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DETERMINATION OF MODAL RESIDUES AND RESIDUAL FLEXIBILITYFOR TIME-DOMAIN SYSTEM REALIZATION

K, F. Alvin*Sandia National Laboratories, Albuquerque, NM 87185

and L. D. PetersontUniversity of Colorado, Boulder, CO 80309

Abstract

A linear least-squares procedure for the determination of modal residues using time-dom ain systemrealization theory is presented. The present procedure is sh own to b e theoretically equivalen t to residuedetermination in realization algorithms such as the Eigensystem Realization Mgorithm (ERA) and Q-Markov CO VER. However, isolating the optimal residue estimation problem from the ge neral realizationproblem affords several advantages over standard realization algorithms for structural dynamics identifi-cation. Primary amo ng these are the ability to identify data se ts with large numbers of sensors using sm allnumbers of reference point responses, and the inclusion of terms w hich accurately model the effects ofresidual flexibility. T he accuracy and efficiency of the present realization theory-based procedure is dem-onstrated for both sim ulated and experimental data.

* Research Fellow, Structural Dynamics and Vibration Control, Dept. 1434;Member AIAAAssistant Professor, Department of Aerospace Engineering ; Senior Mem ber AIAA

OF V H B kENT 1s ~~~~~~~ w-iSTR1

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DISCLAIMERPortions of this document may be illegiblein electronic image products. Images areproduced from the best available originaldocument.

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I. Introduction

Research in structural identification in recent years has lead to a proliferation of algorithms based uponsystem realization theory. These modern system identification techniques24 have important direct ap-plications to structural control, such as dentification and order reduction of input-output models fo r robustcontrol an d adaptive on-line identification fo r nonlinear systems control. These algorithms all realize amodel by minimizing som e measure of the difference between the measured and reconstructed discrete-time impulse response functions, heretofore referred to as Markov parameters. In contrast to many class i-cal m odal identification techniques, the system realization algorithms are time dom ain techniques and aregenerally applicable to multiple-input multiple-output (MIMO) measured data systems.

These algorithms have arguably attracted the m ost attention fo r their use in modal test data analysisand reduction for identification of structu ral parameters5-6. There are a number of reasons for this. First,these method s are fairly simple to understand and implement, requiring only standard matrix manipulationand numerical analysis functions such as those available in Matlab. Secondly, these m ethods are foundedon sam pled data systems theory, which is directly applicable to inexpensive microprocessor-based dataacquisition systems. Finally, system realization theory offered a simplification of the modal identificationprocess by providing a clear indication (at least ideally) of dynamic order and by unifying the pole identi-fication and residue estimation problems into a single step analysis. In other words, these methods werepowerful tools a t the right time; practically a cookbook approach for engineers, including those unfa-miliar with existing modal parameter identification methods and research.

These popular realization algorithms have, however, lacked the practical capabilities inherent in manystandard modal identification software packages. Although these packages use som e multiple referencetime domain identification techniques, such as P ~ l y r e f e r e n c e ~ ,hey also feature separate treatment of poleidentification and residue estimation, and the capability to estimate residual flexibility and inertia which

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improve model reconstructions. By contrast, ERA and other system realization theory-based techniquesgidentify simultaneously the poles and residues in a unified model, and do not generally provide for themodeling of residual effects. Furthermore, many researchers have noted problems in achieving highly ac-curate reconstructions of some types of modal data using discrete time-domain realization algorithms,which has lead, among other things, to the development of frequency domain-based realizationtechniquesg>l0 nd residue re-estimation*1.12.

The purpose of this paper is to develop additional practical capabilities for modern time domain real-ization-based algorithms (such as ERA) through system realization terminology. As such, we intend thepresent paper to provide a natural com plement to existing system realization literature. Our approach isbased on a time domain least squares estimation of the modal residues and residual flexibility, given a prioridentification of the pole information (i.e, frequencies and damping rates) and the modal participation fac-tors of the system inputs. That is, for the discrete-time state space model

~ ( k1) = A X @ ) Bu(k)y(k) = C x ( k )+ D u ( k )

our approach determine s C and D given that A and B have been identified in a prior analysis, using perhapsa subset of the m easured response functions. We w ill show how th is estimation is consistent with, and re-lated to, the residue estimation implicit in existing system realization algorithms. The present procedurealso provides a purely time-domain alternative to the approach of re-estimating of the mode shapes in thefrequency domain using the measured frequency response functions11*12.

