Modal identification of structures from roving input data by means of maximum likelihood estimation of the state space model

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1 Modal idenificaion of srucures from roving inpu daa by means of maximum likelihood esimaion of he sae space model J. Cara, J. Juan, E. Alarcón Absrac The usual way o perform a forced vibraion es is o fix acceleromeers o he srucure, hen apply a known inpu and record he response of he srucure. An ineresing varian consiss of subsequenly repeaing he es bu wih he inpu force applied o differen poins, roving he inpu from es o es bu no he sensors. In a second phase, individual frequency response funcions are combined in a single frequency response funcion which akes ino accoun all he informaion. The advanage of his procedure is double: firs, he mode shapes can be esimaed a all poins where inpus are applied (using he dynamic reciprociy heorem); and second, he measured forces allow o scale he mode shapes o uni modal mass. This problem is solved in echnical lieraure in he frequency domain. In his work we propose a sae space model ha can be used o solve he problem in he ime domain. We also propose a maximum likelihood algorihm o esimae such a model. 1 Inroducion A roving sensor modal es consiss on esimaing he modal parameers using he daa recorded in differen ess changing he posiion of he sensors from one es o he following: some sensors say a he same posiion from es o es, and oher sensors change he posiion unil all he esed poins are covered. The known inpu, if applied, mus be applied a he same poin in all he es seups. This echnique has reached grea populariy in he civil engineering field because of he size of he civil srucures and because he ess can be performed in operaional condiions. The modal parameers can be exraced in he frequency domain (for example, using he Frequency Domain Decomposiion mehod) and in he ime domain (for example, wih he sae space model). Anoher way o run he es is o place sensors a fixed locaions and change he loading poin from one es o he nex covering all he desired poins (roving exciaion modal es). This echnique is quie popular in experimenal modal analysis wih impac exciaions (hammer) for several reasons: srucures are usually small and roving he sensors can affec o he modal parameers; i is easier o change he posiion of he hammer impac han roving he sensors; i is a less expensive approach seen from a hardware poin of view. The modal parameers are exraced in he bibliography using he dynamic reciprociy heorem (or Bei- Rayleigh heorem) in he frequency domain [1]. However, here are no examples in he ime domain. The purpose of his work is o propose a mehod o esimae he modal parameers in he ime domain: firs, we presen a sae space model o deal wih daa recorded in a roving exciaion es; and second, we propose a maximum likelihood algorihm o esimae ha model (he Expecaion-Maximizaion algorihm). The performance of he proposed mehod is analyzed using a simulaed srucure and also using vibraion daa recorded in a real srucure.

2 2 Sae space model for roving inpu daa and is esimaion using maximum likelihood 2.1 Sae space model for roving inpu daa Le be a linear, ime invarian mechanical/srucural sysem. Le us consider n r sample records wih measuremens each, Y = {y 1, y 2,..., y }, r = 1, 2,..., n r (1) U = {u 1, u 2,..., u }, r = 1, 2,..., n r (2) where y R no is he measured oupu vecor for record r, and u R n i is he measured inpu vecor for record r. Assume he inpus vary from record o record (poin of applicaion, number of inpus,... ) bu he oupu sensors remain fixed. The sae space model we propose o use for hese daa is x +1 = Ax + B u + w, w (, Q) (3) y = Cx + D u + v, v (, R) (4) where denoes he ime insan, of a oal number, measured wih consan sampling ime ; x R ns is he sae vecor for record r; n o, n i and n s are he number of oupus, inpus and he size of he sae vecor, respecively; A R ns ns is he ransiion sae marix describing he dynamics of he sysem; B R ns n ir is he inpu marix (n ir is he number of inpus a record r); C R no ns is he oupu marix, which is describing how he inernal sae is ransferred o he he oupu measuremens y ; D R no n ir is he direc ransmission marix. The noise vecors comprise unmeasurable signals: w R ns is he process noise due o disurbances and modelling discrepancies, while v R no is he measuremen noise due o sensor inaccuracy. Boh are assumed o be zero-mean, whie noise sequences wih covariance marices Q and R, respecively. I is imporan o noe ha marix A is invarian because he sysem is ime invarian (i is he same from record o record), and marix C is also invarian because he sensors locaion do no change. On he oher hand, B and D are record-depending because he sysem inpu are differen from record o record. A more pracical way of wriing his model is x +1 = Ax + BL u + w, w (, Q) (5) y = Cx + DL u + v, v (, R) (6) where L R ns n ir is a selecion marix formed by ones and zeros verifying B = L B and D = L D. For example, consider a sysem wih n s = 4, n i = 3 and n o = 1. We can wrie b 11 b 12 b 13 Bu = b 21 b 22 b 23 b 31 b 32 b 33 b 41 b 42 b 43 u 1, u 2, u 3, If we measure now only he inpu u 1, we can wrie b 11, Du = [ ] d 11 d 12 d 13 B (1) u 1, = b 21 b 31 u 1,, D (1) u 1, = d 11 u 1,. b 41 u 1, u 2, u 3,.

