Utilization of Blind Source Separation Techniques for Modal Analysis

Transcription

1 Utlzaton of Blnd Source Separaton echnques for Modal Analyss B. Swamnathan +, B. Sharma +, S. Chauhan * + Structural Dynamcs Research Lab, Unversty of Cncnnat, Cncnnat, OH, USA * Bruel & Kjaer Sound and Vbraton Measurement A/S Skodsborgvej 307, DK 2850, Naerum, Denmark Emal: swamnbr@mal.uc.edu, sharmabr@mal.uc.edu, schauhan@bksv.com Abstract In past few years, there have been attempts at utlzng Blnd Source Separaton (BSS) and Independent Component Analyss (ICA) technques for modal analyss purposes. Most of the early work n ths regard has been promsng, though restrcted to applcaton of these technques to analytcal and laboratory based epermental structures. It s felt that n order to make these technques applcable to more challengng scenaros, they need to be modfed keepng n vew the demands of modal parameter estmaton procedure. hs ncludes makng them more robust and applcable to handle comple scenaros (for e.g. closely coupled modes, heavly damped modes, low sgnal-to-nose rato, etc.). hs forms the motvaton for ths paper whch ams at tunng BSS / ICA methods for modal analyss purposes n an effectve and effcent manner. Amongst other methods, t s shown how to ncorporate sgnal processng technques, modfy BSS technques to handle data n specfed frequency ranges, etract modal parameters from lmted output channels, etc. to derve most benefts out of these algorthms. 1. Introducton Blnd Source Separaton technques emerged from the medcal magng and wreless communcaton felds as mage or sgnal processng technques. Anton [1] demonstrated how these technques can be used for blnd separaton of vbraton components and lad down the assocated prncples. hs was followed by works that attempted at showng the applcaton of these technques for modal analyss [2-5]. hese early works amed at establshng a connecton between BSS technques and Operatonal Modal Analyss (OMA). hese studes showed encouragng results, by means of applcaton of BSS technques to analytcal and smple laboratory structures and estmatng modal parameters n the process. As fundamental prncples of applcaton of BSS technques for OMA became clear and understood, ther lmtatons n ths regard also came nto lght. Mathematcal formulaton of BSS technques shows that they are more sutable to handle systems havng no or neglgble dampng [5]. Early smulatons, however, suggest that ths does not pose serous challenge as long as the modal coordnates [4] have dstnct spectra and are mutually uncorrelated. However, the verdct s stll not clear and more rgourous study needs to be conducted. Yet another lmtaton, on account of mathematcal foundatons of BSS technques, s that they fnd real modal vectors, whch mght not be the case for most real-lfe structures. A framework to etend BSS technques to take ths lmtaton nto account was recently suggested n [6]. Further, the effect of nose on the performance of these technques s not yet evaluated (for modal analyss applcatons). ypcally, these technques work drectly on the raw data wthout much sgnal pre-processng and thus ther performance s more senstve to the qualty of the data acqured. Current formulaton of most BSS algorthms also poses a lmtaton on account of the fact that one can estmate only as many modes as the number of output responses beng measured.

