [sec]

Time history First Mass

0 5 10 15 20 25 30 35-0.04

-0.02

0

0.02

0.04Time history Second Mass

Fig. 10. Experimental displacement time histories of the two masses. Values in (m).


6. Conclusions

For the class of non-linear symmetrical mechanical system under consideration, that is systemswith a cubic forcedisplacement relationship and endowed with similar modes, an analytical

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Fig. 11. Gabor transform of the time history; (a) the rst mass; (b) the second mass.


relationship between the modal frequencies and modal amplitudes were calculated. On the basisof this, an identication procedure based on the Gabor Transform was developed. This procedureis not only able to identify the evolutionary modal parameters, but also the modal frequencyrelationships as a function of the mode amplitudes. In this way one can identify the dynamicbehaviour of the mechanical system under consideration.Analysing the results of the numerical validation it is possible to state that the procedure works

well, within some limitations, specically when the amplitude of oscillation is small, and bothmodal overlap and damping is low. Moreover the results were found to be robust to signicantnoise levels. In all of the simulations a good identication of the linear term was found even whenthe above assumptions were stretched.Finally the procedure was validated experimentally, by taking measurements from a physical

mechanical system exhibiting suitable non-linear behaviour. It was shown that, for the physical

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Table 2

Mean absolute percentage errors for different system congurations

Conguration Mode 1 Mode 2

%el %enl %el %enl

1 1.65 4.50 0.76 5.2

2 2.46 11.5 1.35 14.7

3 1.84 24.9 2.08 5.95

4 1.54 6.12 1.35 31.0

5 1.70 14.0 1.17 12.3

5(a) (b)

10 15 20 25 30-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Modal Components

time[sec]5 10 15 20 25 30

55.5

66.5

77.5

88.5

99.5100.511

time[sec]

Identified frequencies

Fig. 12. (a) Identied modal components Y21 and Y22; (b) comparison between identied (- - -) and analytical ()

modal frequencies.


system under consideration, the modal frequency relationships depend just on two parameters,one related to the linear stiffness and the other one related to the non-linear stiffness. Theadvantage of this representation of the frequency modal relationship is that there are fewerparameters to identify and it leads to more accurate identication. Furthermore, because the twoparameters are the same for every modal frequency, it is possible to employ suitable averaging toachieve more accurate evaluation of the parameters.The extension of the procedure to other non-linear systems (for instance endowed with non-

similar modes) is not trivial. This is because even if the calculation of CNM (and their associatedmodal frequency modulation laws) by Harmonic Balance is always theoretically possible, in mostcases it is not possible to obtain a simple relationship between the modal frequencies and modalamplitudes, such as the quadratic relationship considered here, i.e. Eq. (11). However, anextension in this sense is desirable and is being considered.

Appendix

The Fourier coefcients are [2]

Jij 2Z

Fi cos Fj dU; A:1

where:R: dU 12pn

R 2p0 ?

R 2p0 : dF1 dF2y dFn:

For the linearity of the integral operator, substituting the linear part of the resorting force inEq. (8), yields

Ijl Z

klppxp cos Fj dU X

h;pah

klph

Zxp cos Fj dU

Zxh cos Fj dU

A:2

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10 15 20 25 30-7

-6

-5

-4

-3

-2

-1

0

time [sec]

1st frequency2nd frequency

Fig. 13. Percentage error between the modal frequency calculated analytically and identied experimentally.


substituting Eq. (6) in (A.2) leads to

Ijl Z

klpp cos FjXnj1

aj cos Fj dU

X

h;pah

klph

Zcos Fj

Xnj1

aj cos Fj dU Z

cos FjXnj1

Yhjaj cos Fj dU

!A:3

and because it isZcos Fi cos Fj dU dij ; A:4

where dij is the kroneker symbol, Eq. (A.3) becomes

Ijl klppaj Xhaj

klph1 Yhjaj g0jaj: A:5

Substituting the non-linear part of the restoring force in Eq. (8) leads to

Ijnl Z

knlppx3p cos Fj dU

Xh;pah

knlph

Zxp xh

3 cos Fj dU A:6

and substituting Eq. (6) in (A.5), yields

Ijnl Z

knlpp cos FjXnj1

a3j cos3 Fj dU

X

h;pah

knlph

Zcos Fj

Xnj1

aj cos Fj dU Xnj1

Yhjaj cos Fj dU

!30@1A A:7

and because it isZcos3 Fi cos Fj dU

3

4dij ;Z

cos2 Fi cos2 Fj dU 2 with iaj;Zcos2 Fi cos Fj cos Fh dU 0 with iajah;Zcos Fi cos Fj cos Fh cos Fk dU 0; iajahak; A:8

