Data dependent systems approach to modal analysis Part 1: Theory

Journal of Sound and Vibration (1988) 122(3), 413-422

DATA DEPENDENT SYSTEMS APPROACH TO MODAL ANALYSIS PART I: THEORY S. M. PANDIT AND N. P. MEHTA Department of Mechanical Engineering--Engineering Mechanics, Michigan Tecnological Universit); Houghton, Michigan 49931, U.S.A. (Received 17 December 1986, and in revised form 30 June 1987) The concept of Data Dependent Systems (DDS) and its applicability in the context of modal vibration analysis is presented. The ability of the DDS difference equation models to provide a complete representation of a linear dynamic system from its sampled response data forms the basis of the approach. The models are decomposed into deterministic and stochastic components so that system characteristics are isolated from noise effects. The modelling strategy is outlined, and the method of analysis associated with modal parameter identification is described in detail. Advantages and special features of the DDS methodology are discussed. Since the correlated noise is appropriately and automatically modelled by the DDS, the modal parameters are shown to be estimated very accurately and hence no preprocessing of the data is needed. Complex mode shapes and non-classical damping are as easily analyzed as the classical normal mode analysis. These features are illustrated by using simulated data in this Part I and real data on a disc-brake rotor in Part II. 1. INTRODUCTION With the rapidly increasing emphasis on the accurate modal characterization of elastic structural systems from experimentally derived data, a variety of test and analysis techniques directed towards the identification of modal parameters have been developed over the past decade. The formulation of the digital Fast Fourier Transform (FFT) algorithm, and the growing sophistication of dedicated mini-computer systems, has led to an almost exclusive concentration o f b o t h development and implementation in the area of Frequency Response Function (FRF) based methods [1-3]. However, the recognition of intrinsic shortcomings and specific areas of weakness in the frequency domain approach [3, 4] has created an awareness of the comparative merits of time domain methods as a class, and time-series analysis in particular. An immediate advantage of the direct use of sampled response data lies in the elimination of errors involved in transformation and generation of the FRF, and from signal processing factors such as leakage [1]. Moreover, since the use of stochastic data, or of observations containing a random component, is integral to the time-series formulation, the presence of measurement noise has relatively little effect upon the accuracy of the resulting modal parameter estimates. While this feature effectively renders the need to preprocess data by averaging and smoothing redundant, it also has the potential to facilitate modal resolution and estimation in situations where the individual identification of strongly coupled modes is obscured by interference due to noise. The concept of a time-series analysis [5, 6] is based upon the application of Autoregressive Moving Average (ARMA) difference equation models, or variations thereof, to sets of stationary, discrete-time data. Although extended [6] or hybrid ARMA modeling 413 0022-460x/88/090413+ 10 S03.00/0 9 1988 Academic Press Limited

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schemes have recently been proposed and used specifically in the field of modal analysis [7, 8], the scope of application of this method has generally been limited to stochastic processes and the determination of natural frequencies and damping ratios of vibrational systems subjected to random force inputs [9, 10]. This follows Walker's result [1 l] that the uniformaly sampled response of a linear differential equation describing an N degreeof-freedom structure excited by white noise can be represented by an ARMA model, and the fact that this model duplicates the output covariance function of the system [9]. In practice, the measured motion is assumed to be caused by an unobserved random input, and a statistical best-fit model computed from the covariance of the response. A special case of the ARMA model is used in the complex exponential algorithm [2,3], a collocation procedure often utilized by existing FFT-bascd methods. Digital impulse response function data are derived by inverse transformation of the measured FRF, and curve-fitted to a homogeneous autoregressive (AR) equation model by means of Toeplitz matrix or least-squares formulation; subsequent decomposition of this model directly provides estimates of the system poles and modal residues [2, 3]. While this process is computationally robust, the implicit assumption of strictly deterministic data and the corresponding choice of a purely homogeneous AR representation makes it particularly sensitive to noise. The tendency, especially, of the AR parameter estimation stage to become ill-posed in instances where the noise-to-signal ratio is relatively high [12] invests added importance to effective preprocessing of the data. The purpose of this paper is to present the theory and application of the Data Dependent Systems (DDS) methodology [6, 13] in the context of modal analysis. DDS is a time-series approach to the problem of system identification and analysis that combines the concepts of linear system theory with a rational modelling strategy in a cohesive, analytical framework. Schematically, the method is based upon synthesizing a complete representation of the dynamic system under investigation in the form of a linear mathematical model that is derived from the dependent solely upon the observed response data; this model may take the form of a scalar or vector ARMA difference equation, or of a stochastic differential equation. If the dynamic properties of the system are manifested in its sampled response, whether deterministic or stochastic, these will be reflected functionally in the characteristics of the model, and be identifiable via an appropriate form of analysis. As a result, the model will incorporate the influences of underlying dynamic mechanisms that may not otherwise be apparent. A formulation of this kind has significant strengths in situations where a structural representation is required for system identification, control or simulation, and the generation of idealized models is precluded by complex intrinsic or extrinsic factors whose effects cannot be quantified. Prior applications of DDS [13-16] have consistently demonstrated the accuracy and relevance of this approach in various fields, and recent work has shown it to be a potentially viable time domain technique for modal analysis [17]. State space formulation of the approach may be found in references [18, 19]; in this paper a more or less classical vibration formulation is employed. In Part I of this paper the theoretical background of the DDS methodology as it applies to modal vibration analysis is presented, and the modelling and analysis process is described in detail. The results of a simulation study of a simple lumped parameter system are discussed. In Part II the validity of the theory is established through an experimental study by applying it to the problem of a structural modification of disc-brake rotors aimed at avoiding the brake squeal. 2. MODAL ANALYSIS AND THE DDS MODELS The equations of motion of a viscously damped, linear, lumped parameter system can be expressed in matrix notation as the set of differential equations