To this end, the paper is organized as follows. In Section II, the time dom ain-based system realizationtheory and procedure is presented. For reasons of clarity and conciseness, only the ERA procedure is de-tailed. In Section III, a procedure for optimally computing the m ode shape matrix C using a linear least-squares solution with A and B from Eqn. (1)isdetailed using system realization theory, and itsrelationship

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to the ERA computation of C is examined. In Section IV, the present mode shap e estimation procedure isutilized to develop three useful generalizations of ERA for identification of structural dynamic models.Finally, Section V applies the present procedure to a very realistic simulated data example, and to exper-imental data. Conclusions are offered in Section VI.

11. Review of Time Domain Eigensystem Realization AlgorithmWe begin by presenting the governing equations of motion fo r structural dynamics in their usual forms.

The response of a structure to a set of forces or inputs u(t) is usually modeled as a spatially discretizedsecond order m atrix differential equation of the form:

where Mis the mass matrix, D is the damping matrix, xis the stiffness matrix, and 8 is the force influencematrix. T he vector q(t) includes all the physical degrees of freedom (DOF) of the model. No w define then associated no nna l modes an f Eqn. (2) according to:

x g n =T 2a n m n = diag {anr = l . . .n]T

(3 )

(4)

where an i re the undam ped modal frequencies in radsec and Z is the modal da mping matrix which istypically symmetric but not necessarily diagonal. If we expand &t) in Eqn. ( 2 )as q( t ) = @,q(t), whereq(t)are the m odal displacements, the structural model can be placed into the first order modal state-spaceform, viz.

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2 = A x + B u9 9 99 ,= c x + D u

Here y(t) is a vector of measured responses, x (t ) is the state variable defined as,and the m odal state-space matrices are given by:

in which H a ,Hv nd Ha are the output displacement, velocity and acceleration location influence ar-rays, respectively.

Because allexperime ntal vibration data is sampled in time, all time domain linear system realizationprocedures begin from the presumption that a finite order discrete state-space model of the system existsof the form:

~ ( k1) = A@) + Bu(k)y(k) = Cx(k)+ Du(k) (7 )

in which k is the time sam ple index. Th e procedure by whichan.5 ) s sampled to lead to Eqn. (7) mustbe done carefully to avoid illconditioning due to the transformation from the continuous (s) plane to thediscrete (2)plane. Likewise, the transformation from a realized model of the form of Eqn. (7) back to thecontinuous representation of Eqn. (5 ) requires careful eigenrotation and mass normalization, as describedin [131.

When the model of Eqn. (7) is used as a predictor, the arbitrary response to an input u(k) is given by:

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ky(k) = M ( k - i ) u ( i ) 1 I k < 0

i = 1

in which the system Markov parameters M O re related to the state-space matrices byD k = OM ( k ) = {CAk-'!3 k > O (9)

All state-space time domain realization methods attempt to find the state space matrices A , B, C, D frommeasurements of the sequence M(k). This is the process known as system realization.

Th e essentia l considerations in system realization are the selection of the model order (it is presumedthat the model form is correct) and the determination of the state space parameters from a minimization ofsome prediction error. For ERA, the prediction error is defrned in terms of a Hankel matrix of the Markovparameters, as defined by:

... M ( k + s )... M ( k + s + l )

...r s m =

The ERA realization find s the linear least squares solution to minimize the error in the shift in the Hankelmatrix of the system m odel and the data according to:

k - 1Hrs(k- 1) = V,A Wsin which

vr= I{ w.s = [B AB ... AS-'B]CAr- '

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If the Hankel matrix is formed from the data, then th e factors Vr and Ws are obtained from a singularvalue decomp osition (SVD) of the 0-thHankel matrix according to:

1 /2 1/2 TVr = Prsrs ws = s r s Q sThe m odel order n, is selected (in principle) by exam ining the numerical rank of H,, (0) truncating theSVD as

