ON THE PROBLEM OF STOCHASTIC EXPERIMENTAL MODAL ANALYSIS BASED ON MULTIPLE-EXCITATION MULTIPLE-RESPONSE DATA - PART I: DISPERSION ANALYSIS OF CONTINUOUS-TIME STRUCTURAL SYSTEMS S.D.\\Fassoisl' and J.E. Lee2 Report UM-MEAM-91-14 @1991 by S.D. Fassois and J.E. Lee All rights reserved. 1 Assistant Professor, Mechanical Engineering and Applied Mechanics 2 Graduate Research Assistant, Mechanical Engineering and Applied Mechanics

3d/ ~~ Lt. i{",,,i;''7A

ON THE PROBLEM OF STOCHASTIC EXPERIMENTAL MODAL ANALYSIS BASED ON MULTIPLE-EXCITATION MULTIPLE-RESPONSE DATA PART I: DISPERSION ANALYSIS OF CONTINUOUS-TIME STRUCTURAL SYSTEMS S.D. Fassois* and J.E. Lee** Department of Mechanical Engineering and Applied Mechanics The University of Michigan Ann Arbor, Michigan 48109-2121 ABSTRACT Despite its importance and the undisputable significance of stochastic effects, the problem of multipleexcitation multiple-response Experimental Modal Analysis has thus far been almost exclusively considered within a deterministic framework. In this paper a novel, comprehensive, and effective stochastic approach, that, unlike alternative schemes can also operate on vibration displacement, velocity, or acceleration vibration data, is introduced. The proposed approach is capable of effectively dealing with noise-corrupted vibration data, while also being characterized by unique features that enable~ it to overcome major drawbacks of current modal analysis methods and achieve high performance characteristics by employing: (a) proper and mutually compatible force excitation signal type and stochastic model forms, (b) an estimation scheme that circumvents problems such as algorithmic instability, wrong convergence, and high computational complexity, while requiring no initial guess parameter values, (c) effective model structure estimation and model validation procedures, and, (d) appropriate model transformation and reduction and analysis procedures based on a novel Dispersion Analysis methodology. The Dispersion Analysis methodology introduced as part of the proposed approach is a physically meaningful way of assessing the relative importance of the estimated vibrational modes based on their contributions ("dispersions") to the vibration signal energy. The effects of modal cross-correlations are fully accounted for, physical interpretations are provided in both the correlation and spectral domains, and the phenomenon of negative dis)ersion modes is investigated and physically interpreted. The effectiveness of the proposed approach is finally verified via numerical and laboratory experiments, as well as comparisons with the classical Frequency Domain Method and the deterministic Eigensystem Realization Algorithm. The paper is divided into two parts: The proposed Dispersion Analysis methodology is introduced in the first one, whereas the complete stochastic experimental modal analysis approach is presented in the second [23]. t OCopyright 1991 by J.E. Lee and S.D. Fassois. All rights reserved. * Assistant Professor, Mechanical Engineering and Applied Mechanics. (TO WHOM ALL CORRESPONDENCE SHOULD BE ADDRESSED). ** Research Assistant, Mechanical Engineering and Applied Mechanics.

1. INTRODUCTION Time-domain Experimental Modal Analysis approaches may be classified as either deterministic or stochastic. Deterministic approaches [1-6] are often simpler and computationally attractive, but they face significant difficulties in dealing with noise-corrupted experimental data. Indeed, it is wellknown [7-10] that the quality of modal parameters identified via deterministic methods decreases drastically as the noise-to-signal (N/S) ratio ceases to be negligible, and a number of modes may be even completely impossible to estimate. As a consequence, emphasis has been recently placed on stochastic approaches that can effectively account for the presence of noise in experimental data [7-18]. Most of the available stochastic approaches however are of the single-output type, and are therefore restricted to one vibration measurement location. This is a rather severe limitation for practical applications where multiple-excitation multiple-response [also referred to as Multiple-Input Multiple-Output (MIMO)] methods are in demand because of a number of advantages that they can offer, including significantly reduced total data acquisition and processing time requirements, the use of more consistent data sets, more uniform distribution of the excitation signal energy within the structure, more "consistent" modal parameter estimates due to the larger number of data used, effective separation of multiple or closely-coupled modes, and significantly reduced probability of missing vibrational modes. Despite these potential advantages, very few stochastic MIMO approaches are currently available. Hu et al. [17] discussed an approach based on multivariate AutoRegressive Moving Average (ARMA) models with parameters obtained through the Modified Yule-Walker estimator. The performance of this estimator is however known to vary greatly and result in significant identification scatter and reduced accuracy [19]. In addition, an unnecessarily high number of parameters is estimated, the noise dynamics are not identified, and a number of issues (such as model structure estimation, estimation of the actual number of structural degrees of freedom/model reduction, appropriate selection of the excitation signal type and the required ARMA forms, as well as model validation) are neglected. The approach proposed by Bonnecase et al. [20] for the case of unobservable force excitation is also based on the Modified Yule-Walker estimator (or its overdetermined form) and is of a very similar nature. Part of the reason for the lack of stochastic MIMO experimental modal analysis approaches is due to difficulties related to the discrete estimation stage (algorithmic instability occurrence, wrong convergence, the need for initial guess parameter values, high computational complexity), inappropriate selection of the triple of model structure, excitation signal type, and discrete-to

continuous model transformation, the lack of an effective model structure estimation procedure, and the lack of a methodology for assessing the relative importance of the estimated vibrational modes (necessary for model analysis, reduction, and distinction between "structural" and "extraneous" modes). Although most of these issues (except for the last one) were recently addressed and effectively resolved for the special single-output case [18], those results are not directly extendable to the more general MIMO problem. Indeed, it is a well-known fact that the extension from the single to the multiple-output case is far from being trivial, as stochastic multivariate models have a structure that is much more complicated than that of their univariate counterparts and gives rise to deep identifiability questions that necessitate the use of identifiable parametrizations and special model structure estimation techniques [21]. Additional problems are also encountered as parameter estimation in multivariate models is much more complicated and prone to difficulties related to local extrema/wrong convergence and algorithmic instabilities so that the availability of good initial parameter values is of critical importance [22]. The computational complexity of multivariate estimation algorithms is also excessive, to the point that it is often considered as prohibitive for many practical applications. Regarding the issue of quantitative assessment of the relative importance of a structure's estimated vibrational modes, very little has been done [8,9,18]. This is so despite the issue's apparent significance in model analysis, reduction, and the distinction between actual "structural" and "extraneous" vibrational modes in an estimated structural model. Such extraneous modes primarily appear in conjunction with deterministic methods and are mainly due to the latters' inability to cope with noise-corrupted data. They are, however, encountered in the results of stochastic methods as well (although in drastically reduced numbers); in that case they are due to the dynamics of imperfect instruments and/or statistical errors and estimator inaccuracies. Although some techniques (such as the Modal Amplitude Coherence used in conjunction with the Eigensystem Realization Algorithm [4]) have been suggested for distinguishing structural from extraneous modes in a deterministic setting, those neither address the aforementioned general problem nor work effectively in environments in which the noise is not negligible [10,18]. It is thus the goal of this work to develop an appropriate methodology for the quantitative assessment of the relative importance of a structural system's modes, and also overcome the aforementioned difficulties and develop an effective and realistic stochastic MIMO Experimental Modal Analysis approach. The problem of quantitative assessment of the relative importance of a structural model's vibrational modes is studied first, and a novel and physically meaningful methodology, referred to

