VERIFICATION OF FE MODELS FOR MODEL UPDATING

VERIFICATION OF FE MODELS FOR MODEL UPDATING G. Chen and D. J. Ewns Dynacs Secton, Mechancal Engneerng Departent Iperal College of Scence, Technology and Medcne London SW7 AZ, Unted Kngdo Eal: g.chen@c.ac.uk SUMMARY: Ths paper descrbes a study of FE odel verfcaton. Before undertakng a odel updatng procedure, t s portant to deterne whether the ntal odel can be updated or not. Two ethods for odel verfcaton - a convergence check and a confguraton check - are proposed here to address the probles of dscretsaton and confguraton errors n FE odels, respectvely. The dfferences between the predctons fro a gven FE odel by usng dfferent ass atrx forulatons are seen to be closely related to the convergence range of the odel predcton. The resduals between the experental ode shapes and those obtaned by curve-fttng the experental ode shapes wth egenvectors predcted fro the FE odel are studed here. It s concluded that these resduals can reveal the exstence of confguraton errors n the FE odel. Case studes based on theoretcal odels show that these two ethods are effcent at dstngushng confguraton errors fro paraeter errors and estatng dscretsaton errors and the copensaton for the. KEYWORDS: odel updatng, odel verfcaton, odel convergence, curve-fttng. INTRODUCTION Model updatng s a step n odel valdaton process that odfes the values of paraeters n an FE odel n order to brng the FE odel predcton nto better agreeent wth experental data. In general, there are three knds of error n an FE odel whch cause the dscrepances between the odel predctons and the experental data. Dscretsaton errors coe fro odellng a contnuous structural syste by a dscrete nuercal syste. Confguraton errors arse when the structure s approxated wth soe unsutable knds of eleent. When undertakng an updatng step on an FE odel wth all these types of error, the results fro the procedure are usually not satsfactory ether the paraeters lose ther physcal eanng, or the correlaton wth the experental data does not acheve sgnfcant proveent. Model verfcaton s another step n the process of odel valdaton. In ths step, the odel s "debugged" to verfy that t s odelled accordng to the ntal requreent on the odel. In the vewpont of odel valdaton, odel verfcaton should provde a verfed odel whch can be updated to atch the experental data by only odfyng the paraeters of the odel. Thus, odel verfcaton addresses the dscretsaton and confguraton errors. In ths paper, two ethods of odel verfcaton one addressng the dscretsaton errors and the other the confguraton errors are proposed. CONVERGENCE CHECK In general, an FE odel s a dscrete nuercal odel of a contnuous structural syste. The dfferences between the actual dynac propertes of the structural syste and those predcted by the FE odel, when there s no other error n the odel, are dscretsaton errors of the odel. When ths odel s undertaken an updatng procedure, the obtaned updatng paraeter values are the copensaton for the dscretsaton errors []. If the dscretsaton

errors are large, or the requred dynac property predcton fro the odel s beyond the convergence range of the odel, the copensaton for the dscretsaton errors wll dstort the physcal eanngs of the updatng paraeters. Thus, before undertakng an updatng procedure on an FE odel, a convergence check on the odel s crtcal. Many people have wrtten papers on the estaton of dscretsaton errors of FE odels [, 3, 4]. In ths paper, the authors propose a sple and effcent way to estate the dscretsaton errors.. Luped Mass and Consstent Mass Approaches When perforng a dynac calculaton wth an FE odel, a ass atrx wll be fored for each eleent. There are two dfferent forulatons for the ass atrx of ost knds of eleent luped ass atrx forulaton and consstent ass atrx forulaton [5]. For the luped ass forulaton, the ass of an eleent s sply dvded and dstrbuted around all the grds of the eleent. For exaple, the ass atrx for a rod eleent s: 0 ρ () 0 l [ ] = In ths approach, the oent of nerta propertes of the eleent wll be greater than those of the real structure. If there are no other errors n an FE odel wth only rod eleents, the predcted natural