ol

s

I

tions. Secondly, a full representation of the modal damping matrix

Downloaded Fcould possibly enable parameters of a physical damping model tobe evaluated, thereby revealing the distribution of dampingpresent in the system.

Much work has concentrated on the identification of physicaldamping matrices ~@38# are but a few!. It would then be possibleto obtain the modal damping matrix from the physical dampingmatrix using a normal mode transformation. However, in general,the physical damping matrix may be very large for real systems,rendering the identification difficult. The above references tend toconcentrate on fairly low order lumped parameter models.

However, less attention has been paid to the identification ofmodal damping matrices, particularly for nonproportionallydamped systems. The majority of methods found in the literatureare based on extracting damping information from the complexmodes of the system. For a linear system, mode complexity occursas a direct result of nonproportional damping.

For instance, Hasselman @910# proposed a perturbationmethod for identifying a full modal damping matrix based on

ral frequencies, so leading to a response solely in the normal modeshape of interest by satisfying the phase resonance criterion. Here,the quality of the normal modes can be assessed, and so errors canbe reduced. Rades @16# proposed a method of estimating a non-proportional modal damping matrix based on Phase Resonancetesting. The method uses the complex energy transmitted to thesystem during force appropriation. In fact, soft tuned modes ~apredicted dynamic response of the tuned undamped normal mode!may be used in the method, so eliminating the need for physicallytuning the modes ~so called hard tuning!. It was reported@16# that small errors in locating the undamped natural frequencyproduced large errors in the values of the off-diagonal terms of themodal damping matrix. The method of Rades was analyzed in@17# where it was shown that use of an imperfect force vectorcould lead to bias in the estimated off-diagonal terms of the modaldamping matrix.

In this paper, a method based on Phase Resonance testing @1820# for identifying elements of the modal matrices of non-proportionally damped systems is presented. Given the FrequencyResponse Functions ~FRFs! derived from a modal test, the appro-priated force vectors and natural frequencies allow each normalmode to be excited in isolation using a short burst of sinusoidal

Contributed by the Technical Committee on Vibration and Sound for publicationin the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April 2002;Revised August 2003. Associate Editor: M. I. Friswell.

298 Vol. 126, APRIL 2004 Copyright 2004 by ASME Transactions of the ASMESteven NaylorArvinMeritor Inc., Warton Technical Center,

Hillock Lane, Warton,Preston, Lancashire, PR4 1TP UK

e-mail: steven.naylor@arvinmeritor.com

Michael F. Plattene-mail: michael.platten@man.ac.uk

Jan R. Wrighte-mail: jan.wright@man.ac.uk

Jonathan E. Coopere-mail: jonathan.cooper@man.ac.uk

School of Engineering,University of Manchester,

Oxford Road,Manchester, M13 9PL, UK

IdentificaFreedomNonpropthe ResoThis paper describes aidentification of the mdamped systems usingsystem at each naturaapproach is illustratedto introduce significanand rigid body modeshapes is considered.damping matrix. @DO

1 IntroductionThe identification of accurate dynamic models for systems is of

considerable importance in predicting the response to any inputsignal. While the estimation of modal mass, undamped naturalfrequency and mode shapes is well established for most linearsystems, the general difficulty in modelling energy dissipation hasled to much less reliable results for the damping.

An assumption often made in compiling the damping informa-tion is that the system conforms to the special case of proportionaldamping, i.e. the damping matrix may be expressed as a functionof the mass and stiffness matrices, such that the modal transfor-mation yields a diagonal modal damping matrix. However, whenthe system contains localized discrete sources of damping, as isoften the case in practice, the damping will be nonproportionaland a proportional damping assumption may lead to significanterrors in the predicted dynamic response.

Assuming a nonproportional modal damping matrix yields twoadvantages over an assumed proportional modal damping matrix.Firstly, off-diagonal terms may couple modes when their naturalfrequencies are closely spaced @1#, and close to the excitationfrequency @2#, and such terms may affect the response calcula-rom: http://vibrationacoustics.asmedigitalcollection.asme.org/ on 02/14/2tion of Multi-Degree ofSystems Withrtional Damping Using

nant Decay Methodn extension of the force appropriation approach which permits theodal mass, damping and stiffness matrices of nonproportionallymultiple exciters. Appropriated excitation bursts are applied to thefrequency, followed by a regression analysis in modal space. Theon a simulated model of a plate with discrete dampers positioned

t damping nonproportionality. The influence of out-of-band flexible, imperfect appropriation, measurement noise and impure modeThe method is shown to provide adequate estimates of the modal: 10.1115/1.1687395#

experimentally derived complex modes. The underlying assump-tion is that the complex mode shapes represent only a small per-turbation from the normal mode shapes. Therefore, the methodbecomes less accurate as the amount of nonproportional dampingincreases and is suitable only for systems with weak nonpropor-tional damping. Another limitation is that knowledge of the physi-cal mass matrix is required. For continuous systems this informa-tion is not generally available.

On experimental application of the method, it was found thatthe identified modal damping matrix was in error ~since it was notpositive definite!. Vold et al. @11# showed that the probable sourceof this was errors in the phase of the experimentally derived com-plex modes which obscured the damping information. Other com-plex mode approaches @1214# also suffer from the same prob-lem. Experimentally derived complex mode shapes are usually theresult of a phase separation analysis, where there is no clear mea-sure of the quality of complex modes.