DDS APPROACtt

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ANALYSIS:

PART

I

[m]{s + [c](:~} + [k]{x} = {F},

415

(1)

where {x} is the N-dimensional vector of generalized displacements, and [m], [c], and [k] are real, symmetric, N x N matrices. The free response of the system defined by equation (1) is described, by the modal expansion [20] 2/4

{x} = • cr{~P},e",'.

(2)

Here c, is a complex constant governed by the initial conditions, #,, r = 1, 2 , . . . , 2N, are the complex eigenvalues of the eigenvalue problem corresponding to equation (1), /~, = -~'ro~+iw,.,/l - ~'~= o',+if2,,

r = 1 , 2 , . . . , 2N,

(3)

and {~P}, are 2N complex eigenvectors associated with # , ; since equation (1) has real coefficients, both #, and {6}r appear in complex conjugate pairs. The quantities ~'~ and r in equation (3) denote the damping ratio and natural frequency, respectively, of the rth mode. From equation (2), the motion of the sth lumped mass is written as 2/4

(4)

x , ( t ) = Y c , G , e",'. r=l

If this response is sampied uniformly at intervals of A, where A is selected so as to preserve the controllability and obserability of the system representation after discretization [21], equation (4) takes the form 2/4

x,(kA)=

k=0,1,2,...,

~, c,@,re ~'~ka,

(5)

r=l or 2/4

x~= Y. c,G,A k,

(6)

r=|

where the quantities A, = e ~',a,

r = 1, 2 , . . . , 2N,

(7)

are also complex conjugate, and the subscript t, t = ka,

(8)

refers to sampled time, so that x~ is the kth state of evolution of the deterministic, digitized response G(t). With the backstep notation BPV, = V,_p,

(9)

x~ given by equation (6) is a s'olution of the homogeneous difference equation ( 1 - a~ B - a2B 2 . . . . .

a2NB 2N)x~ = 0.

(10)

The quantities A,, r = 1, 2 , . . . , 2N, are the roots of the characteristic polynomial P(A -= 1 / B ) = A

2N

- a l A 2N-I . . . . .

a2N-IA --a2N.

(11)

If the digitized data is affected by noise such that the actual measured response, denoted by the variable X[, consists ofthe system motion 5c~superposed with a stationary stochastic component, X~ = x~ + n~,

(12)

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the representative model, in accordance with the Fundamental Theorem [6] of DDS, is augmented to the generalized form of an ARMA (n, m) model, (1 - qb,B - ~baB 2 . . . . .

qb, B " ) X ~ : (1 - 01B . . . . .

O,,B')a,,

(13)

where n ~>2N,'m < n, are positive integers, and a, is a zero mean random sequence of residuals, of variance o'2o. The characteristic polynomial L(A = 1 / B ) = A " -

~b,A"-' . . . .

~b,_,A -~b,,

(14)

of the non-homogeneous difference equation (13) contains, as a subset, the roots 3.,, r = 1, 2 . . . . , 2N, associated with the deterministic system response, in addition to the roots A,, r = 2 N + l . . . . , n, which may be real or complex, describing the dynamics of the stochastic portion. The complementary solution y~, of equation (13) is defined by the homogeneous equation

(1

-

c~,,U")y x, = 0

c~,B - d~2B 2 . . . . .