frequences wll be lower than those of the real structure. The estated egenvalue error for the bar eleent s: 4 ωl, π + 0 π = ω 6 N N In the above equaton, the subscrpt l n ω l, s for luped ass, and N s the nuber of eleents per wavelength. For a gven odel, the hgher order of a ode, the saller ths nuber wll be. For a specfc ode, the ore eleents a odel has, the larger ths nuber wll be. Fro the above equaton, t can be seen that the luped ass approach wll produce lower natural frequences. By the consstent ass forulaton, the ass atrx of an eleent has off-dagonal coeffcents that couple the grds of the eleent. For a rod eleent, the ass atrx s: ρ l [ ] = 6 In ths way, not only the ass but also the hgher ass oents of the eleent can be represented accurately by the ass atrx constructon. The estated egenvalue error for a odel wth only rod eleents s: ωc, π + 0 π = + ω 6 N N In the above equaton, the subscrpt c n ω c, s for consstent ass. Ths ass atrx approach has an effect of stffenng the odel. If two noral ode solutons fro the sae odel are obtaned by choosng the two ass atrx forulatons separately, the natural frequency dfference for the sae ode s approxately proportonal to the dscretsaton error of the odel for ths ode. For a odel wth only rod eleents: 4 () (3) (4)

ω ω c, l, π 0 π + + 3 N N As the nuber of eleents n a odel approaches nfnty, the above equaton wll approach one, and the egenvalues predcted for the sae ode by the sae odel wth the two ass atrx approaches wll be the sae and equal to the "true" egenvalue. When the nuber of eleents n a odel s less than nfnty, the predcted egenvalues for a specfc ode by the sae odel wth the two dfferent ass atrx approaches are dfferent fro the "true" egenvalue. The egenvalue obtaned wth the luped ass approach s lower than the "true" egenvalue whle that wth the consstent ass approach s hgher than the "true" egenvalue. The two-ass-atrx-approach can be appled to ost types of eleent, such as bea and plate eleents. It s the sae for a odel wth these types of eleent the two-ass-atrxapproach for gettng noral ode soluton wll result n two sets of egenvalues and egenvectors, and the dfference of the egenvalues of the sae ode fro the dfferent sets s related to the dscretsaton error of the odel for ths ode. The egenvectors obtaned fro the two-ass-atrx-approach for the sae ode are alost the sae wth a lttle dfference n the apltudes because of the ass noralsaton process. Fro the above analyss, t can be seen that the dfference between the natural frequences of the sae ode predcted by the sae odel wth the two dfferent ass atrx approaches can be used as an ndcator for odel convergence. Once a threshold on natural frequency dfferences s set for the convergence check, the frequency range over whch the dfference between the natural frequences of the sae ode fro the two predctons s saller than the threshold can be consdered as the convergence range of the odel. 4 (5). Estate of Copensaton for Dscretsaton Errors The above coparson can be also used to estate the apltude of the copensaton of the updatng paraeters for dscretsaton errors. As an exaple, a odel wth only rod eleents s consdered here. If all the physcal paraeters of the FE odel are correct, and the odel s to be updated n order to reduce the dscrepances of the natural frequences, the Young s odulus (or the ass densty) of the odel needs to be odfed []. When the relatve natural frequency dfferences of the frst odes are taken as coprsng the resdue n the updatng equaton, the objectve of the odel updatng procedure s to nse: R = = δω ω If only one paraeter - the Young s odulus (or the ass densty) of all eleents s selected as the updatng paraeter, the fnal value of the paraeter cannot ake all odes have zero natural frequency dfference, but wll shft the natural frequences of all odes down n the consstent ass case or up n the luped ass case so that the resdue R s nsed. Thus, the odfcaton of the updatng paraeter s: δe E = δω ω Although the dfferences between the natural frequences predcted by the odel and those of the structure are unknown, they are related to the dfferences between the natural frequences predcted by the odel wth the two dfferent ass atrx forulatons. Coparng equatons (, 4) and (5), t can be shown that: (6) (7)

