2. Theoretical development of random decrement technique for nonstationary vibration data

Transcription

1 Journal of Mechancal Scence and Technology 6 (6) (0) 687~696 DOI 0.007/s A modfed random decrement technque for modal dentfcaton from nonstatonary ambent response data only Chang-Sheng Ln,* and Dar-Yun Chang Instrumentaton Development Dvson, Natonal Synchrotron Radaton Research Center, Hsnchu, Tawan Department of Aeronautcs and Astronautcs, Natonal Cheng Kung Unversty, Tanan, Tawan (Manuscrpt Receved October 8, 0; Revsed January 6, 0; Accepted February 0, 0) Abstract Modal dentfcaton s consdered from response data of structural system under nonstatonary ambent vbraton. In a prevous paper, we showed that by assumng the ambent exctaton to be nonstatonary whte nose n the form of a product modelhe nonstatonary response sgnals can be converted nto free-vbraton data va the correlaton technque. In the present paper, f the ambent exctaton can be modeled as a nonstatonary whte nose n the form of a product modelhen the nonstatonary cross random decrement sgnatures of structural response evaluated at any fxed tme nstant are shown theoretcally to be proportonal to the nonstatonary cross-correlaton functons. The practcal problem of nsuffcent data samples avalable for evaluatng nonstatonary random decrement sgnatures can be approxmately resolved by frst extractng the ampltude-modulatng functon from the response and then transformng the nonstatonary responses nto statonary ones. Modal-parameter dentfcaton can then be performed usng the Ibrahm tme-doman technque, whch s effectve at dentfyng closely spaced modes. The theory proposed can be further extended by usng the flterng concept to cover the case of nonstatonary color exctatons. Numercal smulatons confrm the valdty of the proposed method for dentfcaton of modal parameters from nonstatonary ambent response data. Keywords: Modal-parameter dentfcaton; Nonstatonary ambent vbraton; Random decrement technque Introducton Modal dentfcaton from ambent vbraton data has ganed consderable attenton n recent years [, ]. In 97, based upon physcal ntuton rather than metculous mathematcal proof, Cole [3] ntroduced the random decrement technque (RDD) to study the applcaton of correlaton functons n measurng dampng and assessng damage n arcraft structures. The RDD s a technque developed to obtan a randomdec sgnature, whch s an ensemble average computed from random response data wth a common ntal or trggerng condton. Nevertheless, a rgorous mathematcal bass for applyng the RDD to a sngle-nput/sngle-output system was not proposed untl 98 by Vandver et al. [4]. In 986, Bedew [5] extended Vandver's work to a tme-nvarant lnear system wth multple degrees of freedom and concluded that f the nput force vector s a statonary Gaussan whte nose, then the randomdec sgnatures of the dsplacement responses of the system are equvalent to the free decay responses of a system. Ibrahm [6] appled the random decrement technque * Correspondng author. Tel.: Ext.08, Fax.: E-mal address: ln.changsheng@nsrrc.org.tw Recommended by Assocate Edtor Ohseop Song KSME & Sprnger 0 coupled wth a tme-doman parameter dentfcaton method (ITD) [7] to process ambent vbraton data. Although the random decrement technque serves as an alternatve method for estmatng the auto-correlaton and cross-correlaton functons [6] s based on an ntutve theory and does not yet have sound mathematcal bass for general cases [4]. Prevous studes of modal-parameter dentfcaton from ambent vbraton data have usually assumed that the nput exctaton s a broad-band stochastc process modeled by statonary whte or fltered whte nose. However, most ambent exctaton, such as earthquakes, encountered n many engneerng problems s nonstatonary n nature. Hence s desrable to develop applcable methods of modal dentfcaton for nonstatonary ambent vbraton. In the present paper, a theoretcal justfcaton of the random decrement technque s presented for general lnear systems excted by nonstatonary whte nose. It s shown that the nonstatonary random decrement sgnatures evaluated at an arbtrary, fxed tme nstant of structural response are of the same form as free vbraton decay of the structure wth certan ntal condtons. Therefore, by treatng the sample random decrement sgnatures of measured response correspondng to some fxed tme nstant as output from free vbraton decay, a tme-doman modaldentfcaton method, such as the ITD method [7], can then be