On the other hand, phase resonance analysis, or force appro-priation @15# identifies the normal modes of the system, where thequality of the identified mode shapes can be quantatively assessedand errors reduced. Force appropriation involves applying a par-ticular monophase pattern of forces to a system at one of its natu-015 Terms of Use: http://asme.org/terms

excitation. Once the excitation ceases, the free decay of the sys-tem will include a response from any modes coupled by damping

sponse occurs that includes responses in a subset of modescoupled by significant off-diagonal damping terms. Thus, the fully

Downloaded Fforces to the mode being excited. These modal responses, togetherwith the modal force input, form the basis of a least squares fit toobtain an approximate non-proportional modal damping matrix.

This so-called Resonant Decay Method ~RDM! has been ap-plied to several simulated dynamic systems with nonproportionaldamping distributions in previous papers @1819# where encour-aging results were obtained. In this paper, the method is demon-strated on a simulated model of a plate with discrete damperspositioned so as to introduce non-proportional damping. Also, theeffect of out of band flexible and rigid body modes, imperfectappropriation, measurement noise and impure mode shapes isassessed.

2 Theory2.1 Force Appropriation. Force Appropriation involves the

analysis of the Frequency Response Function ~FRF! matrix inorder to obtain a set of multiple exciter force patterns and un-damped natural frequencies so that the normal modes of the sys-tem may be excited in isolation. The phase resonance condition issought such that the displacement or acceleration responses aremonophase, and in quadrature with the excitation. The ModifiedMultivariate Mode Indicator Function method ~MMIF @15#! wasused for this work. In this method, an eigenvalue problem is for-mulated, where the eigenvalues drop to zero at the undampednatural frequencies and the corresponding eigenvectors yield theappropriated force vectors. The tuning of the mode allows mea-surement of the undamped normal ~real! modes of the system,even for a non-proportionally damped system. Such an approachis different to Phase Separation techniques, which attempt tocurve fit the FRF matrix in order to obtain the damped ~complex!modes of the system.

It should be noted that for proportionally damped systems, eachmode of interest may, in principle, be excited by the appropriatedforce pattern at any frequency. This is because the appropriatedforce vector excites the mode of interest but, in essence, intro-duces no modal force into any other modal equation. The systemthen behaves as a single degree of freedom. However, for a non-proportionally damped system, the appropriated force vector isspecifically aimed at counteracting any modal damping couplingforces that are likely to excite other modes. In this case, a normalmode may be excited only at its undamped natural frequency. Atany other excitation frequency, other modes will also be excitedwherever modal damping coupling forces are present.

In the RDM approach, the idea of a burst appropriation isutilized. This involves removing the appropriated forcing functionwhen the system has reached the steady state condition where anormal mode is excited. This allows the system to decay to restwithin the time window of analysis and to show up modescoupled by damping forces.

2.2 Identification of Nonproportionally Damped Systems.Consider the equations of motion expressed in modal space

@M #$ p%1@C#$ p%1@K#$p%5$q% (1)where @M # , @C# and @K# are the (N3N) modal mass, dampingand stiffness matrices respectively, $q% is the (N31) modal forcevector, $p% is the (N31) modal displacement vector, and N is thenumber of degrees of freedom. Differentiation with respect totime is represented by the usual overdot convention. The modalmass and stiffness matrices are diagonal.

In the general case, where the nonproportional modal dampingmatrix is fully populated, each mode would be coupled to everyother mode due to the off-diagonal damping terms. Solving a po-tentially large system of simultaneous equations could prove to beimpractical if broadband excitation were to be used. However,when the system is subjected to a burst appropriation at a givennatural frequency then, once the excitation stops, a decaying re-

Journal of Vibration and Acousticsrom: http://vibrationacoustics.asmedigitalcollection.asme.org/ on 02/14/2populated modal damping matrix will consist of important off-diagonal terms, plus others which can be neglected in the identi-fication process since their coupling contribution to the dynamicresponse is negligible. It is assumed that the modal matrix isknown from a Phase Separation analysis, or from a normal modeexcitation, and thus the modal forces, $q%, can be derived for agiven physical excitation force pattern using

$q%5@f#T$ f % (2)where @f# is the modal matrix and $ f % is the physical force vector.

Consider the burst appropriation of a mode, i , which resultstheoretically in a pure response in mode i ~i.e. a normal moderesponse! during steady state, and a response in, say, mode j ( j.i) that occurs during the decay due to damping coupling forceswhich cease to be suppressed once the appropriated forcing func-tion is removed. Equation ~1! can be written for mode i , at aparticular instant in time, as

M iip it1Ciip it1Ci jp j t1Kiipit5qit (3)This equation can be rewritten as

$ p it p i t p j t pit%H M iiCiiCi jKiiJ 5qit (4)

Equation ~4! can then be expanded to include a number, n , of timepoints over the burst period as

F p i1 p i1 p j1 pi1p i2 p i2 p j2 pi2] ] ] ]p in p in p jn pin

G H M iiCiiCi jKiiJ 5H qi1qi2]

qin

J (5)where the additional subscript refers to the time sample.

Note that if the natural frequencies were assumed known at thisstage from the force appropriation process, then because of therelationship between the modal mass and stiffness terms, Eq. ~5!could be re-written in terms of only three unknowns. Alterna-tively, a modal mass estimate from a different method could beimposed. However, the formulation used in this paper leaves themodal stiffness ~and hence natural frequency! to be identified con-sistently with the other parameters in a single operation, given thatany small nonlinear effects present could influence the estimatedfrequency.