as Dispersion Analysis, that assesses the significance of each mode by determining its contribution to the total vibration signal energy under broad-band excitation conditions, is developed. This methodology fully accounts for all cross-correlation effects (second-order statistical interactions) among the various modes, and provides a definite and objective answer to the question of mode importance. Physical interpretations are also provided, and the important phenomenon of modes characterized by negative contributions considered, existence conditions derived, and clear and meaningful interpretations offered. The complete stochastic MIMO Experimental Modal Analysis problem is considered next, and a novel and effective approach that, unlike alternative schemes is capable of operating om either displacement, velocity, or acceleration vibration data records, and utilizes the Dispersion Analysis methodology for model reduction and analysis, is developed. The performance characteristics of the proposed approach are examined via numerical and laboratory experiments and comparisons with the classical Frequency Domain Method (FDM) and the deterministic Eigensystem Realization Algorithm (ERA) [4]. This paper consists of two parts: The problem of quantitative assessment of the relative importance of a structure's vibrational modes and the proposed Dispersion Analysis methodology are discussed in Part I, whereas the formulation and testing of the MIMO Experimental Modal Analysis approach is presented in Part II [23]. The presentation in Part I is organized as follows: The basic formulation of the proposed Dispersion Analysis methodology is outlined in Section 2, and the existence of modes characterized by negative contributions discussed in Section 3. Various interpretations of the proposed methodology, with emphasis on negative dispersion modes and their physical significance, are discussed in Section 4, and the conclusions of the study are finally summarized in Section 5. 2. ANALYSIS OF DISPERSION FOR CONTINUOUS-TIME STRUCTURAL SYSTEMS Consider a general, linear, viscously-damped n degree-of-freedom structural system described by the differential equation: M v(t) + C - -(t) + K v(t) = f(t) (1) in which M, C,K represent the real and symmetric n x n mass, viscous damping, and stiffness matrices, respectively, (f(t)) the n-dimensional force excitation signal, and (v(t)) the resulting n-dimensional vibration displacement signal. This system is characterized by n vibrational modes,

and hence n pairs of complex conjugate eigenvalues of the form: Si, * = -(ii j~, iWn = -(wi, + jwd, (i = 1,... n) (2) where Wdi represents the i - th damped natural frequency Wdi= - W,i / and j the imaginary unit. Corresponding to these eigenvalues are n pairs of complex conjugate eigenvectors (mode shapes) of the form: (~bi} = ~ba (~57)} = q 9 (i = l..n) (3) Oni i The problem of interest here is that of quantitatively assessing the relative importance of each vibrational mode in the system of Eq.(1), and for its solution a novel and physically-meaningful Dispersion Analysis methodology, according to which the relative importance of a given vibrational mode is judged by its contribution to the vibration energy of the structure under broad-band (uncorrelated) stochastic excitation conditions, is currently proposed. The key idea behind this methodology is in the appropriate decomposition of the vibration energy into modal contributions that will be referred to as (modal) dispersions. The excitation signal is selected as uncorrelated stochastic because it is then characterized by a perfectly flat spectrum that warrants that all frequencies are excited at the same level with no particular weight being assigned to any frequency or frequency range. The foundation of this proposed methodology may be traced to the notion of dispersion used in conjunction with discrete Time Series models [24], which, in turn, has its roots in the statistical Analysis of Variance. The first use of Dispersion Analysis in Structural Dynamics was based on approximate discrete-time models and may be found in the papers by Ben Mrad and Fassois [8,9], Fassois et al. [16], and Lee and Fassois [18]. These earlier attempts to address the problem however suffer from two major drawbacks: They are all based on discrete-time models and are therefore approximate and subject to errors relating to the fact that actual structural systems are inherently continuous and not discretetime. These errors are known to be functions of various factors, including the excitation signal type [8,9], and are very difficult to assess. Furthermore, the utilization of discrete models does not allow for physical insight and a clear understanding of the role of the global and local modal characteristics in determining the contribution of each vibrational mode.

The frequently occurring phenomenon of negative dispersion modes has yet to be addressed. As a consequence a clear understanding of the role of such modes and their physical significance is currently lacking. It is thus the objective of this chapter to develop a proper and ezact Dispersion Analysis methodology based on continuous-time structural models, explicitly relate the relative contribution of each vibrational mode to physical quantities of interest (such as natural frequencies, damping factors, and mode shapes), and also provide clear physical interpretations of the notion of modal dispersion and the phenomenon of negative dispersion modes. The Notion of Modal Dispersion The key idea of the proposed methodology is in the proper decomposition of the vibration displacement energy and the introduction of the notion of modal dispersion. Broadly speaking the modal dispersion is the part of the vibration response energy associated with a given mode under broad-band stochastic excitation conditions. For a precise definition the vibration signal variance for a general viscously-damped structural system needs to be computed and appropriately decomposed. Towards this end the solution of the general structural system equation (1) for a given vector excitation f(t) = [fi(t) *. f,(t)]T is expressed in the form [25]: v(t) = ~Fki(KE ( ai / hi(t-7)fk(r)dT) Okia i-(t - -r)fk(r)dr (4) with: hi(t) = e.it =e-C(wnit(cos dt +jsinwdt) (5) h? (t) e~t = e-wni t (COS di- j sin wd, t) (6) and the constants ai and a< (which are the coefficients of the derivative term in the i-th uncoupled state equation using normal coordinates) are given as: ai -2si. *iTMMi + 4iTCoi (7) a i -- (8) The contribution of each vibrational mode will be defined separately for each one of the system's transfer functions. As a consequence, and without any loss of generality, the ml-th transfer function