2 688 C.-S. Ln and D.-Y. Chang / Journal of Mechancal Scence and Technology 6 (6) (0) 687~696 employed to extract modal parameters, ncludng modal frequences, dampng ratos and mode shapes, of the structure wth complex modes.. Theoretcal development of random decrement technque for nonstatonary vbraton data The orgnal random decrement technque has been studed extensvely for sgnature analyss of vbratng systems. Vandver et al. [4] and Bedew [5] showed that when exctaton force s zero-mean, statonary, Gaussan whte nosehe auto random decrement sgnature δxx( τ ) of response could be denoted as follows: δ R ( ) ( ) xx xx τ = τ x R (0) s () xx where Rxx(0) = Rxx( τ = 0). Note that x s s the threshold level for the acquston of sample tme hstory x() t and generally defned as the root-mean-square value of system s statonary dsplacement response x() t. Eq. () sgnfes that the random decrement sgnature s n proporton to the autocorrelaton functon R xx ( τ ). Snce the correlaton functon has the same mathematcal form as that of free vbraton response [8]he random decrement sgnature can also be treated as a free vbraton sgnal for modal-parameter dentfcaton. The aforementoned random decrement technque was effectve only for statonary processes. In the followng, we extend the random decrement theory to deal wth zero-mean nonstatonary processes. Frsthe nonstatonary cross random decrement sgnature δ X (, ) X t t between two response channels, X and X, can be defned as: δ X (, ) X t t E X ) X ) x = = X P X X X ) x dx = where PX ( ; X ) P X t X ) x, = = PX ( ) (3) where x denotes the threshold level of X ) at a fxed tme nstant t = t, and X () t s the reference channel that beng chosen as a response channel whch contans rcher overall frequency nformaton. In addton, we can deduce from Gaussan dstrbuton assumpton that P X X σ ) = exp X ) X ) πσa ) σa ) σ ) () (4) where σa ) σ ) ρ ) =, and σ ) ρ ) =. σ ) σ ) Note that σ ) and ρ ) are denoted as the covarance and correlaton coeffcent, respectvely, between X ) and X ). Substtutng Eq. (4) nto Eq. (), we fnd σ δ (, ) XX t t = X t σ EX [ ) X )] = X ) EX [ )] ( ) where E [ ] represents the ensemble average operator. Gven t = t and t = t+ τ, Eq. (5) can be rewrtten as: EX [ X ) + τ )] δ (, ) XX tt+ τ = X () t [ ( )] EX t R XX ( τ ). (6) = X () t EX [ )] From Eq. (6), we know that the nonstatonary cross random decrement sgnature s n drect proporton to the nonstatonary cross-correlaton functon under the assumpton of Gaussan processes. From a prevous paper of ours [9], f the nput sgnals can be modeled as nonstatonary whte nose, whch s a product of whte nose and a determnstc tme-varyng functonhe theoretcal nonstatonary correlaton functon R ( τ ) evaluated at an arbtrary fxed tme nstant t of XX structural response has the same mathematcal form as that of the free vbraton of the orgnal system. Therefore, computaton of the random decrement sgnatures δ ( ) X Xt,t + τ between response channels serves as an alternatve approach for modal-parameter dentfcaton from ambent responses correspondng to nonstatonary whte-nose nput n the form of a product model. Note that n practce, computaton of the random decrement sgnatures δ ( ) X Xt,t + τ between response channels s conducted by usng the shfted-sample averagng method. 3. Practcal treatment of nonstatonary data The prevous secton showed that the randomdec sgnatures evaluated at a fxed tme nstant of responses can be used as free-vbraton decay or as mpulse response n tme-doman modal-extracton schemes. However, n engneerng practce, only very lmted data samples are usually avalable, and so evaluaton of the randomdec sgnatures could be a sgnfcant problem. Prevously, we showed [9] that we can try to resolve (5)