(15)

and expressed as the linear combination of constituent solutions y~= ~.. b~,A~.

(16)

t=l

Here bs, are constants determined by the n-dimensional initial state of the sequence y~ in the linear Van der Monde matrix equation [ V]{b~} = {Y'}o,

(17)

where [ V] is the n x n Van der Monde matrix composed of the columns {V},={1

A, A~ . . .

AT-'} r,

r=l,2 .... ,n

(18)

and the elements of {yS}o are the first n values of the deterministic sequence y~ : {yS}o={yg y~

...

y,~_,}V.

(19)

However, since the only non-zero contribution to the deterministic initial state of the time-series X , is provided by the system roots A,, r = 1, 2 , . . . 2N, b.~,= 0,

r=2N+l,...,

n.

(20)

The complementary solution (16) of the model thus becomes 2N

Y~= E bs, 3.k.

(21)

r=l

Equation (21) is structurally identical to equation (6) for the discrete system response x~, so that a term by term comparison yields b~, = C,l~sr,

r = 1, 2 , . . . , 2N.

(22)

The particular solution of equation'(13), which accounts for the stochastic component in the data, is given by the convolution sum Gja,_j,

n t --

(23)

j=o

where Gj is the Green function [6] ofthe ARMA model, which for distinct 3., is ofthe form Gj=

~ g,A~ r=l

(24)

DDS APPROACH TO MODAL ANALYSIS: PART I

417

with

g r = ( Arn-I - ~i=l

Oi}tn-l-il/~(}ir--Ai)'lli=l

r = 1, 2 , . . . , n.

(25)

i~r

The complete solution of the ARMA model (i3) is therefore x , = Z b~,,~, + r=l

Gja,_~.

(26)

j=0

Equation (26) thus expresses the measured response, as represented by the ARMA model, in terms of two distinct components. The first depends upon the deterministic initial values and corresponds to the actual system behavior; the second describes the additive stochastic process as the response of a digital n-pole, m-zero transfer function to the white noise series of residuals. Although in the preceding development distinct eigenvalues are assumed, the theory is readily extended to deal with the general case of systems with multiple eigenvalues, whether or not a spanning set of ordinary eigenvectors exists. 3. MODELLING AND ANALYSIS As stated previously, the DDS modelling procedure consists of developing a representative mathematical model of a dynamic system, in the form of an ARMA difference equation, directly from the time-series data. In principle, this is equivalent to approximating the dynamic content of the data as closely as possible by a linear ARMA model that reduces the dependent, or correlated, time-series to an independent sequence of residuals. The whole process of modelling can thus be summarized as the synthesis of a model that accomplishes this reduction to independence. In practice, this is achieved by fitting progressively higher order ARMA (n, n - 1 ) models to the data by an iterative, leastsquares procedure until the additional parameters introduced by the increase in order fail to provide a significant improvement in fit. The resulting "adequate" model now incorporates all the dynamic information contained in the data and provides an optimal representation of the underlying system. The mechanics of this modelling strategy, along with its rationale and implications, are described in detail in references [6, 13]. The (n, n - 1) scheme can provide a general ARMA (n, m) model, if desired [6], by later eliminating small parameters to obtain a more economical model. Further computational efficiency can be obtained by increasing the model order in steps of two, keeping the autoregressive order even. This is analogous to permitting the systematic introduction of second order modes allowing both real and complex roots, while unit increase in order might falsely constrain the added root to be real. Several statistically based checks of model adequacy, such as the F-test criterion, examination of parameter confidence bounds, and autocorrelation of the a, sequence are available to determine whether or not all dynamic dependence has been removed from the residuals. However, for predominantly deterministic sets of data, a more definitive index of adequacy lies in determining when the variance of the model residuals reaches a level corresponding to the noise floor of the measurement or computational process; increasing the model order beyond this point clearly cannot result in the extraction of further useful information, In situations where both stochastic and deterministic components dominate the data, it is advisable to apply b6th types of adequacy checks. Note that even though only a finite number of residuals are available, as long as that number substantially exceeds the "true" unknown order (i.e., the number of modes) of

