Mode decomposition method for non-classically damped structures using. acceleration responses

Transcription

1 ICCM5, 4-7 th July, Auckland, NZ Mode decomposton method for non-classcally damped structures usng acceleraton responses J.-S. Hwang¹, *S.-H. Shn, and H. Km Department of Archtectural Engneerng, Chonnam Natonal Unversty, Gwangju, Korea. Department of Archtectural Engneerng, Kyungpook Natonal Unversty, Daegu, Korea *Presentng author: Correspondng author: Abstract For structures wth non-classcal dampng or closely dstrbuted modes, t s not easy to apply the tradtonal modal analyss method because the dampng matrx s not dagonalzed by the modal matrx obtaned from the mass and stffness matrces. In ths paper, a new mode decomposton method for structures wth non-classcal dampng rato and structures wth very closely dstrbuted modes s proposed. hs method defnes the generalzed modes n state space, and uses the dfferental state varables estmated from measured acceleraton responses to decompose modal responses. A Kalman flterng s utlzed to calculate the lnear transformaton matrx of governng modes, and the lnear transformaton matrx s updated n the optmzaton process to maxmze the performance ndex cooperated wth a power spectral densty of a target mode. For the verfcaton of the proposed method, a numercal smulaton s performed usng a sngle degree of freedom (SDOF) system coupled wth a tuned mass damper (MD) whch represents a non-classcally damped system wth closely dstrbuted modes. he results from the smulatons show that the proposed method estmates the modal responses more precsely than conventonal mode decomposton methods such as the ndependent component analyss (ICA) method and the proper orthogonal decomposton (POD) method. Keywords: Mode decomposton, Non-classcal dampng, Closely dstrbuted mode, Lnear transformaton matrx, Dfferental state varable, Averaged power spectrum Introducton he response of a lnear mult-degree-of-freedom (MDOF) structure s often estmated usng a few governng mode responses after transformng the system nto sngle-degree-of-freedom (SDOF) systems n the modal space. he transformaton nto the modal space n the modal analyss requres the modal matrx that s composed of mode shapes, and thereby t s necessary to obtan the mode shapes prmarly. he mode shapes or modal matrx s generally obtaned from the egenvalue analyss usng mass and stffness matrces of the fnte element analyss model. he mass and stffness matrces of the actual structure, however, dffer from those of the analyss model yeldng the dscrepancy n the dynamc characterstcs and mode shapes. Further, t s not possble to separate modes usng the mode shapes obtaned from the mass and stffness matrces f a structure has non-classcal dampng that s not proportonal to mass and stffness matrces or the structure has closely dstrbuted modes. In order to estmate the actual mode shapes for accurate modal separaton, the mode decomposton method usng measured structural responses has been studed. In specal, the mode separaton methods usng the measured responses from approprately numbered sensors have been developed because the behavor of a buldng structure subjected to wnd load s governed by a few lower mode responses. he proper orthogonal decomposton (POD) method s one of the mode decomposton methods usng the lnear transformaton of measured responses [Feeny (); Han and Feeny (3)]. he POD method, also called as the prncpal component analyss (PCA) method, Karhunen-Loeve method, or the sngular value decomposton (SVD) method [Gramaand and Subramanan (4); Khall and Sarkar (4)], transforms the hgher order model nto the lower order model wth

