A New Gibbs-Sampling Based Algorithm for Bayesian Model Updating of Linear Dynamic Systems with Incomplete Complex Modal Data

Transcription

1 Proceedng of the Internatonal MultConference of Engneer and Coputer Scentt Vol II, IMECS, March - 5,, Hong Kong A ew Gbb-Saplng Baed Algorth for Bayean Model Updatng of Lnear Dynac Syte wth Incoplete Coplex Modal Data Cheung Sa Hung and Sahl Banal Abtract Model updatng ung eaured yte dynac repone ha a wde range of applcaton n tructural health ontorng, control and repone predcton In th paper, we are ntereted n odel updatng of a lnear tructural dynac yte wth non-clacal dapng baed on ncoplete odal data ncludng odal frequence, dapng rato, and partal coplex ode hape of oe of the donant ode To quantfy the uncertante and plaublty of the odel paraeter, a Bayean approach adopted A new Gbbaplng baed algorth propoed that allow for an effcent update of the probablty dtrbuton of the odel paraeter The effectvene and effcency of the propoed ethod are llutrated by a nuercal exaple nvolvng a lnear tructural dynac yte wth coplex ode Index Ter Bayean odel updatng, lnear, tructural dynac yte, coplex ode, Gbb aplng T I ITRODUCTIO he need of tructural odel updatng uually otvated by the dere to prove the accuracy of predcton of the yte repone and control, and tructural health ontorng In general, the yte aued to be lnear and clacally daped, e, t equaton of oton can be tranfored nto a et of decoupled odal equaton ung the real-valued egenvector and egenvalue Mot vbraton data of tructure are obtaned under low apltude exctaton, thu the aupton that the tructure behave approxately lnearly durng uch vbraton vald However, n any real tuaton a lnear yte can be non-clacally daped Such the cae when a yte ade up of ateral wth dfferent dapng charactertc n dfferent part of the yte For exaple, a ol-tructure yte, a yte wth uppleental vcou daper, or a coupled tructure-equpent (praryecondary) yte non-clacally daped Aung a non-clacally daped yte to be clacally daped ght brng n error n the updatng proce becaue of non-orthogonalty of dapng The uual approach to update a lnear tructural dynac yte to frt dentfy t odal properte and then ue Manucrpt receved Deceber 8, Cheung Sa Hung an Atant Profeor at the School of Cvl and Envronental Engneerng, anyang Technologcal Unverty, 5 anyang Avenue, Sngapore (hcheung@ntuedug) Sahl Banal a PhD canddate at the School of Cvl and Envronental Engneerng, anyang Technologcal Unverty, 5 anyang Avenue, Sngapore (ah4@entuedug) ISB: ISS: (Prnt); ISS: (Onlne) thoe to update the odelng paraeter There are everal abent or forced vbraton baed odal dentfcaton technque avalable [-7] that provde optal etate of the odal paraeter Thee technque can be grouped nto two type: probabltc and deterntc Probabltc technque, partcularly the Bayean approach, provde etate of the optal paraeter along wth ther probablty denty functon (PDF) that can be ued to decrbe the coplete pcture of the uncertanty whle the outcoe of deterntc technque uually a unque et of paraeter Several reearcher [8-] have preented work on updatng of the Fnte eleent odel, baed on the experental odal data However, there are relatvely few paper n tructural odel updatng lterature n whch probabltc odel updatng condered [-4] Chng et al [] propoed a new Gbb apler baed ulaton approach for odel updatng of lnear dynac yte wth clacal dapng In th paper, a tochatc ulaton algorth baed on Gbb apler preented for Bayean odel updatng of lnear tructural dynac yte baed on ncoplete coplex odal data, correpondng to odal frequence, dapng rato, and partal coplex ode hape of oe of the donant ode of a dynacal yte wth non-clacal dapng The propoed ethod robut to the denon of the proble Fnally, to deontrate the effectvene and accuracy of the propoed ethod, a nuercal exaple wth coplex ode hown II BAYESIA MODEL UPDATIG Bayean odel updatng approach provde a robut and rgorou fraework to characterze odelng uncertante Gven the data D and pror PDF p( θ) of the uncertan yte paraeter, by applyng Baye theore the poteror PDF can be wrtten a p( D θ) p( θ) p( θ D) p( D θ) p( θ) dθ p( θ D) often not known explctly and only known up to a noralzng contant However, f aple dtrbuted accordng to PDF p( θ D) are avalable, tattcal etate uch a ean, varance, or PDF of θ can be etated Aung that θ dvded nto G group of uncertan paraeter vector, e, θ[ θ :,, G] The Gbb Sapler [5] one type of Markov chan Monte () IMECS

