FINITE ELEMENT MODEL UPDATING USING BAYESIAN FRAMEWORK AND MODAL PROPERTIES Tshldz Marwala 1 and Sbusso Sbs I. INTRODUCTION Fnte element (FE) models are wdely used to predct the dynamc characterstcs of aerospace structures. These models often gve results that dffer from measured results and therefore need to be updated to match measured results. Some of the updatng technques that have been proposed to date use tme, modal, frequency and tme-frequency doman data [1, ]. In ths Note, we use the modal doman data to update the FE model. A lterature revew on FE updatng [1] reveals that the updatng problem has been manly framed n the maxmum-lkelhood framework. Even though ths framework has been appled successfully n ndustry, t has the followng shortcomngs: t does not offer the user confdence ntervals for solutons t gves; there s no phlosophcal explanaton of the regularzaton terms that are used to control the complexty of the updated model; and t cannot handle nherent ll-condtonng and non-unqueness of FE updatng problem. In ths Note the Bayesan framework s adopted to address the shortcomngs explaned above. Bayesan framework has been found to offer several advantages over maxmum-lkelhood methods n areas closely mrrorng FE updatng [3-5]. Ths Note seeks to address the followng ssues: (1) how pror nformaton are ncorporated nto FE model updatng problem; and () applyng Bayesan framework to update FE models to match expermentally measured modal propertes (.e. natural frequences and mode shapes) to modal propertes calculated from the FE model of a beam. In ths Note Markov chan Monte Carlo (MCMC) smulaton [3] s used to sample the probablty of the updatng parameters n lght of the measured modal propertes. Ths probablty s known as the posteror 1 (Correspondng Author) Assocate Professor, School of Electrcal and Informaton Engneerng, Unversty of the Wtwatersrand, Prvate Bag 3, WITS, 050, South Afrca, e-mal: t.marwala@ee.wts.ac.za Presdent and CEO, CSIR, P.O. Box 395, Pretora, 0001, South Afrca
probablty. Metropols algorthm [6] s used as an acceptance crteron when samplng the posteror probablty. II. MATHEMATICAL FOUNDATION A. Dynamcs All elastc structures may be descrbed n terms of ther dstrbuted mass, dampng and stffness matrces. If dampng terms are neglected, the dynamc equaton may be wrtten n modal doman (natural frequences and mode shapes) for the th mode as follows [7]: ( ω ε (1) [M] + [K]){ φ} = { } Here [M] s the mass matrx, [K] s the stffness matrx, ω s the th natural frequency, {φ} s the th mode shape vector and {ε} s the th error vector. The error vector {ε} s equal to {0} f the system matrces [M] and [K] correspond to the modal propertes. If the system matrces whch are usually obtaned from the FE model do not match measured modal propertes ω and {φ} then {ε} s a non-zero vector. In maxmum-lkelhood method the Eucldean norm of {ε} s mnmzed n order to match the system matrces to measured modal propertes. Another problem that s encountered n many practcal stuatons s that the dmenson of mode shapes does not match the dmenson of system matrces. Ths s because measured modal co-ordnates are fewer than FE modal co-ordnates. To ensure compatblty between system matrces and mode shape vectors, the dmenson of system matrces s reduced by usng a technque called Guyan reducton method [8] to match the dmenson of system matrces to the dmenson of measured mode shape co-ordnates. B. Bayesan Method In ths Note Bayesan method s ntroduced to solve the FE updatng problem based on modal propertes. The fundamental rule that governs the Bayesan approach s wrtten as follows [3]:
P([D] {E})P({E}) P ({E} [D]) = () P([D]) Here {E} s a vector of updatng parameters, P({E}) s the probablty dstrbuton functon of updatng parameters n the absence of any data, and ths s known as the pror dstrbuton, and [D] s a matrx contanng natural frequences ω and mode shapes {φ}. It must be noted that the mass [M] and stffness [K] matrces are functons of updatng parameters {E}. The quantty P({E} [D]) s the posteror dstrbuton functon after a set of data has been seen, P([D] {E}) s the lkelhood dstrbuton functon and P([D]) s the normalzaton factor. Lkelhood dstrbuton functon There are many areas where the lkelhood dstrbuton functon has been appled and these nclude neural networks [3]. In neural network context, the lkelhood dstrbuton functon s defned as the normalzed exponent of the error functon. In ths Note the lkelhood dstrbuton functon, P([D] {E}), s defned as the sum of square of elements of the error vector shown n equaton 1, and can be wrtten n the same way as n neural networks as follows [3]: P([D]{E}) = = 1 Z D 1 Z D ( β) ( β) exp β exp β F N ( εj ) j F N ([ ( ω + φ ] ) [M] [K]){ } j j (3) Here β s the coeffcent of the measured modal property data contrbuton to the error and s set to 1 through tral and error and ε j s the error matrx wth subscrpt representng the th modal propertes and j representng the j th measurement poston. The superscrpt F s the number of measured mode shape coordnates, N s the number of measured modes and Z D s: Z F N D ( ) exp ([ ( [M] [K]){ } ] ) j = β ω + φ d[d] j β (4) It should be noted that n equaton 3 the error ε j s a matrx as opposed to a vector as s the case n equaton 1. Ths s because t takes nto account of all modal co-ordnates. 3