2 he work presented n ths paper attempts at addressng some of these ssues by means of utlzng Second Order Blnd Identfcaton (SOBI) [7, 8] algorthm. In lght of above dscusson, a modfed formulaton of SOBI algorthm s provded n the paper that ams at crcumventng the lmtaton that only as many modes can be found as the number of sensors measurng the output response. hs modfcaton also ncludes use of sgnal processng technques such as averagng and wndowng to reduce the effect of nose. Suggested modfcatons, thus mprove the overall performance of SOBI algorthm and contrbutes towards makng t more sutable for OMA purposes and wdely applcable to more realstc structures. Secton 2 presents SOBI n ts orgnal form and ts modfed form developed n ths work, along wth revstng the lnk between BSS and OMA. Secton 3 demonstrates the postve effect these modfcatons have on overall results, by means of studes conducted on a 15 degree-of-freedom analytcal system and a lghtly damped crcular plate. Fnally, based on these results, conclusons are drawn n Secton Second Order Blnd Identfcaton (SOBI) [7, 8] 2.1 heoretcal Background Mathematcally, an nstantaneous BSS problem, n tme doman, can be formulated as () t As() t = (01) where (t) s a column vector of m output observatons representng an nstantaneous lnear mture of source sgnals s(t), whch s a column vector of n sources at tme nstant t. A s an m X n matr referred to as mng system or more commonly as mng matr. SOBI s a BSS algorthm that separates the sources assumng that they have a temporal structure wth dfferent autocorrelaton functons (or power spectra) and are mutually uncorrelated (havng zero crosscorrelaton). It utlzes the concept of jont dagonalzaton for achevng ths goal. Followng are the steps nvolved n ths algorthm. Note that, number of sources are consdered equal to number of sensors (or observatons).e. m = n. 1. Frst the covarance (mean removed correlaton) matr of the output observatons s estmated N Rˆ 1 ( 0) = ( k ) ( k) (02) N where taken. Rˆ ( 0) k = 1 s the covarance matr at zero tme lag and N s the total number of tme samples 2. Compute EVD (or SVD) of Rˆ ( 0) R ( 0) = U V = V Λ V = V Λ V + V Λ V ˆ (03) where V s s m X n matr of egenvectors assocated wth n prncpal egenvalues of Λ s = dag{λ 1, λ 2,.., λ n } n descendng order. V n s m X (m-n) matr contanng the (m-n) nose egenvectors assocated wth nose egenvalues Λ n = dag{λ n+1, λ n+2,.., λ m }. he number of sources n s thus estmated based on the n most sgnfcant egenvalues (or sngular values n case of SVD). 3. Perform pre-whtenng transformaton ( k) = Λ 1 2 s V ( k) = Q( k) 4. Estmate the covarance matr of the vector (k) other than τ =0. N Rˆ 1 ( ) = ( k) ( k N s k = 1 s s s N N N (04) for a preselected set of tme lags τ, τ,..., τ ) ( 1 2 L τ τ ) (05) 5. Perform Jont Appromate Dagonalzaton on the above set of covarance matrces R U D τ U ( ) ( ) τ = (06)

3 to estmate the orthogonal matr U that dagonalzes a set of covarance matrces. Several effcent algorthms are avalable for ths purpose ncludng Jacob technques, Alternatng Least Squares, Parallel Factor Analyss etc. [9, 10]. 6. he mng matr and source sgnals can be estmated as 1 Aˆ 2 = Q + U = VsΛ s U (07) y ( k) = sˆ ( k) = U ( k) It should be noted that D( τ ) s a dagonal matr that has dstnct dagonal entres. However, t s dffcult to determne a pror a sngle tme lag τ at whch the above crteron s satsfed. Jont dagonalzaton procedure avods ths dffculty by provdng an optmum soluton consderng a number of tme lags. (08) 2.2 BSS and OMA he basc fundamental behnd applcaton of ICA / BSS technques to modal analyss goes back to the concept of epanson theorem [12] and modal flters [14]. Accordng to the epanson theorem [15], the response vector of a dstrbuted parameter structure can be epressed as ( t ) = r = 1 φ η ( t ) (09) r r where Φ r are the modal vectors weghted by the modal coordnates η r. For real systems, however, the response of the system can be represented as a fnte sum of modal vectors weghted by the modal coordnates. In ths manner, mathematcally, epanson theorem yelds smlar formulaton as epressed n Eq. (01), wth source vector s(t) and mng matr A beng analogous to modal coordnate response vector η(t) and modal vector matr [Φ], and hence BSS technques lke SOBI can be appled to obtan modal vectors and modal coordnates (whch defne the modal frequency and dampng). o obtan a partcular modal coordnate η from response vector, a modal flter vector ψ s requred such that ψ = 0, for j (10) φ and 0, so that ψ for = j (11) φ N ψ ( t) = ψ φ η ( t) (12) =ψ φ η r= 1 r r or (13) hus modal flter performs a coordnate transformaton from physcal to modal coordnates. Multplyng the system response wth modal flter matr Ψ results n uncouplng of the system response nto sngle degree of freedom (SDOF) modal coordnate responses (η). 2.3 Suggested Modfcatons Mng model shown n Eq. (1) does not take nto consderaton the addtve nose, whch s often present whle dealng wth real lfe structures. From operatonal modal analyss applcaton pont of vew, one of the drawbacks of SOBI s n dealng wth sgnfcantly nosy output response sgnals. In typcal OMA scenaro, such as response measurements taken over a brdge or a buldng, SNR (Sgnal-to-Nose Rato) mght not be good and ths can deterorate the performance of the algorthm. Apart from whtenng (Eq. 4), SOBI doesn t nvolve any other sgnal pre-processng step and thus errors can creep n due to addtve measurement and process nose that s generally random n nature. One of the ways, to overcome ths ssue and mprove performance of SOBI, s by mnmzng the errors n measurement,.e. effect of nose and bas, by means of technques lke averagng and use of wndowng