2 Proceedng of the Internatonal MultConference of Engneer and Coputer Scentt Vol II, IMECS, March - 5,, Hong Kong Carlo (MCMC) algorth that allow aplng fro an arbtrary ultvarate PDF f aple ulaton accordng to the PDF of each group of uncertan paraeter condtoned on all the other group poble In Gbb aplng algorth, the full condtonal PDF * * θ θ: θ : G p (,, D) are requred for,, G Uually oe ntal porton of Markov chan aple are dcarded before the tatonary tage reached After the burn-n perod the Markov chan aple obtaned are dtrbuted a the target PDF p( θ D) Stattcal etate uch a the ean, varance, or PDF can be etated ung the reanng aple A et of experental odal data dentfed fro the tructure under conderaton, condered for the Bayean odel updatng proble The odal data are denoted by Dˆ ˆ, ˆ, ˆ :, (),,, Where ˆ ˆ,,,, and ˆ o, are the oberved odal frequency, dapng rato, and coplex ode hape vector of the -th ode n the -th odal data et Here, the nuber of ode dentfed and o the nuber of oberved DOF (degree of freedo) III LIEAR SYSTEM UPDATIG MODEL FOR COMPLEX MODAL DATA In tate-pace, the equaton of oton of a general d - DOF te nvarant yte can be expreed by a frt order dfferental equaton a follow X ( t) = AX ( t) + BU ( t) () X () = X (4) A= I - - -M K -M C where X(t) and U(t) denote the tate and exctaton vector at te t, repectvely, and X denote the ntal condton The yte atrx A a functon of a, dapng, and tffne atrce M, C, and K Coplex egenvalue and egenvector, for,,,, can be obtaned fro the oluton of the egenvalue proble correpondng to the yte atrx A a A (6) (8) The egenvalue and egenvector occur n coplex conjugate par Ung (5)-(6) and rearrangng the ter gve the relatonhp between odal data and dynac odel paraeter M C K (9) (5) (7) Replacng yte egenvalue wth oberved egenvalue ˆ, gve ˆ M ˆ C K ε (),,, where the yte ode hape related to the oberved ode hape ˆ, through a electon atrx Γ that pck the oberved DOF fro the yte ode hape ˆ e (),, In the above equaton ε, and e, are the coplex rando vector repreentng the odel predcton error, e, the error between the repone of the tructure under conderaton and that of the aued odel The a, dapng and tffne atrce n () are repreented a a lnear u of contrbuton of the correpondng a, dapng, and tffne atrce fro the ndvdual precrbed ubtructure M( ) M M () C( ) C C () K( ) K K (4) where [ α, β, η ] are the contrbuton paraeter to be updated ([ α, β, η ] [,,] gve the nonal atrce) Dapng atrx can alo be repreented n ter of a and tffne atrx (a n the cae of clacal dapng), and contrbuton fro other dapng ource (a n the cae of vcou dapng) Other paraeter whch are unknown n ()-() and need to be updated are the yte ode hape and the paraeter defnng the probabltc odel of the odel predcton error Separatng the real and agnary part, ()-() are tranfored to Re( ˆ M( ) ˆ C( ) K( ) ) Re( ε ) (5),,, I( ˆ M( ) ˆ C( ) K( ) ) I( ε ) (6),,, Re( ˆ ) Re( e ) (7),, I( ˆ ) I( e ) (8),, Baed on the Prncple of Maxu Entropy [6], the PDF for vector Re( ε, ),I( ε, ),Re( e, ),I( e, ) are zero-ean Gauan PDF and ther covarance atrce are aued to be equal to caled veron of the dentty atrx I of approprate order, Re, Re( ε ) (, I ) (9), I, I( ε ) (, I ) () ISB: ISS: (Prnt); ISS: (Onlne) IMECS