3 C.-S. Ln and D.-Y. Chang / Journal of Mechancal Scence and Technology 6 (6) (0) 687~ the problem by frst extractng the ampltude-modulatng functon from the response and then the orgnal nonstatonary responses can be transformed nto statonary ones. If the exctaton can be modeled as nonstatonary whte nose as represented by the product modelhen the responses of the system can also be modeled approxmately as a product model wth the same ampltude-modulatng functon as that assocated wth the exctaton tself. Consder a dscrete lnear system subjected to exctaton resulted from a sngle source wt ( ), whch s assumed to be statonary whte nose. The equaton of moton can be expressed as ) + ) + ) = w) Mv Cv Kv l (7) where v ), v ) and ) v are the statonary dsplacement, velocty and acceleraton responses, respectvely, and l s a vector for whch the elements are the nfluence factors for each degree of freedom (DOF) and may be thought of a measure of the extent to whch the ( ) wt partcpates n the total exctaton on the structure. Multplyng both sdes of Eq. (7) by a slowly-tme-varyng ampltude-modulatng functon Γ() t we can obtan ) + ) + ) = f ) Mu Cu Ku l (8) where f ) s a nonstatonary whte nose as represented by the product model f () t = Γ() t w() t (9) u v. In dervng Eq. (8), we assumed that Γ t s a slowly tme-varyng functon [.e., v u and v u. The results ndcate that f the exctaton can be modeled as nonstatonary whte nose as represented n Eq. (9) wth a slowly-tme-varyng envelope functon Γ ) hen the nonstatonary responses of the system can also be modeled approxmately as a product model wth the same envelope functon. To transform the orgnal nonstatonary responses nto statonary ones and to crcumvent the practcal problem of evaluatng nonstatonary randomdec sgnatures from very lmted data samples, we extract the ampltude-modulatng functon from the orgnal nonstatonary data, whch can be done by evaluatng the temporal root-mean-square functons from the real data. The theoretcal background s gven as follows [0]. and () t = Γ() t () t () Γ ) 0, Γ ) 0 ], and so Γ () t () t ) Γ() t () t () t Denote the tme average of u ( τ ) as u ), whch s defned as t+t u () t = u ( τ ) dτ. (0) T t-t Recall that we have assumed the Γ( τ ) to be a slowly varyng functonhen from Eq. (0) u () t can be approxmated as t+t u t = v d T t-t Γ t+t () t v ( τ) dτ T t-t () Γ ( τ ) ( τ) τ () for T beng a short tme nterval. The temporal mean-square functon u () t s practcally estmated by averagng over short tme ntervals of the record. If we assume that v ( τ ) s an ergodc processhe ntegral on the rght-hand sde of Eq. () s just an approxmaton to E v and so u () t Γ () t E v. () Then the temporal root-mean-square functon denoted as Ψ ) can be evaluated by tme-averagng over a sngle sample record as Ψ () t u () t = Γ() t C (3) where ( ) C E v =. Note that the temporal root-meansquare functon Ψ ) of each DOF s proportonal to the same envelope functon of tme Γ() t. The precedng result ndcates that the temporal root-meansquare functons Ψ ) of the response hstores descrbe the same tme varaton as gven by the envelope functon Γ ). Ths suggests that f the orgnal nonstatonary data could be represented by the product model wth a slowly varyng envelope functonhe temporal root-mean-square functons of the data also have the same nonstatonary trend as that of the orgnal data. The temporal root-mean-square functon Ψ ), and so the envelope functon Γ() t, can thus be determned by usng nterval average and then applyng curve-fttng technque. We can then acqure the approxmate statonary responses by dvdng the nonstatonary responses of each DOF wth the same envelope functon Γ() t. The randomdec sgnatures of the statonary response data can then be obtaned, whch are n turn treated as the free-decay responses correspondng to each DOF. The modal parameters of the system can then be obtaned va a tme-doman modal-dentfcaton method, such as the ITD method [7], as descrbed next. 4. Ibrahm tme-doman modal-dentfcaton method The Ibrahm tme-doman (ITD) method uses free decay responses of a structure under test to dentfy ts modal parame-