2 orthogonal bass mnmzng the loss of hgher order model nformaton. he ndependent component analyss (ICA) method s another mode decomposton method usng the lnear transformaton of measured responses based on the assumpton that modes are ndependent each other [Roberts and Everson (); Kerschen and Poncelet (7)]. It s also possble to perform the mode decomposton usng the output-only method such as the stochastc subspace dentfcaton (SSI) and frequency doman decomposton (FDD) methods, whch estmates the modal characterstcs of a structure usng the measured responses [Van Overschee and De Moor (996); Brncker et al. (); Ku et al. (7)]. hese mode decomposton methods are applcable to structures wth classcal dampng, whch s proportonal to mass and stffness matrces. Further, they yeld relable results when buldngs have very small dampng rato and thereby possess the characterstcs of structures wth classcal dampng. he ICA method, whch s mostly close to the method proposed n ths paper, assumes that modes are separated enough to be ndependent each other. However, the dampng matrx s not proportonal to mass and stffness matrces of actual structures and t s not approprate to assume that the closely dstrbuted modes are ndependent each other. herefore, there exsts a lmt when the prevous mode decomposton methods are appled to structures wth non-classcal dampng or wth closely dstrbuted modes. In ths paper, the new mode decomposton method usng only measured responses s presented for structures wth non-classcal dampng or wth closely dstrbuted modes. hs mode decomposton method apples the lnear transformaton to measured response for calculatng the modal responses smlarly to the ICA method. he lnear transformaton matrx dffers from that of the ICA method such that t s obtaned by optmzng the objectve functon. he objectve functon s gven to maxmze the energy at the certan mode and to mnmze the dfference between averaged modal response spectrum and the lnear transformaton matrx assumng that each mode possesses unque pole wth one natural frequency and one dampng rato. For the verfcaton of the proposed mode decomposton method, the numercal smulaton of a two DOF system wth a tuned mass damper (MD) that s a representatve system wth non-classcal dampng and very closely dstrbuted modes. It s assumed n the numercal smulaton that the external load has wde spectral range lke wnd loads and the only responses of the man structure and MD are measurable. he mode shapes and modal responses obtaned from the measured responses are compared to the analytcal ones to verfy the proposed mode decomposton method. Mode decomposton n state-space doman Estmaton of unmeasured state varables he mode decomposton of a structure wth non-classcal dampng s not possble because the dampng matrx cannot be dagonalzed usng the mass and stffness matrx. hs requres havng lnear combnaton of state-space varables to construct modal responses. he mode shapes of an MDOF system whose governng equaton s gven n Eq. () are defned as the lnear combnaton as n Eq. (). he mode separaton s possble only when the dampng matrx, C, s dagonlazed by the mode shape matrx, Φ, n Eq. (3), whch s obtaned from the egenvalue analyss of mass matrx, M, and stffness matrx, K. M x + Cx + Kx = Ef () x = Φη () η + Φ CΦη + Ωη = Φ Ef (3) where f s the external force, E s the force locaton matrx, Ω s the dagonal matrx wth entres of squared natural frequences, and x and η are the response vectors n tme doman and modal space, respectvely. If the structure has non-classcal dampng, the term Φ CΦη n Eq. (3) s not a dagonal matrx, and thereby the mode decomposton s not attanable. Consequently, t s requred to expand the modal responses nto the state-space doman for the mode decomposton. Eq. () s transformed nto Eq.

3 (4) n state-space doman, and the state varable, z, can be transformed nto modal space usng the newly defned modal responses n state-space doman, p, as n Eqs. (5) and (6). where z = Az + Bf (4) z = Ψp (5) p = Ψ AΨΨ+ Ψ Bf (6) M A = K = M E B I M C and the mode shape n state-space doman, Ψ, satsfes I Aˆ = Ψ AΨ = Ω Ζ where Ζ = dag(ξ ω ), dag() s the dagonalzaton functon, and ξ and ω are the dampng rato and natural frequency of the -th mode, respectvely. All of state varable z of dsplacement and velocty or dfferental varable z of velocty and acceleraton are necessary n order to obtan the modal responses n state-space doman of Eq. (5). However, t s not practcal to measure every state and t s often to measure acceleraton responses n practce. herefore, t s assumed n ths paper that the number of sensors s equal to the number of governng modes and velocty and dsplacement responses are obtanable from the measured acceleraton usng the Kalman flter. Gven that the order of Kalman flter s twce the number of sensors, the ntal estmate of the system matrx A s gven as = SSdv (7) (8) (9) A () where A s the ntal estmate of the system matrx A, S = E[ z z ] and dv E[ z z ] S =. Multplyng Eq. (4) by z and averagng yelds Eq. (). he external force term s gnored snce t s not known or measurable. Snce the velocty and dsplacement are requred n Eq. (), the followng smple ntegratng flter s ntroduced. q = q q + q x m where x m s the measured acceleraton, and q and q are dsplacement and velocty ntegrated from the measured acceleraton, respectvely. If the measured acceleraton n Eq. () s based, the ntegrated dsplacement and velocty have consderable amount of errors and often dverge durng ntegraton. In order to avod the dvergence and to mnmze the errors, the control varable, u, s added to Eq. () as n Eq. () where the control gan, G, of sze x s decded to mnmze the squared dsplacement and squared control varable n Eq. (3). q = q q + x q m + u () (.a) 3