3 Using he selecion marices: b 11 b 11 b 12 b 13 B (1) u 1, = b 21 b 31 u 1, = b 21 b 22 b 23 1 b 31 b 32 b 33 u 1, = BL (1) u 1,. b 41 b 41 b 42 b 43 D (1) u 1, = d 11 u 1, = [ ] 1 d 11 d 12 d 13 u 1, = DL (1) u 1,. Therefore, L (1) = [ 1 ] T. The locaion marices are known for each seup r. The unknown parameers of model (5)-(6) are θ = {A, B, C, D, Q, R, x, P }, r = 1, 2,..., n r (7) where x and P are he mean and variance of he iniial sae x respecively (which is assumed o be normal disribued). 2.2 Maximum likelihood esimaion: EM algorim We assume all he oupus Y = {Y (1), Y (2),..., Y (nr) }, inpus U = {U (1), U (2),..., U (nr) } and saes X +1 = {X (1) +1, X(2) +1,..., X(nr) +1 } are known (noe ha Y = {y 1, y 2,..., y }, U = {u 1, u 2,..., u } and X +1 = {x 1, x 2,..., X +1 } are he oupus, inpus and saes for one individual record r, respecively). The densiy funcion for record r is given by (see [2]) f θ (X +1, Y U ) = f where under Gaussian assumpion: f (x ) = 1 (2π) ns/2 P (x ) =1 f 1 (X +1 X, U ) =1 f 2 (Y X ( exp 1 ( ) ) 1/2 2 (x x )T P 1 (x x ),, U ), (8) f 1 (X +1 X =, U ) = 1 (2π) ns/2 exp Q 1/2 ( 1 2 (x +1 Ax ) BL u ) T Q 1 (x Ax 1 BL u ), f 2 (Y X =, U ) = 1 (2π) no/2 exp R 1/2 ( 1 2 (y Cx ) DL u ) T R 1 (y Cx DL u ). Thus, if we consider n r independen regisers, he join densiy funcion f θ (X +1, Y U ) will be he produc of he individual ones: n r f θ (X +1, Y U ) = f θ (X +1, Y U ). (9)

4 The complee daa likelihood is defined by L(θ) = f θ (X +1, Y U ). In pracice we generally work wih he log-likelihood, so informaion is combined by addiion and i can be wrien as a sum of he log-likelihood of each individual record: n r log L(θ) = log f θ (X +1, Y nr ) = l (θ), (1) U where l (θ) is he log-likelihood of record r, ha is, l (θ) = log f θ as he sum of hree uncoupled funcions: where, ignoring consans: l (X +1, Y U ). I can be wrien l (θ) = 1 2 [l 1 ( x, P ) + l 2 (A, B, Q) + l 3 (C, D, Q)], (11) l 1 ( x 2 (A, B, Q) = log Q + l 3 (C, D, R) = log R + ( ), P ) = log P + (x x )T P 1 (x (x +1 Ax =1 (y =1 Cx x ), (12) BL u ) T Q 1 (x Ax 1 BL u ), (13) DL u ) T R 1 (y Cx DL u ). (14) The objecive now is o maximize he log-likelihood given by equaion (1). In his work we propose o use a maximizaion procedure based on he Expecaion Maximizaion algorihm (see [2]). The EM algorihm is simple o apply since a each ieraion he opimal soluion for he unknown parameers can be obained from explici formulas. I consiss on wo seps: he E-sep and he M-sep. E-sep (expecaion sep): Given he measured oupus in all he records Y, he measured inpus in all he records U and a value for he parameers θ, he log-likelihood (1) canno be compued because he saes X +1 are unknown (in fac, he saes are unobserved quaniies). The mehod proposes o replace hem wih heir expeced values: n r E[log L(θ) Y, U, θ ] = E[l (θ) Y, U, θ ] = n r + E[l 2 (A, B, Q) Y, U M-sep (maximizaion sep):, θ n r ] + E[l 3 n r E[l 1 ( x (C, D, R) Y, U, P, θ ) Y, U, θ ] ]. (15) Maximizing E[log L(θ) Y, U, θ ] wih respec o he parameers θ, consiues he M-sep. This is he srong poin of he EM algorihm because he maximum values, obained equaing o zero he corresponding derivaives, are obained from explici formulas: θ E[log L(θ) Y, U, θ ] = θ = ˆθ (16) A new E-sep can be now performed considering θ = ˆθ wha gives, applying he M-sep, o a new value for he parameers, ˆθ. This leads o an ieraive procedure in which he wo seps, expecaion and maximizaion, are repeaed unil he likelihood is maximized. This procedure is called he EM algorihm.