4 functons. hus, nstead of workng drectly on raw data, SOBI algorthm can be modfed to work wth correlaton functons that are obtaned after nverse Fourer transformng the averaged power spectra. hs means that n step 5, nstead of applyng jont dagonalzaton procedure to covarance matrces calculated usng complete data, t s performed on covarance matrces obtaned after averagng procedure s carred out for nose mnmzaton. Welch Perodogram method [11] s one such method that can be used for obtanng averaged output response power spectra. Along wth averagng the power spectra, wndowng and overlappng can also be used to reduce the leakage (bas) errors. hs process of nose mnmzaton, usng power spectra averagng based technque, also provdes a mechansm to etend the effectveness of SOBI algorthm by makng t possble to apply ths technque wthn frequency band of nterest. hs s a sgnfcant mprovement over SOBI n ts orgnal form, as t overcomes the lmtaton of dentfyng only as many sources as number of measured responses. hus step 5, n the prevous secton, can be preceded by selectng averaged power spectra n the frequency band of nterest, say between ω 1 and ω 2, Gˆ ( ω) ω. hs s followed by nverse Fourer transformng the power spectra n 1, ω2 selected frequency range to obtan correspondng Covarance matr. ~ˆ 1 R τ ˆ = I G ω (14) ( ) ( ) ( ) ω ω 1, 2 Step 5 then nvolves performng jont dagonalzaton of these covarance matrces, thus restrctng SOBI to estmate modes n the specfed frequency range. Advantages of these modfcatons are shown n the net secton where modfed verson of SOBI s appled to a 15 Degree-of-Freedom analytcal system and a lghtly damped crcular plate. 3. Case Studes Degree-of-Freedom Analytcal System A 15 DOF analytcal M-C-K system s consdered for analyss usng the modfed verson of SOBI algorthm (Fgure 1). hs system has some moderately damped (1-4 %) modes; wth a par of closely spaced modes around 53.3 Hz and also some locally ected modes n Hz frequency range. Fgure 1: Analytcal 15 Degree of Freedom System heoretcal modal frequency (n Hz) and dampng (n % Crtcal) values for the system are shown n able 1. Response data s obtaned at each DOF by ectng the system at these DOFs by uncorrelated random forces. Smulated response data s sampled at a frequency of 1024 Hz and a total of response samples are collected. Complete Frequency Range Based Analyss In ths analyss, data s analyzed n complete frequency range from Hz. able 1 lsts the estmates of frequency and dampng obtaned usng SOBI n ts orgnal form and ts modfed form. he estmates of modal frequency and dampng are obtaned by applyng SDOF frequency doman methods on the power spectra of the obtaned sources [12]. hese are compared aganst the theoretcal values of frequency and dampng. It should be noted that whle usng modfed verson of SOBI, frequency band of nterest s chosen

5 to cover the complete frequency range. hus, effectvely ths s smlar to SOBI orgnal, ecept that modfed verson of SOBI works wth covarance functons obtaned after nverse Fourer transformng the averaged complete power spectra, where as orgnal form of SOBI works drectly on the data wthout performng any pror sgnal processng. Fgure 2 shows the plots of power spectra of SDOF modal coordnates obtaned usng orgnal SOBI. All 15 modes are easly obtaned usng orgnal SOBI. Smlar plots are obtaned for modfed verson of SOBI as well. Fgure 2: Power Spectra of SDOF Modal Coordnates (Orgnal SOBI) able 1 shows that the comparson on the bass of modal frequency and dampng estmates s qute satsfactory though dampng values obtaned usng SOBI (both algorthms) are slghtly overestmated. Both SOBI approaches, orgnal and modfed, show farly smlar results. able 1: Comparson of Modal Parameter Estmates for the 15 DOF System heoretcal Orgnal SOBI Modfed SOBI Freq (Hz) Dampng (%) Freq (Hz) Dampng (%) Freq (Hz) Dampng (%)