4 690 C.-S. Ln and D.-Y. Chang / Journal of Mechancal Scence and Technology 6 (6) (0) 687~696 ters n complex form [7]. From the measured free responses at n statons on a structure under test, each wth q samplng ponts, we defne a system matrx A such that A X = Y (4) where X s an n q matrx of measured response, and Y s a matrx of tme-delayed response. Generallyhe number q s chosen larger than the number of measurement channels n. Thereforehe system matrx A can be estmated by the leastsquares method. It can be shown that the natural frequences and the dampng ratos of the orgnal vbratng system are drectly related to the egenvalues of the system matrx, A, and the egenvectors of A correspond to the mode shapes of the orgnal system. Thus, once the system matrx A s obtaned va leastsquares analyss from measured datahe modal parameters of the structure system of nterest can then be determned by solvng the egenvalue problem assocated wth the system matrx, A. samplng nterval s chosen as Δt = 0.0s, and the samplng perod s T = Nt Δ t = 30.7s, where N t was chosen as 7 to assure that more than 500 good samples of shfted sgnals can be obtaned for performng the randomdecrement averagng. The statonary whte nose smulated s then multpled by an ampltude-modulatng functon 0.00t 0.004t Γ() t = 4 (e e ) to obtan the nonstatonary whte nose, whch serves as the exctaton nput actng on the 6 th mass pont of the system. A typcal plot of the ampltudemodulatng functon s shown n Fg., n whch we can clearly see that Γ ) vares wth t slowly. The tme sgnal of a smulated sample of the nonstatonary whte nose and the power spectrum of the correspondng statonary part are shown n Fg. 3 and Fg. 4, respectvely. The dsplacement responses of the system obtaned through Newmark s method [] are shown n Fg. 5. By examnng the Fourer spectra assocated wth each of the response chanx x x 3 x 4 x 5 x 6 Fg.. Schematc plot of the 6-DOF chan system. 5. Numercal smulaton To demonstrate the effectveness of the proposed method, we frst consder a lnear 6-DOF chan model wth vscous dampng. A schematc representaton of ths model s shown n Fg.. The mass matrx M, stffness matrx K, and the dampng matrx C of the system are gven as follows: M = N s /m, K = 600 N/ m, C= 0.05M+ 0.00K+ 0. N s/ m. 66 Note that the dampng matrx C s not a lnear combnaton of M and K, so that the system has nonproportonal dampng (and so complex modes). Consder that the ambent vbraton nput can be modeled as nonstatonary whte nose as represented by the product model. The statonary whte nose s generated by usng the spectrum approxmaton method [] as a zero-mean band-pass nose, whose standard devaton s 0.0 N s/rad wth a frequency range from 0 to 50 Hz. The Fg.. A typcal plot of the ampltude-modulatng functon.

5 C.-S. Ln and D.-Y. Chang / Journal of Mechancal Scence and Technology 6 (6) (0) 687~ Table. Results of modal-parameter dentfcaton of the 6-DOF chan system subjected to nonstatonary whte-nose nput. Mode Natural frequency (rad/s) Dampng rato (%) Exact ITD Error (%) Exact ITD Error (%) MAC Fg. 3. A sample functon of nonstatonary whte nose n tme doman. Fg. 4. Power spectrum assocated wth the statonary part of the nonstatonary whte nose. nel, we chose the st channel s response X ), whch contans rcher overall frequency nformaton, as the reference channel to compute the random decrement sgnatures of the system. Accordng to the theory presented n the prevous sectonshe nonstatonary problem may reduce to a statonary problem f we can extract the ampltude-modulatng functon from the orgnal nonstatonary data. Therefore, we can follow the same procedures as those for statonary problems, and the random decrement sgnatures thus obtaned, as shown n Fg. 5, are treated as free-vbraton data. The Ibrahm tme-doman method could then be appled to dentfy modal parameters of the system. The results of modal parameter dentfcaton are summarzed n Table, whch shows that the errors n natural frequences are less than % and the maxmum error n dampng ratos s about 40%. The exact modal dampng ratos lsted n Table are actually the equvalent modal dampng ratos