FINITE ELEMENT MODEL UPDATING OF AN EXPERIMENTAL VEHICLE MODEL USING MEASURED MODAL CHARACTERISTICS

COMPDYN 009 ECCOMAS Thematic Confeence on Computational Methods in Stuctual Dynamics and Eathquake Engineeing M. Papadakakis, N.D. Lagaos, M. Fagiadakis (eds.) Rhodes, Geece, 4 June 009 FINITE ELEMENT MODEL UPDATING OF AN EXPERIMENTAL VEHICLE MODEL USING MEASURED MODAL CHARACTERISTICS Dimitios Giagopoulos, Evangelos Ntotsios, Costas Papadimitiou, Sotiios Natsiavas Aistotle Univesity of Thessaloniki Depatment of Mechanical Engineeing, Thessaloniki 544, Geece dgiag@auth.g, natsiava@auth.g Univesity of Thessaly Depatment of Mechanical Engineeing, Volos 38334, Geece entotsio@uth.g, costasp@uth.g Keywods: Modal Identification, Model Updating, Stuctual Identification, Multi-Objective Optimization, Stuctual Dynamics. Abstact. Methods fo modal identification and stuctual model updating ae employed to develop high fidelity finite element models of an expeimental vehicle model using acceleation measuements. The identification of modal chaacteistics of the vehicle is based on acceleation time histoies obtained fom impulse hamme tests. An available modal identification softwae is used to obtain the modal chaacteistics fom the analysis of the vaious sets of vibation measuements. A high modal density modal model is obtained. The modal chaacteistics ae then used to update an inceasingly complex set of finite element models of the vehicle. A multi-objective stuctual identification method is used fo estimating the paametes of the finite element stuctual models based on minimizing the modal esiduals. The method esults in multiple Paeto optimal stuctual models that ae consistent with the measued modal data and the modal esiduals used to measue the discepancies between the measued modal values and the modal values pedicted by the model. Single objective stuctual identification methods ae also evaluated as special cases of the poposed multiobjective identification method. The multi-objective famewok and the coesponding computational tools povide the whole spectum of optimal models and can thus be viewed as a genealization of the available conventional methods. The esults indicate that thee is wide vaiety of Paeto optimal stuctual models that tade off the fit in vaious measued quantities. These Paeto optimal models ae due to uncetainties aising fom model and measuement eos. The size of the obseved vaiations depends on the infomation contained in the measued data, as well as the size of model and measuement eos. The effectiveness of the updated models and the pedictive capabilities of the Paeto vehicle models ae assessed.

INTRODUCTION Stuctual model updating methods have been poposed in the past to econcile mathematical models, usually discetized finite element models, with expeimental data. The estimate of the optimal model fom a paameteized class of models is sensitive to uncetainties that ae due to limitations of the mathematical models used to epesent the behavio of the eal stuctue, the pesence of measuement and pocessing eo in the data, the numbe and type of measued modal o esponse time histoy data used in the econciling pocess, as well as the noms used to measue the fit between measued and model pedicted chaacteistics. The optimal stuctual models esulting fom such methods can be used fo impoving the model esponse and eliability pedictions [], stuctual health monitoing applications [-7] and stuctual contol [8]. Stuctual model paamete estimation poblems based on measued data, such as modal chaacteistics (e.g. [-6]) o esponse time histoy chaacteistics [9-0], ae often fomulated as weighted least-squaes poblems in which metics, measuing the esiduals between measued and model pedicted chaacteistics, ae build up into a single weighted esiduals metic fomed as a weighted aveage of the multiple individual metics using weighting factos. Standad optimization techniques ae then used to find the optimal values of the stuctual paametes that minimize the single weighted esiduals metic epesenting an oveall measue of fit between measued and model pedicted chaacteistics. Due to model eo and measuement noise, the esults of the optimization ae affected by the values assumed fo the weighting factos. The model updating poblem has also been fomulated in a multi-objective context [] that allows the simultaneous minimization of the multiple metics, eliminating the need fo using abitay weighting factos fo weighting the elative impotance of each metic in the oveall measue of fit. The multi-objective paamete estimation methodology povides multiple Paeto optimal stuctual models consistent with the data and the esiduals used in the sense that the fit each Paeto optimal model povides in a goup of measued modal popeties cannot be impoved without deteioating the fit in at least one othe modal goup. Theoetical and computational issues aising in multi-objective identification have been addessed and the coespondence between the multi-objective identification and the weighted esiduals identification has been established [-3]. Emphasis was given in addessing issues associated with solving the esulting multi-objective and single-objective optimization poblems. Fo this, efficient methods wee also poposed fo estimating the gadients and the Hessians [4] of the objective functions using the Nelson s method [5] fo finding the sensitivities of the eigenpopeties to model paametes. In this wok, the stuctual model updating poblem using modal esiduals is fomulated as single- and multi-objective optimization poblems with the objective fomed as a weighted aveage of the multiple objectives using weighting factos. Theoetical and computational issues ae then eviewed and the model updating methodologies ae applied to updating the finite element models of an expeimental vehicle model using acceleation measuements. Emphasis is given in investigating the vaiability of the Paeto optimal models and the vaiability of the esponse pedictions fom these Paeto optimal models. MODEL UPDATING BASED ON MODAL RESIDUALS ( k) ( ) 0 Let { ˆ, ˆ k N D= ω φ R, =,, m, k =,, N D} be the measued modal data fom a ( ) stuctue, consisting of modal fequencies ˆ k ( ) ω and modeshape components φ ˆ k at N 0 measued degees of feedom (DOF), whee m is the numbe of obseved modes and N D is the

