ANALYSIS OF REAL-TIME DATA FROM INSTRUMENTED STRUCTURES

Transcription

1 3 h World Conference on Earhquake Engineering Vancouver, B.C., Canada Augus -6, 004 Paper No. 93 ANAYSIS OF REA-IME DAA FROM INSRUMENED SRUCURES Erdal SAFAK SUMMARY Analyses of real-ie daa fro insruened srucures require ools and echniques ha are differen han hose used for riggered daa. Real-ie onioring involves analysis of vibraion daa ha are low in apliudes and high in noise. Moreover, real-ie analysis requires ehods ha can be applied in real ie and recursively. his paper presens soe of he basics in processing and analyzing real-ie vibraion daa. he opics discussed include uilizaion of running ie windows, racking ean and ean-square values, derending, filering, and syse idenificaion. he ehods presened are based on he concep of opial adapive filering and recursive leas-squares esiaion. INRODUCION In ajoriy of srucures insruened for vibraion onioring, he recorders are se o rigger only during large-apliude oions, such as hose generaed by oderae o large earhquakes. Recenly, a few srucures have been insruened o provide coninuous daa in real ie, recording no only large-apliude oions bu also sall-apliude oions generaed by abien loads. he ain objecive in coninuous onioring is o rack any changes in srucural characerisics in order o deec daage. his ype of onioring is coonly known as Srucural Healh Monioring. Fourier-based specral analysis ehods have been he priary ool o analyze daa fro insruened srucures. Mehods ha are used o analyze large-apliude vibraion daa are no always appropriae o analyze abien vibraion daa. In ers of daa analysis, large-apliude records (e.g., earhquake records) have uch beer signal-o-noise raios han abien records. However, hey are ypically ransien, exhibi ie varying (i.e., nonsaionary) eporal and frequency characerisics, and can have nonlinear properies. In coparison, abien records have he advanages of being infiniely long duraion, saionary, and in os cases linear. However, hey conain high level of noise. he long duraion and saionariy of abien daa allow us o uilize saisical signal processing ools, which can copensae for he adverse effecs of low signal-o-noise raios. his paper presens soe of he basics in real-ie daa analysis, including he uilizaion of running ie windows, racking ean and ean-square values, derending, filering, and syse idenificaion. he ephasis is on ehods ha can be applied recursively, which are ore appropriae for real-ie analysis. Research Srucural Engineer, U.S. Geological Survey, Pasadena, California, USA. E-ail: safak@usgs.gov

2 RACKING IME VARIAIONS IN SIGNA PROPERIES Coninuous onioring requires coninuous and auoaed daa processing and analysis. he ehods ha are used for analysis should be able o adap and accoun for any changes in signal characerisics. he siples and os sraighforward approach o analyze coninuous daa is he block-daa approach. In his ehod, he records are handled in blocks of specified lengh. Each block is processed and analyzed as soon as i is full, and while he daa for he nex block are being acquired. More efficien ways o analyze coninuous daa can be developed by uilizing running ie windows. Running windows are in essence weighing funcions ha ephasize recen daa, while gradually deephasizing pas daa. he windows ensure ha any propery calculaed fro daa conains easureens ha are relevan o he curren sae of he srucure. he wo widely used weighing funcions are exponenially decaying windows and sliding recangular windows. he exponenially decaying window is defined as λ i wi (, ) wih i 0,,,, and wi (, ) λ = = = () λ where λ is known as he forgeing facor wih 0.0 < λ <.0. he window applies exponenially decaying weighs o pas daa poins. he eory ie consan, N 0, of he window is defined as he nuber of sapling poins over which he characerisics of he srucure can be assued o reain consan. I can be approxiaed as: N 0 λ () his equaion provides a siple crierion for he selecion of λ.. ypically, λ=0.900 ~ Sliding recangular windows consider only a liied nuber of pas daa poins wih equal weighs. I is defined as: wi (, ) = for i wih 0,,,, and wi (, ) = (3) where is he window lengh and he eory ie consan. RACKING OF MEAN VAUE he ean value in a vibraion record represens he saic coponen of he srucure s response. Norally, he ean value of vibraion records should be zero. In real-ie onioring, he ean value ay flucuae around zero due o various facors, such as iperfecions in sensors, environenal facors (e.g., wind, rain, ec.), naural changes in srucure (e.g., srucural odificaions, change in loads, aging, ec.), or peranen daage afer an exree even (e.g., inelasic deforaions during an earhquake). herefore, i is iporan ha he changes in ean value are racked accuraely. We can consider he calculaion of ean value of a signal as a proble of a weighed leassquares fi of a consan o he record by iniizing he following error funcion:

3 [ ] (4) ε() = wi (,) xi () () where x(i) is he signal, w(,i) is he weighing funcion, and () is he ean value. For he weighing funcions described above, he iniizaion gives he following expressions for he ean value a ie : λ i For exponenially decaying window: ( ) = λ x() i λ For sliding regangular window: ( ) = xi ( ) + (5) Above expressions can be pu ino a recursive for such ha he ean value a ie is calculaed fro he ean value a ie -. he recursive fors are (Safak, 004) For exponenially decaying window: ( ) = λ ( ) + ( λ) x( i) For sliding regangular window: ( ) = ( ) + y() y( ) [ ] (6) Copuaionally, he recursive fors are ore appropriae for real-ie daa. RACKING OF MEAN-SQUARE VAUE Mean-square (MS) value is anoher iporan paraeer ha needs o be racked in real-ie onioring. MS value provides inforaion on average vibraion apliudes and is one of he key paraeers o deec changes in srucural response and exciaion. Siilar o he ean value, he MS value, s(), a ie can be calculaed by iniizing he following equaion: ε ( ) = wi (, ) x ( i) s ( ) (7) which resuls in he following recursive expressions for s() (Safak, 004): For exponenially decaying window: s( ) = λ s ( ) + ( λ) x( i) For sliding regangular window: s( ) = s( ) + y ( ) y ( ) (8) DERENDING Derending, also known as baseline correcion, reoves a linear rend fro records. Again, for real-ie daa i should be applied by uilizing running ie windows, so ha he rend ha is being reoved is he curren one. he equaions for derending are developed by deerining he bes sraigh-line fi, i.e., y()=a x() + b, o daa and incorporaing he ie windows of Eqs. and 3. he coefficiens a and b of he sraigh line are deerined by iniizing he following error funcion wih respec o a and b:

4 ε() = wi (,) xi () a xi () b [ ] (9) he explici expressions for a and b can be found in Safak (004). Noe ha because of he er b in he equaion, derending also reoves he ean value fro he daa. FIERING Band-pass filering Filering is required o reduce noise and eliinae unwaned frequency coponens in records. he sandard approach o noise reducion has been o use band-pass filers. Band-pass filers reove he frequency coponens ha are believed o be doinaed by noise (ypically he very low and very high ends of he frequency band). Alhough he noise is sill presen in he reaining frequency band, i is assued ha he signal-o-noise raio is high enough so ha he noise can be ignored. Band-pass filering, as well as low- or high-pass filering, can be done by using recursive iedoain filers of he following for (e.g., reer, 976): n k l (0) k= l= 0 y() = a y ( k) + b x ( l) where y() is he filered signal and x() is he original signal. he filer paraeers a k and b l and he filer orders and n are deerined by he corner frequencies, and he rae of decay of filer apliudes near he corner frequencies. A unique se of paraeers can be calculaed for any specified filer. For exaple, for a second order band-pass Buerworh filer wih corner frequencies of 0. Hz and 0 Hz and a sapling inerval of 0.0 second, he filer paraeers, calculaed by using MAAB (MahWorks, 003), are: a= , a= , b0=0.05, b= 0, b = () Opial filering Mos of real-ie vibraion daa consis of abien vibraions, which are characerized by low apliudes and low signal-o-noise raios. Analyses of such noisy signals require he applicaions of sophisicaed filers so ha he aoun of inforaion exraced fro he daa is axiized. A general class of such filers is known as opial filers. he key difference beween band-pass filers and opial filers is ha opial filers reduce he noise over he enire frequency band. We will presen he general concep of opial filering based on he principle of Recursive eas Squares (RS) approxiaion. Siilar filers can be developed by using he eas Mean Square (MS) approxiaion, as well as by uilizing sae-space forulaions, leading o Kalan Filers (KF) and Exended Kalan Filers (EKF). Due o space liiaions, hese alernaive fors will no be discussed; hey can be found in a large nuber of references ha are available in he lieraure (e.g., Widrow and Searns, 985; Braer and Siffling, 989). he basic philosophy in opial filering is ha he recorded response is coposed of a signal (i.e., he acual response) plus noise. Since he signal represens he response of a dynaic syse o soe exciaion, i is reasonable o expec ha response values a discree ies will follow a paern (e.g., a daped sinusoid), and herefore will be correlaed wih each oher. For linear syses, his paern can be represened by wriing he response a ie as a linear cobinaion