Equation ~5! is in the form

@A# i$x% i5$b% i (6)and can thus be solved in a least squares sense @21# for the un-known parameter vector, $x% i , typically using a Singular ValueDecomposition approach. The solution yields the diagonal modalmass, stiffness and diagonal/off-diagonal modal damping termscorresponding to the ith row of the modal matrices. If couplingsto other modes are also present, then the process may be extended,simply by adding further damping coupling terms into Eq. ~3! andEq. ~5!, until a goodness of fit criterion is satisfied.

The process described above seeks to identify the cross-coupling terms from a combination of the steady state and decaydata. It would be possible to solve for the cross coupling terms formode i , using just the steady state part of the data. However, thisrelies on perfect appropriation, which is unrealistic. By includingthe decay, an indicator of coupling is provided and also a responsein other modes is included.

It should be noted that in practice only the physical accelera-tions and forces are usually measured and Eq. ~5! will thereforerequire integration of the physical acceleration data to obtain the

APRIL 2004, Vol. 126 299015 Terms of Use: http://asme.org/terms

physical velocity and displacement prior to transformation tomodal space. If the integration is performed in the frequency do-

Downloaded Fmain rather than the time domain, the burst nature of the forcingfunction will ensure a leakage free Fourier Transformation, pro-vided the response has decayed sufficiently, and integration driftwill therefore be minimized. The physical force and response datathen need to be transformed to modal space using the modal ma-trix, prior to performing the least squares curve fit.

It is possible to take advantage of reciprocity in the formulationof the solution for a series of modes. Consider next the appropria-tion of an adjacent mode, j , being coupled to modes i and kwhere i, j,k . The modal equation for mode j , at a particularinstant in time, is

M j jp j t1C jip it1C j jp j t1C jkpkt1K j jp jt5q jt (7)and if the modal damping matrix is symmetric ~i.e. the systemobeys the principle of reciprocity!, then Ci j5C ji , and Eq. ~7!may be rewritten as

M j jp j t1C j jp j t1C jkpkt1K j jp jt5q jt2Ci jp it (8)where Ci j is known from the solution of Eq. ~5!. Equation ~8! maybe formulated in a similar way to that used earlier, so as to permita solution of the modal parameters.

It is clear that the process may be repeated on a mode-by-modebasis to identify the modal matrices row by row. The principle ofreciprocity may be used in order to solve for the upper triangularsection of the modal damping matrix but any error in the identi-fication of off-diagonal damping terms in the early modes willultimately be carried forward to the other modes. For this reason ithas been decided to solve for the upper and lower triangular partsof the modal damping matrix independently by avoiding the sub-traction of reciprocal terms shown in Eq. ~8!. The symmetry of theresulting modal damping matrix may then be used as an indicationof reciprocity and hence, for a linear system, the accuracy of theestimation. However, if a nonsymmetric solution is obtained, thisis not necessarily because of non-linearity; it could be due to noiseetc.

If the entire identification of all the modes was carried out in asingle step, a more consistent estimate might be obtained; how-ever, the reason that the mode-by-mode approach is used here isthat it allows the number of damping cross coupling terms usedfor each mode to be minimized.

In principle, Eq. ~5! could also be written in physical co-ordinates and physical parameters could be identified, but this hasits limitations. Providing excitation at one co-ordinate and mea-suring the response at other co-ordinates would provide the datafor the first solution of Eq. ~5!. However, the estimation of theremaining physical equations of motion would not be possiblesince the right hand side of these equations would be zero, unlessa physical force were to be applied at every co-ordinate, which isnot a practical approach. This problem is overcome @7# if thesystem displays symmetry in its physical damping matrix. Theright hand side of Eq. ~8!, expressed in physical co-ordinates,would then consist of known symmetric damping forces butwould depend on the accurate estimation of the earlier parameters.

Expressing Eq. ~5! in modal co-ordinates avoids such problemssince qi is a modal force and will always be non-zero for theperiod of normal mode excitation, thereby avoiding the necessityfor the modal damping matrix to display symmetry. For largesystems with perhaps hundreds of measurement stations, the so-lution of the equations in modal space involves far fewer un-known parameters than the equivalent model in physical space.

3 Simulations Based on a Plate Model With Nonpro-portional Damping

In this section, the robustness of the RDM is tested further bythe introduction of the following types of measurement errors:measurement noise, mode shape errors, inclusion of flexible and

300 Vol. 126, APRIL 2004rom: http://vibrationacoustics.asmedigitalcollection.asme.org/ on 02/14/2rigid body modes outside of the bandwidth of interest, and imper-fect force appropriation. Each of these sources of error, likely tobe manifest when applying the method to a real system, is simu-lated in turn. The effects of these imperfections on the identifica-tion are discussed.

3.1 Plate Model. A Finite Element ~FE! dynamic model ofa freely suspended plate was used throughout the following simu-lations. The aluminum plate had dimensions of 0.830.4830.012 m and included 10 flexible and 6 rigid body modes ofvibration, although only the first 5 flexible modes were includedin most of the cases. The natural frequencies for these modes are48.6, 66.4, 130.4, 171.5 and 192.7 Hz. A proportional dampingdistribution of 0.5% critical damping was assumed for each mode.Additionally, nonproportional damping was introduced in theform of two viscous dampers ~damping coefficient 450 Nsm21)linking the plate to earth. The dampers were positioned at oppositecorners of the plate. A schematic diagram of the plate arrangementcan be seen in Fig. 1.

In all the estimations performed, the modal mass and stiffnessmatrices were identified accurately so the focus will be on thedamping matrix results. The exact modal damping matrix for thefirst 5 flexible modes of the simulated system is shown in Table 1below and indicates that modes 1, 2 and 5 are coupled and so aremodes 3 and 4, the couplings dictated by the symmetry or asym-metry of the mode shapes.