numbe of modal data sets available. Conside a paameteized class of linea stuctual models used to model the dynamic behavio of the stuctue and let θ R θ be the set of fee N stuctual model paametes to be identified using the measued modal data. The objective in a modal-based stuctual identification methodology is to estimate the values of the paamete N set θ so that the modal data { ω ( ), ( ) d θ φ θ R, =,, m }, whee N d is the numbe of model DOF, pedicted by the linea class of models best matches, in some sense, the expeimentally obtained modal data in D. Fo this, let ( ) ( ) ˆ ( ) ˆ β θ Lφ θ φ φ ωˆ φˆ ω θ ω ε ( θ) = and ε ( θ) = ω =,, m, be the measues of fit o esiduals between the measued modal data and the model pedicted modal data fo the -th modal fequency and modeshape components, espectively, whee z = z z is the usual Euclidean nom, and ( ) ˆT β θ = φ Lφ( θ) / Lφ ( θ) T is a nomalization constant that guaanties that the measued modeshape φ ˆ at the measued DOFs is closest to the model modeshape β ( θ) Lφ ( θ ) pedicted by the paticula value of θ. N0 Nd The matix L R is an obsevation matix compised of zeos and ones that maps the Nd model DOFs to the N0 obseved DOFs. In ode to poceed with the model updating fomulation, the measued modal popeties ae gouped into two goups [3]. The fist goup contains the modal fequencies while the second goup contains the modeshape components fo all modes. Fo each goup, a nom is intoduced to measue the esiduals of the diffeence between the measued values of the modal popeties involved in the goup and the coesponding modal values pedicted fom the model class fo a paticula value of the paamete set θ. Fo the fist goup the measue of fit J ( ) θ is selected to epesent the diffeence between the measued and the model pedicted fequencies fo all modes. Fo the second goup the measue of fit J ( ) θ is selected to epesents the diffeence between the measued and the model pedicted modeshape components fo all modes. Specifically, the two measues of fit ae given by m m ω J = = () J ( θ) ε ( θ) and ( θ) = ε ( θ) () = The afoementioned gouping scheme is used in the next subsections fo demonstating the featues of the poposed model updating methodologies.. Multi-objective identification The poblem of identifying the model paamete values θ that minimize the modal o esponse time histoy esiduals can be fomulated as a multi-objective optimization poblem stated as follows [5]. Find the values of the stuctual paamete set θ that simultaneously minimizes the objectives y= J( θ) = ( J ( θ), J ( θ )) (3) subject to paamete constains θ low θ θ uppe, whee θ = ( θ,, θ Nθ ) Θ is the paamete vecto, is the paamete space, y= ( y,, y ) Y is the objective vecto, Y is the objec- Θ n φ 3