P , = Pr(:, 1:nJ

and realizing Hrs(0) Vr and Ws as

- -1/2 -1/2- Tv, = P,Srs W, = Srs QsFrom this, the system realization problem is solved by:

- -1/2C = EIPrSy sD = M(0)

where E , = [5, , ... 0 an d E l = kl 0 ... 0 .Reference [14Jdiscusses the computationallyI ' 1Tmore efficient approach of factoring Hrs(0)Hrs(O)instead of Hrs (0 ) to obtain the fac tors of Eqn. (13). Inthis case, it is more computationally efficient to calculate the factors using a symmetric eigensolver in

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place of the SVD. By only computing the largest n , eigen vau es and vectorsof the Hankel matrix product,it is possible to determine realizations using very large values of Y and s without calculating he entire spec-tral decomposition.

111. Time Dom ain Residue EstimationUsing the term inology consistent with system realization theory and outlined in Section II,we now

develop the time domain residue estimation as follows.A. Least-SquaresSolutionfor Residues

Suppose the state space matrices A and B have been determined from a data set using, for example,ERA/DC or Q-Markov COVER. Then, using Eqn. (9), we have

CB = M ( l )C A B = . M ( 2 )

C A S - ' B = M(s)Hence,

where

.,CWs = {M(l:s)}

-ws = [B AB ...As-'B]

{M(l:s)} = [M(l) M(2) ...M(s)]>^ I

The least squares solution for C using Eqn. (18)is given by

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C = { M ( l : s ) } W i= { M ( l : s ) $[kS~2-'

The solution is well-defined as long as s>n ,where n is the system o rder (dimension of A).Note that thesolution for D,ha t is >

D = M ( 0 ) (21)holds under the present thmry, so long as additional terms, such as residual flexibility, are not add ed tothe problem.

The implementation of Eqn. (20) is equally straightforward. This is because we can utilize the singularvalue decomposition of Hrs (0 ) used in the previous analysis to realize A and B in order to determine- +ws , iz.

. . 2w; QsSrsThus,

B. Relationship to Residue Estimation in ERAAs reviewed in Section 11, in ERA the solution for C is given by

".. -c = Elvr =[I . 0 and vr s the ge neralized observability m atrix, which is realized from the sin-here El = I x z 0 1

gular value decom position of the measured Hankel matrix Hrs (0) ,viz.

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Then, from Eqn. (24) we have

= { M ( l : s ) } W ;Thus, comparing Eqn. (20) and Eqn. (26), the new leas t squares solution for C is fully consistent with thesystem realization theory-based residue determination. The fundamental distinction is that the ERA solu-tion for C is optimal for the realized Markov parameters; that is, the approximated Markov parametersas expressed by the realized or approximated Hankel matrix f i r s (0 ) ,whe reas the least squares solu tionis optim al for the actual measured M arkov parameters.

IV. Algorithms Based on Present Theory:A. An Eigensystem Realization Algorithm using Reference Point Responses(ERA-RP)

The residue estimation algorithm presented in Section II eads to a very useful generalization of ERAfor stru ctural dynam ics identification. Since the m ode s which *are dentifiable from the data are limited tothose which are disturbable from the system inputs, it is only necessary to include a small number of ref-erence point responses from which the same mod es are observable. In the case of structural dynam icswhen the system is reciprocal (Le. symm etric mass, stiffness and damping properties), the logical senso rcomplements are d riving point measurements, that is sensors co-located with the system input degrees offreedom. ...

The prime adv antage of this approach is that it enab les the use of longer data records for the same Han-kel matrix dimension, or allows the reduction of the Hankel matrix dim ension to increase overall compu-tational efficiency. The use of longer data records is important for obtaining accurate and consistent

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frequency and dam ping estimates from real data, although they must be weighed against possible biasesdue to system nonlinearities. Reducing the size of the Hankel matrix is also important because the m ajorcom putational overhead in the system realization procedure is strongly dictated by the minimum dimen-sion of the matrix.

