Ph.D. Scholar, Indian Institute of Technology Kharagpur, Kharagpur, WB, b,

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1 Proceedgs of ICTACEM 2014 Iteratoal Coferece o Theoretcal, Appled, Computatoal ad Expermetal Mechacs December 29-31, 2014, IIT Kharagpur, Ida ICTACEM-2014/398 No-teratve egestructure assgmet based fte elemet model updatg of a Mdl Resser plate Duca form of state space usg ambet vbrato respose Subhamoy Se a ad Badurya Bhattacharya b* a Ph.D. Scholar, Ida Isttute of Techology Kharagpur, Kharagpur, WB, b, Professor, Ida Isttute of Techology Kharagpur, Kharagpur, WB, ABSTRACT Egestructure assgmet (ESA) based model updatg s a cotrol based techque for systematc calbrato of fte elemet models usg measured respose from real structure. Applcato of ths techque physcal space restrcts smultaeous updatg of stffess ad dampg matrces of ay mechacal system. O the other had ESA whe used state space doma demads state space egestructure to be detfed whch s a challegg job. It s ot certa that the detfed state space egestructure wll be the same order ad oretato as desred by the ESA algorthm. I ths paper we used Duca form of state space model of the mechacal system so that assgable egestructure ths form ca be easly costructed usg modal propertes of the system ts physcal space ad thus problems regardg oretato s avoded. To acheve compatblty betwee assgable state space egestructure ad state space model the later has bee reduced usg structural equvalet reducto expaso program (SEREP). Assgable egestructure s the used alog wth ESA algorthm gve by B.C. Moore to update the reduced prmary model of the mechacal system to smultaeously update the stffess ad dampg matrces. Proposed algorthm s tested o a Plate modeled usg Mdl-Resser plate elemet ad updated model demostrated a good agreemet wth the desred result. Keywords: Fte elemet model updatg, Subspace detfcato, Egestructure assgmet. 1. INTRODUCTION Fte elemet models of real lfe structures fal to replcate the realty owg to Improper modelg approach ad assumptos towards boudary codto, parameter values ad model order. Systematc calbrato of the prmtve model s therefore requred before usg t as a relable predctor model. Ths ca be acheved by combed use of system detfcato ad model updatg. System detfcato whch mostly cosders the real system as a black box, tres to detfy mportat characterstc features (modal propertes, olearty etc.) of the real system through whch the system ca be terpreted. Ths s doe by costructg parametrc or oparametrc models usg lttle aalytcal sese. Thus resultg model may or may ot be physcally uderstadable. Model updatg take ths effort oe step further by usg these characterstc features to alter a prmary model costructed takg physcs of the system to cosderato. Updated model therefore retas the physcal * Further author formato: (Sed correspodece to B. Bhattacharya) S. Se: E-mal: subhamoyse@cvl.tkgp.eret., Telephoe:

2 sgfcace whle ts respose coforms to that of the real system. Fte elemet model updatg therefore ca be foud as a terestg feld of research especally the felds of structural health motorg for the past few decades. Exstg methods for fte elemet model updatg are mostly vbrato based where modal propertes whch are bascally physcal space egestructure are used to update a model. Dfferet optmzato algorthms ragg from gradet or hessa based, sestvty based, perturbato based to ature mmckg types (GA, PSO, Hybrd techques) are tred by dfferet researchers ths edeavor. Apart from the regular requremet of matchg the modal propertes of the FE model wth the detfed structure, codtos to qualfy as a good FE model updatg techque clude ablty to reta the explotable propertes (postve, symmetrc, baded, ad sparsty) of stffess ad mass matrx. Ufortuately most of the exstg techques suffer from problems regardg fulfllmet of the secod codto. Besdes some follow sequetal updatg techque (frst stffess the mass or vce versa) ad most do t gve much atteto to update dampg matrx. Besdes these optmzato algorthm has ther ow drawbacks cludg mproper covergece, multple possble solutos ad computatoal expese. Cotrol theory based egestructure assgmet s o the other had s a good approach to update ay fte elemet model updatg. I ths method desred egestructures (modal parameters) are embedded to the system model so that updated model has the same egestructure as desred by the desger. Egevalue assgmet or pole placemet has always bee a terestg feld of research for cotrol egeers. Geerally pole placemet techques are used to cotrol a system wth mmum cotrol effort possble. There are several pole placemet techques exst lterature. Arbtrary assgmet of egevalues for a closed loop system has bee dscussed by Woham [2]. B. C. Moore [3] was the frst perso to detfy the flexblty offered by state feedback multvarable systems beyod closed loop egevalue assgmet. He further demostrated hs paper that a specfc umber of elemets of each egevector of a closed loop MIMO system ca be freely assged. Kautsky, et. al. [4], Srathkumar [5] dscussed robust pole assgmet techque lear tme varat system. Several other researchers (Sobel et. al. [6]) also developed algorthm to place egestructure for closed loop system. Egestructure assgmet for model updatg s however relatvely ew feld. Geerally vbrato data ad modal propertes have bee used for model updatg most of the B. Bhattacharya.: badurya@cvl.tkgp.eret., Telephoe: , Address: Ida Isttute of Techology 2