δω δω δω, ω cl, c, l (8) cl, ω ω The bar over ω ndcates as ean value. Fro equatons (7) and (8), the copensaton of the updatng paraeter for the esh sze can be estated as: δe E = δωc ω, δωcl ω, = cl, In general, the relaton between δω /ω and δω cl, /ω cl, s not as clear as n the case of a odel of only rod eleents and wth the luped ass and the consstent ass approaches. For soe knds of eleent n NASTRAN, the ass atrx forulatons for the coupled ass approach are not the sae as for the consstent approach. Accordng to [5], the predcted natural frequences fro the coupled ass approach have hgher precson than those fro the luped ass approach for ost types of eleent. Thus, as a conservatve estaton, the copensaton can be calculated by the followng equaton. δe E δωcl ω, = cl, The subscrpt c n the above equaton s for "coupled ass" whch n NASTRAN takes the place of "consstent ass" for ost knds of eleent. If several paraeters are selected n the updatng procedure, the copensaton for the dscretsaton errors wll be dfferent fro the above estaton. In ths case, the estaton for the copensaton can be obtaned by an updatng procedure that takes the predcton fro the odel wth one ass atrx approach as the target and updates the odel wth the other ass atrx approach. It s worth entonng that the convergence check ethod proposed here s based on the dscretsaton errors that are caused by the constructon of ass atrces. The dscretsaton errors that are caused by the constructon of stffness atrces wll ake the estated egenvalues hgher than the "true" egenvalues. However, the dscrepances caused by these errors are usually saller than those caused by the constructon of ass atrces, as shown n equatons (, 4) n the case of rod eleents. The dfference between the natural frequences of the sae ode predcted by a odel wth the two dfferent ass atrx forulatons wll cover the natural frequency dscrepancy on the sae ode caused by the constructon of stffness atrces. Thus, the convergence check and estaton of the copensaton proposed here are conservatve. 3 CONFIGURATION CHECK When an FE odel s constructed for predctng the dynac propertes of a structure, there are usually soe splfcatons ade when representng coplcated parts n the structure by standard eleents n the odel. Although n ths constructon process the splfcatons are ade accordng to the experence of the odellng engneer, n general, the effects of the splfcatons on the dynac propertes of the odel are unknown, or at least not clear for soe of the. If a splfcaton ade to the odel has the capablty to provde all the key features of the correspondng part of the structure for predctng the requred dynac propertes, even f the predcted propertes are not accurate, ths splfcaton does not cause confguraton errors. However, f a splfcaton results n a loss of soe key features and akes the odel unable to predct the requred propertes accurately, even by odfyng paraeters n the odel, ths splfcaton results n the odel havng confguraton errors. In general, f soe key features n the structure are ssng n the FE odel, and ths akes the odel unable to (9) (0)

predct the dynac propertes n a specfc frequency range wth the requred precson, the dscrepances of the predcted propertes fro those of the structure cannot be reduced sgnfcantly even by odfyng the paraeters n the odel, and the odel s sad to have confguraton errors. 3. Egenvector (Mode Shape) Curve-fttng Functon Any structure, no atter how sple or coplcated, can be consdered as a syste that has partcular dynac propertes n a specfc frequency range. Suppose that there s an FE odel that can produce the sae dynac propertes n the range, ths odel s called the structural odel n ths paper. An FE odel for predctng the dynac propertes of that structure, called the analytcal odel, can also be consdered as a syste that has ts own propertes. In general, ths analytcal odel would not be the sae as the structural odel that s defned above. The dfferences between the dynac propertes of the structural odel and those of the analytcal odel are caused by uncertantes n the odel