Excitation was provided at two positions, corresponding to thedamper positions at opposite corners numbered 4 and 21 in Fig. 1.This choice allowed the forces to be applied in line with thedampers, thus enabling their coupling effect to be counteracteddirectly.

FRFs were calculated in the range 0250 Hz using 1024 spec-tral lines for the MMIF calculation. The burst appropriation timeresponses were obtained by transforming the input to the fre-quency domain, multiplying by the FRF, and transforming theresult back to the time domain. The FRFs used for the responsecalculation were generated with the same frequency resolution inthe range 02000 Hz ~using 8192 spectral lines!; this satisfied aminimum of 16 time points per cycle for all of the modes, soproviding a visually clear response. Each of the excitation timesignals was chosen to have a 50% burst, including 25% ramp up

Fig. 1 Plate model dimensions and measurement positions

Table 1 Exact modal damping matrix for the first 5 flexiblemodes of the system

52.07 24.28 0 0 26.5924.28 11.67 0 0 23.10

0 0 5.36 4.45 00 0 4.45 4.19 0

26.59 23.10 0 0 0.98

Transactions of the ASME015 Terms of Use: http://asme.org/terms

and 5% ramp down. These ramps were given a half period cosineenvelope in order to avoid severe transients, as will be seen later

3.3 Case A Results: Datum 2 Exciters and No Other Er-ror Sources

Journal of Vib ol. 126 301

Downloaded From: http://vibratFig. 3 Modal forces and responses for the burst appropriation of mode 2 Case Ano error

ration and Acoustics APRIL 2004, Vin the burst time histories.

3.2 Summary of Test Cases. The RDM was applied to thenonproportionally damped plate model with only the first 5 flex-ible modes, for a total of 6 different cases. Strictly speaking, theno experimental error case should correspond to using as manyexciters as there are modes in the model. This would result inperfectly pure normal modes. However, the purpose of force ap-propriation is to determine the undamped natural frequencies andappropriated force pattern that will isolate, as near as possible, thenormal modes of a system with the given number of excitersavailable. Thus, force appropriation allows the number of excitersto be less than the total number of modes while still achievinghigh quality normal modes. This is a more realistic case. For thisreason, the case of using 2 exciters will be taken as the datum~Case A! for comparison with other test cases, each representing adifferent source of experimental error. The errors associated withusing 2 exciters positioned at the damper positions are sufficientlysmall so as not to obscure the effects of the other sources of error.

The sources of error considered for application of the RDMwere as follows. Case A: Datum ~2 Exciters!, Case B: Out of BandFlexible Modes, Case C: Out of Band Rigid Body Modes, Case D:Force Appropriation Errors, Case E: Measurement Noise, and fi-nally Case F: Mode Shape Impurities.

It should be noted that in all cases shown, the natural frequen-cies are estimated accurately, even if small errors occur on themodal mass and stiffness terms.ionacoustics.asmedigitalcollection.asme.org/ on 02/14/2Force Appropriation Results. The calculated FRFs were usedin a Modified MMIF analysis to yield the eigenvalues in Fig. 2,giving the undamped natural frequencies ~48.6, 66.4, 130.4, 171.5and 192.7 Hz! and appropriated force vectors. The primary ~small-est magnitude! eigenvalue can clearly be seen to drop close tozero for all five modes, indicating that good appropriation can beachieved.

Modal Forces and Responses. Figure 3 shows the modalforces for the burst appropriation of mode 2. The modal forcesand responses are zero for modes 3 and 4. Because damping cou-pling is present between modes 1, 2 and 5, some modal force isapplied to modes 1 and 5 in order to appropriate mode 2. Theseforces counteract the inter-modal damping coupling forces duringthe steady state phase of the burst. The modal velocity responsesdue to the burst appropriation of mode 2 can also be seen in Fig.3. The steady state response is an almost pure response in mode 2,confirming a successful force appropriation. During the ramp upthere is a response in mode 1 but this dies away as the steady statedevelops. During the decay following the burst, mode 1 can beseen to be responding significantly, indicating that there is strongdamping coupling between modes 1 and 2. There is no visibleresponse in mode 5 during the decay, indicating that although theC25 coupling term exists in the modal damping matrix, it only hasa small effect due to the relatively large frequency separation be-tween these modes.Fig. 2 Modified MMIF eigenvalues for two exciters and non-proportional damping Case A015 Terms of Use: http://asme.org/terms

When mode 3 is excited by a burst appropriation ~Fig. 4! it canbe seen that modes 3 and 4 are coupled less obviously than modes

Estimation of Modal Matrices. The modal responses suggestfairly strong nonproportional damping coupling between modes 1

Downloaded FFig. 5 Modal forces and responses for the burst appropriation of mode 5 Case Ano error

302 Vol. 126, APRIL 2004 Transactions of the ASME1,2 and 5 in Fig. 3 and there is no modal force or response inmodes 1, 2 and 5. The behavior of these responses however issimilar to that for the excitation of mode 2.

Figure 5 shows the modal forces for the burst appropriation ofmode 5, with modal forces present in modes 1 and 2 to counteractcoupling forces. The modal velocity responses show a steady stateresponse dominated by mode 5. There is no noticeable response inany other mode during the decay, again suggesting that the effec-tive coupling to modes 1 and 2 is small.