3 lterature [7]. Quadratc Partal Egevalue Assgmet ad Partal Egestructure Assgmet techque to update models are dscussed by Datta [8]. J. Carvalho [9] showed how state estmates ca be used to update a FE model usg optmzato techques. However, symmetry ad other explotable propertes of updated stffess ad dampg matrx have always bee a major cocer. Several optmzato techques are used wth dfferet metaheurstcs dfferet lterature to mata these propertes ESA for model updatg thus has bee extesvely used both physcal ad state space doma for aerospace ad vehcular moto cotrol where the objectve has bee to cotrol the path of a movg body wth mmum cotrol effort possble. However these types of problems do ot have ay specal kds of structure whle state space model ows a very specfc structure whch ca be exploted to ga lots of other formato about the health of the system. Thus use of ESA moto cotrol problems s characterstcally dfferet from the use case of FEM updatg. Use of ESA based FEM updatg s although exsts the lterature. However most of the work that has bee performed ths effort s cast physcal space doma whle state space doma offers a greater flexblty of smultaeous updatg of stffess ad dampg matrces rederg the updatg method to be more practcal. But updatg a state space model of a mechacal system matag ts basc explotable structure has ts ow challeges. Frst of all ths demads that the egestructure state space doma eeds to be detfed from the real structure proper order ad oretato compatble wth the prmary model whch s supposed to be updated. However t may happe that detfed state matrx ca be rotated by a trasposto matrx ad t s ot certa that the detfed state matrx wll be the same oretato as the system model. Beg coordate depedet egevalues ca be detfed easly, but problem arses whle detfyg egevectors assgable oretato. I order to avod ths problem a dfferet form of state space modelg amely Duca form, has bee adopted ths paper. Egestructure of state matrx Duca form of state space model has clear relato wth modal propertes of the system ts physcal space. Therefore stead of detfyg state space egestructure, modal propertes of the system have bee detfed. Usg these modal propertes desred egestructure for Duca form of state space model has bee recostructed. Ths egestructure s the used as the desred egestructure to update the system state matrx usg ESA method. Kharagpur, WB, Ida,

4 2. THORY 2.1 Dscrete tme stochastc subspace detfcato As proposed damage detfcato techque s fte elemet model updatg based t starts wth a detfcato step to extract the modal parameters of the system from ts respose. State space modelg s good approach to detfy the system ths regard. Ay order dfferetal equato of a system ca be defed as 2 umber of coupled frst order equato, termed as state space form of the system. Cosderg a mechacal system cotuous tme of mass M, stffess K ad dampg D, state trasposto matrx A c, vector x t ad output matrx C c ca be defed as: 1 T 0 I qt Cd CaM K c ; x ; C = 1 1 t c 1 M K M D q t Cv CaM D A 1 Where yt Caqt Cvqt Cd qt s the output vector ad C, C, C are the output matrx for a v d accelerato q t, velocty q t ad dsplacemet q t cotuous tme respectvely. Usg these terms system model cotuous tme ca be expressed as: x A x w t c t t y C x v t c t t 2 The model structure s cosdered here to be a stochastc wth ukow put. w t ad v t are process ose ad measuremet ose respectvely. Dscrete tme stochastc subspace detfcato algorthm gve by Vaoverschee & Demoor [10] has bee used to detfy ths cotuous system usg samplg. Ths s a o-teratve approach of state space modelg. The dscrete tme state space model of the system ca be wrtte state space form as: x( k 1) Ax( k) w( k) y( k 1) Cx( k) v( k) 3 Where A s state trasposto matrx, C s output matrx relatg state vector to output, x(k) s the dscrete tme state vector at k th tme stat, y(k) s output vector or measuremet terms. Usg Kalma s [11] forward ovato techque the same system s descrbed as: 4