relating the force fi(t) to the vibration displacement response v,(t) will be considered in the sequel. Based on Eq.(4) the autocorrelation function of the vibration signal {vm(t)} may be computed as: RV m(T) = E [Vm(t)m(t+ r)] = —, [. j. fi(t - rl)hi(rl)drl +,ii fi(t - l)h h (rx)dr,'i a? i=1 k=1 aI Jo fJ " o ]fi@ f(t + - r2)hk(r2)dr2 + f -(t + - r2)h(r2)d2 (9) where E[.] denotes statistical expectation. The first term in this expression may be rewritten as: 1-st term = [( ij (t - rl)hi(-l)dr1) ( (a!k L fi(t + - r2)hk(r2)dr2) a)k jj~k X E[fi(t - r1)fz(t + r - -2)]hi(rld)hk(7r2)drl dr2 mialimk lk Jf J (- r rlf - + r- r 2)hi(l)hk(rhr2)drdr2 (10) aiak o ~ with Rf(.) representing the autocorrelation function of the force excitation signal {fi(t)}. In the case of uncorrelated stochastic excitation Rf(.) is of the form: Rf(r + r1 - 2) = Rfo 6(r + r1 - 2) (11) where S(.) represents the Dirac's delta function, and Rfo its intensity. The substitution of (11) into (10) then yields for r > 0: 1-st term = RIo.~ mialimkrnklk hi(rl)hk(r + rl)drl = e- (kL'k (COSswdr + jsinwdkr) [Rf mj k hi(rl)hk(rl)drl] (12) By similarly computing the remaining three product terms within the square bracket of Eq.(9), the autocorrelation function Rvm(T) may be finally expressed in the following decomposition form: RVmT)= Dk() [exp-(kw,,kr) i COSWdkT + (k) sinwdk,)] (13) k=l k=l with Dk(T) obviously defined and: (k) = Rfoa jr. E ( mk hm(t)hk(t)dt+ k k j hi(t)hk(t)dt + akak [ hi(t)hk(t)dt + ia *Ik 0 hk(t)hk(t)dt") (14) ~+ ~ qi b~mkilk [00 h*(t)hk(t)dt + mi kmkP.k [ h (t)h;(t)dt) (15) aiak J0 aa J

Observe that both 7 d(k) (and Ok) (k = 1, 2,..., n) are real-valued since their first and fourth, as well as their second and third, terms are complex conjugate. From Eq.(13) it is evident that the autocorrelation function Rm(r7) has been decomposed into n decaying trigonometric terms Dk(r) (k = 1,2,...,n), each one of which is associated with each one of the system's vibrational modes and represents that mode's contribution to R,,m(7). The energy associated with the vibration displacement signal {vm(t)} may be then evaluated from (13) by setting r = 0, and is therefore expressed as: Rvm(O) = > [2Rfo Re (qmiqti mk'klk f; hi(t)hk(t)dt + m imkl j ht)h(t)dt) aiak aiak (16) with Re(.) denoting the real part of the indicated quantity. Since: j| h( t) h,( t )d t jAoe-(.w+C. )t [cos(wd, + Wdk)t + j sin(Wdj + Wd)t] dt ((iW, + (kWnk) + i(wdi + Wdk ) Wo+niW + 2(ikWniWnk + 2WniWnk, /(- ~)(1 - = (i,k) (17) and: o hi(t)hi(t)dt = ((iw, + (kWnk) + j(wd, -Wdk) w +Uw. + 2(i kWniWnk - 2wniWnk i1 -( )(1 - (i,k) (18) Rvm(O) may be rewritten in the decomposition form: Rvm (0)- E Dk(O) E 52Rfo Re ( mi iimkrn lkl ](i'k) + "mi-.'I)1 (19) k-k=1 1 aiak akak Based on this the k-th modal dispersion is now defined as follows: Definition: The k-th modal dispersion is defined as the k-th mode's contribution to the vibration signal energy given by the expression: Dk(0) = 2Rio Re aja' k;+ aia* N 2 Dl(O) _2Rfo * ~ Re (.~ mi4li.~k4ck ~n(,k) + (20 ) In addition, the dispersion percentage (or normalized dispersion) of the k-th mode is defined as the dispersion of that mode normalized by the total vibration signal energy, that is: k- n ( ) x 100% (k = 1,2,...,n) (21) /,j 1 Dj(9

Note that expressions (20) and (21) represent the k-th modal dispersion and dispersion percentage, respectively, for the ml-th transfer function. The additional subscripts indicating this have been dropped for the sake of simplicity, but should be used in cases of potential ambiguity. The dispersion percentage of a given mode within a particular transfer function therefore is a "measure" of that mode's relative significance within that particular transfer function. For an n degree-of-freedom structural system characterized by n modes, an n x n' transfer matrix incorporating all transfer functions may be defined, and the dispersion percentages of a given mode may be correspondingly written in the form of a dispersion percentage matrix. Example: Consider the two degree-of-freedom proportionally-damped system of Figure 1 with physical parameters: M 1.0 0.0 C 2.5 -0.8 K[ 110.0 -11.0 = [0.0 1.0 C -0.8 2.5 -11.0 110.0 The modal parameters and complete Dispersion Analysis results, indicating the importance of each vibrational mode in each one of the system's transfer functions, are given in Table 1 in the form of dispersion percentage matrices. As it is evident from these results, mode 1 is a heavier contributor to the vibration energy, and hence of higher importance than mode 2 for this particular system. Another interesting observation is that the dispersion of mode 2 is negative in both transfer receptance cases (transfer functions vi/f2 and v2 /fi), a phenomenon that deserves further attention and will be therefore discussed in detail in the sequel. 3. MODES CHARACTERIZED BY NEGATIVE DISPERSIONS: THE CASE OF PROPORTIONALLY-DAMPED SYSTEMS The definition of a structural mode's dispersion, and its interpretation as the mode's contribution to the vibration response energy under broad-band stochastic excitation conditions, provides an objective and physically meaningful approach for assessing the relative importance of vibrational modes within a given structural system. As it has been, however, already observed, a problem arises in that the dispersions of some modes become negative for certain structural systems, and this creates difficulties, as it is not immediately evident either how negative contributions can be interpreted within this context, or what their exact physical significance and role are. As a consequence, and in order for the relative importance of such modes to be properly judged, satisfactory answers to these questions have to be provided.

In achieving this objective alternative interpretations of the notion of modal dispersion and the Dispersion Analysis methodology, allowing for additional insight into the underlying physical mechanisms, will be sought in the rest of this paper. In this section our attention will be focused on the special, but interesting, case of proportionally-damped systems, for which somewhat simpler and easier to interpret expressions can be derived. The more general non-proportionally-damped case will be examined in Section 4. Based on expression (20) the contribution Dk(0) of the k-th mode in the vibration signal energy may be decomposed as follows: Dk(O) = /kk + 3k (22) where: /3kk "- 2Rfo Re (mk qIkklk r(kk) + (mkl lk kk) (23) aR lakli2 (3 n 3k = > /3ik (24) i=1,iqk Iik 2RfO* Re ( 1mf mklk (k) + mik'lk ik (25) aiak + * 4 The term f3kk in this decomposition depends upon the k-th modal parameters and will be therefore referred to as the k-th modal autocorrelation, whereas the term /3k depends upon the remaining modal parameters as well, and will be referred to as the k-th modal cross-correlation. This latter term represents the part of the energy due to the interactions between the k-th mode and the rest of the modes present in the system. It is interesting, as well as instructive, to consider the special case of proportionally-damped structural systems first. In that case the above expressions for the k-th modal auto and crosscorrelations reduce to: 3kk = RfO. k Ik/4 (26) 4(k26) i=l,ik (wLn -Wnk) + 4[(i(wnink(wn2 +wnk)+ (a +'()wwnk] where {i} represents the i-th mass-normalized eigenvector, and the fact that the ai's of Eqs.(7)(8) can, in this case, be simply expressed as ai = j2wd, (i = 1,2,..., n), was used. Based on these expressions the following corollaries follow: Corollary 1: The modal autocorrelation is nonnegative for any mode of a proportionally-damped system. The dispersion of any particular mode will thus be negative if and only if the mode's modal 11