It should be noted that the relative amplitudes of the modalresponses depend on both the physical response of the system andon the relative scaling of the mode shapes used in the transforma-tion between physical and modal space. In this study the modeshapes were normalized to unity maximum displacement and, forthis particular plate model, the contribution of each normalizedmode to the general system response is of the same order of mag-nitude. This means that visual inspection of the burst modal re-sponses, combined with knowledge of the exact modal dampingmatrix, gives a good indication of the coupling terms whichshould be included in the curve fit ~see below!. For more complexstructures under non-ideal measurement conditions these assump-tions may not hold. In that case a more suitable approach may beto include in the curve fit damping coupling terms to all modes inthe vicinity of the mode of interest. These terms may then beranked in order of their contribution to the response and the leastsignificant term discarded until a desired goodness-of-fit isachieved.rom: http://vibrationacoustics.asmedigitalcollection.asme.org/ on 02/14/2and 2, with both of these modes effectively uncoupled from mode5. There is also evidence for moderate coupling between modes 3and 4. In fitting the modal time histories, the symmetry of themodal damping matrix was ignored ~i.e. each row of the modalmatrices was identified independently! and three variations ofcoupling were considered:

Case A-1: all true coupling terms includedCase A-2: as for Case A-1, but with coupling to mode 5

omittedCase A-3: no coupling included, i.e. diagonal modal damp-

ing matrixIn the following results, the estimated modal damping ratio is

used to compare the accuracy of the different assumed dampingconfigurations. This can be found from a complex mode analysisof the estimated modal matrices.

For Case A-1, where all relevant damping coupling terms areincluded in the fit, a modal damping matrix which is identical tothe exact modal damping matrix ~Table 1! is identified. The modaldamping ratios and modal mass and stiffness matrices are alsoidentified exactly.

The weak coupling of modes 1 and 2 to mode 5 suggests thatthese terms may be neglected in the least squares fit and Case A-2considers the effect of this assumption. Table 2 shows the esti-mated modal damping matrix, which when compared to Table 1shows good agreement for all but the mode 5 coupling terms. Thedamping ratios for modes 1, 2 and 5 are found with 0.0, 1.8 and3.8% error respectively. The estimated modal mass and stiffnessFig. 4 Modal forces and responses for the burst appropriation of mode 3 Case Ano error015 Terms of Use: http://asme.org/terms

Table 2 Identified modal damping matrix with mode 5 cou-pling terms omitted

Table 4 Identified modal damping matrix with out-of-bandrigid body modes

Downloaded Fvalues are identified more accurately ~maximum error 0.1%! thanthe modal damping matrix, and this is the case throughout thesimulations.

Since the coupling between modes 1 and 2 has been seen to bestrong, then neglecting such coupling terms in the least squares fitis expected to cause large errors. The effect of omitting couplingbetween modes 3 and 4 is not expected to be as severe. For CaseA-3, the estimated diagonal of the modal damping matrix is@52.07, 11.70, 5.33, 4.22, 1.00#, which when compared to theexact matrix in Table 1 shows good agreement for diagonal ele-ments However, the absence of off-diagonal terms produces largeerrors in the modal damping ratios ~percentage errors of 15.9%and 60.4% for mode 1 and 2 respectively!. It is clear thereforethat, while some insignificant damping coupling terms may beneglected, the assumption that all coupling terms may be ne-glected ~Case A-3! leads to unacceptable errors in this case. Themaximum error on the modal mass and stiffness terms is 0.7%.

3.4 Case B Results: Out-of-Band Flexible Modes. A realsystem will have modes outside of the frequency range of interestand these will affect the identified modal model. The effect ofthese out-of-band modes was considered in this case but willobviously depend upon factors such as frequency separation anddamping. A further five flexible modes were included in themodel, having natural frequencies of 266.61, 270.61, 363.94,463.67 and 496.81 Hz.

The modal forces and responses are used, as before, in a mode-by-mode least squares fit to obtain the coefficients of the modalmass, damping and stiffness matrices for the first five modes withno assumption of reciprocity. A modal matrix which includes onlythe first five modes is used in the modal transformations but theresidual effects of the other modes are present in the FRF andtherefore in the response data. All damping coupling terms be-tween modes 1, 2 and 5 and modes 3 and 4 are included in theidentification to show the effect that the out-of-band modes haveon the estimation of off-diagonal modal damping terms. Table 3shows the estimated modal damping matrix. The errors on thediagonal terms happen to be zero and those for the coupling termsbetween modes 1 and 2 and modes 3 and 4 are small, but thecoupling terms corresponding to mode 5 are in much greater error,typically over 100% with one term being of the wrong sign. Themodal mass and stiffness terms are identified within 0.04%.

The errors in the coupling terms for the fifth mode are largestbecause it is affected by the presence of the out-of-band modesmore than the lower frequency modes in the model, due to thereduced frequency separation. The out-of-band modes affect theaccuracy of the force appropriation and also the modal transfor-mation. Also, the modal responses in the burst appropriation cor-responding to the coupling terms involving mode 5 are very smalland therefore more sensitive to error. A practical way of dealing

52.07 24.58 0 0 024.02 11.68 0 0 0

0 0 5.36 4.45 00 0 4.45 4.19 00 0 0 0 1.00

Table 3 Identified modal damping matrix with out-of-band flex-ible modes

52.07 24.26 0 0 213.9924.27 11.67 0 0 26.26

0 0 5.36 4.54 00 0 4.34 4.19 0

3.39 210.76 0 0 0.98

Journal of Vibration and Acousticsrom: http://vibrationacoustics.asmedigitalcollection.asme.org/ on 02/14/2with these issues is simply to include in the analysis any modesjust above the frequency range of interest which may be inducingthese errors.