For exam ple, suppose a typical mod al test of a complex structure is performed fdr the purpose of char-acterizing the normal modes. If the te st da ta consists of 100 accelerometers and 3 force inputs, an ERAanalysis might utilize Hankel block dimensions of r = 50 , = 2000 , eading to a Hankel matrix of size5000 x 60 00 . O n the other hand, a reference point ERA analysis might instead use r = 500, s = 1600 ,for a Hank el matrix of size 1500 x 4800. In the latter case, the length of the Marko v sequence actuallyused in the H ankel matrix is slightly greate r than in the ERA analysis, but the m inimum matrix dimensionis reduced by 70%and the com putational effort requkd to decompose the m atrix is reduced by approxi-mately 97%. I *B.

5

Recomputing Residues after Eliinination of InaccuratePolesTh e experimental study in Reference [151 found two main problems with determining structural poles

from tim e dom ain realization algorithms: a ,Man y stru ctural poles converge on ly after massive overspecification of the model order (n, in Eqn.(16)). Overspecification of m odel order, however, engenders additional computational or noise m odeswhich should n ot be retained for subsequen t analysis usingthe identified model.Poles w hich have converged can occasionally split into two nearly repeated (but nonphysical) modes asmodel order overspecificationis increased to converge other less observable poles.

In view of these patho logies, it is usually necessary to use one or several quantitative model quality indi-cators (MQI) to de tect convergence and discriminate unwanted or unreliable modes from the system real-

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ization (e.g. see Reference [121).Unfortunately, because all the global mode shapes are extracted through the realization process sim ul-

taneously, only the mode shapes of the full realization can be considered to be op timal in any sense. If,however, some mod es are not retained for further analysis, the remaining mode shapes cannot be said tobe optim al with respect to either the measured o r the realized response parameters. This is often not a prob-lem if the mo des are well-spaced and orthogonal via the measurement points. It can be a seriou s problem,however, when computational mode splitting, asdescribed above, occurs in the realization analysis. In thiscase, the mode shape information can split between the two nearly identical poles. Hen ce, when splittingis detected and one pole is removed from the modal set, important mode shape information is also lost.

The application of the linear least-squares solution fo r C is straightforward in this case. S imply per-form the system realization analysis to obtainA, B and C in their decoupled modal form. Then, after com-puting various M QI and removing unreliable po les firom A and B, C is recomputed using Eqn. (20). Itshould be noted that the generalized controllability matrix W, ust be recomputed using the reduced A

I

and B matrices, rather than using the singular values and vecto rs of the Hankel matrix a s in Eqn. (23). Thisis not a significant computational burden, however, as powers of A are inexpensive to compute in the de-coupled block mod al form, and the largest matrix inverse operation is of the order of the retained modes(which is relatively small in m ost instances).C. Inclusionof Residual Flexibility Terms

On e serious deficiency of the discrete-time state space model form com mon to ERA and other algo-rithms is that it cannote-. lways account for the residual flexibility effects of modes outside the measurementbandwidth. In particular, when using velocity sensors or accelerometers (arguably the most popular trans-ducer types for modal testing), the modes above the measurement bandwidth contribute a sum term pro-portional to the Laplace terms s and s2, respectively (see [12] and [16]). Such terms cannot be properly

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expressed in the discrete-time state-space model form , however, even though their influence is capturedin the measured FRFs (and thus the Marko v parameters).

One possible corrective approach is to compute a residual flexibility term by fitting the trend of thefrequency domain error between the measured FRF and its model-based reconstruction. This approach isgenerally effective but ignores the weak coupling at all frequenc ies between the contribution of the iden-tified modes and residues and the residual flexibility. Th e result also mixes least-squ ares time-domain andfrequency domain comp utations, obscuring the op timality criterion of the com plete model response.

The present linea r least-squares algorithm for estim ating mode sh ape s of the system realization is eas-ily extend ed to include the residual flexibility contribution in a consisten t manner. Sta rting from the properexpressio n of the discrete FRF including residual term s, we have12

where F is the residu al flexib ility matrix and p is the differentiation order of the sensor type with respectto d isplacem ent (i.e. p=l for velocity, p=2 fo r acceleration)."Here he Laplace term s has been evaluatedalong the frequency axisio at the discrete frequency'values i0, = 2nfk,where f = k / (NAt) and k ,N, nd Af are the sam ple index, total num ber of sam ples and sam pling rate, respectively.