6 he mode shapes of ths system are comple. Ecept for the frst mode at Hz, whch has farly real mode shape, other modes have comple mode shapes. Snce, BSS technques (such as SOBI n ts orgnal form) are typcally suted to gve real mode shapes (although modfed SOBI does produce comple mode shapes), t s epected that results mght not be as accurate as epected. However, as shown by able 2, when mode shapes obtaned usng the two algorthms are compared wth theoretcal mode shapes, the comparson between theoretcal mode shapes and SOBI Orgnal mode shapes s pretty good. MAC (Modal Assurance Crteron) [12] comparson for the mode shapes correspondng to the two closely spaced modes (around 53.3 Hz) s not as good as other modes, whch hghlghts that BSS technques such as SOBI fnd t dffcult to handle effectvely the cases nvolvng closely spaced modes. Modfed SOBI s performance, on the other hand, s nferor n comparson (Hghlghted n able 2). One of the possble reasons can be the fact that the output response data obtaned for ths analytcal system s free from nose and use of sgnal processng technques such as averagng, wndowng and overlappng nvarably ntroduces some errors nto the processed power spectra. Further, the modal vectors obtaned usng modfed SOBI are comple (due to dscrete Fourer transformaton) and t s observed that f only the real part of the modal vectors s compared wth the theoretcal modes, MAC s sgnfcantly mproved. able 2: MAC Comparson heoretcal Modes heoretcal vs. Orgnal SOBI heoretcal vs. Modfed SOBI heo. vs. Mod. SOBI Mode Shapes (Real Part Only) Analyss n a Lmted Frequency Band able 3 hghlghts the real advantage of Modfed SOBI whch s not apparent when t s appled to complete frequency range lke Orgnal SOBI. In ths case Modfed SOBI s appled to the data n three dfferent frequency ranges of Hz, Hz and Hz. MAC values (hghlghted n able 3) for the two closely spaced modes at 53.3 Hz and also 59.4 Hz and 61.6 Hz mode show sgnfcant mprovements n comparson to results obtaned usng Modfed SOBI n complete range. MAC number for these four modes mproves from range to around 1 (around 0.95 for the closely spaced modes). In fact, the results are even superor to those obtaned usng Orgnal SOBI n whch case MAC s 0.91 and 0.88 (see able 2). A typcal plot of power spectra of SDOF modal coordnates (for a chosen frequency range of Hz) s shown n Fgure 3. here are only two proper estmates of modal coordnates n the selected frequency range (as ndcated n Fgure 3) and these estmates correspond to the modes at Hz and Hz. Rest of the 13 estmates can be neglected.

7 Fgure 3: Power Spectra of SDOF Modal Coordnates (Modfed SOBI Hz) able 3: Frequency Band based Analyss usng Modfed SOBI heoretcal Modes Modfed SOBI (Appled n Freq. Band) Frequency (Hz) Dampng (%) Frequency (Hz) Dampng (%) MAC Analyss conducted on 15 DOF analytcal system verfes that, when appled n frequency bands, Modfed SOBI s as effectve (and n certan cases better) than orgnal SOBI algorthm and compares well wth the theoretcal modal parameters.

8 hs study s now followed by analyss on an epermental structure to assess the utlty of Modfed SOBI further, especally n terms of assessng ts performance n lmted response sensors scenaro (Number of sensors measurng output response s less than number of modes of nterest). 3.2 Lghtly Damped Crcular Plate A lghtly damped alumnum crcular plate (Fgure 4) s consdered for epermental valdaton of Modfed SOBI method. 30 responses are taken on the plate n a confguraton shown n Fgure 5. Plate s randomly ected by tappng t wth fngers all over ts surface. Data s acqured for a perod of 5 mns. at a samplng rate of 1600 Hz, thus provdng 4,80,000 samples. Fgure 4: Epermental Set Up for Lghtly Damped Crcular Plate Fgure 5: Lghtly Damped Crcular Plate (Sensor Locatons) Snce there are 30 responses observed over the plate, due to algorthmc lmtatons one can at most dentfy 30 modes n the frequency range of nterest (0-700 Hz) usng orgnal SOBI. For comparson purposes, an EMA test s also performed by ectng the plate by means of random ectaton at three locatons usng