3.5 Case C Results: Out-of-Band Rigid Body Modes. Inthe case where a system is suspended to give free-free boundaryconditions, the suspension system will allow rigid body motion.There are 6 rigid body modes corresponding to translation androtation about a 3-dimensional Cartesian axis system. The fre-quencies of the rigid body modes are a function of the stiffness ofthe suspension system and will ideally be well below the frequen-cies of the flexible modes so as to have a minimal effect.

The purpose of this case was to investigate the effect that rigidbody modes have on the results of the RDM, again with only thefirst 5 flexible modes used in the modal transformation. It is dif-ficult to quantify the effect since it will be a function of the prox-imity in frequency of the rigid body and flexible modes of vibra-tion. As a general rule of thumb, rigid body modes areconsidered to have a small effect on the flexible mode results ifthe natural frequency of the highest rigid body mode is less than1020% of the natural frequency of the first flexible mode ofvibration.

The mode shapes and modal masses for the rigid body modeswere obtained directly from the FE model of the plate. Since onlyout-of-plane motion was considered in the plate model, only thethree rigid body modes corresponding to heave, pitch and rollwere considered, with arbitrary frequencies of 4, 8 and 12 Hzrespectively ~the highest rigid body mode having a natural fre-quency approximately 25% of the natural frequency of mode 1 at48.6Hz!. Table 4 shows the identified modal damping matrixwhen the rigid body modes were included in the system responsebut not used in the subsequent modal transformations and curvefit.

The diagonal damping terms are found with good accuracy, asare the coupling terms between modes 1 and 2 and modes 3 and 4.However, the off-diagonal damping terms corresponding to mode5 are more in error but not as severely as for the higher frequencyout-of-band modes ~Case B!. Again, the modal response of mode5 due to the burst appropriation of modes 1 and 2 is small, and sois prone to error. Also, the very low frequency of the rigid bodymodes means that their contribution is amplified in the integra-tions performed to estimate velocity and displacement. The modalmass and stiffness values are identified with errors in the range0.44.7% because the rigid body modes are not included in thefitted model.

3.6 Case D Results: Errors in Force Appropriation.There are a number of ways in which errors in the force appro-priation may be introduced, but here an error of 5% was arbitrarilyadded to the natural frequencies identified from the MMIF to in-fluence the effectiveness of the appropriation of each mode. Themodal responses in Fig. 6 show a departure from the datum casein that the steady state response is not purely in the appropriatedmode but responses of other modes are also present, implying asexpected that the system is not responding in the normal modecondition. However, the least squares fit to the data still reveals anaccurate model ~mass, stiffness and damping! provided that all themodes associated with the response of interest are included. Onthe other hand, when all coupling terms are omitted from the fit,the modal damping matrix exhibits severe errors, with two diag-

51.91 26.52 0 0 227.5525.01 11.65 0 0 29.19

0 0 5.33 4.02 00 0 4.39 4.18 0

28.79 20.87 0 0 0.99

APRIL 2004, Vol. 126 303015 Terms of Use: http://asme.org/terms

modes 3 and 4 (;5%) and significant for mode 5 (;15%). Theerror on the modas the mode num

1% NoiseAdamping matrix.unaveraged casesensitive term C

t

Table 5 Identiunaveraged

51.9224.11

00

20.53

28.12 21.84 0 0 0.99

th 5% noise

17.5521.38

00

0.99

5% noise25

2.1922.09

00

0.99

304 Vol. 126 of the ASME

Downloaded From: http://vibratal mass and stiffness terms increases from 111%ber increases.

veraged. Table 6 shows the estimated modalThese results are far more accurate than for the

, even for the mode 5 coupling terms, with the15 now having an error of only 3%. The damping

fied modal damping matrix with 1% noise

24.51 0 0 0.4811.65 0 0 22.17

0 5.35 4.47 00 4.46 4.19 0

27.90 0 0 0.99

Table 7 Identified modal damping matrix wiunaveraged

51.90 24.10 0 024.16 11.65 0 0

0 0 5.35 4.730 0 4.50 4.19

1.15 29.07 0 0

Table 8 Identified modal damping matrix withaverages

51.92 24.61 0 024.11 11.65 0 0

0 0 5.35 4.450 0 4.47 4.19

21.88 26.82 0 0

, APRIL 2004 Transactionsonal terms identified with a negative sign. This occurs because thesteady state response includes other modes not accounted for inthe fit.

3.7 Case E Results: Measurement Noise. All physicalmeasurements made on a real test are subject to unwanted distur-bances superimposed on the desired signals that tend to obscuretheir information content. Because the RDM is based upon dataobtained at the resonance condition, the signal to noise ratio willbe larger than that from a broadband random excitation test,though if force drop out occurs, it will increase noise on theforce signals at resonance.

For this case, uncorrelated broadband white noise signals withzero mean and 1% or 5% rms amplitude were added to the noise-free acceleration signals prior to the identification. If noise is sus-pected to be a problem in practice, one approach is to average thedata from several nominally identical burst appropriations using atrigger at the start of each burst to synchronise the signals. Eachlevel of noise was considered with both unaveraged and averaged~25 times! data. Note that the unaveraged case was performed foronly one noise sample and so is not a statistically general result.However, the averaged cases used independent noise signals foreach sample and so will demonstrate how well the RDM is able todeal with random noise.

1% NoiseUnaveraged. Table 5 shows the estimated modaldamping matrix. Apart from the off-diagonal terms for the cou-pling of mode 5, all modal damping terms are found accurately(,1% error!; the coupling terms for mode 5 are significantly inerror, with term C15 having the incorrect sign. The errors in thedamping ratio are small for modes 1 and 2 (;1%), moderate for

Fig. 6 Modal forces and responses forDappropriation errorionacoustics.asmedigitalcollection.asme.org/ on 02/14/2ratios of the estimated model all have a small error (;1%). Theerror on the modal mass and stiffness terms has a maximum valueof 0.9%.