4 q u = G q = J ( Q q + Ru ) dt (.b) where Q and R are weghts. Note that the control varable u s equal to the dfference between the actual measured acceleraton and the estmated one. he control gan, G, can also be obtaned by modfyng weghts Q and R such that the dfference between the actual measured acceleraton and the estmated one s n a certan range. he more detaled process for control gan calculaton ncludng the Kalman flter method s omtted here snce t has been wdely ntroduced n many references [Hwang et al. ()]. Objectve functon for mode decomposton Once the state varables are estmated from the measured acceleraton responses usng the Kalman flter, the relatonshp between the state varables and the modal responses can be defned usng Eq. (5). Because the purpose s the mode decomposton usng the measured responses, Eq. (5) s rewrtten as (3) p = z (4) where = Ψ. Snce t s assume that the number of the sensors, n, s equal to the number of governng modes, the transformaton matrx,, s a square matrx of n x n and ts nverse matrx exsts. It can be noted that the each column of the transformaton matrx s the combnaton of lnear transformaton coeffcents that separate certan modes from the measured state varables. Snce the measured acceleraton and ts ntegral value, velocty, are mostly used, Eq. (4) can be rewrtten usng dfferentated state varables as p = z (5) where the entres of the transformaton matrx are constant and are not affected by dfferentaton. From Eq. (5), t can be noted that the number of dfferentated state varable, z, s n and the number of correspondng generalzed modes s also n. he frst n modes obtaned from Eq. (5) have relatonshp wth the rest of modes defned as n Eq. (6). If the effect of external force s neglgble n Eq. (6), the relatonshp becomes velocty to acceleraton. hat s, the relatonshp between -th mode and (+n)-th mode s velocty to acceleraton, f n and the effect of external force s neglgble. he -th mode can be presented usng the -th column of of Eq. (5) as ( ) z p = (6) In order for the -th mode obtaned from Eq. (6) to be decomposed nto a true vbraton mode wth sngle pole that conssts of natural frequency, ω, and dampng rato, ξ, the modal power spectrum obtaned from Eq. (6) needs to have only one peak near the natural frequency when there s no specal poles n the external force. hat s, the effect of other modes should not be appeared showng no peaks near other modes. In ths paper, the followng necessary condtons are defned for true mode decomposton descrbed above. hese condtons also are the precondtons to defne the objectve functon for mode decomposton. Note that these condtons are not necessary and suffcent condtons for mode decomposton and that other necessary condtons based on other dea can also be adopted. Necessary condton : he total energy of decomposed modes s always constant. hs condton s satsfes by settng the ntegral value of modal response spectrum, whch s equal to the varance value, to be. 4