Taking the inv erse discrete fourier transform of Eqn. (27) lead s to the following relationships

i-r .

M ( 0 ) = D + M (0)FM(l) = CB -I-M ,(1)FM ( 2 ) = CA B + M ( 2 ) F

sps

sp

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where M (i) are the real-valued IDFT coeffi cients of the discrete functionS P

evaluated at

N Nk = [I-!!! ... -1 0 1 ... --12 2LThe order of values for k given in Eqn. (30) depends on the numerical algorithm for computing the IDm,the above ordering is consistent with that used for the inverse fast Fourier transform in MatlabI7. The time-domain coefficients M (i) of the residual term are essentially Marko v parameters of the sum contributionS Pof the modes above the m easurement bandwidth to the estimated FFWs, normalized such that the constantcoefficient F is the residual flexib ility. Figure 1 show s the coefficientsofs and s2for a small sample recordwith unit sample time. Using this result, we can then fbrm the line ar least-squares problem

M ( 0 ) = D + M (0)Fswhere

n- . I

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and Im is the rn x rn identity matrix, where rn are the number of inputs. The so lution of Eqn. (31) is then

[c4 M { 1 :s} WT(WWT)-'= M ( 0 ) M s ( 0 ) F

(33)

Note that when F and its coefficients are dropped from Eqn. (31), the solutions fo r C and D re given byEqn. (18) and Eqn. (21), respectively.

The above formulation can o ften lead to an illconditioned matrix due to the m ixture of continuous anddiscrete frequency dom ain terms. I n order to avoid these problems, we replace Sk in Eqn. (29) and F inEqn. (3 1) by frequency-normalized counterparts

Then the estimated term F ismultiplied by ( A t ) to ybtain the co m c t residual flexibility term consistenti .

with the con tinuous equations of motion.

V. Applications Sin@Examples'.,A. Numerical Example using Modal Test Simulator

In order to properly understand the behavior of any system identification algorithm, it is important toperform simulations using data which ishighly characteristic of the actual data the algorithm will ultimate-ly be app lied to. Often, in time dom ain system identification research, this realism is limited to the additionof gaussian noise to M arkov parameters of displacement outputs, which are generated in the time domainby the discretized system equations. Unfortunately, this approach neglects the process by which Markov

.

parameters are usually ob tained in testing; that is, frequency domain FRF stimation of acceleration datawith signal conditioning, ensem ble averaging and dig ital signal processing. T he signal processing and re-

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sidual flexibility effects engendered can be far more significant on the performance of system realizationalgorithms than the level of no ise which is typically encountered, at least in controlled modal testing en-vironments. Therefore, a mod al testing simu lator was developed which includes all of the afo rementionedeffects, in add ition to assumed measurement noise and burst random excitation.

Figure 2 show s a planar truss example. The model includes 36 unconstrained b O F , 18 accelerationsensors and 3 externally applied force inputs; 3 of the sensors axe collocated with the 3 inputs. Th e modaltesting simulato r wa s used to generate the mcFs fo r all 54 input-outpu t pairings. The simulator used 8 192samples per ensemble, sampled at 1000 Hz with anti-alias filtering set at 400 Hz. The FRFs were obtainedusing 10 ensem ble averages and 1%noise was added to the measu rements of the forces and accelerations.

Th e first stage of the time-domain system realization process cons ists of the estimation of the reliab lesystem poles. For this exam ple, the 3 driving-point (i.e. collocated o utput) measurements were retained asreference responses or a to tal of 9 FRFs. An efficient

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1 1

ing, we make the following observations.First, the transfer FRFshown in Figure 4 annot be obtained from the ERA analysis alone, as the Mark-

ov parameters for this input-output pair were not included in the Hankel matrix. That is,although the ERA-derived mode shapes did not include this response location, they were effectively estimated using thepresent procedure. Second , because we chose to eliminate 3 modes of the ERA realiiation, it was also use-ful to reestimate the mode shapes fo r the reference point responses, as the ERA mode shapes were extract-ed simultaneou sly with the inaccu rate modes. In this particular case, because the inaccurate poles were notclosely coupled w ith any of the retained poles, the original ERA mo de shapes and the re-estimated modeshapes at the referenc e points w ere essentially identical.