obtaned by utlzng ITD from the free vbraton response data of the nonproportonally damped structure. The dentfed mode shapes are compared wth exact values n Fg. 6, where good agreement s observed. The errors of dentfed dampng ratos and mode shapes are somewhat larger than those of modal frequences. Ths s because the system response generally has lower senstvty to these modal parameters than to the frequences. It s also noted that the hgher modes are not dentfed so accurately as the lower ones, because the contrbuton of hgher modes to the system response s somewhat less than that of the lower modes. It should be mentoned that the proposed method s vald n dentfyng modes wth close frequences, as long as the modal dampng ratos of each mode of a system are dfferent and small. Ths s because that, when a system has lght dampnghe nterference between modes would, n general, not lead to large errors n separatng the closely spaced modes of a system. It should be mentoned that the selecton of reference channel for computng random decrement sgnatures s also mportant to the dentfcaton results. The rcher frequency content the reference channel hashe better results of modal parameters dentfcaton can be acheved. It s remarkable that for the random decrement technque to be effectve, we need generally more than 500 samples of tme hstory for each response channel. We can then perform averagng over the samples to obtan good quas free-vbraton data for further modal parameters dentfcaton. For ths purposehe random decrement technque generally requres response data to be much longer n tme, and therefore s practcally less effcent when compared wth the correlaton technque n a prevous paper of the authors [9]. In the above, we consdered the nonstatonary exctaton to be nonstatonary whte nose modeled as the product of statonary whte nose and an ampltude-modulatng functon. Ths restrcton could be removed by treatng the nonstatonary exctaton as nonstatonary color nose or fltered whte nose. A nonstatonary color nose s modeled as the product an ampltude-modulatng functon and a statonary color nose, whch s, n turn, obtaned as the output of a certan system (actng as a flter) to an nput of statonary whte nose. A smulated sample of nonstatonary color nose s shown n Fg. 7, where the flter system was assumed to be a second-order system wth a frequency of.69 rad/sec and a dampng rato of 5%. It s remarkable that the modes dentfed generally nclude the vbratng modes of the structural system and the fcttous modes due to numercal computaton. From dentfcaton

6 69 C.-S. Ln and D.-Y. Chang / Journal of Mechancal Scence and Technology 6 (6) (0) 687~696 Fg. 5. Typcal dsplacement responses and the correspondng randomdec sgnatures of the 6-DOF chan system subject to nonstatonary whte nput.

7 C.-S. Ln and D.-Y. Chang / Journal of Mechancal Scence and Technology 6 (6) (0) 687~ Fg. 6. Comparson between the dentfed mode shapes and the exact mode shapes of the 6-DOF chan system subjected to nonstatonary whtenose nput.

8 694 C.-S. Ln and D.-Y. Chang / Journal of Mechancal Scence and Technology 6 (6) (0) 687~696 Table. Results of modal-parameter dentfcaton of the 6-DOF chan system subjected to nonstatonary whte color nput. (a) Mode Natural frequency (rad/s) Dampng rato (%) Exact ITD Error (%) Exact ITD Error (%) MAC The 4 th mode s a fcttous mode of exctaton, and has no correspondng exact mode shape for computng the value of MAC. Modal parameters estmated of the 7 th mode from RDD-sgnatures by ITD method s not dentfed accurately. Fg. 8. Truss structure wth 8 unrestraned degrees of freedom [6] (one horzontal and one vertcal for each of the nodes denoted by,,3, and 4). results, as shown n Table can be seen that the system s characterstcs as well as the nput characterstcs were both dentfed. We can dstngush the fcttous modes from the vbratng modes of the structural system f the mass matrx or the stffness matrx of the structural system s avalable accordng to the orthogonalty condtons, whch show that vbratng shapes are orthogonal wth respect to the stffness matrx as well as to the mass