5 x( k 1) Ax( k) K e( k) y( k 1) Cx( k) e( k) g 4 Where ek ( ) s the ovato vector ad K g s called Kalma ga matrx. Usg stochastc subspace detfcato algorthm gve by Vaoverschee ad Demoore state trasposto matrx A, output matrx C ad ga matrx K g ca be easly detfed usg output sgal. Here we used output or measuremet vector Y(k) whch s the tme hstory of accelerato respose obtaed from sesors placed at approprate locato of the structure. Idetfed system s the trasformed to cotuous system usg zero-order-hold techque yeldg ew set of state ad output matrces cotuous doma. Post multplyg egevector of ths ew state matrx wth output matrx yelds array of mode shape coordates of the system physcal space the predefed sesor locatos. Egevalues however, beg sestve towards oretato of the state matrx, ca be detfed easly from the state matrx. These detfed physcal space egestructure s the used to costruct egestructure for the Duca form of state space model descrbed the followg secto. 2.2 Duca form Duca form was frst gve by Duca hs paper [12] I ths state space form the dyamcs of the cotuous system s descrbed by the followg equatos: R x ( t) K x( t) F( t) 5 where M D 0 K 0 R ; K ; F( t) 0 M M 0 f ( t) 6 M, K, D are the system mass, stffess ad dampg matrces. To obta the homogeeous soluto of ths frst order system we assume a soluto of ths form: t x() t e Whch gves a soluto the form: R K [0] egevalue problem of a matrx term U as:. Ths equato s mapulated as a 7 1 U 8 5

6 1 1 1 K M K D Where U K R ad egevectors of ths problem ca be descrbed usg 0 I egevector of the system ts physcal space as: 9 Where s the egevector the physcal space.e. egevector of the quadratc pecl: 2 M D K 0 10 I coecto to the prevous secto t ca be show that egevectors of system descrbed Duca form of state space.e. egestructure of the system ts physcal space.e. ca easly be costructed ts desred oretato usg ad whch are actually mode shapes ad atural frequeces. Ths approach has bee tred ths paper. The egestructure physcal space detfed usg subspace detfcato algorthm descrbed prevous secto s used to costruct desred egestructure for Duca form. After updatg s performed stffess, mass ad dampg matrces are aga extracted from the updated state matrx usg followg equato. However, oe has to cosder oe of these matrces to be stadard ad uchaged eve after updatg. Mass matrx beg most relable ths regard has bee cosdered to be stadard most other lteratures dealg wth these kd of smultaeous updatg stuato. We here adopt that same strategy to extract the other two system matrces usg followg equatos: 2.3 Egestructure assgmet Aul Aur Adetfed Adl A dr K MA ; C MA ; detfed dl detfed dr Egestructure assgmet s a cotrol based techque to properly place desred egestructure to a system. Traset respose of a system s a fucto of ts egestructure ad to alter the system s traset respose egestructure has to be altered. ESA uses feedback to calculate a cotroller or ga matrx to update the egestructure of the system. Cosder dyamcs of a system has bee descrbed as: x( k 1) Ax( k) Bu( k) y( k) Cx( k)