tive space and θ low and θ uppe ae espectively the lowe and uppe bounds of the paamete vecto. Fo conflicting objectives J ( θ ) and J ( ) θ thee is no single optimal solution, but athe a set of altenative solutions, known as Paeto optimal solutions, that ae optimal in the sense that no othe solutions in the paamete space ae supeio to them when both objectives ae consideed. The set of objective vectos y = J ( θ) coesponding to the set of Paeto optimal solutions θ is called Paeto optimal font. The chaacteistics of the Paeto solutions ae that the esiduals cannot be impoved in one goup without deteioating the esiduals in the othe goup. The multiple Paeto optimal solutions ae due to modelling and measuement eos. The level of modelling and measuement eos affect the size and the distance fom the oigin of the Paeto font in the objective space, as well as the vaiability of the Paeto optimal solutions in the paamete space. The vaiability of the Paeto optimal solutions also depends on the oveall sensitivity of the objective functions o, equivalently, the sensitivity of the modal popeties, to model paamete values θ. Such vaiabilities wee demonstated fo the case of two-dimensional objective space and one-dimensional paamete space in the wok by Chistodoulou and Papadimitiou [].. Weighted modal esiduals identification The paamete estimation poblem is taditionally solved by minimizing the single objective J( θ; w) = wj ( θ) + w J ( θ ) (4) fomed fom the multiple objectives Ji ( θ ) using the weighting factos wi 0, i =,, with w+ w =. The objective function J( θ ; w) epesents an oveall measue of fit between the measued and the model pedicted chaacteistics. The elative impotance of the esidual eos in the selection of the optimal model is eflected in the choice of the weights. The esults of the identification depend on the weight values used. Conventional weighted least squaes methods assume equal weight values, w = w =. This conventional method is efeed heein as the equally weighted modal esiduals method. The single objective is computationally attactive since conventional minimization algoithms can be applied to solve the poblem. Howeve, a sevee dawback of geneating Paeto optimal solutions by solving the seies of weighted single-objective optimization poblems by unifomly vaying the values of the weights is that this pocedue often esults in cluste of points in pats of the Paeto font that fail to povide an adequate epesentation of the entie Paeto shape. Thus, altenative algoithms dealing diectly with the multi-objective optimization poblem and geneating unifomly spead points along the entie Paeto font should be pefeed. Fomulating the paamete identification poblem as a multi-objective minimization poblem, the need fo using abitay weighting factos fo weighting the elative impotance of the esiduals Ji ( θ ) of a modal goup to an oveall weighted esiduals metic is eliminated. An advantage of the multi-objective identification methodology is that all admissible solutions in the paamete space ae obtained. Special algoithms ae available fo solving the multi-objective optimization poblem. Computational algoithms and elated issues fo solving the single-objective and the multi-objective optimization poblems ae biefly discussed in the next Section. 4