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the model, the residual variance will continue to drop significantly until the true model order is reached and the drop will be statistically insignificant beyond, when the residuals are practically white or uncorrelated. With known noise floor, the procedure can be short-cut at a lower order before reaching the true order. As is evident from the formulation in section 2, the presence of a stochastic process in the measured response necessitates assigning more degrees of freedom to the ARMA model than the underlying system actually possesses. This feature is common to FFT based techniques, where artificial modes must usually be inserted into the mathematical model to enhance the accuracy of the real parameter estimates. However, the degree of overspecification of the ARMA model, rather than being governed by qualitative or ad hoc factors, is controlled on a strictly quantitative basis; when the entire dynamic structure of the data has been accounted for by a model, this is accurately indicated by the tests of adequacy. 3.1. M O D A L P A R A M E T E R I D E N T I F I C A T I O N From equation (21), the free response of the sth lumped mass, as derived from the complementary solution of the adequate ARMA model for the measurement X~, is of the form 2/4

x~= Y~ b~,Ak.

(27)

t=l

The identification problem thus consists of determining, from the model, the quantities b , and A,, and obtaining the poles and modal vectors of the system from this information. Since the roots A, appear in conjugate pairs, 2N-l

[b,,eAr + b , ( A , ) ].

(28)

odd

The 2 N discrete-time roots A, can be evaluated directly from the characteristic equation (14) of the ARMA model, and converted to their continuous-time analogs (3) with use of the relation (7) as o'~ = (1/2A)In (A,A*), 12, = ( I / A ) arctan [Im (A,)/Re (A~)],

r = 1, 3 , . . . , 2 N - 1.

(29a) (29b)

The corresponding natural frequencies and damping ratios (3) are then obtained as

to~=.J~,+g22,,

2 2 ~,=-o'Jx/o',+J'l,,

r= 1,3,...,2N-1.

(30a,b)

The complex coefficients be, in equation (28) are explicitly defined by the matrix equation (17): (b~} = [ V]-~{ye}o.

(31)

The n x n matrix [V] is constructed as per equation (18) using the discrete roots A~. The elements of {Y'}o, describing the deterministic initial state of the time-series X~ are obtained by using equation (26), by removing the stochastic component of the first n data values. P

)'~ = X~ - ~, G~ap_j,

p = 0, 1. . . . . n - 1.

(32)

j-0

The Van der Monde matrix equation (31) is linear in the quantities b,, and can be solved in a computationally efficient manner. For noisy data, it is preferable to solve

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419

directly for b,r by equation (26), with use of a set of n stochastic component subtracted data values beginning not at zero but at say m # 0 where the deterministic component is still large. For very large noise, non-linear least squares recommended in reference [6] for deterministic trends may be employed. From equation (22), one has bsr = c,r

vs~~,~, h* - ~~*-,-*

r= 1,3,...,2N-1,

(33)

so that the elements of the modal vectors {~}, can be determined to within a complex multiplicative constant by modelling and decomposing the responses of all N lumped masses. Since no input measurement is taken, and making N simultaneous measurements poses practical difficulties, free response data can be collected simultaneously from two locations at a time in a series of tests that retains one co-ordinate as a common reference. In other words, the modal displacements of the reference are arbitrarily assigned the value ul, = 1,

r = 1, 3 . . . . . 2 N - 1,

(34)

and the quantities u,r, r = 1, 3 , . . . , 2 N - 1, defined by

u~,=b~,/bl,,

=~bsff~p~,,

s=2,3,...,

N.

(35,36)

It is convenient for the purposes of analysis to express bsr in polar form as b,~ = C,r eir`',

r = 1, 3 , . . . , 2 N - 1.

(37)

Equations (35) and (36) then take the form

u,e7(C~ffCl~)elC~,,-~,,)=tp, fftpl,,

s = 2 , 3 . . . . . N.

(38)

An ensemble of paired sets of ARMA models, derived from the N spatial co-ordinates, will therefore permit construction of the vectors {u},, r = 1 , 3 , . . . , 2 N - 1 ; it is evident from equation (36) that {u}~ is the rth complex eigenvector {ff}~ scaled such that its first element is unity. Note that, by definition, {u},+t = {u}*,

r = 1, 3, . . . , 2 N - 1.

(39)

The fact that only a finite set of residuals a, is made available by the modelling process may perturb the calculation of y~ by equation (32), so that the estimates of b,~ could become non-zero for r = 2 N + 1 , . . . , n, in contradiction of condition (20). This creates the problem of distinguishing between actual system modes and those arising purely from the computational error caused by this constraint. However, this difficulty is resolved by inspection of the global distribution of the characteristic roots A,--those associated with the system should be present in the decompositions of the entire ensemble of ARMA models--and by disregarding the modes for which b,, are relatively small in magnitude that are likely to arise from the computational noise. 4. MODELLING RESULTS In order to demonstrate, in an expository manner, the applicability of the preceding theory, the DDS methodology was applied to a lumped parameter, two degree of freedom system with the coefficient matrices 1

[m]=[50

1~]'

[c]=[-0.7

-0.7

1.5]' . - [ k ] = [ _ ~

-26].