5 Necessary condton : he modal energy s max near ts natural frequency. he correspondng natural frequency can be obtaned from the system matrx A n Eq. (). Necessary condton 3: he effect by neghborng modes s mnmzed. hs condton can be satsfed by mnmzng the dfferences between the modal power spectrum and averaged power spectrum at neghborng modal frequences. he objectve functon satsfyng the above necessary condtons and can be defned as J ω + ω = + S λ S ω ω where λ s a Lagrange multpler for constranng the necessary condton, ω s the nfntesmal change of frequency, and S (ω) s the power spectrum of the decomposed mode. S (ω) s one-sded spectrum gven as va (7) S ( ω ) = S (8.a) S va = z( ω) z( ω) (8.b) where z(ω) s the Fourer transformaton of dfferental state varable, z (t) and z (ω) s the complex conjugate of z(ω). Substtutng Eq. (8) nto Eq. (7) smplfes the objectve functon of Eq. (7) as J S + λ ( S ) (9.a) S = peak var ω + ω peak = Sva ω) ω ω ( (9.b) Svar = Sva (9.c) S peak and S var are readly obtanable from the dfferentated state varables drectly. Consequently, the transformaton matrx,, for the -th mode that satsfes the condton and can be derved by dfferentatng J of Eq. (9.a) wth respect to and settng the resultng value to be. he result of dfferentaton s gven as ( S λ S ) = (.a) S peak + var peak = λs (.b) var It can be noted from Eq. () that the value of ( λ) s the egenvalue of two matrces, S peak and S var, whle s the correspondng egenvector. hs means that the largest egenvalue becomes the maxmum value of objectve functon and the correspondng egenvector becomes the lnear transformaton matrx. If modes are separated enough to affect each other margnally, t s possble to perform the mode decomposton accurately usng the transformaton matrx obtaned from Eq. (). hen modes are closely dstrbuted, the recprocal effect between modes becomes sgnfcant. In that case, the objectve functon that satsfes the necessary condtons and only cannot yelds the accurate mode decomposton. In order to mnmze the effect of neghborng modes, the followng objectve functon that satsfes the necessary condtons 3 as well as and s defned. J ω + ω S ω ω = + l ωk + ω S ( ) ω log ωk ω S H 3 S where S H s the averaged power spectrum gven as () 5

6 S ( ω ) S H ( s) (.a) H = o s H ( s) = s + ξ ω + ω (.b) S ω + ω ω ω o = ω + ω ω ω S H ( s) (.c) where s s the Laplace varable, H(s) s the transfer functon of velocty response from the external force of a SDOF system, and S o s the constant that represents the rato of the -th modal power spectrum to the transfer functon near the -th mode frequency. he dfference between Eqs. (7) and () s that the logarthmc rato of the -th modal spectrum to the averaged spectrum near the frequency of neghborng mode, ω k, s ncluded n the denomnator. Mnmzng the rato n the denomnator maxmzes the objectve functon, whle the logarthmc rato accentuates the dfference between two spectrums. he objectve functon n Eq. () can be smplfed usng as where J ( log( q ) ( S ) peak 3 = + l ω + var k ω ωk ω S S q = (4) S H he natural frequency, ω, n Eq. (.b) can be drectly obtaned from the system matrx of Eq. () whle ehe dampng rato, ξ obtaned from the system matrx has large error. herefore, the dampng rato needs to be selected such that S o of Eq. (.c) satsfes the followng relatonshp derved from the necessary condton. So ( ξ ) H ( s) = S = (5) Once the values of all varables n Eq. (3) are calculated, the transformaton matrx,, can be obtaned by dfferentatng the objectve functon wth respect to. However, the closed-form smlar to one n Eq. () cannot be derved due to the nonlnearty. herefore, the senstvty of objectve functon s utlzed n the optmzaton process to obtan the transformaton matrx,. Valdaton of the proposed method Example structure and ts modal characterstcs A numercal smulaton usng an example structure wth non-classcal dampng and very closely dstrbuted modes s carred out to verfy the proposed mode decomposton method. he example structure s a two-dof system wth a MD whch s a representatve system wth non-classcal dampng and very closely dstrbuted modes. he dynamc characterstcs of the structure and external load are summarzed n able. It s assumed that the low-pass fltered whte nose s appled to the man structure only. In ables and 3, the mass and stffness matrces along wth the correspondng mode shapes are presented n tme and state-space domans, respectvely. hese mode shapes wll be compared to ones obtaned usng the proposed mode decomposton method. It can be noted that the dampng matrx s not dagonalzed by the mode shape obtaned from the egenvalue analyss of mass and stffness matrces n tme doman from able, whle modes are apparently separated n state-space doman from able 3. (3) 6