3 COMPUTATIONAL ISSUES IN MODEL UPDATING The poposed single and multi-objective identification poblems ae solved using available single- and multi-objective optimization algoithms. These algoithms ae biefly eviewed and vaious implementation issues ae addessed, including estimation of global optima fom multiple local/global ones, as well as convegence poblems. 3. Single-objective identification The optimization of J( θ ; w) in (4) with espect to θ fo given w can eadily be caied out numeically using any available algoithm fo optimizing a nonlinea function of seveal vaiables. These single objective optimization poblems may involve multiple local/global optima. Conventional gadient-based local optimization algoithms lack eliability in dealing with the estimation of multiple local/global optima obseved in stuctual identification poblems [,6], since convegence to the global optimum is not guaanteed. Evolution stategies (ES) [7] ae moe appopiate and effective to use in such cases. ES ae andom seach algoithms that exploe bette the paamete space fo detecting the neighbohood of the global optimum, avoiding pematue convegence to a local optimum. A disadvantage of ES is thei slow convegence at the neighbohood of an optimum since they do not exploit the gadient infomation. A hybid optimization algoithm should be used that exploits the advantages of ES and gadient-based methods. Specifically, an evolution stategy is used to exploe the paamete space and detect the neighbohood of the global optimum. Then the method switches to a gadient-based algoithm stating with the best estimate obtained fom the evolution stategy and using gadient infomation to acceleate convegence to the global optimum. 3. Multi-Objective Identification The set of Paeto optimal solutions can be obtained using available multi-objective optimization algoithms. Among them, the evolutionay algoithms, such as the stength Paeto evolutionay algoithm [8], ae well-suited to solve the multi-objective optimization poblem. The stength Paeto evolutionay algoithm, although it does not equie gadient infomation, it has the disadvantage of slow convegence fo objective vectos close to the Paeto font [5] and also it does not geneate an evenly spead Paeto font, especially fo lage diffeences in objective functions. Anothe vey efficient algoithm fo solving the multi-objective optimization poblem is the Nomal-Bounday Intesection (NBI) method [9]. It poduces an evenly spead of points along the Paeto font, even fo poblems fo which the elative scaling of the objectives ae vastly diffeent. The NBI optimization method involves the solution of constained nonlinea optimization poblems using available gadient-based constained optimization methods. The NBI uses the gadient infomation to acceleate convegence to the Paeto font. 3.3 Computations of gadients In ode to guaantee the convegence of the gadient-based optimization methods fo stuctual models involving a lage numbe of DOFs with seveal contibuting modes, the gadients of the objective functions with espect to the paamete set θ has to be estimated accuately. It has been obseved that numeical algoithms such as finite diffeence methods fo gadient evaluation does not guaantee convegence due to the fact that the eos in the numeical estimation may povide the wong diections in the seach space and convegence to the local/global minimum is not achieved, especially fo intemediate paamete values in the vicinity of a local/global optimum. Thus, the gadients of the objective functions should 5

be povided analytically. Moeove, gadient computations with espect to the paamete set using the finite diffeence method equies the solution of as many eigenvalue poblems as the numbe of paametes. The gadients of the modal fequencies and modeshapes, equied in the estimation of the gadient of J( θ ; w) in (4) o the gadients of the objectives Ji ( θ ) in (3) ae computed by expessing them exactly in tems of the modal fequencies, modeshapes and the gadients of the stuctual mass and stiffness matices with espect to θ using Nelson s method [5]. Special attention is given to the computation of the gadients and the Hessians of the objective functions fo the point of view of the eduction of the computational time equied. Analytical expessions fo the gadient of the modal fequencies and modeshapes ae used to ovecome the convegence poblems. In paticula, Nelson s method [5] is used fo computing analytically the fist deivatives of the eigenvalues and the eigenvectos. The advantage of the Nelson s method compaed to othe methods is that the gadient of the eigenvalue and the eigenvecto of one mode ae computed fom the eigenvalue and the eigenvecto of the same mode and thee is no need to know the eigenvalues and the eigenvectos fom othe modes. Fo each paamete in the set θ this computation is pefomed by solving a linea system of the same size as the oiginal system mass and stiffness matices. Nelson s method has also been extended to compute the second deivatives of the eigenvalues and the eigenvectos. The fomulation fo the gadient and the Hessian of the objective functions ae pesented in efeences [4, 0]. The computation of the gadients and the Hessian of the objective functions is shown to involve the solution of a single linea system, instead of N θ linea systems equied in usual computations of the gadient and N ( N + ) θ θ linea systems equied in the computation of the Hessian. This educes consideably the computational time, especially as the numbe of paametes in the set θ incease. 4 APPLICATION TO AN EXPERIMENTAL VEHICLE MODEL 4. Expeimental set up and modal identification Expeimental data fom a laboatoy small scale vehicle model, shown in Figue, ae used to demonstate the applicability of the poposed model updating methods and the pediction accuacy of the Paeto optimal models. The vehicle stuctue is housed at the Machine Dynamics Laboatoy of the Depatment of Mechanical Engineeing in Aistotle Univesity. Figue also shows an oveview of the expeimental set up. In paticula, the mechanical system tested consists of a fame substuctue (pats with ed, gay and black colo), simulating the fame of a vehicle. The main expeimental instuments used fo pefoming the expeimental measuements include the following: acceleometes Piezobeam 863C0, 8690C0, 8634B5 and K-beam 83A fom Kistle Instumente AG, load cell type 97Β50 fom Kistle Instumente AG, impulse foce hamme type 974Α5000 fom Kistle Instumente AG, analog to digital convete cads, PCI -455, PCI -455 Dynamic signal acquisition and PCI-655 E-seies fom National Instuments, data acquisition and signal pocessing softwae Labview 7.0. Moe details can be found in efeence []. Figue pesents details and the geometical dimensions of the fame subsystem alone. The fame substuctue is made of steel with Young s modulus E=. 0 N m, Poison s 6