7 Where x(k) s the state vector at k th stat, A s state trasposto matrx, B s put matrx, u(k) s put vector, C s output matrx ad y(k) s measured output vector. If a put sequece s selected a such a way that u( k) K x( k) ; the equato ca be rewrtte as: Whch the ca be mapulated as: c x( k 1) Ax( k) BK x( k) x( k 1) ( A BK ) x( k) Ax( k) Ths yelds altogether a ew system wth trasposto matrx A whch has the same egestructure as desred by the desger. Ths s termed as full state feedback where every state has bee used. We here used algorthm gve by B.C Moore [3] to assg desred egestructure to the prmary state matrx to update t such a way that egestructure of the updated state matrx cocdes wth desred egestructure. The algorthm s descrbed below. Algorthm: 1. Defe S [ I A B] c ad partto ts bass vector as: R N M N has the same order as A, where s desred egevalue. c so that 2. Defe dyamcs wth desred egestructure[, v] wth a ga matrx Kc as: 3. Compare these two equatos: Ths comparso sgfes that Kvspas aother vector space. c ( ABK ) v Iv c N v I A B 0 ad I A B 0 M Kv c 3. Calculate z that relates two vector spaces of N ad v spas the same vector space whereas N ad v ; ad also vector spaces of M ad M ad Kvusg c ths equato as: z N v where symbolzes Moore-Perose pseudo verse. 4. Calculate ga matrx as: K M z v 1 c 5. Update the state matrx as: A ( A BK c ) 7

8 I ths method we vrtually use a array of put vector whch stablzes the system ( ths case match systems egestructure to the desred oe). We here used a arbtrary B matrx whch oly esures that Rak ( I A B ). Usg ths B matrx we follow the algorthm gve by B. Moore to obta a ga matrx egestructure to the desred values. K c whch updates the state matrx A to match ts 3. FINITE ELEMENT MODEL UPDATING To update the prmary model frstly assgable egestructure eeds to be detfed from the respose hstory of the real structure. Ths s doe by state space detfcato of the real system usg subspace detfcato algorthm as descrbed the prevous secto. Idetfed system state matrx ad output matrx (A ad C) s the coverted to cotuous tme doma by usg Zero order hold techque. Modal parameters are the extracted from these two matrces usg followg equatos: Egestructue of A state space { ; } Egevector physcal space C Thereafter usg equato (9) assgable egestructure for the Duca form of state space model s recostructed. Prmary FE model of the system s the reduced dow to oly measured degrees of freedom usg Structural equvalet reducto expaso program (SEREP) algorthm to fulfll the order crtera of the assgable egestructure. ESA s the appled o the Duca form of state space model costructed usg reduced order system matrces. Updated stffess ad dampg matrces are the detfed usg equato (11). 4. NUMERICAL VALIDATION To valdate the proposed method a umercal expermets have bee performed o a alumum plate. A fte elemet model of the plate s prepared wth assumed parameter value lsted Table 1. Ths has bee cosdered as the real system for whch accelerato respose hstory s smulated usg Newmark-beta method wth a samplg frequecy 500Hz. To demostrate the ose sestvty of the proposed method umercally obtaed tme sgal has bee cotamated usg 10% ose. The plate s excted wth a whte ose sequece of zero mea ad ut stadard devato to replcate ambet vbrato exctato codto whch s obvous large sze structural system detfcato. Smulated tme hstory sgal s the put through the stochastc subspace algorthm to obta egestructure of detfed state matrx. I ths process we developed state space 13 8

9 model of dfferet order ad the usg some huma terveto based o practcal costrats o the detfed egevalues (e.g. practcal value of dampg.e. egatve ad less tha 20%, removg ustable egevalues) we collected oly feasble egestructure of the detfed state matrx. I the ext step the egevalues of the detfed state matrx ad egevector of the system physcal space s extracted usg equato (13). Desred egestructure for the Duca form of state space model s the costructed from outcome of the last step usg equato (9). A prmary FE model s also prepared wth a set of assumed elastcty values dfferet from the orgal oe ad ths model s cosdered as prmary model (lsted table 1). The detfed physcal space egevector s havg the same order as of the umber of sesor pots ad ts coordate correspods oly to vertcal degrees of freedom (DOF) (cosderg structure has bee strumeted wth accelerometer for vertcal movemet oly). For that reaso udamaged FE model of the system s reduced to the specfed degrees of freedom by usg SEREP algorthm so that detfed egestructure ca be used to update the udamaged model. Usg ths reduced stffess, mass ad dampg matrces Duca form represetato for udamaged state of the system s costructed ad updated usg ESA method. Usg equato (11) updated stffess matrx are the extracted ad compared wth the tal stffess matrx of the system. Fgure 1lsts the tal, target ad updated frequeces of the fte elemet model whch demostrates very close coformato wth the desred result. Fgure 2 shows the MAC values for frst fve modes s very close to the desred value of 1. Thus we ca coclude that the updated model s represetg the real system better tha the tal model. Table 1: Assumed materal propertes for FE model Structure Alumum plate Elastcty 63 GPa Posso s rato Dmeso 0.25 m x 0.45m x m Desty Kg/m 3 Elemet Mdl-Resser Plate elemet Boudary codto Catlever (Clamped-free-freefree) Total elemets 5x5=25 Assumed elastcty 70% of orgal.e GPa 5. CONCLUSION I ths paper we tred usg ESA based fte elemet model updatg techque to update a prmary model of a Mdl-Resser plate. We also evsaged robustess of the proposed 9