9 electrodynamc shakers. Usng EMA algorthms, such as Polyreference me Doman (PD) [13], a total of 21 modes are dentfed. hs s n agreement wth the correspondng Comple Mode Indcator Functon (CMIF) plot [12] (Fgure 6) whch also ndcates the presence of 21 modes. hese modes are lsted n able 4 and are consdered the reference aganst whch performance of both SOBI algorthms (orgnal and modfed forms) wll be evaluated. Fgure 6: CMIF Plot for EMA Case Fgure 7: Cross MAC Plot (EMA modes vs. Modfed SOBI modes)

10 able 4 lsts the modal parameters obtaned usng the three approaches; EMA, orgnal SOBI and modfed SOBI. It should be noted that n case of modfed SOBI, modal parameters are obtaned by performng analyss n varous frequency bands where as orgnal SOBI estmates the modes by analyzng complete frequency range (0-800 Hz). Further, modal frequency and dampng, n case of SOBI algorthms, s obtaned by applyng SDOF frequency doman methods to power spectra of obtaned sources (as done n case of 15 DOF system). Advantage of modfed SOBI s apparent n ths analyss as t s able to dentfy all the epected modes n comparson to orgnal SOBI whch s able to dentfy only 19 modes, though overall estmates for these modes are n good agreement wth the values obtaned usng EMA. Modfed SOBI, on the other hand, provdes satsfactory estmates for all the modes ecept for the dscrepancy n frequency estmates whch can be eplaned n terms of dfferent frequency resoluton used n the two approaches. MAC plot between modes obtaned through EMA and those obtaned usng modfed SOBI, shown n Fgure 7, ndcates that mode shapes correspondng to the two closely spaced around 133 Hz are not matchng well. However, f real part of mode shapes obtaned from modfed SOBI s consdered and compared wth EMA, the MAC mproves sgnfcantly. hese MAC values are lsted n able 4. hs behavour s smlar to that observed whle analyzng closely spaced modes n 15 DOF system. Some hgher modes (above 650 Hz) show smlarty to some of the other lower modes. hs s, perhaps, due to lmted spatal resoluton due to whch these modes appear to have smlar mode shape as the lower modes. able 4: Comparson of Modal Parameter Estmates for the Crcular Plate EMA Modes Orgnal SOBI Modes Modfed SOBI Modes Freq (Hz) Damp (%) Freq (Hz) Damp (%) MAC Freq (Hz) Damp (%) MAC (Real Part)

11 Lmted Sensor Based Analyss One of the man contrbutons of ths paper s to demonstrate mproved ablty of modfed SOBI algorthm to deal wth stuatons where number of modes of nterest eceeds the sensors measurng the response. hs analyss brngs to fore ths advantage of modfed verson of SOBI. As mentoned before, one of the lmtatons of SOBI s that the number of modes that can be dentfed s at most equal to the number of sensors measurng the response. Followng analyss shows that ths lmtaton can be overcome by applyng the proposed modfed verson of SOBI n a number of frequency bands, nstead of complete frequency range as s the case wth orgnal SOBI. For ths case, eghteen channels (out of thrty) are selected; other channels are not consdered (See Fgure 8). Goal s to dentfy all twenty one modes, gven the response nformaton correspondng to these eghteen channels only. Fgure 8: Selected Channels

12 Fgure 9: Power Spectra of SDOF Modal Coordnates (Orgnal SOBI Lmted Channel Study) When orgnal SOBI s appled on ths lmted dataset, t s known beforehand that at most eghteen modes could be dentfed due to algorthmc lmtatons. However, power spectra of estmated modal coordnates (Fgure 9); ndcate that only nne modes are properly dentfed. hs underlnes the drawback on part of orgnal SOBI and severely affects ts utlzaton for OMA purposes. Modfed SOBI takes care of ths lmtaton effectvely snce t can be appled n lmted frequency ranges. Before dscussng the results obtaned usng modfed SOBI algorthm, effect of reducng the number of sensors from orgnal thrty to eghteen on the estmaton of modes s dscussed. One of the potental dangers of reducng the number of sensors s that some of the modes mght not be observable due to lmted spatal resoluton. hs s also ndcated by the auto MAC plot for the EMA modes shapes defned by the selected ponts, as shown n Fgure 10. Indeed, some of the modes appear to be dentcal to some of the other modes, for e.g. closely spaced modes around 133 Hz are very dentcal, whch s also the case for closely spaced modes around 571 Hz. On nspectng the mode shapes for these two par of modes, the reasons for ths observaton becomes even clearer. It turns out that these modes are torsonal modes and each closely spaced mode dffers from the other only wth respect to relatve moton between the ponts, torsonal moton beng shfted by 45 deg. n the two cases. Removng mmedately adjacent ponts to the ones selected has resulted n affectng the observablty of these modes and thus t s epected that these modes mght not be estmated. It s mportant to understand that, n ths analyss, EMA mode shapes for comparson purposes are obtaned by truncatng the orgnal EMA mode shapes (mode shapes obtaned durng analyss n prevous secton where complete data from all 30 sensors s used). It s observed that even EMA algorthms are not able to estmate these modes f data correspondng to the selected 18 channels s used. CMIF plot based on power spectra of the 18 selected channels also supports ths (Fgure 11). Some of the other modes also show smlarty to other modes due to observablty ssues (Modes around 57 Hz and 233 Hz, 355 Hz and 670 Hz). he observablty can be mproved by alterng the selecton of channels on the physcal structure, and results for a second set of 18 response ponts on the structure (dscussed n Append A) ndcate mproved estmaton (all the modes are dentfable) and better resoluton between some of the closely lyng modes.