paraeters, the dscretsaton errors and possbly the wrong confguraton of the analytcal odel. After a convergence check on an analytcal odel, the dscretsaton errors can be lted to a range and reduced by the copensaton of updatng paraeters. Thus, the object of the odel confguraton check s to dstngush confguraton errors fro paraeter errors of the analytcal odel. The as of the ethod proposed here are () to deterne whether there s any confguraton error, and () to try to fnd out the unsutable splfcatons n the FE odel that are consdered to be confguraton errors. The dynac propertes of a structural odel n a specfc frequency range can be descrbed by the egenvalues and egenvectors n that range (plus the resdual of the odes outsde the range). Suppose that all the egenvectors n the range can be expressed by a knd of functon wth the coordnate data of the DOFs as the varables. For the th eleent n the j th egenvector, ( x y z ) a j T a X j φ j = { f ( x, y, z )} () M a Nj f,, called egenvector (ode shape) curve-fttng functon for the sake of convenence, s the functon wth the coordnate data of the DOFs as the varables. a kj (k =,,, N) are constants used to curve-ft all eleents n the j th egenvector. For the th eleent n the k th egenvector, the functon has the sae forat but dfferent constants. For all DOFs and all egenvectors, t follows that: ak X T ak φ k = { f ( x, y, z )} () M a X [ ] = F {}{}{} x, y, z Nk φ [ ( )] [ A] (3)

where [ F( {}{}{} x, y, z )] = T { f ( x )}, y, z T { f ( x, y, z )} and [ A] M { ( )} T f x, y, z n n n a a = M a N For exaple, the egenvectors for the out-of-plane dsplaceent of a rectangular plate can be expressed by polynoal functons wth the varables as the two coordnates that deterne the poston of a pont n the plane of the plate. In another exaple, the egenvectors for the dsplaceent of an axsyetrc rng n the radal drecton can be expressed by trgonoetrc functons wth the spatal phase coordnate as the varable. N n equaton () s the nuber of the constants needed for the equaton to express an egenvector. It s also the order of the egenvector curve-fttng functon, and t depends on the coplexty of the egenvector and the forat of the functon used. Generally, an egenvector of a hgher order needs a greater value of N than that of a lower order. Actually, the egenvector curve-fttng functons wth constant vectors have the sae bass as the functons used n the Raylegh-Rtz ethod, [6]. In the Raylegh-Rtz ethod, the functons are also functons of coordnates. When the constants n the functons are separated n the for of equaton (), the egenvector curve-fttng functons are obtaned. 3. Confguraton Check wth Egenvector Curve-fttng Functons As one or ore paraeters of an FE odel are changed, ths wll alter the egenvectors predcted by the odel. The egenvector curve-fttng functons wll change as well. If the change n the paraeters s sall, t wll not change the basc features of the egenvectors. Ths eans that the forat and the order of the egenvector curve-fttng functons ay, possbly, rean un-changed, and only the constants of the functons wll be changed. However, f soe key features of the odel are changed or ssng, t wll affect the egenvectors sgnfcantly. Therefore, not only the constants but also the forat or the order of the egenvector curve-fttng functons wll change. Ths phenoenon s used to develop a ethod for verfyng FE odels wth confguraton errors. Consder a structure and an FE odel ("the analytcal odel") that s constructed to predct the dynac propertes of the structure. Fro a odal test on the structure, a set of ode shapes can be obtaned. Fro the analytcal odel, egenvectors of soe odes can be predcted. Suppose that egenvector curve-fttng functons can be found to express the ode shapes fro the experent and the egenvectors fro the analytcal odel. Let the ode shapes fro the experent be expressed as n equaton (3), and the egenvectors fro the analytcal odel be expressed as: A [ ] = G {}{}{} x, y, z [ ( )] [ B] a a M an L a a M a N. φ (4) where [G] s a functon atrx wth the coordnates of DOFs as the varables and [B] s a constant atrx wth the sae for as [A] n equaton (3). If the analytcal odel can predct the dynac propertes of the structure by havng a sall change for the values of soe of the paraeters n the odel, the egenvector curve-fttng functons for both the ode shapes fro the experent and the egenvectors fro the FE odel should have the sae forat and the sae order. Ths s because a sall change n only the paraeters would not change sgnfcantly the basc features of the egenvectors. Thus, we have: [ F ({}{}{} x y, z )] [ G( {}{}{} x, y, z )], (5)