cross-correlation is negative and of magnitude larger than that of its modal autocorrelation. O Corollary 2: In a proportionally-damped system no negative dispersion modes can occur within a point receptance transfer function (that is a transfer function relating identical excitation I and response m coordinates). Negative dispersion modes can be, however, encountered in transfer receptance cases, in which the excitation and response coordinates are different (I 4 m). 0 Corollary 3: A necessary condition for the k-th mode to be characterized by negative dispersion within the ml-th (m $ 1) transfer function of a proportionally-damped system, is that at least one of the products of mode shape elements O$miq)ljmk<$lk for i = 1, 2,, n be negative. O As it will be later shown, the validity of Corollary 1 can be extended to the case of nonproportionally-damped systems as well. In order to gain further insight into this problem, the natural frequency ratio between two modes is defined as: ~w k = i (28) Then, the k-th modal cross-correlation 3k of the proportionally-damped structural system [Eq.(27)] may be rewritten as: n E 2Rfo, k mi',lidmkclk((iwik + (k) (i) (ii) It is interesting to observe that both the modal autocorrelation I3kk [Eq.(26)] and cross-correlation /k [Eq.(29)] have the term w3k as a common factor in their denominators. By using appropriate natural frequency ratios, the quantity w3k may be thus made a multiplicative factor in the total vibration signal energy expression, and, as a consequence, the dispersion percentages 6k (k = 1,2,.., n) calculated from (21) will be independent of it, and, in fact, of any one of the exact natural frequency values. This observation is organized into the following corollary: Corollary 4: The k-th mode dispersion percentage 6k of a proportionally-damDed structural system does not depend upon the exact value of any one of the system's natural frequencies Wni (i = 1,..., n), but only on the natural frequency ratios wik (i = 1,. -, n). 0 For the rest of this section we will further limit our analysis to the case of proportionally-damped systems satisfying the following assumptions: Al. The structural system is lightly-damped (i << 1,V i) A2. The mass-normalized mode shape elements are all of the same order of magnitude. 12

Apart from being of obvious practical importance, this case allows for some additional calculations that provide insight into the physical conditions under which negative dispersion modes may appear. Towards this end let us now examine the dispersion percentage 8k as a function of the natural frequency ratios ~ik (i = 1, 2,..., n).'If Ewik,, > 1, i, i $ k, the terms (i) and (ii) in expression (29) may be approximately expressed as: (i) eWik (ii). 4(ikek (30) and thus (i) > (ii); a fact further strengthened by the light damping assumption Al. If, on the other hand, ~,,k 1, by neglecting terms of order higher than two we have: (i) 1 (ii). 4(i(keWik (31) and (i) > (ii) in this case as well. As a consequence, if the lightly-damped structural system has well-separated modes (e >k ~ 1 or w..k < 1, Vi, i $ k), the term (i) is dominant to (ii), and the k-th mode dispersion Dk(0) may be thus approximated as [see (26), (27)]: Rf4 02klk2 n 275mjijlimkj/kq(lk (iw;, + (k) Dk(0) = Pkk + P4k + 2 ) ] (32) (iii) (iv) with the denominator of (iv) approximated as E4, (if >Wjk ~ 1) or 1 (if E,ik < 1). In both cases, however, and under the assumptions Al and A2, the term (iii) will be dominant, in magnitude, to (iv). Due to the difference in magnitude between these two terms, and for reasonably small number of degrees of freedom n, (iii) will be dominant when compared to the sum of the terms (iv) as well. That is, the modal autocorrelation will be significantly larger, in magnitude, than the modal cross-correlation, and we, therefore, arrive at the following corollary: Corollary 5: For a proportionally-damped system satisfying Al and A2, and characterized by wellseparated modes (Ewik 1 or Ewik < 1, Vi,i S k), the modal autocorrelation terms dominate, in magnitude, the corresponding modal cross-correlations, and thus, by virtue of Corollary 1, no negative dispersion modes exist. 0 Based on this result, it is evident that a necessary condition for the k-th mode of a proportionallydamped structural system satisfying Al and A2 to be characterized by negative dispersion is that, at least, one additional natural frequency be close to Wnk, so that the term (ii) in Eq.(29) can dominate (i), and the modal autocorrelation PIkk not necessarily dominate (in magnitude) in (32).

This is a somewhat expected result, since for 6k to be negative the k-th modal cross-correlation needs to be negative and of magnitude larger than that of its modal autocorrelation (Corollary 1), with the former (modal cross-correlation) becoming maximum as another damped natural frequency approaches Wdk (wd, -k wdk) (see last remark in subsection 4.1); a fact that, in the case of lightly-damped systems, occurs only when w,, approaches wnk. In order to investigate how closely two modes should be in order for the term (i) not to be necessarily dominant in the denominator of expression (29), the case of a two degree-of-freedom system is now considered. By defining the quantity A through the expression: EW2-1+A A<<l (33) and substituting it into the terms (i) and (ii) of Eq. (29) we obtain: (i) = ((1+A)2-l) = 4A2 + 4A3 + A4 (34) (ii) = 4[(1(2(1 + A)[(1 + )2 + 1] + ( + 22)(1 + A)2] = 4[(V1 + (2)2 + (1C2(4L + 4A2 + A3) + ((12 + (2)(2A + A2)] (35) By neglecting terms involving orders of A higher than two, and also, in view of Al, the terms 161(1;2A2 and 4(j12 + (22)A2, we have: (i) 4 422 (36) (ii), 4((1 + (2)2 + 1612 + ( + 4( + ()2 (37) In order for term (ii) to be dominant: 4A2 < 4(~1 + (2)2 + 16(1~2A + 4((1 + (22)z2 m ~a2 - 2((1 + (2)2a - (1 + (2)2 < 0 (38) or, equivalently: (1+2)2- (C1 2)4 (1 2)2 < A < (1 + 2)2 + /(1 + 2)4- ( 2)2 Al (39) where, obviously, Al > O and A2 < 0. By approximating the quantity under the square root as (C1 + C2)2, and neglecting terms of order higher than two, (39) may be simplified as: Il <1+422 (40) 1~~~~~~~~~~~~40