7 Structure External load able. Dynamc characterstcs of the example structure Descrpton Value Remark Man structure mass (m s ) kg Natural frequency of the man structure (f ) =. Hz MD mass (m t ) kg Mass rato of MD mass to man structure mass =. Man structure stffness (K s ) N/m MD stffness (K t ).5635 N/m Man structure dampng.533 N s/m Man structure dampng rato (ξ s ) =. MD dampng.5 N s/m MD dampng rato (ξ t ) =.5 Flter Low pass flter Zero to 4 Hz Samplng tme.5 s. Samplng frequency = Hz Duraton 36 s. able. Analytcally obtaned modal propertes n tme doman Matrx Symbol Value Mass matrx M Dampng matrx C Stffness matrx K Natural frequences ω st mode:.95 (.898 Hz) ω nd mode:.377 (.97 Hz) Mode shapes Φ ransformed dampng matrx Φ CΦ Dampng rato of dagonal terms = (.86.36) It can be noted that even though the transformed dampng matrx n able s not a dagonal matrx, the dagonal entres, (.68.83), are very close to those of dampng part of mode transformed system matrx, Â, n able 3. It can be also notced that the natural frequences n tme doman,.898 Hz and.97 Hz, and those n state-space doman,.9 Hz and.95 Hz, are very close each other, whle the dfference between the frst and second modes s only. Hz ndcatng the very closely dstrbuted modes. In able 3, the modal matrx, Ψ, n state-space doman s shown n the ascendng order of natural frequences, and ts nverse matrx,.e. the lnear transformaton matrx,, s also provded. Consderng that the frst two rows of load partcpaton matrx, B, are zeros, t can be noted that the frst and second columns of modal responses n state-space doman are ntegral values of the thrd and fourth columns, respectvely, ndcatng the dsplacement-velocty and velocty-acceleraton relatonshps. 7

8 able 3. Analytcally obtaned modal propertes n state-space doman Matrx Symbol Value System matrx A Load partcpaton matrces B [ ] Modal matrx Lnear transformaton matrx Mode transformed system matrx Egenvalues Ψ Â ω and ξ Natural frequency Dampng rato st mode. (.9 Hz).85e- nd mode.3 (.85 Hz) 3.7e- Characterstcs of mode decomposton dependng on the objectve functon A numercal analyss of the coupled man structure-md s performed to obtan the acceleraton responses. he external load presented n able s appled n the numercal analyss. he dsplacement and velocty responses are obtaned usng the ntegral flter gven n Eq. (), and the ntal estmate of the system matrx s calculated usng Eq. (). able 4 presents the covarance matrces used for calculatng the ntal estmate of the system matrx along wth the modal characterstcs. able 4 ndcates that the natural frequences obtaned from the ntal estmate of the system matrx present nsgnfcant error compared to the exact natural frequences gven n able. he dampng ratos are, however, negatve values ndcatng sgnfcant error. he modal matrx and lnear transformaton matrces also dffer from the exact ones whle the correlatonshp of sgn between matrces s very large. he lnear transformaton matrces obtaned usng the proposed mode decomposton method are compared to the exact one n able 5. Frst, the frst mode lnear transformaton matrx that maxmzes the objectve functon J n Eq. (9) s obtaned usng Eq. (). he vector wth norm value of s also presented n able 5 for easer comparson. It can be seen that the lnear transformaton matrx obtaned from the ntal estmate of the system matrx s closer to the exact one than one obtaned usng the objectve functon J. he values n the frst three rows show very close results to exact ones whle the value of the last row s about.5 tmes to that of exact one. 8