5% NoiseUnaveraged. Table 7 shows the estimated modaldamping matrix. The direct terms are found with very small errorand the off-diagonal terms near to the diagonal are found with arelatively small error of between 0.5% and 6.3%. The couplingterms for mode 5 have a much greater error, terms C15 and C51both having the incorrect sign. The error on the modal mass andstiffness terms ranges from 1376%. Clearly, the method is highlysensitive for this unaveraged case.

5% NoiseAveraged. Table 8 shows the estimated modaldamping matrix. The errors are negligible, apart from the mode 5coupling terms that are still significantly in error, though thesehave been reduced by averaging. The error on the modal mass andstiffness terms increases from 0.911% as the mode numberincreases.

The errors on the estimated damping ratios are between 1.2% to6.5% for the first four modes, and 16.2% for mode 5.

It is clear that averaging the noisy data reduces the effect of the

he burst appropriation of mode 1 Case

Table 6 Identified modal damping matrix with 1% noiseaveraged

51.93 24.20 0 0 26.3824.18 11.65 0 0 22.87

0 0 5.35 4.45 00 0 4.44 4.19 0015 Terms of Use: http://asme.org/terms

A contribution of 5% of the normal mode shape of mode 2 wasadded to the normal mode shape of mode 1, and vice versa formode 2, i.e.

where r represen0.05 in this casedependent on theand so it is not cshape corruptioto demonstrate tcorruption.

Figure 7 showburst appropriatthose in case Astate response no

as a multi-exciter approach to the identification of nonproportion-ally damped systems on a mode-by-mode basis, using an exten-

ethod has beenng a simulatedof different im-of out of band

on errors, mea-the whole the

ptably accuratefrequency sepa-ling effect wasto errors in theodal mass andrors.

and EPSRC in

t

Journal of Vib ol. 126 305

Downloaded From: http://vibrat$f%1corr5$f%11r$f%2

$f%2corr5$f%21r$f%1 (9)

ts the proportion of mode shape corruption, being. It should be noted that mode shape corruption is

choice of mode shape scaling used in the modelorrect to read the above as a universal 5% moden. Nevertheless, this case is included purelyhe effect of an arbitrary amount of mode shape

s the modal forces and velocity responses for theion of mode 2. The modal forces are similar to. However, the modal responses show a steadyt only in the appropriated mode, but also in mode

sion of the force appropriation method. The mdemonstrated and its robustness investigated usinonproportionally damped plate model. A numberperfections were considered, namely the presenceflexible and rigid body modes, force appropriatisurement noise and mode shape impurities. Onmethod performed very well and yielded an accemodal damping matrix. Because of the significantration between these modes, the damping coupsmall and so the parameters were more sensitiveestimation process. In all cases, the estimated mstiffness values were identified with only small er

AcknowledgmentsThe authors acknowledge the support of Airbus

this work.

ration and Acoustics APRIL 2004, Vnoise and hence allows more accurate estimates of the modalmodel to be made. By examining the error in the estimated damp-ing ratio against the number of averages, it has been found thatabout 10 averages are sufficient for errors to be acceptable at the5% noise level.

Note that errors due to noise may be additionally reduced byapplying a bandpass filter prior to integration so as to reduce theamplitude of any noise outside the relatively narrow frequencyrange of the burst.

3.8 Case F Results: Mode Shape Impurities. A fundamen-tal principle of the RDM is that the system is excited such that thesteady state response is essentially an isolated normal mode. Thephysical forces and responses are converted to modal co-ordinatesusing the estimated normal mode shapes, obtained from a softor hard tune or from a phase separation analysis. In practice,these normal mode shapes will not be perfectly pure but will,especially in the presence of nonproportional damping, be cor-rupted by other modes if the force appropriation is imperfect, e.g.insufficient exciters are used or exciters are not optimally placed.Thus, if another mode is excited during the tuning process, acombination of mode shapes will be measured. It is this effect thatis the subject of this section.

Adding a small amount of one mode to another will corrupt themodal matrix used in the conversion of physical data to modaldata. Since the system is nonproportionally damped such thatmodes 1 and 2 are strongly coupled, mode 1 was corrupted withpart of mode 2 and vice versa. The same principle was applied tothe coupled modes 3 and 4.

Fig. 7 Modal forces and responses for the burserrorionacoustics.asmedigitalcollection.asme.org/ on 02/14/21. This leaking of modal information is due to the pollutednature of the modal matrix used to convert from physical to modalspace.

Table 9 shows the estimated modal damping matrix. For modes1 and 2, it can be seen that the errors for the diagonal and off-diagonal terms are significant and of the same order. The resultingestimated damping ratios are in error by 14.3% and 11.1% formodes 1 and 2 respectively. The errors on the modal mass andstiffness terms were 1 and 10% for modes 1 and 2 respectively.

Errors in mode shape mean that the underlying basis functionsof the model are not normal modes and therefore that the modalmass and stiffness matrices are not strictly diagonal. Includingmass and stiffness coupling terms in the fit would overcome thisproblem but defeat part of the object of using a modal represen-tation unless a re-orthogonalization was performed to transformthe model back to uncoupled form.