5 3 Compuaion of modal parameers aural frequencies, modal damping raios, mode shapes and modal masses can be rerieved from marices A, B and C using he following equaions (see [3] for a general overview): The eigenvalues of A come in complex conjugae pairs and each pair represens one physical vibraion mode. Assuming proporional damping, he jh eigenvalue of A has he form ) ) λ j = exp (( ζ j ω j ± iω j 1 ζj 2 (17) where ω j are he naural frequencies, ζ j are damping raios, and is he ime sep. aural frequencies ω j and he damping raios ζ j are hen given by ω j = ln (λ j) ζ j = Real [ln (λ j)] ω j (18) (19) The jh mode shape φ j R no evaluaed a sensor locaions can be obained using he following expression: φ y,j = Cψ j (2) where ψ j is he complex eigenvecor of A corresponding o he eigenvalue λ j. Finally, he modal mass corresponding o φ y,j can be compued using m m,j = eλ j 1 λ j (λ j λ j )φt u,jγ j (21) where sands for he pseudo-inverse, γ j R n i 1 sands for he j-h row of marix Γ = V 1 B, wih V meaning he eigenvecors marix of A; φ u,j is he modal vecor φ y,j seleced a he DOFs where he measured inpus are applied (considered as a column vecor). oe ha Equaion (21) can be only used if he measured loads are applied o DOFs wih sensors. 4 umerical examples 4.1 Simulaed daa We have considered a simply suppored beam wih he following characerisics: lengh, L = 2. m; Young modulus, E = /m 2 ; secion momen of ineria, I = m 4 ; densiy, ρ = 785. kg/m 3 ; siffness a boh ends: k 1 = k 2 = 1EI. For he purpose of numerical simulaions, he beam was modelled by 18 Hermiian beam elemens. For he mass marix, he consisen formulaion was considered. The firs four naural frequencies of vibraion are calculaed o be 1.17 Hz, 4.69 Hz, 1.55 Hz and Hz. Viscous damping is assumed wih 2% of criical for all modes. The mode shapes are shown in Figure 1, where hey are scaled o he maximum componen equal o one. The modal mass corresponding o hese mode shapes are calculaed o be 314.6, , and We have generaed nineeen differen sample records (n r = 19) using: Sampling frequency f s = 5 Hz. Toal duraion of signals, 1 seconds ( = 5). An i.i.d. Gaussian whie noise wih variance equal o 1 was used as he measured inpu load for each seup, u. This inpu has been applied node 1 in seup 1, o node 2 in seup 3, and so on.

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9 Figure 5: Some esimaed mode shapes. Acknowledgemens This research was suppored by he Miniserio de Educación y Ciencia of Spain under he research projec Prognosis and inegraed analysis of he human-induced vibraions in srucures (BIA C2-1-R). The financial suppor is graefully acknowledged. References [1].J. Jacobsen, Modal parameer esimaion from inconsisen daa, in Proceedings of he Inernaional Conference on oise and Vibraion Engineering - ISMA 212. Leuven, Sepember 212, pp [2] F. J. Cara, J. Carpio, J. Juan, E. Alarcón, An approach o operaional modal analysis using he expecaion maximizaion algorihm, Mechanical Sysems and Signal Processing, Volume 31, Augus 212, pages [3] J. Cara, Compuing he modal mass from he sae space model in combined experimenaloperaional modal analysis, Journal of Sound and Vibraion, Volume 37, pages 94-11, 216. hp://dx.doi.org/1.116/j.jsv [4] F. J. Cara, J. Juan, E. Alarcón, Esimaion of modal parameers in srucures using muliple ime hisory records, in Proceedings of he Inernaional Conference on oise and Vibraion Engineering - ISMA 212. Leuven, Sepember 212, pp