Finally, other than the resonance which w as not identified at approximately 312Hz, he reconstructedFFWs are highly a ccurate at the resonance peaks. The zeros of the FRFs, however, show varying deg reesof error, particularly the driving po int response. These errors are due to the exclusion of a residual flexi-bility term in the ERA model and in th e new mode shape estimation via Eqn. (19). If, however, we include

:

a term to mod el residual flexibility, as n Eqn. (33), the reconstpxtion is significantly improved, as shownin Figure 5. Further im provem ent at low frequency could possibly be obtained by changing the number oftime points used, o r by app lying constraints to Eqn. (27).Th e accuracy of the estimated mode shapes from4 , ,

Eqn. (20) and Eqn. (33) with respect to the exact mode shapes of the examp le model is shown in the lasttwo columns of Table 1, in terms of the modal assurance criteria (normalized vector correlation). Whilethe mo de shape estimate witho ut including the residual term is nearly exact, there is an improvement inthe mo de shape by sim ultaneously estimating the residual flexibility. This result is consistent with the ex-istence of a w eak coupling between the two response contributions..*

B. Application to Experimental DataAlthough the preceding numerical example was ealistic via use of the modal testing simulator, it is

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often helpful to verify the accuracy accrued by the cu rve fitting procedure through its application to actualexperimentally measured data. Figure 6 shows a photograph of the three-dimensional cantilevered trussstructure tested. Th e modal testing used one force input and 6 1 accelerometers (including a driving poin tlocations), with a sam pling frequency of 500 Hz and 50 ensem ble averages.

As in the numerical example, the poles were estim ated using ERA witli the single driving point rnea-suremen t; after selecting nx=lo0 (50modes), 28 modes were retained for the final model. The 61 responsemeasurements were then re-estimated to yield the desired mode shapes and residual flexibility. A repre-sentative transfer FRF and the driving point are shown in Figures 7 and 8. One important lesson learnedwith this data was hat it was important to include the last 20 time sam ples of the impulse response in theleast squares equation in order to obtain good estimates of the residual flexibility. This is because of themagnitude increase of M as k +N n Figure 1.S

A linear least-squares mode shape estimation algorithm using time domain system realization theoryhas been presented. The present procedure enhance s existing time dom ain system realization algo rithmssuch as ERA, ERA/DC and Q-Markov COVER by adding'the ability to compute (or reestimate) glo balmod e shapes when performing reference point-based pole estim ation and unreliable pole elim ination. Fur-thermore, the procedure can be generalized to estimate residual flexibility terms which cannot be m odeledwithin the discrete state space form. These capabilities have been demonstrated via num erical and exper-ime ntal data. _..

ACKNOWLEDGEMENTS

We would like to thank our colleagues Drs. John Red-Horse and G eorge James at Sandia National Lab-

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oratories, and Prof. K. C. Park and Dr. Scott Doebling at University of Colorado for their help and encour-agement. The present research has been sponsored by Sandia National Labo ratories under DO E Contract[==I. This work was supnnrted by the United

Ststes Departkentof Energy underContract BE-ACS4-94AL850#,REFERENCES

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* .pp. 308-319.Vold, H., Kundrat, J., Rocklin, G. T., and Russell, R., A M ultiple-Input Modal Estimation Algorithmfor Mini-Com puters, SAE Paper 820194, SAE Transactions, Vol. 91/1, 1982, pp. 815-821.

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8. Juang, J.-N., Mathematical Correlation of M odal Parame ter Identification Methods via System Re-alization Theo ry, Int. J. ofAnd and Expt. ModalAnalysis, Vol. 2, No. 1, pp. 1-18.

9. Jacques, R. N., and M iller, D. W.,Multivariable Mo del Identification from Frequency Response Da-ta, submitted to 32nd IEEE Con$ Decision and Control.

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12. Peterson, L. D., and Alvin, K. F., A Time and Frequency Domain Procedure for Identification of

proach, Proc. 1993AIAA Guidunce, Navigation and Controls Con,, Monterey, CA, 1993,pp. 1236-1242.