If there are enough odes ncluded n equaton (4), the atrx [G] can be obtaned by: A [ G( {}{}{} x, y, z )] = [ φ ] [ B] + (6) where, [ ] + s the pseudo-nverse of the atrx. Cobnng the above two equatons and puttng the nto equaton (3) gves the followng: X A + [ φ ] = [ φ ] [ B] [ A] A = [ φ ] [ C] The condton for achevng equalty n the above equaton s that the FE odel can be ade to predct the dynac propertes of the structure just by odfyng soe of ts paraeters. Thus, ths equaton gves a necessary condton for an FE odel to be confguraton error free. If the above equaton does not hold, the FE odel has soe confguraton errors that ake the odel unable to predct the dynac propertes of the structure descrbed by the ode shapes, even after odfyng the paraeter values n the odel. In ths equaton, the ode shapes fro the experent on the structure are expressed as lnear cobnatons of the egenvectors predcted by the FE odel. If the nuber of the egenvectors equals to the nuber of DOFs and the egenvector atrx s of full rank, the atrx [C] ust exst to satsfy the equaton. Ths wll gve false nforaton for detectng confguraton error. In real cases, there ust be soe nose n the experental ode shapes, and ths wll ake the equaton not stand even when there s no confguraton error. Consderng these two factors, the practcal use of the ethod s descrbed below. For the experental ode shapes of all odes n a gven frequency range, the egenvectors predcted by an FE odel n a lttle wder frequency range than the experent frequency range are selected. Puttng the experental ode shapes and analytcal egenvectors nto equaton (7), the atrx [C] can be obtaned as, by the least-squares ethod: A + X [ C] = [ φ ] [ φ ] Then, a set of curve-ftted ode shapes can be calculated by the equaton: X A [ ] = [ φ ] [ C] (7) (8) φ ~ (9) The atrx [C] s called the "curve-fttng atrx" because t serves to curve-ft the ode shapes. The correlaton between the experental ode shapes and the curve-ftted ode shapes s obtaned usng the MAC, as follows: [ ] ~ T X X ({ φ } { φ }) ~ X T ~ X X T X ({ φ } { φ }){ φ } { φ } MAC ~ XX = (0) ( ) The values of the dagonal eleents of the MAC atrx gve the curve-ft results. By exanng both the MAC atrx and the curve-fttng atrx, [C], t s possble to verfy the odel wth or wthout confguraton errors. Frst, consder the stuaton where there s no nose on the experental ode shapes. If the value of the th dagonal eleent of the MAC atrx s close to.0, and there are only one or a few eleents n the th colun of [C] that have ther absolute values uch greater than other eleents n the sae colun, t can be sad that the ode shape of the th ode can be curveftted wth the selected egenvectors. Further, the analytcal odel, by whch the egenvectors are predcted, has no confguraton error or the confguraton errors n the odel do not affect the ablty of the odel to predct ths ode wth a hgh accuracy. If the value of the th dagonal eleent of the MAC atrx s close to 0.0, that eans that the

th ode shape cannot be curve-ftted at all by the selected egenvectors. Thus, the odel s unable to predct ths ode, even by odfyng paraeters n the odel, and the odel has confguraton errors. If the value of the th dagonal eleent of the MAC atrx s soewhere between 0.0 and.0, the value reflects the degree to whch the th ode shape can be curve-ftted by the selected egenvectors. Usually, a low MAC value, say lower than 0.8, eans that even f any odes predcted by the analytcal odel are ncluded n curve-fttng the th experental ode shape, the curveftted shape s stll not the sae as the orgnal one. Therefore, t s concluded that the odel has confguraton errors that ake the odel unable to predct ths ode. If the value of the th dagonal eleent of the MAC atrx s between 0.8 and.0, and there s only one eleent n the th colun of [C] that has a larger absolute value than the other eleents n the sae colun, there s a possblty that not enough egenvectors have been selected to curve-ft ths ode. When there s nose on the experental ode shapes, the values of the dagonal eleents n the MAC atrx wll drop to soe extent, and the nuber of eleents n coluns of the atrx [C] that have larger absolute values than others wll ncrease. Thus, when usng ths ethod n practcal cases, these effects should be taken nto account. 