9 able 4. Covarance matrces and modal characterstcs of ntal estmate of system matrx Matrx Symbol Value Covarance matrx of state S varable dv Cross covarance matrx S Intal estmate of system matrx Modal matrx Lnear transformaton matrx Egenvalues A o Ψ ω and ξ Natural frequency Dampng rato st mode.9 (.894 Hz) -.8e-4 nd mode.3 (.85 Hz) -.74e-4 able 5. Comparson of the frst mode lnear transformaton matrx: values n parenthess are normalzed vectors Method Symbol Value Exact soluton Intal estmate of system matrx Objectve functon J n Eq. (9) Objectve functon J 3 n Eq. (3) ICA [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) 9

10 he frst mode lnear transformaton matrx that maxmzes the objectve functon J 3 n Eq. (3) s also presented n able 5. It can be seen that the values of the frst three rows are almost dentcal to exact ones whle the value of the last row has error of about %. he optmzaton process for maxmzng the objectve functon J 3 s presented n Fg.. he ntal values used n the optmzaton teraton are the values that maxmze the objectve functon J. It can be seen from Fg. that the value of the objectve functon ncreases gradually as the teraton number ncreases, and t converges to a certan value as the teraton number s about. Among the varous optmzaton methods, the smple gradent method s used n ths paper. he lnear transformaton matrx s updated at the -th teraton as where + = +.δ (6) 3 δ = (7) J Fgure also presents the teraton results of the denomnator and numerator of Eq. (3) along wth the error between the estmated frst mode lnear transformaton matrx and exact one. he error s calculated as ( ) ( ) 4 j exact j e = (8) j= exact where exact s the exact frst mode lnear transformaton matrx presented n able 5. It can be seen that the error approaches to zero as the teraton numbers ncreases. Fgure shows the dampng rato estmaton process for the averaged power spectrum of Eq. () used n the optmzaton of the objectve functon J 3. It can be notced that the area of the power spectral functon becomes almost same to that of the averaged spectrum near the dampng rato of.6. Usng ths dampng rato and the frst mode frequency, the objectve functon J 3 s optmzed. he frst modal spectrums obtaned from the dfferent mode decomposton method are compared to the exact one n Fgure 3. It can be seen that the modal spectrums decomposed usng the ntal estmate of the system matrx and the objectve functon J are dstorted consderably near the second mode frequency. In specal, the decomposed modal spectrum obtaned usng the objectve functon J s contnuously smaller above the second mode frequency. On the contrary, the decomposed modal spectrum obtaned usng the objectve functon J 3 matches the exact one closely J 3 5 log( q ) Iteraton No Iteraton No.. Eq. (5).5. S peak e Iteraton No. Fgure. Iteraton result 5 5 Iteraton No Dampng rato Fgure. Dampng rato estmaton

11 Exact Intal estmate J J 3.4. Intal estmate J J 3 Modal spectrum Modal spectrum rato Frequency (Hz) Frequency (Hz) Fgure 3. Modal spectrum comparson Fgure 4. Modal spectrum rato to exact one Exact J 3 ICA J 3 ICA Modal spectrum Modal spectrum Frequency (Hz) Frequency (Hz) Fgure 5. Modal spectrum comparson Fgure 6. Modal spectrum rato to exact one In order to compare the decomposed modal spectrum more closely, the ratos of decomposed modal spectrums to exact one are presented n Fgure 4. It can be notced the more dstngushed error n the decomposed modal spectrum obtaned usng the ntal estmate of the system matrx and the objectve functon J. he decomposed modal spectrum obtaned usng the objectve functon J 3 shows the rato near one meanng almost dentcal result except near the second mode frequency. herefore, t can be concluded that the objectve functon defned n ths paper yelds the decomposed mode wth mnmum effect from the neghborng modes even when the structure has very closely dstrbuted modes. For the comparson of the proposed method to the prevous mode decomposton methods, the decomposed frst modal spectrum usng the ICA method s compared n Fgure 5. he modal spectrum ratos to exact one are also compared n Fgure 6. he correspondng lnear transformaton matrx for the frst mode s presented n able 5. It can be notced that the modal spectrum rato obtaned usng the ICA method s close to unty only near the frst mode frequency. However, the rato abruptly decreases near the second mode frequency and ncreases contnuously above that frequency. hs s because the ICA method matches the spectral area n average sense tryng to maxmze the modal ndependency from the neghborng modes. hs feature of the ICA method leads the decrease or ncrease of the rato where the modal frequences do not exst. he other decomposton methods such as the POD and