4 ConclusionsThis paper has presented the Resonant Decay Method ~RDM!

appropriation of mode 2 Case Fmode shape

Table 9 Identified modal damping matrix with corrupted modeshapes

57.33 31.04 0 0 210.3227.37 14.32 0 0 23.30

0 0 5.36 4.45 00 0 4.45 4.19 0

26.75 23.43 0 0 0.99015 Terms of Use: http://asme.org/terms

Nomenclature@C# 5 modal damping matrixCii 5 direct modal damping coefficient for mode i

(Nsm21)Ci j 5 modal damping coupling coefficient between modes i

and j (Nsm21)K 5 modal stiffness matrix

Kii 5 modal stiffness of mode i (Nm21)@M # 5 modal mass matrixM ii 5 modal mass of mode i ~kg!

N 5 number of degrees of freedomn 5 number of sample points

$p% 5 modal displacement vector$ p% 5 modal velocity vector$ p% 5 modal acceleration vectorpit 5 modal displacement of mode i at sample point t ~m!p it 5 modal velocity of mode i at sample point t (ms21)p it 5 modal acceleration of mode i at sample point t

(ms22)q 5 modal force vector

qit 5 modal force of mode i at sample point t ~N!@f# 5 modal matrix

$f% i 5 mode shape of mode ir 5 proportion of mode shape corruption

References@1# Hasselman, T. K., 1976, Modal Coupling in Lightly Damped Systems,

AIAA J., 14~11!, pp. 16271628.@2# Park, S., Park, I., and Fai, M., 1992, Decoupling Approximations of Nonclas-

sically Damped Systems, AIAA J., 30~9!, pp. 23482351.@3# Caravani, P., and Thomson, W. T., 1974, Identification of Damping Coeffi-

cients in Multidimensional Linear Systems, ASME J. Appl. Mech., 41, pp.379382.

@6# Minas, C., and Inman, D. J., 1991, Identification of Non-proportional Damp-ing Matrix from Incomplete Information, ASME J. Vibr. Acoust., 113, pp.219224.

@7# Mohammad, K. S., Worden, K., and Tomlinson, G. R., 1992, Direct Param-eter Estimation for Linear and Non-linear Systems, J. Sound Vib., 152~3!, pp.471499.

@8# Pilkey, D. F., and Inman, D. J., 1997, An Iterative Approach to ViscousDamping Matrix Identification, Proceedings of the 15th International ModalAnalysis Conference, pp. 11521157.

@9# Hasselman, T. K., 1972, Method of Constructing a Full Modal DampingMatrix from Experimental Measurements, AIAA J., 10~4!, pp. 526527.

@10# Hasselman, T. K., and Chrostowski, J. D., 1993, Estimation of Full ModalDamping Matrices from Complex Test Modes, AIAA Paper, Paper 93-1668-CP.

@11# Vold, H., Melo, A., and Sergent, P., 1992, Phase Errors in Component ModeSynthesis, Proceedings of the 10th International Modal Analysis Conference,pp. 11321134.

@12# Zhang, Q., and Lallement, G., 1985, Simultaneous Determination of NormalEigenmodes and Generalized Damping Matrix from Complex Eigenmodes,Proceedings of the 2nd International Symposium on Aeroelasticity and Struc-tural Dynamics, pp. 529535.

@13# Placidi, F., Poggi, F., and Sestieri, A., 1991, Real Modes Computation fromIdentified Modal Parameters with Estimate of Generalized Damping, Pro-ceedings of the 9th International Modal Analysis Conference, pp. 572579.

@14# Alvin, K. F., Park K. C., and Peterson, L. D., 1993, Extraction of UndampedNormal Modes and Non-diagonal Damping Matrix from Damped System Re-alization Parameters, AIAA Paper 93-1653-CP.

@15# Nash, M., 1991, A Modification of the Multivariate Mode Indicator Functionemploying Principal Force Vectors, Proceedings of the 9th InternationalModal Analysis Conference, pp.688693.

@16# Rades, M., 1981, On Modal Analysis of Systems with Non-proportionalDamping, Rev. Roumaine Sci. Tech. Ser. Mec. Appl., 26~4!, pp. 605622.

@17# Naylor, S., 1988, Identification of Non-proportionally Damped Systems usinga Force Appropriation Technique, PhD Thesis, University of Manchester,UK.

@18# Naylor, S., Cooper, J. E., and Wright, J. R., 1997, On the Estimation ofModal Matrices with Non-proportional Damping, Proceedings of the 15thInternational Modal Analysis Conference, pp. 13711378.

@19# Naylor, S., Wright, J. R., and Cooper, J. E., 1998, Identification of Non-proportionally Damped Systems using a Force Appropriation Technique, Pro-ceedings of the 23rd International Seminar on Modal Analysis, pp. 481488.

Downloaded F@4# Hanagud, S., Meyuppa, M., Cheng, Y. P., and Craig, J. I., 1984, Identificationof Structural Dynamic Systems with Non-proportional Damping, Proceed-ings of the 25th SDM Conference, Palm Springs, pp. 283291.

@5# Fritzen, C.-P., 1986, Identification of Mass, Damping and Stiffness Matricesof Mechanical Systems, ASME J. Vibr. Acoust., 108, pp. 916.

306 Vol. 126, APRIL 2004rom: http://vibrationacoustics.asmedigitalcollection.asme.org/ on 02/14/2@20# Naylor, S., Wright, J. R., and Cooper, J. E., 1999, Identification of a Non-proportionally Damped Truss System, International Forum on Aeroelasticityand Structural Dynamics, pp. 847856.

@21# Golub, G. H., and Van Loan, C. F., 1989, Matrix Computations 2nd Edition,Johns Hopkins University Press, Baltimore.

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