4 CASE STUDIES 4. Case study for convergence check An FE odel for a flat plate s shown n Fgure. There are 50 4-grd shell eleents n the odel. 0 flexble odes were predcted by ths odel whch are n the frequency range up to 400Hz. Usng the convergence check proposed n ths paper, the frequency dfferences of the odes by the two ass atrx forulatons are plotted n Fgure. Fro ths fgure, t can be easly seen that only the frst fourteen odes predcted by the odel have the natural frequency dfferences saller than 0%, whle the natural frequency dfferences for the other odes are greater than 0%. If we set the threshold to 6%, the odel can only be consdered as a converged odel to predct the frst eght odes. It cannot be used to predct other odes n the sense of odel convergence. Equaton (0) was used to estate the copensaton of updatng paraeters for the dscretsaton errors. For the frst eght odes t resulted n the followng: δe E 8 8 δω ω cl, = cl, = 0.07 Ths value s the estaton for the case wth one updatng paraeter representng the Young s odule of all bjp JG Fgure. FE odel for a plate SHIS Fgure. Convergence Chek

eleents n the odel. Another FE odel for the sae plate was fored wth the sae geoetrcal and physcal paraeters as the odel n Fgure but wth 3750 4-grd shell eleents, whch s 5 tes fner n each planar drecton than the odel shown n Fgure. Takng the frst eght odes predcted by ths fne odel as an "experental" data set, two odel-updatng procedures were undertaken on the odel shown n Fgure. In the frst procedure, there was one updatng paraeter, whch represented the Young s odules of all eleents. The obtaned updatng paraeter value was 0.057. It s reasonably close to the estated value of 0.07. In the second procedure, three updatng paraeters were selected. An updatng procedure for estatng the copensaton of these paraeters was undertaken as descrbed n Secton of ths paper. Ths procedure resulted n an estaton of the updatng paraeter copensatons as (0.56, -0.0, 0.07). The updatng procedure, whch took the egenvalues fro the fne odel as the target, resulted n the updatng paraeter values (0.4, -0.008, 0.088). Coparng wth the estaton, t can be seen that the estated values are relatvely close to the obtaned updatng paraeter values. Ths s consstent wth the analyss descrbed n ths paper. 4. Case studes for confguraton check Each case study for the confguraton check uses two FE odels wth the sae esh sze for avodng dscretsaton errors. Fgure 3 shows the odels. There are two flat plates (odelled by 4-grd shell eleents) joned by soe connectng eleents. The structural odel provdes a set of "experental" ode shape data for checkng the confguratons of the analytcal odels. Fgure 3. FE odel for confguraton check The analytcal odel n each case ay have dfferent paraeter values for eleents or a dfferent type of eleent for the jonts n order to ntroduce paraeter errors or confguraton errors nto the odel. The paraeter values (and the type for the connectng eleents) n all odels are lsted n Table. Copared wth the structural odel, Analytcal odels and have no confguraton error but do have paraeter errors, whle Analytcal odel 3 has a confguraton dfferent fro the structure odel but no paraeter errors. Group A Group B Connectng Eleents Table. Paraeters for case studes of confguraton check Para Structure odel Analytcal odel Analytcal odel Analytcal odel 3 E.06*0 3.09*0.06*0.06*0 ρ 7800 7800 7800 7800 E.06*0.06*0.06*0.06*0 ρ 7800 7800 7800 7800 kx.0*0 6.0*0 6.0*0 6 ky.0*0 6.0*0 6.0*0 6 MPC kz.0*0 4.0*0 4 8.0*0 4