10 method uder presece of ose whch ehaces ts applcablty the real feld scearo. Whle exstg methods uses gradet or hessa based or evolutoary algorthm based optmzato techque to update FE models whch are teratve ature ad most ofte leads to wrog result, proposed method s o-teratve ature whch updates the prmary model state space doma to embed desred state space egestructure the updated FEM. Table 2: Comparso of atural frequeces betwee tal, target ad updated FE models for frst fve modes Modes Target FE Ital FE Updated FE Frst Mode Secod Mode Thrd mode Fourth mode Ffth mode Fgure 1: Frequecy respose fucto of tal, actual or target ad updated FE models Ulke other optmzato based FEM updatg techque ths method uses lear algebra applcato to have a computatoally expesve soluto of the sad problem. Because of ts o-teratve ature ad less computato requremet ths method s also justfed to be take up as a ole damage detfcato techque. I ths paper we employed the proposed method a Mdl-Resser Plate ad for practcalty whte ose exctato has bee used to smulate ambet vbrato codto. Numercal expermets show that proposed method s capable of capturg damage features eve the presece of up to 10% 10

11 ose suffcetly. Ths same s also testfed from umercal expermetato ad results are satsfactorly close to actual scearo. The goodess of the method les the fact that t uses huma terveto durg state space detfcato of the damaged structure to separate real physcal roots from the o-physcal roots through the use of stablzato plots. These ophyscal roots are attrbuted to ether computatoal outcome or presece of ose the sgal. Removg these o-physcal roots by huma terveto leads to removal of ose fluece as well as makg the job of workg wth osy sgals less complcated. Thus ths method combes lear algebra applcato to have a o-teratve soluto alog wth huma decso whch makes the method more practcal to be used as damage detfcato for real lfe structure. Fgure 2: MAC value for frst fve modes REFERENCES 1. M. I. Frswell, & J. E. Mottershed, Fte elemet model updatg structural dyamcs. Dordrecht: Kluwer Academc, W. M. Woham, O pole assgmet multput cotrollable lear systems, IEEE Tras. Automat. Cotr., vol. AC-12, pp , B. C. Moore, O the flexblty offered by state feedback multvarable systems beyod closed loop egevalue assgmet, IEEE Tras. Automat. Cotr., vol. 21, , J. Kautsky, N. K. Nchols, & P. V. Doore, Robust pole assgmet lear state feedback, Iteratoal Joural of Cotrol, vol. 41, pp , S. Srathkumar, Robust egevalue/ egevector selecto lear state feedback systems, Proceedgs of the 27th Coferece o decso ad Cotrol, pp , K. M. Sobel, Y. Waglg, & J. L. Frederck, Egestructure assgmet wth ga suppresso usg egevalue ad egevector dervatves, Joural of Gudace, vol. 13, No-6, pp , A. Bagch, Updatg the Mathematcal Model of a Structure Usg Vbrato Data, Joural of Vbrato ad Cotrol, vol. 11(12), pp , B. N. Datta, Fte-elemet model updatg, egestructure assgmet ad egevalue embeddg techques for vbratg systems, Mechacal Systems ad Sgal Processg, vol. 16, pp ,

12 9. J. Carvalho, State-estmato ad fte-elemet model updatg for vbratg systems, Ph. D. dssertato, Norther Illos Uversty, DeKalb, IL, V. P. Overschee & D. B. Moor, Subspace detfcato for lear systems: Theory mplemetato applcatos, Dordrecht: Kluwer Academc, Kalma, Rudolph Eml. "A ew approach to lear flterg ad predcto problems." Joural of Fluds Egeerg 82.1 (1960):