Pror dstrbuton functon of parameters to be updated The pror dstrbuton functon conssts of the nformaton that s known about the problem. In FE updatng t s generally accepted that FE updatng s usually vald f the model s close to the true model. In ths Note, t s known that not all parameters to be updated have the same level of modelng errors. Ths means that some parameters are to be updated more ntensely than others. For example, parameters next to jonts should be updated more ntensely than those wth smooth surface areas and are far from jonts. In ths Note the pror dstrbuton functon for parameters to be updated may be wrtten by usng Gaussan assumpton as follows [3]: 1 α = Q P ({E}) exp {E} (5) ZE ( α) Here Q s the number of groups of parameters to be updated, α s the coeffcent of the pror dstrbuton functon for the th group of updatng parameters. The pror dstrbuton functon n equaton 5 ensures that large updatng of parameters s less lkely than small adjustments of updatng parameters. The Gaussan pror has been successfully used [3] to dentfy a large number of weghts n neural networks, and therefore t s assumed that t should be successful on dentfyng a small number of updatng parameters n ths Note. The hgher the α the lower s the degree of updatng of the th group of parameters and s the Eucldean norm of. In equaton 5, f α s constant for all the updatng parameters, then the updated parameters wll be of the same order of magntudes. Equaton 5 may be vewed as a regularzaton parameter [9]. In equaton 5, Gaussan prors are convenently chosen because many natural processes tend to have Gaussan dstrbuton. In Bayesan framework regularzaton method s vewed as a mechansm of ncorporatng pror nformaton whereas n maxmum-lkelhood method they are vewed as mathematcal convenence. The functon Z E (α) s a normalzaton factor gven by [3]: Q α Z E ( α) = exp {E} dα (6) 4
Posteror dstrbuton functon of weght vector The dstrbuton of the weghts P({E} [D]) after the data have been seen s calculated by substtutng equatons 3 and 5 nto equaton to gve: ([ ] ) = α ( ) β F N ω + φ Q 1 exp [M] [K]){ } j {E} Zs α, β j P ({E}[D]) (7) where Z Q F N α S (, β) = P([D]) = exp β ([ ( ω [M] + [K]){ φ} ] ) j {E} d{e} α j (8) In equaton 7, the optmal weght vector corresponds to the maxmum of the posteror dstrbuton functon, whch s the soluton as the one obtaned from a maxmum-lkelhood approach. Ths mples that Bayesan method at least gves the soluton that s gven by the maxmum-lkelhood method but n addton gves probablty dstrbutons. C. Markov chan Monte Carlo method The applcaton of Bayesan approach to FE model updatng usng Monte Carlo approach, results wth a set of updated parameter vectors {E} that are statstcal rather than determnstc. As a result, the FE model updatng wll gve dstrbutons of the predcted modal propertes and from these dstrbutons averages and varances of modal propertes may be constructed. Followng the rules of probablty theory, the dstrbuton of vector {y}, representng measured modal propertes may be wrtten n the followng form: P ({y} [D]) = P({y} [E])P({E} [D])d{E} (9) Equaton 9 depends on equaton 7, and s dffcult to solve analytcally due to relatvely hgh dmenson of updatng parameter vector. As a result, Markov chan Monte Carlo (MCMC) method s employed to determne the dstrbuton of updatng parameters, and subsequently, of predcted modal propertes. The ntegral n equaton 9 s solved, usng Metropols algorthm [6], through generatng a sequence of vectors 5