5 of pas responses; ha is x () = ak x ( k) + n () () k= he firs er on he righ-hand side represens he correlaed porion of he record, i.e., he signal, and he second er he uncorrelaed porion of he signal, i.e., he noise. Hence, he proble becoes deerining he opial values of a k such ha he prediced value of x() is as close o is recorded value as possible. Assuing ha n() is a zero-ean rando process and all he pas values of x() are known, he bes esiae, E[x()], of x() can be wrien as Ex [ ( )] = ak x ( k) (3) k= he error in he esiae, ε(), is defined as he difference beween he recorded value and he esiaed value of x(); and he oal error, V, is he su of squared errors over he record lengh, N: N k ε (4) k= = ε() = x() + a x( k) and V = () where N is he nuber of sapling poins. By aking V / ak = 0 for k =,, we obain he following arix equaion for a k : xx K xx xx a r (0) r ( ) r () M = M O M M a rxx( ) rxx(0) r xx( ) (5) where rxx ( j) = xkxk ( ) ( j) (6) k= is he auocorrelaion funcion of x(). By defining he following vecors and he arix, we can wrie i in a ore copac for as: a rxx () rxx (0) K rxx ( ) θ = R f where θ =, f, R M = M = M O M a rxx( ) r xx( ) rxx(0) (7)

6 A key paraeer ha needs o be seleced o apply hese equaions is, he filer order. A filer wih oo sall does no accuraely represen he signal, whereas a filer wih oo large ay ry o represen noise as well as he signal. here several crieria available in he lieraure o selec (see, Sodersro, 987). A sipler and ore sraighforward selecion can be ade by ploing he variaion of V = ε () wih (Safak, 004). his su ypically shows a fas drop wih increasing, and hen level off. he value where he su sars o level off can be aken as he opial filer order. Eq. 4 is based on he iniizaion of forward squared predicion errors because we used he pas values of x() o predic is curren value. Siilar equaions can be developed by iniizing he backward predicion errors. We can also iniize he su of forward and backward predicion errors, which resuls in he so-called Burg s ehod (Burg, 968). he relaionships beween he forward and backward iniizaion forulaions lead o very fas recursive algorihs for he soluion of filer coefficiens, such as evinson-durbin algorih (evinson, 947; Durbin, 959) and laice filers (Griffihs, 977; Makhoul, 977). An iporan assupion ade in he derivaion of Eq. 4 is ha x() is a saionary signal. In oher words, he eporal and frequency characerisics of x() does no change significanly wih ie, and herefore he auocorrelaion funcion r xx (Eq. 5) is he funcion of he ie lag only beween he wo coponens. he assupion of saionariy is appropriae for vibraions under abien forces and wind loads, bu no for vibraions under ransien loads such as earhquakes or blas loads. Adapive filering Adapive filering is required when he eporal and frequency characerisics of he signal change wih ie, i.e., he assupion of saionariy is no longer valid. Equaions for adapive filering can be developed by siple incorporaing a weighing funcion (e.g., Eq. or 3) in he error funcion for he leas-squares esiaion. (Eq. 4). ha leads o he following error funcion: k (8) k= V() = w(,) i x() i + a x( i k) Noe ha V is now a funcion of ie. By denoing φ () = [ x( ), x( ),, x( )] (9) and using he exponenially decaying for for w(,i), he paraeer vecor θ = [ a, a,, a ] can be calculaed recursively by he following se of equaions (jung, 999):

7 θ() = θ( ) + () x() φ () θ( ) P ( ) φ ( ) () = λ + φ () P ( ) φ( P ( ) φ( ) φ ( P ) ( ) P () = P ( ) λ λ + φ () P ( ) φ( (0) As indicaed, he filer paraeers now are ie varying. SYSEM IDENIFICAION Syse idenificaion refers o he exracion of srucural characerisics fro recorded signals. he basic approach o syse idenificaion is very siilar o ha used for opial filering. We search for an opial filer ha convers he recorded exciaion signal ino he recorded response signals. he filer idenified represens he dynaic characerisics of he srucure. As will be shown laer, here is a one-o-one correspondence beween he filer paraeers and he odal frequencies and daping raios of he srucure. If we do no have access o he exciaion signal (e.g., ground acceleraions in he case of an earhquake-induced vibraions), we assue ha he exciaion is a zero-ean wind-band rando process. his case is known as oupu-only idenificaion. We presen he basics of syse idenificaion for boh oupu-only and inpuoupu cases below. Oupu-only syse idenificaion: he forulaion for oupu-only syse idenificaion is idenical o ha for opial filering. We search for a filer ha, when applied o daa, will give a residual ha is as close o a zero-ean whie-noise sequence as possible. Maheaically, his leads o he sae error funcion and he iniizaion proble defined by Eq. 4 (for saionary signals) or by Eq. 8 (for nonsaionary signals). he filer paraeers are calculaed fro he resuling equaions, Eq. 5 or Eq. 0. In order o calculae he dynaic properies of he srucure fro he filer coefficiens, we firs calculae he roos of he following equaion: + az + + a z = 0 () he roos of he equaion are called he poles, p k, of he ransfer funcion for he srucure. For srucures wih posiive daping, all he roos are in coplex conjugae pairs, resuling in / disinc odal frequencies, f k, and daping raios, ξ k. he f k, and ξ k are calculaed fro he following equaions (Safak, 99): ( p ) + ( p ) / ( p ) ln / od( ) ln / od( ) ξk =, fk = in Hz, k =,, / () arg( p ) ln / od( ) πξk k where od( ) and arg( ) denoe he odulus and arguen of he coplex poles, and is he sapling inerval in seconds. Inpu-oupu syse idenificaion: If he inpu (i.e., he exciaion) signal is also recorded, we deerine filer paraeers by aking