3 Proceedng of the Internatonal MultConference of Engneer and Coputer Scentt Vol II, IMECS, March - 5,, Hong Kong, Re, Re( e ) (, I ) (), I, I( e ) (, I ) () Re, I, The varance paraeter and are aued to be known or are drectly etated fro the aple varance of the experental odal data Re, Re( ˆ, ) () o j ˆ I, I( ˆ, ) (4) o j ˆ where varance paraeter the averaged ode hape for -th ode The Re,, I, are left for updatng In total, the paraeter to be updated are the contrbuton paraeter [ α, β, η ], ode hape [Re( ),I( ),,Re( ),I( )], and predcton Re, I, Re, I, error varance [,,,, ] IV PRIOR PDFS To defne the Gbb Sapler algorth, three group of paraeter are condered θ [ α, β, η], θ θ [Re( ),I( ),,Re( ),I( )], Re, I, Re, I, [,,,, ] It wll be ore convenent to chooe Bayean conjugate pror whch wll allow exact aplng fro the full condtonal PDF θ, θ, D), θ, θ, D), and p( θ θ, θ, Dˆ ) Thu, the ntal PDF for the yte paraeter θ taken to be the product of ndependent Gauan PDF, () () P () θ ( θ, ) wth ean θ and dagonal covarance atrx P () to expre the ntal uncertante Slarly, the ntal PDF for the yte ode hape θ taken to be the product of ether ndependent Gauan PDF n cae any pror nforaton avalable, or ndependent unfor PDF n cae of no pror nforaton (a for the cae wth unknown coponent of the ode hape) The ntal PDF for predcton error varance θ taken to be the product of ndependent nvere gaa PDF, IG(ρ,κ ) V CODITIOAL PDFS Equaton (5)-(6) are lnear wth repect to θ, e, they can be wrtten n the followng for Y - Aθ Ε (5) Then, the full condtonal PDF θ, θ, D) Gauan whoe frt two oent are gven by E( θ θ, θ, Dˆ ) () T () () T P A A ( P θ A Y) Cov( θ θ, θ, Dˆ ) () T A P A (6) (7) Slarly, the PDF θ, θ, D) alo a Gauan and t frt two oent can be obtaned n a lar anner θ, θ, D) an nvere gaa gven by ˆ d T p( θ θ, θ, D) IG, Ε Ε (8) j VI GIBBS SAMPLIG ALGORITHM ) Intalze aple, drawn fro pror or chooe nonal value and let k= ) Saple contrbuton paraeter θ fro θ, θ, D) ) Saple ode hape θ fro p( θ θ, θ, Dˆ ) 4) Saple predcton error varance θ fro θ, θ, D) 5) Let k=k+ and go to tep, untl aple are obtaned It can be een that the propoed approach a generalzaton of what wa propoed by Chng et al [] wth tep to 5 beng the ae and A, E, Y, θ, θ and θ beng dfferent For a clacally daped yte, (9) reduce to r ( K M) and the denon of the proble r reduce by half a all the agnary coponent are equal to zero VII ILLUSTRATIVE EXAMPLE The yte elected for the llutratve exaple a 4- DOF echancal yte condered n [6] a hown n Fg, and ha the followng properte: = = = 4 = kg, k =k =k 5 =7 /, k =k 4 =8 /, c =c =c 5 =7 /, c =c 4 =8 / The odal data for the odel updatng proble ung Gbb apler for th yte cont of et of odal data ( =) wth the frt two odal frequence, odal dapng rato, and partal coplex ode hape (correpondng to DOF,, and 4, e, o =) for each data et ( =) Varablty n odal data ntroduced by perturbng the ae by 5%, and tffnee and dapng by % For Bayean dentfcaton purpoe, a dynac odel tructure baed on ae 4-DOF echancal yte condered and the uncertan paraeter to be updated for th odel cla are contrbuton paraeter [,,, 4], [,,, 4, 5], [,,, 4, 5], predcton error varance [ Re,, I,, Re,, I, ], and coplete coplex ode hape [, ] Mae are aued to be known wth uffcent accuracy and thu the ntal PDF for ISB: ISS: (Prnt); ISS: (Onlne) IMECS

4 Proceedng of the Internatonal MultConference of Engneer and Coputer Scentt Vol II, IMECS, March - 5,, Hong Kong [,,, 4] are choen wth ean value equal to and coeffcent of varaton (cov, e, the rato of tandard devaton to ean) for each equal to %; and pror ean value for [,,, 4, 5] and [,,, 4, 5] are aued equal to, wth pror cov for each equal to % Flat ndependent pror are taken for [, ] Independent nvere gaa pror PDF wth, are taken for [ Re,, I,, Re,, I, ] (Jeffrey non-nforatve pror) The total nuber of paraeter to be updated 4 (4 for the contrbuton paraeter, 6 for the two ode hape, and 4 predcton error varance) c k c k (a) c k c 4 k 4 4 c 5 k 5 Fg The 4-DOF echancal yte Paraeter Mean θ TABLE I BAYESIA IDETIFICATIO RESULTS Standard Devaton cov (%) θ θ c c c c c k 6 64 k k k k (b) Ung the propoed Gbb aplng baed algorth, =5, aple of contrbuton paraeter, ode hape and predcton error varance are obtaned The burn-n perod le than aple Table I how the tattcal properte of the contrbuton paraeter aple It how the etated poteror ean value θ (colun ), (c) ISB: ISS: (Prnt); ISS: (Onlne) IMECS