Eq. (A.7) becomes

Ijnl Xnh1

ghjaja2h:

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References

[1] R.M. Rosenberg, The normal modes of nonlinear n-degree of freedom systems, J. Appl. Mech. 169 (1962) 319347.

[2] S.W. Shaw, C. Pierre, Normal modes for non linear vibratory systems, J. Sound Vib. 164 (1) (1993) 85124.

[3] S.W. Shaw, C. Pierre, Normal modes for non-linear continuous systems, J. Sound Vib. 169 (3) (1994) 319347.

[4] A.H. Nayfeh, Non Linear Interactions Analytical, Computational, and Experimental Methods, Wiley Series in

Nonlinear Science, Wiley, New York, 2000.

[5] A.F. Vakakis, et al., Normal Modes and Localization in Nonlinear Systems, Wiley, New York, 1996.

[6] W. Szemplinska-Stupnicka, The Behavior of Nonlinear Vibrating Systems, Vol. II: Advanced Concepts and

Applications to Multi-Degree-of-Freedom Systems, Kluwer Academic Publishers, Dordrecht, 1990.

[7] S. Bellizzi, A. Bouc, Analysis of multi-degree of freedom strongly nonlinear mechanical systems with random input

Part I: non-linear modes and stochastic averaging, Probab. Eng. Mech. 14 (1990) 229244.

[8] S. Bellizzi, P. Gulliman, R. Kronland-Martinet, Identication of coupled non-linear modes from free vibration

using timefrequency representations, J. Sound Vib. 243 (2) (2001) 191213.

[9] R. Camillacci, N.S. Ferguson, P.R. White, Identication of nonlinear normal modes and coupled nonlinear

modes, Institute of Sound and Vibration Research, Memorandum No. 884, 2002.

[10] R. Camillacci, F. Brancaleoni, N.S. Ferguson, C. Valente, Third Joint Conference of Italian Group of

Computational Mechanics and Ibero-Latin America Association of Computational Methods in Engineering,

Analisi e Identicazione di sistemi meccanici discreti dotati di rigidezze cubiche, 2002.

[11] D. Spina, C. Valente, G.R. Tomlinson, A new procedure for detecting nonlinearity from transient data using

Gabor transform, Nonlinear Dynamics 11 (1996) 235254.

[12] D. Spina, C. Valente, M.P. Petrangeli, Sulluso della Trasformata di Hilbert nella Identicazione della Risposta

Dinamica di Sistemi non Lineari, University of Roma La Sapienza, Internal Report l/91, 1991.

[13] N. Delprat, B. Escudie, P. Guillemain, R. Kronland-Martinet, P. Techamitchian, B. Torresani, Asymptotic

wavelet and Gabor analysis: extraction of instantaneous frequencies, IEEE Trans. Inform. Theory 38 (2) (1992)

644664.

[14] W. Lacarbonara, G. Rega, A.H. Nayfeh, Resonant non-linear normal modes, Part I: analytical treatment for

structural one-dimensional systems, Nonlinear Mech. 38 (2003) 851872.

[15] W. Lacarbonara, G. Rega, Resonant non-linear normal modes, Part II: activation/orthogonality conditions for

shallow structural systems, Nonlinear Mech. 38 (2003) 873887.

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Simulation and experimental validation of modal analysis for non-linear symmetric systemsIntroductionAnalytical CNM for non-linear systemsIdentification procedureFirst level of identificationSecond level of identificationNumerical validation

Description and mathematical model of an experimental systemResultsConclusionsReferences

Documents

Simulation and Experimental Validation of Modal Analysis for Non-linear Symmetric Systems