13 Fgure 10: Auto MAC Plot (EMA Mode Shapes defned by Selected Channels) Fgure 11: CMIF Plot for Lmted Channel Confguraton

14 here are stll, however, nneteen vald modes that need to be dentfed by means of nformaton avalable from the eghteen channels. Estmates of these modes usng modfed SOBI algorthm along wth correspondng MAC values (compared wth EMA mode shapes defned by selected ponts) are shown n able 5. All the estmates compare well wth the EMA estmates, underlyng the mproved performance of SOBI algorthm wth proposed modfcatons. Cross MAC plot between EMA and modfed SOBI estmates, shown n Fgure 12, s very smlar to the auto MAC plot for EMA estmates (Fgure 10),.e. certan EMA modes that looked smlar to each other due to lmted spatal resoluton, look smlar to correspondng modes obtaned usng modfed SOBI as well (for e.g. modes around 232 Hz and 57 Hz, and 355 Hz and 670 Hz). able 5: Modal Parameter Comparson for Lmted Sensor Case EMA Modes Modfed SOBI Modes Freq (Hz) Damp (%) Freq (Hz) Damp (%) MAC

15 Fgure 12: Cross MAC (EMA vs. Modfed SOBI) Lmted Channels Confguraton 6. Conclusons It has been shown, by means of work presented n ths paper, how modfyng the orgnal Second Order Blnd Identfcaton algorthm can mprove ts effectveness and utlty for Operatonal Modal Analyss purposes. hese modfcatons take nto consderaton more elaborate sgnal processng technques, such as averagng, wndowng etc., to reduce the effect of nose that s generally present whle acqurng response data on real lfe structures, there by mprovng performance of SOBI. One of the man advantages of the suggested modfcaton s that t enables SOBI to be appled even to stuatons where number of sensors measurng the response s lesser than the number of modes to be estmated. It s, however, mportant to note that ths does not avod observablty related ssues that mght stll be nherently present due to lmted spatal resoluton. In a more practcal scenaro (epermental analyss of the crcular plate), t s shown that the modfed form of SOBI outperforms ts orgnal verson. hs s attrbuted to ncorporaton of better nose reducton sgnal processng capabltes n the algorthm and etendng ts applcablty to lmted frequency ranges. Results presented n ths paper are encouragng and form good foundaton for future research n ths area. One of the obvous research drecton s to assess the performance of modfed SOBI on real lfe structures that are typcal OMA applcatons, lke buldngs, brdges etc. SOBI utlzes the concept of modal epanson theorem, accordng to whch response of a system can be decomposed nto several sngle degree of freedom systems (defned by resonant frequences of the system) by means of modal vectors of the system whch act as modal flters. ypcally, modal epanson theorem s vald for systems havng dstnct modes and neglgble or very lght dampng. Dampng constrants also mply that epanson theorem s defned for systems wth real modal vectors. Although one should not draw conclusons only on the bass of work presented n ths paper, yet t s nterestng to note that SOBI can dentfy heavly damped modes as well as modes wth comple modal vectors (as shown n secton 3.1, 15 DOF system). However, estmaton of closely spaced modes does pose some challenges. In vew of ths dscusson, future work n ths research should concentrate on mprovng SOBI further. hs requres makng SOBI capable of handlng closely spaced modes, and assessng ts performance more rgourously for practcal structures and for systems wth heavly damped and comple modes.