matrx. Furthermoreo keep track of the target modes, a modetrackng method utlzng the MAC [3] s proposed. The modal assurance crteron has been extensvely used n the expermental modal analyss. The defnton of MAC s MAC( φ, ) (b) Fg. 7. A sample of nonstatonary color nput: (a) tme hstory; (b) power spectrum(statonary part). T φ A φ jx A φ jx = T T φa φa φ jx φ jx (5) where φ A and φ jx represent two mode shape vectors of nterest, and the superscrpt * denotes the complex conjugate. The value of MAC vares between 0 and. When the MAC value s equal to he two vectors φ A and φ jx represent exactly the same mode shape. Results of modal parameter dentfcaton of the 6-DOF system subjected to nonstatonary whte color nput are summarzed n Table, whch shows that the 4th mode s a fcttous mode of exctaton, and has no correspondng exact mode shape as well as the value of MAC, and modal parameter estmated of the 7 th mode from RDDsgnatures by ITD method s not dentfed accurately. To examne the effectveness of the present method for more complex structural systems, we consder a twodmensonal truss model wth proportonal dampng [4]. Ths system, shown n Fg. 8, has a total number of 8 nodes of whch 4 are fully restraned, and hence the total number of actve DOFs s 8 (one horzontal and one vertcal per each node). The mass, dampng and stffness matrces for ths system are gven as follows: M = N s /m,

9 C.-S. Ln and D.-Y. Chang / Journal of Mechancal Scence and Technology 6 (6) (0) 687~ Table 3. Results of modal-parameter dentfcaton of the 8-DOF truss system subjected to nonstatonary whte nput. Mode Natural frequency (rad/s) Dampng rato (%) Exact ITD Error (%) Exact ITD Error (%) MAC K = N/ m, C = 0.7M K N s/ m In ths example, we stll use the prevous nonstatonary whte-nose n the form of a product model as nput actng horzontally and vertcally at all actve DOFs of the truss model, and then the correspondng dsplacement responses obtaned by Newmark s method are used for modal-parameter dentfcaton. By examnng the Fourer spectra assocated wth each of the response channel, we chose the 7 th channel s response X 7 ),.e.he 7 th actve DOF s response, whch contans rcher overall frequency nformaton, as the reference channel to compute the random decrement sgnatures of the system. The results of modal parameter dentfcaton are summarzed n Table 3, whch shows that the modal dentfcaton n ths case s satsfactory. The errors n natural frequences are less than % and the maxmum error n dampng ratos s about 0%. Observng the MAC values, whch sgnfy the consstency between the dentfed and the theoretcal mode shapes, we found that all modes were dentfed accurately (MAC 0.9). 6. Conclusons For the purpose of dentfyng dynamc characterstcs of structureshe modal-analyss method of usng measured responses to ambent nonstatonary exctaton s studed. It can be shown that f the nput sgnals can be modeled as nonstatonary whte nose, whch s a product of whte nose and a determnstc tme-varyng functonhe theoretcal nonstatonary random decrement sgnatures of structural response wll have the same mathematcal form as free vbraton of the structure. The practcal problem of nsuffcent data samples avalable for evaluatng nonstatonary random decrement sgnatures can be approxmately resolved by frst extractng the ampltude-modulatng functon from the response and then transformng the nonstatonary responses nto statonary ones. The random decrement sgnatures of the statonary response are treated as free-vbraton response, and so the Ibrahm tmedoman method can then be appled to dentfy modal parameters of the system. In addtonhe choce of the reference channel s sgnfcant to compute the random decrement sgnatures. The reference channel s chosen as a response channel whose Fourer spectrum has rch frequency content around the structure modes of nterest. The rcher frequency content the reference channel hashe better results of modal-parameter dentfcaton can be acheved. Modal-parameter dentfcaton usng ambent data excted by nonstatonary color nose s also consdered. Ths s accomplshed va addng, n cascade, a pseudo-force system to the