12 PCA methods are also examned, but ther results are provded here because ther decomposton resolutons are far less than the ICA method. In addton to the frst mode decomposton, the second to fourth mode decompostons are also performed and ther results are compared to exact ones. he results show that the mode decomposton usng the objectve functon J 3 also yelds very close modal spectrums to exact ones for hgher modes. Snce the results are almost dentcal to that of the frst mode, they are not presented here. Conclusons he new mode decomposton method s proposed and valdated numercally. he proposed method can mprove the decomposton resoluton for structures wth non-classcal dampng and closely dstrbuted modes whose mode decomposton s dffcult due to non-dagonalzaton of dampng matrx and strong correlaton between neghborng modes. he proposed method defnes a generalzed mode n state-space doman and performed the mode decomposton usng the state varable estmated from the measured responses. he numercal smulaton usng a SDOF-MD system ndcates that the objectve functon usng the averaged spectrum wth sngle pole yelds the best mode decomposton results. Further, t s shown that the proposed method yelds the decomposed mode wth mnmum effect from the neghborng modes even when the structure has very closely dstrbuted modes compared to results to those of the ICA and POD methods. References L, Y., Lu, G. R., Luan, M.., Da, K. Y., Zhong, Z. H., L, G. Y. and Han, X. (7) Contact analyss for solds based on lnearly conformng radal pont nterpolaton method, Computatonal Mechancs 39, Grama, S. N. and Subramanan, S. J. (4) Computaton of Full-feld Strans Usng Prncpal Component Analyss, Expermental Mechancs 54(6), Khall, M. and Sarkar, A. (4) Independent Component Analyss to enhance performances of Karhunen Loeve expansons for non-gaussan stochastc processes: Applcaton to uncertan systems, Journal of Sound and Vbraton 333(), Hwang, J.-S., Kareem, A., and Km, H. () nd load dentfcaton usng wnd tunnel test data by nverse analyss, Journal of nd Engneerng and Industral Aerodynamcs (99), 8-6. Feeny, B. () On Proper Orthogonal Coordnates as Indcators of Modal Actvty, Journal of Sound and Vbraton (55), Han, S. and Feeny, B. (3) Applcaton of Proper Orthogonal Decomposton to Structural Vbraton Analyss, Mechancal System and Sgnal Processng (7), Roberts S. and Everson.R. () Independent Component Analyss: Prncples and Practce, Cambrdge Unv. Kerschen G, Poncelet F., and Golnval J.-C. (7) Physcal nterpretaton of ndependent component analyss n structural dynamcs, Mechancal Systems and Sgnal Processng (), Van Overschee, P. and De Moor, B. (996) Subspace dentfcaton for Lnear Systems, Kluwer Academc Publshers. Brncker, R., Zhang, L., and Andersen, P. () Modal dentfcaton of output-only systems usng frequency doman decomposton, Smart Materals and Structures (), Ku, C.J., Cermark, J.E., and Chou, L.S. (7) Random decrement based method for modal parameter dentfcaton of a dynamc system usng acceleraton responses, Journal of nd Engneerng and Industral Aerodynamcs (95),