5 Proceedng of the Internatonal MultConference of Engneer and Coputer Scentt Vol II, IMECS, March - 5,, Hong Kong of the paraeter to be updated effcently even f there are a large nuber of uncertan paraeter Lke ot proble n dentfcaton, the qualty of the updated reult can only be a good a the qualty of the data avalable (d) Fg Poteror ean and cov etate for tffne (a, b), and dapng (c, d) contrbuton paraeter a functon of the nuber of Gbb aple, tablzng after about aple poteror tandard devaton (colun ), the poteror cov (colun 4), and the noralzed dtance (colun 5) The noralzed dtance repreent the abolute value of dfference between the etated ean value and the true ean value θ of tructural paraeter, noralzed wth repect to the etated tandard devaton It can be een that the etated poteror ean paraeter are cloe to the true ean value of the yte and ther cov uch aller than that what wa ntally aued Table II how the dentfed ng ode hape coponent (correpondng to DOF-) Agan the Bayean dentfcaton reult are qute cloe to the true ean value Table II IDETIFIED MODE SHAPE COMPOET, DOF- True ean Poteror ean Mode (ab, angle) (ab, angle) The ulaton reult tablzed at around aple, a hown n Fg where the poteror ean and cov etate of the tffne and dapng contrbuton paraeter are reported a functon of the nuber of Gbb aple REFERECES [] R Brncker, L Zhang and P Anderen, "Modal dentfcaton fro abent repone ung frequency doan decopoton," Proc, Internatonal Modal Analy Conference, 65 [] B Peeter and G De Roeck, "Stochatc yte dentfcaton for operatonal odal analy: a revew," Journal of Dynac Syte, Meaureent, and Control, vol, pp 659, [] L S Katafygot and K V Yuen, "Bayean pectral denty approach for odal updatng ung abent data," Earthquake Engneerng & Structural Dynac, vol, pp -, [4] K V Yuen and L S Katafygot, "Bayean te-doan approach for odal updatng ung abent data," Probabltc Engneerng Mechanc, vol 6, pp 9-, [5] J Yang, Y Le, S Pan and Huang, "Syte dentfcaton of lnear tructure baed on Hlbert Huang pectral analy Part : noral ode," Earthquake Engneerng & Structural Dynac, vol, pp , [6] J Yang, Y Le, S Pan and Huang, "Syte dentfcaton of lnear tructure baed on Hlbert Huang pectral analy Part : Coplex ode," Earthquake Engneerng & Structural Dynac, vol, pp 5-554, [7] E Parloo, "Applcaton of Frequency-doan Syte Identfcaton n the Feld of Operatonal Modal Analy," PhD the, Departeent of Mechancal Engneerng, Vrje Unvertet Bruel, [8] B Jah and W X Ren, "Structural fnte eleent odel updatng ung abent vbraton tet reult," Journal of Structural Engneerng, vol, pp 67-68, 5 [9] S Modak, T Kundra and B akra, "Coparatve tudy of odel updatng ethod ung ulated experental data," Coputer & tructure, vol 8, pp , [] J Motterhead and M Frwell, "Model updatng n tructural dynac: a urvey," Journal of Sound and Vbraton, vol 67, 47-75, 99 [] Beck and Katafygot, "Updatng Model and Ther Uncertante I: Bayean Stattcal Fraework," Journal of Engneerng Mechanc, vol 4, pp , 998 [] J L Beck and S K Au, "Bayean Updatng of Structural Model and Relablty ung Markov Chan Monte Carlo Sulaton," Journal of Engneerng Mechanc, vol 8, pp 8-9, [] J Chng, M Muto and J L Beck, "Structural odel updatng and health ontorng wth ncoplete odal data ung Gbb apler," Coputer Aded Cvl and Infratructure Engneerng, vol, pp 4-57, 6 [4] M Vank, J Beck and S Au, "Bayean Probabltc Approach to Structural Health Montorng," Journal of Engneerng Mechanc, vol 6, pp , [5] S Gean and D Gean, "Stochatc Relaxaton, Gbb Dtrbuton, and the Bayean Retoraton of Iage," IEEE Tranacton on Pattern Analy and Machne Intellgence, vol 6 pp 7-74, 984 [6] E T Jayne, Where do we tand on axu entropy? MIT Pre, Cabrdge, Ma, 978 VIII COCLUSIO A new Gbb aplng baed approach for odel updatng of a lnear tructural dynac yte wth nonclacal dapng baed on ncoplete odal data ncludng odal frequence, dapng rato and partal coplex ode hape of oe of the donant ode propoed The reult fro the exaple deontrate that all the updated paraeter are reaonable when copared wth the true ean value, ndcatng the effectvene of the procedure The propoed ethod alo allow the uncertanty ISB: ISS: (Prnt); ISS: (Onlne) IMECS