Expression (39), or its approximate version (40), therefore composes a necessary condition for a negative dispersion mode to appear in a two degree-of-freedom proportionally-damped system satisfying Al and A2. As is evident from this condition, the lighter the damping of the system the smaller the magnitude of A (and thus the smaller the distance between the two natural frequencies) that is necessary for a negative dispersion mode to appear. These facts are illustrated in the example that follows: Example (Continued): Consider the two degree-of-freedom structural system of Figure 1. With mode shapes fixed (same as in Table 1), the dispersion percentages of the transfer function V2/f are shown in Figure 2 as functions of the natural frequency ratio 6W12, for four cases, as follows: (a) (1 = 0.35, (2 =0., (b) (1 = 0.07, (2 = 0.1, (c) ( = 0.007, ( = 0.01, and (d) C = 0.0007, (2 = 0.001. As is evident from this Figure, the lighter the damping of the system the closer the two natural frequencies should be in order for a negative dispersion mode to occur. Although condition (40) is, of course, not applicable to the high damping case (a), it is easy to verify that it is satisfied in all of the remaining cases for the intervals of A for which a negative dispersion mode occurs. Indeed, in case (b) mode 2 has negative dispersion for 0.9 < Ewl2 < 1.05 (-0.1 < A < 0.5), whereas (1 + (2 = 0.17. Similarly in case (c), mode 2 has negative dispersion for 0.993 < s42 < 1.006 (-0.007 < \ < 0.006), and (I + (2 = 0.017, and in case (d), mode 2 has negative dispersion for 0.9994 < ec,2 < 1.0006 (-0.0006 < A < 0.0006) and 1(i + (2 = 0.0017. 0 4. INTERPRETATIONS AND PHYSICAL SIGNIFICANCE OF DISPERSION ANALYSIS: THE GENERAL NON-PROPORTIONALLY-DAMPED SYSTEM CASE In this Section the non-proportionally-damped system case is considered, and various interpretations of the notion of modal dispersion that offer additional physical insight into the proposed Dispersion Analysis methodology and also clarify the issue of negative dispersion modes are derived. 4.1 Interpretation in terms of Modal Impulse Response Functions A key interpretation in this context is based on the currently introduced notion of the modal impulse response function for general viscously-damped structural systems. For this purpose consider the function x(t) defined as: From the examination of the response equation (4) of a non-proportionally-damped structural system it can be shown that x(t) may be interpreted as the impulse response function of the ml-th

transfer function and be decomposed as: n rL n x(t) - [zi(t) + z'(t)] = 2 Re[zi(t)] - xi(t) (42) i=l i=l i=l with obvious definitions for zi(t) and xi(t) (i = 1, 2,..., n). From these expressions it is evident that xi(t) is real-valued and represents that part of the impulse response function x(t) that is associated with the i-th vibrational mode, and will be thus referred to as the i-th modal impulse response function within the ml-th transfer function. Based on this, the vibration signal energy given by Eq.(16) may be rewritten as: RVm (O) 2f= b [2Ro * j Re [zi(t) Zk(t) + i z*) (t)] dt] k=1 i=1 - E [4Ri~'o * j Re[zi(t)] Re[zk(t)] dt - = [Rfo J j xi(t)xk(t)dt (43) and the contribution Dk(O) of the k-th vibrational mode to the total vibration signal energy expressed as: P00 n, Dk(O) - /3ikk +(t)dt + Rf (t)dt + xi(t)(t)Xk(t)dt (44) i=,iZk From this form it is evident that the k-th modal autocorrelation Okk is proportional to the deterministic autocorrelation function of the k-th modal impulse response {Xk(t)} evaluated at lag r = 0, and the k-th modal cross-correlation 13k is proportional to the sum of the deterministic cross-correlation functions between {Xk(t)} and {zi(t)} (i = 1,2,..., n; i $ k) also evaluated at lag r = 0. The coefficient of proportionality is, in both cases, equal to the intensity of the uncorrelated stochastic excitation signal. Based on these observations the following corollary (that is an extension of Corollary 1 to the case of non-proportionally-damped structures) follows: Corollary 6: All modal autocorrelations are necessarily nonnegative. As a consequence, in order for the k-th vibrational mode to have negative dispersion, its modal cross-correlation has to be negative and of magnitude larger than that of its modal autocorrelation. 0 Based on Eq.(44) the dispersion of the k-th mode to the vibration signal energy may be alternatively expressed as: Dk(O)= Rfo' j i(t) x k(t). dt = Rfo. x(t) xk(t) dt (45) 16

and the corresponding dispersion percentage rewritten in the form: 6k = fooo x(t) Xk(t). dt (46) fxo x2(t).dt x The following interpretations in terms of deterministic correlation functions are now immediate: Corollary 7: The contribution (dispersion) of the k-th mode to the vibration signal energy is equal to the product of the uncorrelated stochastic excitation intensity and the deterministic crosscorrelation between the total impulse response {z(t)} and the modal impulse response {Xk(t)} evaluated at lag r = 0. The corresponding dispersion percentage is hence equal to the crosscorrelation between {z(t)} and {zk(t)} at lag zero normalized by the autocorrelation of {z(t)} at that same lag. o Corollary 8: A mode characterized by negative dispersion has an associated modal impulse response function f{Xk(t)} that is negatively correlated with the total impulse response {z(t)} in the time interval [0, oo). This means that the k-th mode is, in average, acting in a way that opposes {x(t)}, and therefore reduces its magnitude. C Remark: The degree of cross-correlatedness between {z(t)} and {xk(t)} that determines the k-th modal dispersion may be also assessed in terms of the normalized quantity: xo' x(t). Xk(t). dt Vro:x2(t)' dt*. fXo2x(t.dt Due to the Cauchy-Buniakovski inequality [26], Pk is indeed normalized in the interval [-1, 1]. 0. These ideas are now illustrated through an example: Example (continued): Consider the two degree-of-freedom proportionally-damped system of Figure 1 with modal parameters and Dispersion Analysis results presented in Table 1. As it may be observed, and in accordance with Corollary 2, the dispersion percentages are both positive in the point receptance cases (transfer functions vl/fi and V2/f2), but of opposite signs in the transfer receptance cases (transfer functions v2/fi and vl/f2). The transfer receptance case v2/fi is further examined in Figure 3. The modal impulse responses xz(t) and X2(t) are shown in part (a) of the Figure and are apparently negatively correlated in the observation time interval. Indeed, the 2nd modal cross-correlation is (assuming Rf, = 1): 1 12a =j zx(t)x2(t)dt = J xl(t)x2(t)dt + xl(t)x2(t)dt + xl(t)Z2(t)dt +... Y1 "Y2 3 In the first time interval [O, tl] the modal cross-correlation is negative (?y < O), then (time interval [tl, t2]) positive (72 > 0), and so on. As time grows, the impulse responses decay, and their overall