The displacement responses of the co-ordinates xt(t)and x2(t) of the system to the initial condition x2(0)= 1 were generated by using an A C S L (Advanced Continuous

420

s . M . DANDIT AND N.

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MEHTA

Simulation Language) package. The resulting two data sets (500 observations, A = 0.13274), denoted X~ and X,:, were modelled by following the scheme outlined in section 3; based upon the adequacy criteria described previously, ARMA (I0, 9) models were chosen for analysis. The decompositions of the complementary solutions (16) of these models are presented in Tables l(a) and (b). The first column of the tables lists the characteristic autoregressive roots A,, and the next two the natural frequency and damping ratio of their continuous-time analogs. The last two columns show the displacement coefficients C,, and y,, associated with each root. The characteristics of both models, as evidenced by these decompositions, show that only the first two modes contribute significantly to the deterministic component of the data; the others relate to the noise introduced by round-off and truncation errors. It is therefore concluded that the system modes are to~ =0.63256, ~'~ =0.06323, and to3 = 0.99991, st3 = 0.14008. By using equations (33) and (37), the complex modal vector associated with the first mode is found to be

C|le ir,,]

[0.33811 e -i~

]

C2~ ei~'J = c~{ff}~= L 0.33660 e i''328s J" Following the normalization scheme in equation (38) yields the scaled mode shape

1 {u}' = [0.99348 e,,.88, ] 9 TABLE 1

Decompositions of the ARMA (10, 9) models Discrete roots A, (a) X~

(b) X,z

to,

~,

C,,

0.9912 9 i0.0832 0.9731 +i0.1286 -0.1504 ~:i0.9003 -0.6651 +i0.4813 -0.7362 0.9993

0.63248 0.99989 13.098 19.006

0.063265 0.14007 0.052474 0.078211

0-33811 0.67072 0-18759 x 0-50443x 0.14022 x 0.12461 x

0.9912 • 0.9731 +i0-1286 -0.1168 +i0.8935 -0.6883 + i0.6010 -0.8650 0"9867

0.63263 0.99993 12.836 18.272

0.063181 0.14008 0.061148 0.037162

0"33660 0-33867 0.70917 x 0.34089• 0.70762 • 0.46911 x

7~, (deg)

10-6 10-6 10-6 10-6

10-7 10-7 10-7 10-8

-0.5523 9.3830 -138-3726 -137.7316

1.3288 -173.3879 3"7928 -34.3123

TABLE 2

Comparison of iheoretical and modelling results ~o,

~',

{uh

Theoretical

0.63226

0.06312

{0.99381eii.799} 0"99984 0.14073

ARMA(10,9)

0.63256

0.06323

{0.99348eit.ss,}

1 1

'~3

6

{uh

1

{0.50562e_,S2.o3 }

1

0.99991 0-14008 {050494e_.,s2.77 }

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421

Similarly, for the second mode, /' 1 {u}3 = [0.50494 e-i,82-77] 9 A comparison of these results with iheoretically derived values is presented in Table 2.