8 he exciaion signal as he inpu o he filer and aching he oupu-signal in he leas-squares sense. he filer equaion for he inpu-oupu case is: n k l (3) k= l= 0 x () = a x ( k) + b u ( l) + n () where u() denoes he inpu signal. he filer paraeers, a k and b l can be deerined recursively fro Eq. 0, by siply changing he definiions of he vecors θ() and φ() as follows: θ ( ) = [ a, a,, a, b0, b,, bn] φ( ) = [ x ( ), x ( ),, x ( ), u ( ), u ( ),, u ( n)] he naural frequencies and daping raios are calculaed fro a k coefficiens as discussed above; b l coefficiens are relaed o he odal paricipaion facors (for deails, see Safak, 988; 99). Since θ() is ie-varying, his algorih can rack any changes in he odal frequencies and daping raios. SUMMARY AND CONCUSIONS Analyses of real-ie daa fro insruened srucures require fas and efficien algorihs. Coninuous records differ fro riggered records in ha hey are ypically very low in apliudes and high in noise. Mehods used o analyze real-ie daa should be able o exrac he signal buried in noise, and iniize he effecs of noise over he enire frequency band. he ehods should also be in a fora ha can be applied recursively in real ie, and be able o adap o rack any ie variaions in signal properies. Opial adapive filers offer one of he bes ools o analyze real-ie daa. hese filers exrac he signal fro record by searching hidden correlaion paerns in he daa. Such filers are used no only for noise reducion over he enire frequency band, bu also for syse idenificaion, because he forulaion for boh leads o he iniizaion of sae error funcions. Opial adapive filers can be expressed in a large nuber of differen fors, and known under differen naes depending on he forulaion and error iniizaion algorihs used in he developen, such as RS filers, MS filers, and Kalan Filers. REFERENCES. Braer, K. and Siffling, G. Kalan-Bucy Filers, Arech House, Norwood, MA, Burg, J.P. A new analysis echnique for ie-series daa, NAO Advanced Sudy Insiue on Signal Processing, Enschede, he Neherlands, Durbin, J. Efficien esiaors of paraeers in oving average odels, Bioerica, 46:306-36, Griffihs,.J. A coninuously adapive filer ipleened as a laice srucure, Proc. 977 IEEE In. Conf. on Acousics, Speech and Signal Processing, , evinson, N. he Wiener rs error crierion filer design and predicion, J. Mah. (4)

9 Phys., 5:6-78, jung,. Syse Idenificaion: heory for he User, Prenice Hall PR, Upper Saddle River, NJ, Makhoul, J. Sable and efficien laice ehods for linear predicion, IEEE rans. Acousics, Speech and Signal Processing, ASSP-5:43-48, he MahWorks, Inc. MAAB he anguage of echnical Copuing, Version 6, Naick, MA, Safak, E. (988). Analysis of recordings in srucural engineering: Adapive filering, predicion, and conrol, Open-File Repor , U. S. Geological Survey, Menlo Park, California. 0. Safak, E. (99). Idenificaion of linear srucures using discree-ie filers, Journal of Srucural Engineering, ASCE, Vol.7, No.0, Ocober 99, pp Safak, E. (004). Analysis of real-ie vibraion daa (in preparaion).. Sodersro,. Model srucure deerinaion, in Encyclopedia of Syses and Conrol (M. Singh, ed.), Pergaon Press, Elsford, NY, reer, S.A. Inroducion o Discree-ie Signal Processing, John Wiley and Sons, New York, NY, Widrow, B. and Searns, S.D. Adapive Signal Processing, Prenice-Hall, Inc., Englewood Cliffs, NJ, 985.