16 Acknowledgements We would lke to thank Dr. Randall J. Allemang, SDRL, Unversty of Cncnnat, for hs epert gudance and feedback whch were crtcal for completon of the paper. References [1] Anton, J.; Blnd separaton of vbraton components: prncples and demonstratons, Mechancal Systems and Sgnal Processng, Vol. 19 (6), pp , November [2] Kerschen, G., Poncelet, F., Golnval, J.C. (2007), Physcal Interpretaton of Independent Component Analyss n Structural Dynamcs, Mechancal Systems and Sgnal Processng (21), pp [3] Poncelet, F., Kerschen, G., Golnval, J.C. (2006), Epermental Modal Analyss Usng Blnd Source Separaton echnques ; Proceedngs of ISMA Internatonal Conference on Nose and Vbraton Engneerng, Katholeke Unverstet Leuven, Belgum. [4] Chauhan, S., Martell, R., Allemang, R. J. and Brown, D. L. (2007), Applcaton of Independent Component Analyss and Blnd Source Separaton echnques to Operatonal Modal Analyss, Proceedngs of the 25th IMAC, Orlando (FL), USA. [5] Anton, J., Chauhan, S. (2010); Second Order Blnd Source Separaton (SO-BSS) and ts relaton to Stochastc Subspace Identfcaton (SSI) algorthm, o be presented at 28th IMAC, Jacksonvlle (FL), USA. [6] McNell, S.I., Zmmerman, D.C. (2008), A Framework for Blnd Modal Identfcaton Usng Jont Appromate Dagonalzaton, Mechancal Systems and Sgnal Processng (22), pp [7] Cchock, A., Amar, S.; Adaptve blnd sgnal and mage processng, John Wley and Sons, New York, [8] Belouchran, A., Abed-Meram, K.K., Cardoso, J.F., Moulnes, E.; Second order blnd separaton of correlated sources, Proceedngs of Internatonal Conference on Dgtal Sgnal Processng, pp , [9] Cardoso, J.F., Souloumac, A.; Jacob angles for smultaneous dagonalzaton, SIAM Journal of Matr Analyss and Applcatons, Vol. 17, Number 1, pp , January, [10] Hor, G.; A new approach to jont dagonalzaton, Proceedngs of 2 nd Internatonal Workshop on ICA and BSS, ICA 2000, pp , Helsnk, Fnland, June [11] Stoca, P., Moses, R.L.; Introducton to Spectral Analyss, Prentce-Hall, [12] Allemang, R.J.; Vbratons: Epermental Modal Analyss, Structural Dynamcs Research Laboratory, Department of Mechancal, Industral and Nuclear Engneerng, Unversty of Cncnnat, CN /664, Revson - June [13] Vold, H., Kundrat, J., Rockln,., Russell, R.; A mult-nput modal estmaton algorthm for mncomputers, SAE ransactons, Volume 91, Number 1, pp , January, [14] Shelly, S.J.; Investgaton of dscrete modal flters for structural dynamc applcatons, PhD Dssertaton, Department of Mechancal, Industral and Nuclear Engneerng, Unversty of Cncnnat, [15] Merovtch, L.; Analytcal Methods n Vbratons, Macmllan, 1967.

17 Append A hs secton detals results from a lmted-sensors based study for an altered combnaton of response ponts on the lghtly damped crcular plate. hs study s amed at evaluatng the effect of the selecton ponts on the structure, on estmaton of all the modes usng the Modfed SOBI methods. Eghteen ponts are chosen out of the avalable thrty responses, wth the selecton shown n Fgure A.1. Fgure A.1: Selected Channels Modal estmates for ths set of channels are lsted n able A.1. It s observed that ths combnaton of channels s able to resolve the torson modes around 133 Hz wth better dstncton. However, ths selecton of ponts too s unable to dstnctly observe the closely lyng modes around 571 Hz, although results are better n comparson to confguraton shown n Fgure 8, Secton 3.2. hs ndcates the mportance of selectng proper measurement locatons n order to observe and estmate all modes of nterest.

18 able A.1: Modal Parameter Comparson for Lmted Sensor Case EMA Modes Modfed SOBI modes Freq (Hz) Damp (%) Freq (Hz) Damp (%) MAC