5. C O N C L U S I O N S

The concept of Data Dependent Systems, and its applicability in the context of modal analysis, has been presented. An expository demonstration of the validity and relevance of the theoretical framework has been provided by means of a modelling study with simulated data. The ability of the DDS difference equation models to provide accurate representations of sampled response data forms the basis of the approach. When an ensemble of paired sets of these models is synthesized and analyzed, a complete global characterization of a linear, non-classically damped, vibratory system can be obtained. Moreover, the capacity to isolate strictly the system dynamics from extraneous stochastic effects is a particularly strong feature; in consequence, the accuracy of parameter estimates is unaffected by measurement noise and the need to preprocess observed data completely eliminated. Since quantitative checks of adequacy can be applied, the dimensions of the correct model are clearly defined. As a result, a qualitative, a priori, assessment of the number of modes possessed by the system is rendered unnecessary. In support of the theoretical aspects of the method, the modelling routines used arecapable of efficiently generating models of up to 100th order, so that the analysis of large, elastic structures is feasible. Simulation results indicate that the modal parameter estimates obtained by the method are remarkably precise: natural frequencies deviate from their theoretical values by 0.05% or less; what is even more remarkable is that even the damping ratios deviate only by 0.5% or less, the corresponding numbers for modal amplitudes and phase angles being 0.1% and 5%, respectively. Since only the finite word length of the data entered in the computer and single precision computational noise effects degrade the estimates, this accuracy is attained only at a moderate increase in model order enough to account for the round-off and computational noise modes: the theoretical model should be 4th order whereas the actual model is 10th order; moreover, the results faithfully return the amplitudes of the added modes of the order of 10 -7 , nearly the single precision accuracy. This is in accord with the theory outlined in the paper which further suggests that similar accuracy should be attainable even with the added external noise. This has indeed been empirically confirmed in reference [17], where it has been shown that comparable accuracies are obtained for a three-degree-of-freedom system theoretically requiring a 6th order model, by an actual 12th order model when no external noise is added, and by a 26th order model when external noise is added so that the noise to signal ratio is 17%. It is well known that for stationary noise, the accuracy can be enhanced by increasing the number of observations, the standard deviation of the error being inversely proportional to ~ [6]. REFERENCES 1. M. RICHARDSON 1975 Seminar on Understanding Digital Control and Analyxis (Part l I), 43-64. Modal analysis using digital test systems. 2. D. BROWN, R. ALLEMANG, R. ZIMMERMANand M. MERGEAY 1979 S A E Paper No. 79022. Parameter estimation techniques for modal analysis.

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3. M. MERGEAY 1979 Modal Analysis. Leuven: Katholiek Universiteit. Theoretical background of curve-fitting methods used by modal analysis. 4. G. F. LANG 1983 Sound and Vibration, 20-22. Modal density--a limiting factor in analysis. 5. E. A. ROBINSON 1981 Time Series Analysis With Applications. Houston: Goose Pond Press. 6. S. M. PANDIT and S. M. Wu 1983 Time Series and System Analysis With Applications. New York: John Wiley. 7. Z. USHIJIMA 1984 Proceedings of the 2nd International Modal Anab'sis Conference, 437-442. Modal parameter estimation by z-transformation method. 8. J. LEURIDAN and H. VOLD 1983 Modal Testing and Model Refinement Symposium, ASME Winter Annual Meeting. A time domain linear model estimation technique for multiple input modal analysis. 9. W. GERSCH and R. S.-Z. L1u 1976 Journal of Applied Mechanics 43, 159-165. Time-series methods for the synthesis of random vibration systems. 10. W. GERSCH and D. A. FOUTCH 1974 Institute of Electrical and Electronic Engineers Transactions on Automative Control AC-I9, 898-903. Least squares estimates of structural parameters using covariance function data. 11. A. M. WALKER 1950 JounTal of the Royal Statistical Society, Series B, 12, 102-107. Note on a generalization of the large sample goodness of fit test for linear autoregressive schemes. 12. D. G. DUDLEY 1979 Radio Science 14, 387-396. Parametric modeling of transient electromagnetic systems. 13. S. M. PANDIT 1977 Transactions of the American Society of Mechanical Engineers Journal of Dynamic S),stems, Measurement and Control 99G, 221-226. Stochastic linearization bY data dependent systems. 14. S. M. PANDIT 1977 Shock and Vibration Bulletin 47, 161-174. Analysis of vibration records by data dependent systems. 15. S. M. PANDIT, H. SUZUKI and C. H. KAHNG 1980 American Society of Mechanical Engineers Journal of Mechanical Design 102,233-241. Application o f d a t a dependent systems to diagnostic vibration analysis. 16. S. M. PANDIT and D. R. KING American Society of Mechanical Engineers Paper No. 81WA/DSC-6. Data dependent systems linearization of nonlinear pendulums. 17. S. M. PANDIT and N. P. MEHTA 1984 Proceedings of the 2nd International Modal Analysis Conference, 35-43. Data dependent systems approach to modal parameter identification. 18. S. M. PANDIT and N. P. MEHTA 1985 Transactions of the American Society of Mechanical Engineers, Journal of Dynamic Systems, Measurement and Control 107, 132-138. Data dependent systems approach to modal analysis via state space. 19. S. M. PANDIT 1989 (to appear) Modal and Spectntm Analysis: Data Dependent Systems in State Space, New York: Wiley Interscience. 20. L. MEIROVITCH 1967 Analytical Methods in Vibrations. MacMillan. 21. R. E. KALMAN, Y. E. H o and K. S. NARENDRA 1963 Contributions Differential Equations 1, 189-213. Controllability of linear dynamical systems.