structural system under consderaton. Identfcaton results are then sorted out as ether structural parameters or nputs force(s) characterstcs usng the orthogonalty condtons of structure modes. Acknowledgment Ths research was supported n part by the Natonal Scence Councl of the Republc of Chna under the grant NSC-00- -E The authors would lke to thank the anonymous revewers for ther valuable comments and suggestons n revsng ths paper. Nomenclature A : System matrx f ) : Nonstatonary whte nose n the form of a product model E [ ] : Ensemble average operator M : System mass matrx C : System dampng matrx K : System stffness matrx l : Vector for whch the elements are the nfluence factors for each degree of freedom T : Short tme nterval X : Matrx of measured response x s : Threshold level for the acquston of sample tme hstory Y : Matrx of tme-delayed response u ) : The temporal mean-square functon v () t : The statonary dsplacement response wt ( ) : Statonary whte nose Γ ) : Determnstc envelope functon (or ampltudemodulatng functon) φ A : th theoretcal mode shape φ : j th dentfed mode shape jx References [] D. M. Srngorngo and Y. Fujno, System dentfcaton of suspenson brdge from ambent vbraton response, Engneerng Structures, 30 (008) [] D. Y. Chang and C. S. Ln, Identfcaton of modal parame-

10 696 C.-S. Ln and D.-Y. Chang / Journal of Mechancal Scence and Technology 6 (6) (0) 687~696 ters from nonstatonary ambent vbraton data usng the channel-expanson technque, Journal of Mechancal Scence and Technology, 5 (5) (0) [3] H. A. Jr. Cole, Method and apparatus for measurng the dampng characterstcs of a structure, Unted States Patent No. 3, 60, 069 (97). [4] J. K. Vandver, A. B. Dunwoody, R. B. Campbell and M. F. Cook, A mathematcal bass for the random decrement vbraton sgnature analyss technque, ASME Journal of Mechancal Desgn, 04 (98) [5] N. E. Bedew, The mathematcal foundaton of the auto and cross-random decrement technques and the development of a system dentfcaton technque for the detecton of structural deteroraton, Ph.D thess, Unversty of Maryland College Park (986). [6] S. R. Ibrahm, Random decrement technque for modal dentfcaton of structures, Journal of Spacecraft and Rockets, 40 () (977) [7] S. R. Ibrahm and E. C. Mkulck, A Method for the drect dentfcaton of vbraton parameters from free response, Shock and Vbraton Bulletn, Vol. 47, Pt. 4, Sept. (977) [8] G. H. James, T. G. Carne and J. P. Lauffer, The natural exctaton technque (NExT) for modal parameter extracton from operatng structures, Modal Analyss, 0 (4) (995) [9] D. Y. Chang and C. S. Ln, Identfcaton of modal parameters from nonstatonary ambent vbraton data usng correlaton technque, AIAA Journal, 46 () (008) [0] Newland, D. E., Applcaton notes:nonstatonary processes, An Introducton to Random Vbratons, Spectral Analyss & Wavelet Analyss, 3 rd ed., Longman, London, 993, pp. -7. [] M. Shnozuka and C. -M. Jan, Dgtal smulaton of random processes and ts applcatons, Journal of Sound and Vbraton, 5 () (97) -8. [] N. M. Newmark, A method of computaton for structural dynamcs. Journal of Engneerng Mechancs, ASCE, 85 (EM3) (959) [3] R. L. Allemang and D. L. Brown, A correlaton coeffcent for modal vector analyss, Proceedngs of the st Internatonal Modal Analyss Conference, Socety for Experment Mechancs, Bethel, CT, (983) 0-6. [4] M. De Angels, H. Lus, R. Bett and R. W. Longman. Extractng physcal parameters of mechancal models from dentfed state space representatons, ASME Journal of Appled Mechancs, 69 (5) (00) modal analyss. C. S. Ln receved hs Ph.D from the Department of Aeronautcs and Astronautcs of Natonal Cheng Kung Unversty, Tawan n 0. He s currently a Post-doctoral research fellow at Natonal Synchrotron Radaton Research Center, Hsnchu, Tawan. Hs research nterests are n random vbraton and D. Y. Chang receved hs Ph.D n Appled Mechancs from Calforna Insttute of Technology, USA, n 99. He joned the faculty of Natonal Cheng Kung Unversty, Tawan n 993 where he s currently a professor at the Department of Aeronautcs and Astronautcs. Hs research nterests are n system dentfcaton and plastcty.