(in the time interval [O,oo)) cross-correlation is negative. This is also illustrated in part (b) of the Figure, where the curves xl(t). x2(t) and x2(t) are shown. The area between the former curve and the time axis represents the modal cross-correlation term (P2 = P12), whereas the area between the latter curve and the time axis represents the 2nd modal autocorrelation term (P22). It is apparent that P2 + P22 < O, SO that the second mode is characterized by negative dispersion. Similar observations may be made from parts (c) and (d) of the Figure that depict the impulse responses z(t), x2(t), and the product x(t) s x2(t), respectively. Indeed, it is evident that the two responses are negatively correlated, so that on average (in the time interval [0, oo)) z2(t) tends to suppress x(t). o Remark: The case of systems with closely-spaced frequencies Based on the discussion thus far and the above example, it is evident that for general (nonproportionally-damped) structural systems with adjacent damped natural frequencies (wd, wdd), one mode will necessarily have negative contribution (dispersion) provided that the modal impulse responses {xz(t)} and {X2(t)} start out in opposite directions. In addition, it is also evident that the modal cross-correlation has maximum effects when wd1 = Wd2. 0 4.2 Interpretation in terms of Modal Responses due to Uncorrelated Stochastic Excitation Due to the duality between the deterministic delta function and uncorrelated stochastic signals, it is expected that the foregoing results and interpretations should be (appropriately) extendable to the case of stochastic force excitation by considering corresponding modal responses. That is indeed the case, and may be shown as follows: Consider a structural system excited by an uncorrelated stochastic excitation {fi(t)} and define the part of the response associated with mode k, referred to as the k-th modal response, by the following convolution integral: Vmk (t) Xk(t - r)fl(r)dr (48) The total system response may be similarly expressed as: Vm(t) = x(t - r)f(r)dr (49) and the cross-correlation between the signals {Vm(t)} and {vmk(t)} at lag r = O then is: E[Vm(t)Vmk(t)] = J J xzt - )(tt-R) -2) * E[fi(ri)fi(r2)]drldr2 J x(t- 1)xk(t - -). Rhoda -oo~~1

Rf0 j x(A)xk(A)dA (50) By comparing this expression to (45), the following interpretation in terms of stochastic crosscorrelation functions is obtained: Corollary 9: The contribution (dispersion) of the k-th mode to the vibration signal energy is equal to the cross-correlation between the corresponding total vibration response and the k-th modal response evaluated at lag zero, that is: Dk(0) = E[vm(t)v,mk(t)] (51) In case that mode k is characterized by negative dispersion E[Vm(t)Vmk(t)] will be negative, so that at every time t the k-th mode is, in an ensemble average sense, acting in a way that opposes the total response {Vm(t)}, and therefore reduces its magnitude. O 4.3 Spectral Interpretation The notion of modal dispersion is now examined in the spectral domain. Towards this end define as Fk(jw) the Fourier Transform of the part of the output autocovariance associated with the k-th vibrational mode: Fk(jw) =2 i _ Dk(r)e"-3dr (k = 1,2,,n) (52) The spectrum of the vibration signal {vm(t)} may be then expressed in the following decomposition form: SVm(W) = 27 Vm(r)e-jwdr = E Fk(jw) = n S>Svm(W6) = E Sk(W) (53) k=1 where Sk(w) is defined as the real part of Fk(jw): Sk(w) Re[Fk(jw)] (54) Based on these expressions it is evident that Sk(w) represents the contribution of the k-th vibrational mode to the vibration signal energy at each particular frequency w, and will be thus referred to as the spectral contribution of the k-th mode. The relationship between the modal dispersion and Sk(w) may be readily established by considering the inverse Fourier Transform of (52) for r = 0: roo Dk(O)= i Sk(w)*d (55) 1-9

The dispersion of the k-th vibrational mode is thus equal to the integral of its spectral contribution, that is the area between the latter curve and the frequency axis. Dk(0) will therefore be negative exactly when the integral of Sk(w) is negative, and the following interpretation of negative dispersion modes follows: Corollary 10: A mode is characterized by negative dispersion if and only if its overall [in the frequency range (-oo, oo)] contribution to the power spectrum of the vibration signal produced by an uncorrelated stochastic excitation is negative. This result is complementary to the time-domain results of Corollaries 8 and 9, and indicates that a negative dispersion mode acts in a way that suppresses the spectrum of the vibration signal, and therefore its energy, in the frequency range (-Oc, ). 0 For the computation of the spectral contribution Sk(W) of the k-th mode to the vibration signal energy in the general case of a non-proportionally-damped structural system we proceed as follows: Rewrite the modal contribution Dk(r) to the vibration signal autocovariance in the form [see. Eqs.(9), (10), and (13)]: Dk(T-) = E(mikwkmk $k J J Rf(- + 7- - r2)hi(rl)hk(r2)dridr2 i=i atak aiak o J-00 *jmi0Oli__ mk~lk 00 nj+k~mk'l J J Rj(v + il - T2)h'(rl)hk(T2)dld i2 + mkl j. J Rf(r + T - T2)hf(T1)h;(T2)dT-dT2) (56) The Fourier transform of the first term in this expression may be written as: Omitlitmk(lk _ r 1 0aaikebjir 0/ 0hi(Tr)hk(r2)dTldT2 - /- Rf(r + T1 - T2)e- dT ai aaik IJ-O00 J- oo Jqmi5liqbmkklk hi(rl)ewT dr1. hk(r2)e- wr2dr2 Rf(A)e-jwAdA aiak -oo 0-oo 2 or o0 where the substitution A = r + T1 - r2 was made. Next notice that: 2f | Rf(A)e-jwAdA = / Rfo. 6()e-JwAdAX - o _ (57) R-J X- 2-r j 2ir where the constant Sfo represents the spectrum of the uncorrelated stochastic excitation. Also, by using Eqs.(5), (6): |foo ~ wlT = - Hi(jw) (58) f-oo si - jw ]_ hi(rl)eiwfldrl = += H1(jw) (59) 20

I h(r)e-wrldr = 1 1(jw) (60) 00oo s? - j00 ] hT(r)e3wrldr - 1 H,(jw) (61) based on which [as well as (53) and (54)] Sk(w) may be finally expressed as: Sk(w) = Re[Fk(jw)] n Sk(S) = Re[Fk(jw)] = > i) Re[G(j) Gki(jw)] (63) i=1 akia (i hlj Hm(imkjw) 1 Gt(i = ( H -(j2)Hk(jw) + =' (2(j) (j) (62) Rearmple: The case of proportionally-dampedm- ped systems functions. The pcialint receptanc case isof proportionally-damped systems thin Figure 4a, whexpression the decomposition ofreduces to:he (Wthe)= Re[Fk(jm)] r Spone s pmctlimk Slk Re[S ( jw). spt S k(w)] (63) with. Gi (j?) + 2 _ j (64) ai a a -W, Example (continued): Consider again the two degree-of-freedom proportionally dam- ped system of functions. The point receptance case is examined in Figure 4a, where the decomposition of the the decomposition of the vibration response spectrum S,,(w) in terms of Sl(w) and S2(w). As it may be readily observ ed both modal impulse responses are positively correlated with x(t), and are therelated with thse response x(t), as Figuresult which is in agreement with the seond mode is 21