{E} 1, {E}, that form a Markov chan wth a statonary dstrbuton P([D] {E}). The ntegral n equaton 9 may be thus approxmated as follows: ~ 1 y G({E} ) (10) L R+ L 1 = I Here G s a fnte element model whch takes vector {E} and predcts the average output {} s the vector contanng the modal propertes, R s the number of ntal states that are dscarded n the hope of reachng a statonary dstrbuton descrbed by equaton 7 and L s the number of retaned states. Several methods have been proposed to smulate the dstrbuton n equaton 7 such as Gbbs samplng [10], Metropols algorthm [6] and hybrd Monte Carlo method [11]. Hybrd Monte Carlo, whch has been shown to be the most effcent of the Monte Carlo methods thus far, s not used n ths Note because t requres gradent nformaton, whch s not avalable n exact form n FE updatng problem. As a result, MCMC method s used to dentfy the posteror dstrbuton functon of the updatng parameters. In ths Note MCMC method s mplemented by samplng a stochastc process consstng of random varables {{E} 1,{E},,{E} n } through ntroducng random changes to updatng parameter vector {E} and ether acceptng or rejectng the sample accordng to Metropols algorthm [6]. Metropols crtera can be wrtten as follows: f Pnew ({E}[D]) > P else accept {E} new old ({E}[D]) accept state {E} wth probablty P P new old new ({E}[D]) ({E}[D]) (11) In ths Note we vew ths procedure as a way of generatng a Markov chan wth transton from one state to another conducted usng the crteron n equaton 11. III. EXAMPLE: EXPERIMENTALLY MEASURED BEAM To test the proposed procedure a freely suspended alumnum beam s used. The beam, whch s shown n Fgure 1, has the followng dmensons: length: 1.0m; wdth: 5.4mm and thckness: 13.4mm. 6
Acceleraton measurements are taken at 13 equdstant postons and the beam s excted at a poston located 40mm from the end of the beam (see Fgure 1). Further detals of ths beam are found n [1]. The FE model wth 1 elements s constructed usng modulus of elastcty of 800 10 8 Nm - and densty of 700 kgm -. Usng conventonal sgnal processng analyss [7], the measured data are transformed nto frequency response functons (FRFs) and from the FRFs, natural frequences and mode shapes are extracted usng modal extracton technques [7]. Usng the extracted natural frequences and mode shapes the FE model s updated usng Bayesan framework. When applyng Bayesan framework equaton 7 s used and pror nformaton s dvded nto four parts and each part has ts own coeffcent of pror dstrbuton (α 1, α, α 3 and α 4 ). These coeffcents are also shown n equaton 7 by settng Q equals to 4. The coeffcent α 1 s assocated wth the densty of the beam and s known to be unform for all elements and s also known to be farly accurate. The coeffcent α 1 s set to 10 to ensure that the densty of the beam s not updated sgnfcantly. The coeffcent α s assocated wth the modul of elastcty of all elements. All elements are known to have unform modulus of elastcty whch s known farly accurately. The coeffcent α s set to 10 to ensure that the modulus of elastcty s not updated sgnfcantly. The coeffcent α 3 s assocated wth the cross-sectonal areas of elements 1-4 and 7-1, whch are known farly accurately. The coeffcent α 3 s set to 10 to ensure that cross-sectonal areas of these elements are not updated sgnfcantly. The coeffcent α 4 s assocated wth cross-sectonal areas of elements 5 and 6, whch are not known accurately because they enclose the area whch was drlled to mount the exctaton devce. The coeffcent α 4 s set to 0.1 to ensure that cross-sectonal areas of these elements are updated sgnfcantly. The MCMC method s mplemented by employng Metropols acceptance crteron (see equaton 11) and 1000 samples are retaned to form a posteror dstrbuton functon ndcated by equaton 7. 7
IV. DISCUSION When natural frequences from the updated FE model are compared to those calculated from the ntal FE model as well as those from the measured natural frequency data, the results n Table 1 are obtaned. Table 1 also shows standard devatons of the dstrbutons obtaned through the use of the MCMC method to sample dstrbuton n equaton 7. The updated natural frequences are calculated usng equaton 10. Ths table shows that for all the modes the updated model s more accurate than the ntal model. Furthermore, t s observed n Table 1 that the hgher the mode the hgher s the standard devaton ndcatng that hgher modes are less certan than lower modes. Ths s consstent wth the knowledge that n general hgh frequency modes are less certan than low frequency modes. To compare analytcal mode shapes to measured mode shapes, the modal assurance crteron (MAC) s used [13]. The MAC s a crteron that represents how well two mode shapes are correlated. Two perfectly correlated mode shapes gve an dentty matrx. As a result, n ths Note the dagonals of the MAC whose elements are supposed to be equal to 1 for smlar mode shapes are used to assess the effectveness of the proposed updatng method. The dagonal of the MAC between mode shapes from experment and from the updated FE models are shown n Table. Ths table shows that the updated FE model gves more accurate mode shapes than the ntal FE model. Tables 1 and show standard devatons, and these are used to construct error bars that measure confdence ntervals of updated models. The results showng the dstrbutons of the frst natural frequency and mode shape co-ordnate are shown n Fgure. From these dstrbutons error bars may be constructed for confdence ntervals. V. CONCLUSIONS In ths Note an updatng procedure, whch uses Bayesan framework and modal propertes s mplemented usng Markov Chan Monte Carlo method. The method takes nto account of pror nformaton and has an advantage of gvng dstrbutons of predcted modal propertes. When the method s tested on expermental data t s found to sgnfcantly mprove the accuracy of fnte element models. 8