1 1. Mayes, R. L., AMulti-Degree-of-Freedom Mode S hape Estimation Algorithm Using Q uadrature Re-sponse, Proc. of the 1 th International Modal Analysis Con., February 1993.

Structural Dy nam ic Models, Proceedings of the 36th Structures, Structural Dynamics and Materialsi1

Conference,Hilton Head, SC, 1994; submitted to Journal of Sound and Vibration.

13. Alvin, K. F., and Park, K. C., Second-Order Structural.Identification rocedure via State-Space-LBased System Identification, A I . ournal,Vol. 32 , No. 2, 1994,pp. 397-406.

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Guidance, Control and Dynamics.15. Doehling, S. W., Alvin, K. F., and Peterson, L. D., Lim itations of State-Space System Identification

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16. Ewins, D. J., Modal Testing: Theory and Practice, Research Studies Press, Ltd., 1984.17. Matlab Users Guide,The Mathworks Co., 1993.

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Table 1: -ccuracy of Identified M odes using RP-ERA with and without Residual FlexibilityMAC using ERA/RpAC usingl

ERA/RP - Res. Flex.Mode ## fexact (Hz) fikd (Hz) Error%123456789

10

23.06254.70081.56692.457132.26162.94171.20205.43235.15237.93

23.06554.70081.56692.457132.26162.94171.20205.43235.15237.93

0.01130.00 110.00040.00030.00050.00280.00020.00020.00060.0003

1.00001oooo1 oooo1oooo1 oooo0.99991oooo1.00001.00001.0000

1.00001oooo1.00001.00001.00001oooo1oooo1.00001.00001oooo

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I I I I . I I

h N = 32 samples ,

9 2 erm IDFTI ., i

5 10 15 20IndexL

25 30 35

Figure 1: Time Dom ain Representation of, the Laplace Terms s and s2

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U1Y6 U3,Y14

Figure 2:2-D russ Numerical Example

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Frequency Response (Bode Diagram)2006 03it-200

0)Q)-0

0 50 100 150 200 250 300 350 400Frequency, Hz80 I I I I I I I

60

-20 ' I-40' I I I I I I I I0 50 100 150 200 250 300 350 400Frequency, Hz

_* Rgure 3: Driving Point FRF Reconstruction

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Frequency Response (Bode Diagram)2006 03n-200

ma,vc

0 50 100 150 200 250 300 350 400Frequency, Hz60 I I I I I I I

I40t

I-40' I I I I I I I0 50 100 150 200 250 300 350 400Frequency,Hz

Figure 4: Transfer FRJ?Reconstruction

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Frequency Response (Bode Diagram)200

c3)a,'c3$ 0(dra-2000 50 100 150 200 250 300 350 400Frequency, Hz80 I I I 1 I I I

I

- -0 50 100 150 200 250 300 350 400Frequency, Hz-.Rgure 5: Driving Point FR F with Residual Flexibility

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*^ Figure 6: Photograph of Truss Modal Testing

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I

Frequency Re spon se (Bode Diagram)200wa,Uu) 0(dSa-2000 50 100 150Frequency, Hz

0 --10-20

",-30

--m

a,U

ce- 4 0 -3-50 :2

3 .1

Figure 7: Drivig-Point FRF' and Reconstructionwith Residual Flexibility for Truss Tower

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Frequency Response (Bode Diagram)

Frequency, Hz0 I I

-loot '0 50 100 150Frequency, Hz

-120'

Figure 8: Transfer FRF'and Reconstruction with Residual Flexibility for Truss Tower.^

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LIST OF F'IGURES

Figure 1: Time Domain Representation of the Laplace Terms s and s2Figure 2: 2-D Truss Numerical ExampleFigure 3: Driving Point FRF ReconstructionFigure 4: TransferFRFReconstructionFigure 5: DrivingPointFRFwith Residual FlexibilityFigure 6:Photograph of Truss Modal TestingFigure 7: Driving PointFRF nd Reconstruction with R esidual Flexibility for Truss TowerFigure 8: TransferFRF nd Reconstruction w ith Re sidual Flexibility for Truss Tower

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