therefore characterized by negative dispersion. This is in agreement with the dispersion analysis results of the system (see the discussion in the previous subsection) and also the spectral results, as it is evident that S2(w) has a negative overall contribution to the vibration signal spectrum S(w) (although for a particular frequency range the contribution of the second mode to the spectrum is in fact positive). 0 5. CONCLUSIONS In this paper the problem of quantitatively assessing the relative importance of a structural system's vibrational modes was addressed, and an appropriate and novel Dispersion Analysis methodology introduced. The proposed methodology assesses the significance of each mode in an objective and physically meaningful way by determining the mode's contribution (modal dispersion) to the total vibration signal energy under broad-band stochastic excitation conditions and fully accounting for all cross-correlation (interaction) effects among the various modes. Physical interpretations of the modal dispersion have been derived in both the correlation and spectral domains based on the notion of modal response functions, and the dispersion of a mode has been shown to be equal to the deterministic (stochastic) cross-correlation between the corresponding modal and the total vibration responses due to a deterministic delta function (uncorrelated stochastic signal). The modal dispersion may be thus clearly interpreted as a measure of the mode's influence in shaping the total vibration response. Explicit and physically significant expressions that relate the modal dispersion to the structure's global and local characteristics have been also derived, and the phenomenon of modes characterized by negative dispersions investigated. Conditions for the existence of such modes were derived, and physical interpretations provided. Negative dispersion modes were thus shown to be encountered in cases of strong modal interference, specifically in cases where the modal cross-correlation effects dominate over those of the modal autocorrelation; something that can happen in cases of structural systems characterized by closely-spaced modes. Once encountered, negative dispersion modes have a negative overall contribution to the vibration response power spectrum, and act in a way that tends to suppress the structural system's total response. In the second part of the paper [23] the problem of stochastic MIMO Experimental Modal Analysis will be considered, and a realistic and effective approach that uses the Dispersion Analysis methodology for model analysis and reduction/distinction between structural and extraneous modes, developed.

REFERENCES 1. S.R. IBRAHIM and E.C. MIKULCIK 1973 Shock and Vibration Bulletin 43(4). A time domain vibration test technique. 2. D.L. BROWN, R.J. ALLEMANG, R. ZIMMERMAN and M. MERGEAY 1979 SAE Transactions 88, 828-846. Parameter estimation techniques for modal analysis. 3. H. VOLD and T. ROCKLIN 1982 Proceedings of the 1st International Modal Analysis Conference Orlando, Florida, 542-548. The numerical implementation of a multi-input modal estimation method for mini-computers. 4. J.N. JUANG and R.S. PAPPA 1985 AIAA Journal of Guidance, Control, and Dynamics 8, 620-627. An eigensystem realization algorithm for modal parameter identification and model reduction 5. J.M. LEURIDAN, D.L. BROWN and R.J. ALLEMANG 1986 ASME Journal of Vibration, Acoustics. Stress, and Reliability in Design 108, 1-8. Time domain parameter identification methods for linear modal analysis: a unifying approach. 6. *H. KANO 1989 ASME Journal of Dynamic Systems, Measurement, and Control 111, 146-152. An identification method of multiinput multioutput linear dynamical systems for the experimental modal analysis of mechanical structures. 7. C.P. FRITZEN 1986 ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design 108, 9-16. Identification of mass, damping, and stiffness matrices of mechanical systems. 8. R. BEN MRAD and S.D. FASSOIS 1991 ASME Journal of Vibration and Acoustics 113, 354-361. Recursive identification of vibrating structures from noise-corrupted observations, part I: identification approaches. 9. R. BEN MRAD and S.D. FASSOIS 1991, ASME Journal of Vibration and Acoustics 113, 362-368. Recursive identification of vibrating structures from noise-corrupted observations, part II: performance evaluation via numerical and laboratory experiments. 10. J.E. LEE and S.D. FASSOIS 1990, Proceedings of the 8th International Modal Analysis Conference, Kissimmee, Florida, 2, 1424-1433. A stochastic suboptimum maximum likelihood approach to structural dynamics identification. 11. K.J. KIM, K.F. EMAN, and S.M. WU 1984 International Journal of Machine Tool Design and Research 24(3), 161-169. Identification of machine tool structures by the dynamic data system approach. 12. P. DAVIES and J.K. HAMMOND 1984 ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design 106, 40-48. A comparison of fourier and parametric methods

for structural system identification. 13. S.M. PANDIT and N.P. MEHTA 1988 Journal of Sound and Vibration 122, 413-422. Data dependent systems approach to modal analysis, - part I: theory. 14. S.M. PANDIT and N.P. MEHTA 1988 Journal of Sound and Vibration 122, 423-432. Data dependent systems approach to modal analysis, - part II: application to structural modification of a disc-brake rotor. 15. Y.C. SHIN, K.F. EMAN and S.M. WU 1989 ASME Journal of Engineering for Industry 111, 116-124. Experimental complex modal analysis of machine tool structures. 16. S.D. FASSOIS, K.F. EMAN and S.M. WU 1990 ASME Journal of Vibration and Acoustics 112, 98-106. A linear time-domain method for structural dynamics identification. 17. S. HU, Y.B. CHEN and S.M. WU 1989 Proceedings of the 12th Biennial Conference on Mechanical Vibration and Noise, Montreal, Quebeck, Canada, 259-265. Multi-output modal parameter identification by vector time series modeling. 18. J.E. LEE and S.D. FASSOIS 1991 ASME Journal of Vibration and Acoustics, in press. Suboptimum maximum likelihood estimation of structural parameters from multiple-excitation vibration data. 19. S.M. KAY and S.L. MARPLE 1981 Proceedings of IEEE 69(11), 1380-1419. Spectrum analysis - a modern perspective. 20. D. BONNECASE, M. PREVOSTO and A. BENVENISTE 1990 Proceedings of the 8th International Modal Analysis Conference, Kissimmee, Florida, 1, 382-388. Application of a multidimensional arma model to modal analysis under natural excitation. 21. M. GEVERS and V. WERTZ 1987 Control and Dynamic Systems: Advances in Theory and Applications, edited by C.T. Leondes, Academic Press, 26, 35-86. Techniques for the selection of identifiable parametrizations for multivariable linear systems. 22. R.H. JONES 1985 Time Series Analysis of Irregularly Observed Data, edited by E. Parzen, Springer-Verlag Notes in Statistics, 25, 158-188. Fitting multivariable models to unequally spaced data. 23. S.D. FASSOIS and J.E. LEE 1992 Journal of Sound and Vibration this issue. On the problem of stochastic experimental modal analysis based on multiple-excitation multiple-response data- part II: the modal analysis approach. 24. S.M. PANDIT and S.M. WU 1983 Time Series and System Analysis with Applications. John Wiley and Sons. 25. L. MEIROVITCH 1967 Analytical Methods in Vibrations. Macmillan Publishing Company.