REFERENCES 1) Mottershead, J.E., and Frswell, M.I., Model updatng n structural dynamcs: a survey Journal of Sound and Vbraton, Vol. 167, No., 1993, pp. 347-375. ) Marwala, T., Fnte element model updatng usng wavelet data and genetc algorthm Journal of Arcraft, Vol. 39, No. 4, 00, pp. 709-711. 3) Neal, R.M., Probablstc nference usng Markov chan Monte Carlo methods, Unversty of Toronto, Techncal Report CRG-TR-93-1, Toronto, Canada, 1993. 4) Beck, J.L., and Katafygots, L.S., Updatng models and ther uncertantes. I: Bayesan statstcal framework Amercan Socety of Cvl Engneerng, Journal of Engneerng Mechancs, Vol. 14, No.4, 1998, pp. 455-461. 5) Katafygots, L.S., Papadmtro, C., and Lam, H.F., A probablstc approach to structural model updatng Sol Dynamcs and Earthquake Engneerng, Vol. 17, No. 7-8, 1998, pp. 495-507. 6) Metropols, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., and Teller, E., Equatons of state calculatons by fast computng machnes, Journal of Chemcal Physcs, 1, 1953, pp.1087-109. 7) Ewns, D.J., Modal testng: theory and practce, Research Studes Press, Letchworth, U.K, 1995, pp. 30-35. 8) Guyan, R.J., Reducton of stffness and mass matrces Amercan Insttute of Aeronautcs and Astronautcs, Vol. 3, No., 1965, p. 380. 9) Vapnk, V.N., The nature of statstcal learnng theory, nd edton, Sprnger-Verlag, New York, 1999, p. 314. 10) Geman, S., and Geman, D., Stochastc relaxaton, Gbbs dstrbutons and the Bayesan restoraton of mages Insttute of Electrcal and Electronc Engneers Transactons on Pattern Analyss and Machne Intellgence, Vol. 6, 1984, pp. 71-741. 11) Duane, S., Kennedy, A.D., Pendleton, B.J., and Roweth, D., Hybrd Monte Carlo, Physcs Letters, Vol. 195, 1987, pp. 16-. 1) Marwala, T., Mult-crtera method for determnng damage on structures, Masters of Engneerng Thess, Department of Mechancal and Aeronautcal Engneerng, Unversty of Pretora, Pretora, South Afrca, 1997. 13) Allemang, R.J., and Brown, D.L., A correlaton coeffcent for modal vector analyss, Proceedngs of the 1 st Internatonal Modal Analyss Conference, Socety of Expermental Mechancs, 1983, pp. 110-116. 9
Tshldz Marwala Table 1. Measured natural frequences, those calculated from an ntal FE model, updated FE model and assocated standard devatons Key: FE: fnte element Mode Experment (Hz) Intal FEM Average Standard Devaton Number (Hz) Updated (Hz) (Hz) 1 64 70 67.8 184 193 183 7.6 3 349 379 360 16.1 4 599 68 590 8.7 5 898 94 893 76.4 10
Tshldz Marwala Table. Modal assurance crteron between the measured mode shapes and FE model calculated mode shapes as well as assocated standard devatons. Key: MAC: modal assurance crteron. Mode number MAC Average MAC Standard Devaton Experment/ntal Experment/ Updated 1 0.9961 0.999 0.0011 0.9895 0.9974 0.0019 3 0.9799 0.9958 0.009 4 0.9703 0.9981 0.0011 5 0.971 0.9949 0.0044 11
Tshldz Marwala Exctaton Elastc Bands FE Element 1 3 4 5 6 7 8 9 10 11 1 13 13.4 mm 1.0 m 5.4 mm Fgure 1. A dagram showng a beam showng, ts cross-sectonal area, elastc bands used for suspenson and postons where acceleraton measurements were taken 1
100 80 Frequency 60 40 0 0 6 63 64 65 66 67 68 69 70 Natural Frequency 100 80 Frequency 60 40 0 0 1.5 1 0.5 0 0.5 1 1.5 Mode Shape Co ordnate x 10 11 Fgure. Sample dstrbutons of the natural frequences and mode shape co-ordnate of the updated FE model predcton. Key: FE: fnte element 13
LIST OF TABLES: Table 1. Measured natural frequences, those calculated from an ntal FE model, updated FE model and the assocated standard devatons. Table. Modal assurance crteron between the measured mode shapes and FE model calculated mode shapes as well as assocated standard devatons. Key: MAC: modal assurance crteron. LIST OF FIGURES: Fgure 1. A dagram showng a beam showng, ts cross-sectonal area, elastc bands used for suspenson and postons where acceleraton measurements were taken Fgure. Sample dstrbutons of the natural frequences and mode shape co-ordnate of the updated FE model predcton. 14