The confguraton check for the three analytcal odels descrbed above was undertaken wth the results shown n Fgures 4 to 9. Fgures 4, 6, and 8 are the MAC atrces between the egenvectors predcted by the structural odel and those predcted by the analytcal odels. Fro these MAC atrces, t s dffcult, f not possble, to dentfy whch analytcal odel has confguraton errors. Fgures 5, 7, and 9 are the MAC atrces between the ntal egenvectors fro the structure odel and those curve-ftted by egenvectors fro the analytcal odels. Fro Fgure 5 t can be seen that the frst 36 "experental" ode shapes can be curve-ftted by the egenvectors predcted by Analytcal odel wth MAC values hgher than 80%. Thus, for these 36 odes, Analytcal odel has no confguraton error. The reason why the MAC values for the last few odes are sall s that only 40 analytcal egenvectors were used n the curve-fttng process. The sgnfcant change n the stffness of the connectng sprngs n Analytcal odel causes poor correlaton wth very low MAC values of soe odes when copared wth those fro the structural odel as shown n Fgure 6. However, the correlaton between the ntal "experental" ode shapes and the those curve-ftted by egenvectors fro ths analytcal odel, as shown n Fgure 7, has hgh MAC values n the dagonal eleents of the MAC atrx. Ths ndcates that the analytcal odel has no confguraton errors when ths odel s used to predct the frst 40 odes of the Structure Model. Fgure 4. Mode shape correlaton (Structural odel vs. Analytcal odel ) Fgure 5. Mode shape correlaton (curve-ftted by Analytcal odel ) Fgure 6. Mode shape correlaton (Structural odel vs. Analytcal odel ) Fgure 7. Mode shape correlaton (curve-ftted by Analytcal odel )

*MKYVISHIWLETIGSVVIPEXMSR 7XVYGXYVEPQSHIPZW%REP]XMGEPQSHIP Fgure 9. Mode shape correlaton (curve-ftted by Analytcal odel 3) Fro Table t s known that the confguraton of Analytcal odel 3 s dfferent fro that of the structural odel. In the MAC atrx shown n Fgure 9, there are soe dagonal eleents wth values lower than 80% and soe of the are even lower than 40%. Fgure 0 shows the "experental" ode shape of Mode 3 and the one curve-ftted by the egenvectors fro Analytcal odel 3. It can be seen that the analytcal odel cannot predct the relatve dsplaceents of the two plates around the jonts. Thus, the MAC atrx shown n Fgure 9 ndcates that the analytcal odel has confguraton errors. By coparng the "experental" ode shapes and the curve-ftted ones that have low MAC values, t s possble to pont out unsutable eleents n the analytcal odel. Fgure 0. The ntal (left) and the curve-ftted (rght) ode shapes for Mode 3 5 COMCLUSION Estatng dscretsaton errors and dstngushng confguraton errors fro paraeter errors are two portant requreents before an FE odel can undertake the odel-updatng procedure n a odel valdaton process. Two ethods are proposed n ths paper to address these requreents separately. Through case studes based on theoretcal odels, t can be seen that the ethods proposed here are effcent to deterne whether an FE odel fulfls these requreents. Furtherore, by usng the ethod for confguraton check, t s possble to dentfy unsutable eleents n the odel that contrbute to confguraton errors of the odel. 6 ACKNOWLEDGEMENT The authors of the paper gratefully acknowledge the fnancal and techncal support fro Rolls-Royce plc. for ths project. 7 REFERENCES [] G. Chen & D.J. Ewns A Perspectve on Model Updatng Perforance, 8 th IMAC,

February 000 [] J.E. Mottershead, M.I. Frswell & Y. Zhang On Dscretsaton Error Estates for Fnte Eleent Model Updatng, Modal Analyss: The Internatonal Journal of Analytcal and Experental Modal Analyss, (3&4) Deceber 996, p 55-64 [3] H. Ahadan, M.I. Frswell & J.E. Mottershead, Mnsaton of the Dscretsaton Error n Mass and Stffness Forulatons by an Inverse Method, Internatonal Journal for Nuercal Methods n Engneerng, Vol. 4, 998, p 37-387 [4] O.C. Zenkewcz & J.Z. Zhu, A Sple Error Estator and Adaptve Procedure for Practcal Engneerng Analyss, Internatonal Journal for Nuercal Methods n Engneerng, Vol. 4, 987, p 337-357 [5] R. H. MacNeal, The NASTRAN Theoretcal Manual, The MACNEAL-SCHWENDLER Corporaton, Deceber 97 [6] M. Geradn and D. Rxen, Mechancal Vbraton Theory and Applcaton to Structural Dynacs, second Edton, John Wley & Sons, NY, USA 997