26. S.I. BASKAKOV 1986 Signals and Circuits. MIR Publishers. 25

APPENDIX: NOMENCLATURE C viscous damping matrix Dk(T) part of Rm(r) associated with the k-th mode Dk(O) k-th modal dispersion f(t) (vector) force excitation Fk(jW) Fourier Transform of Dk(r) Gi(jw) frequency response on the i-th normal coordinate system j imaginary unit (if not an index) K stiffness matrix M mass matrix n number of degrees of freedom x(t) impulse response of a scalar receptance transfer function xi(t) i-th modal impulse response function [part of x(t)] Rf(r) autocorrelation of the excitation {fl(t)} Rfo intensity of an uncorrelated force excitation signal Rivm(r) autocovariance of the m-th vibration displacement signal Si system eigenvalue Sf, uncorrelated stochastic force excitation spectrum Sk(w) k-th mode spectral contribution [real part of Fk(jw)] Sv,, (W) spectrum of {v, (t)} v(t) (vector) vibration displacement Vm(t) the m-th component of the vibration displacement vector vmk(t) k-th modal response due to uncorrelated force excitation [part of Vm(t)] /3kk k-th modal autocorrelation,6k k-th modal cross-correlation bk k-th mode dispersion percentage (normalized dispersion) A quantity equal to 1- Ew12 eWik natural frequency ratio: Wni/Wnk (k k-th damping factor nk k-th natural frequency Wdk k-th damped natural frequency Obk k-th mode shape Oik the i-th element of the k-th mode shape Conventions E{.}) denotes expectation Re(.) denotes real part

xi the i-th scalar component of the vector x x(t) the value at time. t of the analog signal x {x(t)} the signal x * (superscript) denotes complex conjugate T (superscript) denotes transpose capital bold-face denotes matrix quantity lower-case bold-face denotes vector quantity Abbreviations ARMA Autoregressive Moving-Average ERA Eigensystem Realization Algorithm FDM Frequency Domain Method MIMO Multiple-Input Multiple-Output N/S noise-to-signal 27

Mode 1 Mode 2 wnk (Hz) 1.5836 1.7507 _ k 0.08543 0.15000 Ok{ 1 1 1'~k v 1 v2 -1 Dispersion Percentage [ 61.61 132.28 3839 -32.28 Matrices 132.28 61.61 J -32.28 38.39 (%) Table 1. Modal parameters and Dispersion Analysis results for the two degree-of-freedom proportionally-damped system. 28

LIST OF FIGURES Figure 1. Two degree-of-freedom structural system. Figure 2. The dispersion percentages of a proportionally-damped system as a function of the natural frequency ratio EW2 = wn /wn2. *: mode 1, - - -: mode 2. (a) (1 = 0.35 (2 = 0.50; (b) (1- = 0.07 (2 = 0.10; (c) (1 = 0.007 (2 = 0.01; (d) (1 = 0.0007 (2 = 0.001. (Two degree-of-freedom system; transfer function v2/fl). Figure 3. (a) The first [xl(t)] and second [x2(t)] modal impulse response functions; (b) the products xl(t) x2(t) and x22(t); (c) the total [x(t)] and second modal [x2(t)] impulse response functions; (d) the product x(t) x2(t). (Two degree-of-freedom system; transfer function v2/fl). Figure 4. (a) The first [xi(t)], second [x2(t)], and total [x(t)] impulse response functions of the transfer function vl/fl; (b) the decomposition of the vibration signal spectrum Sl1(w) into the modal components S1(w) and S2(w) for the transfer function vl/fl; (c) The first [xi(t)], second [x2(t)], and total [x(t)] impulse response functions of the transfer function v2/fi; (d) the decomposition of the vibration signal spectrum Sl(w) into the modal components S1(w) and S2(W) for the transfer function v2/fi. (Two degree-of-freedom system).

1, (j ON VI ~V2 kl k 2 k ~3 gC eC C 1L f 2 3 2 Figure 1. Two degree-of-freedom structural system.

DISPERSION PERCENTAGES 6k DISPERSION PERCENTAGES 6I C) a aaa ~- ~~~~~ CD CDa II~~~~~~~~~~~~~~~~~~~~~~~~~ II I 2~~~~~~ OO O Or Ob O, O (O~~ ~~~ OI O' O OO ~ O ~ ~ ~ ~~..... O......JA b ~ ~~~~~~~~~~~' vl II~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I o r Mt.< 0 cn 9LCn ~~~~P U E~~~~~~~~~~~~~ ~~~~~~~~~ II Y " II u U o tc II o~~~ O~~ o oI o r~ ~~~o o (~ II ~ ~ O ~1 E....,.... o~~ 3) 3)., ~4,1r't ~~~~~~P v, CdC.'. p~ m ~. ~, ~~-t II e a, ~ ~ ~ - o D1 d c~ ~ r fu 4 ~ ~ ~ ~ ~~~~~~~~~ 45 r -C r'~~~~~~~~~~~~~~~~~~~~~~~~~~ 0ao

(t//za uo:p)unj jajswreil'uiwalss uIopaaij-jo-oaap omI,) (;)ZZ* (i)x plnpoid aill (p)!suoilDunj asuodsai asIndmi [()Zx] repowu puoDas pue [(j)x] rejol ail (:) (j)Z'x pue (;)Zx (;l)Tx sinpad all (q)!suorlaunj asuodsal asludum repotu [(;)zz] puovas pue [()ixz] isI a); (e) S nL (P) (:) (3os) 3W1y (30s)3fll O'z'L O'l O 0'0 9'0 980'tO z'O 0'0 t000'o -,90'0U) 9 00000 O- 00, 9;OO) Zi ~-.0', *0'0 (Z)ZZ* (;)xw - - (t)xw': *00ooo0 900o (q) (e) (39s) 3V41 (39s)3NU 9 0 0 +'O zro 010 go0 910 *'o Z'O QO IOO'O- 90,0U) -Q E'I C) tO'O0 c:000' f 0'::I'0 U)' 00' ofi~t2 oo- - - tOO ~~~/'-A' T,

(matsis mopaal —jo-aawap omL) tJ/za uotplunj Iajsu'el aq, loJ (e)&S pue (e)'S sluauodmoo repom aql o0u! (en)l'.5 mnlpads leui!s uo.IeJq! alq Jo uo!lTsodmo:ap aql (p) ITf/Za uo!l3unj ~eJsuejl aqp jo suotpunj aeuodsal aslndml [())xJ eolo puv'[(j)zz] pUo)as'[(t)1z slx aSqj, (v)'1/la uo!ijunj laJsuv aq ~oj (e)tr pu-e (en)gS sluauodmo~ repomu aq omu! (r)t14S mnlvads reui!s uo!ieJq!A aql. jo uo!r!sodmo3ap aql (q) tf/tla uo!13un Jajsueil aql jo suo!lunj asuodsaJ aspidm! [(j)x] reio puo'[(I)tZ] puo3as [(I)tlz xSzj aqjL (e)' ailut t/fn uo!T1Un JaJ ueT, (q) (ZH) kON3n.O3JI (o3S) 3WIL 0 O' O' O' 0~ O'0 O': O' o 0'o zo0o I 90' 0i%! 9 0 0' 09000' O 90 / "v, I - f/ n uovuu /',! ro eo_ (z~~H) ADN\flOJ2J (:)s) 3Il t000 00 00'"I....90' 0I -..'t O't I'O n0'0 I00'0 _00'0 OL 0,, "'~~~1 Tr~~~ri ~

39015 0 9 1549