Figue : Scaled vehicle model and expeimental set up. Figue : Dimensions of the fame substuctue. 7

3 atio ν = 0.3 and density ρ = 7850 kg m. Moeove, the measuement points ae indicated in Figue 3. Measuements ae collected fom 7 locations. Senso locations have been chosen in such a way so as to gathe as much infomation as possible about the stuctue s modal esponse. Figue 3: Measuement points on the fame substuctue. Using the available acceleation sensos, to measue the vibations induced by an applied impulse foce, the fequency esponse functions (FRF) of the measued DOFs ae estimated. These fequency esponse functions ae used to estimate the modal popeties using the Modal Identification Tool (MITool) [] developed by the System Dynamics Laboatoy in Univesity of Thessaly. The values of the modal fequencies, modal damping atios, modeshape components and modal paticipation factos wee estimated fom the softwae in the 0 to 70 Hz fequency bandwidth. Figue 4 compaes the measued FRFs with the FRFs pedicted by the identified optimal modal model fo a epesentative senso. As it is seen a high modal density modal model is obtained. Moeove, the fit of the measued FRF is vey good which validates the effectiveness of the modal identification softwae. 0 0 measued modal fit 0 0 FRF 0-0 - 0-3 0-4 0 40 60 80 00 0 40 60 Fequency (Hz) Figue 4: Compaison between measued and optimal modal model pedicted FRF. 8

The identified values of the modal fequencies and the modal damping atios ae epoted in Table. Twenty modes wee clealy identified in the fequency ange 0 to 70 Hz with values of modal damping atios of the ode of 0.% to.3%. Mode Fequency Damping FEM (Hz) atio (%) (Hz) 3.46.3 5.39 4.98 0.48 3.73 3 4.54 0.5 40.4 4 48.5 0.46 48.6 5 58.9 0.6 58.70 6 69. 0.7 66.73 7 69.60 0.7 70.34 8 80.0 0.4 8.94 9 86.5 0.3 84.99 0 00.3 0.09 99.8 0.7 0.4 0.35 0.50 0. 09.00 3 5.8 0. 5.8 4 3.77 0.08 5.7 5 7.8 0. 6.9 6 3.6 0.3 3.5 7 35.3 0. 33.60 8 39.4 0.09 4.80 9 48.9 0.6 5.34 0 64.46 0.0 57.34 Table : Identified and nominal FE model pedicted modal fequencies and damping atios. 4. Updating of the finite element vehicle model Detailed finite element models wee ceated that coespond to the model used fo the design of the expeimental vehicle. The stuctue was fist designed in CAD envionment and then impoted in COMSOL Multiphysics [3] modelling envionment. The models wee constucted based on the mateial popeties and the geometic details of the stuctue. The finite element models fo the vehicle wee ceated using thee-dimensional tiangula shell finite elements to model the whole stuctue. In ode to investigate the sensitivity of the model eo due to the finite element discetization seveal models wee ceated deceasing the size of the elements in the finite element mesh. The esulted twelve finite element models consist of 886 to 44985 tiangula shell elements coesponding to 6 to 36074 DOF. The convegence in the fist eleven modefequencies pedicted by the finite element models with espect to the numbe of models DOF is given in Figue 5. Accoding to the esults in Figue 5, a model of 5468 finite elements having 4636 DOF was chosen fo the adequate modelling of the expeimental vehicle. This model is shown in Figue 6 and fo compaison puposes, Table lists the values of the modal fequencies pedicted by the nominal finite element models. Compaing with the identified modal fequency values it can be seen that, with the exception of the second modal fequency, the nominal FEM-based modal fequencies ae faily close to the expeimental ones. Repe- 9

sentative modeshapes pedicted by the finite element model ae shown in Figue 7 fo the fist and the fifth mode. 0 0 0 0 Relative eo (%) 0-0 - 0-3 0-4 0 4 6 8 0 numbe of degees of feedom x 0 4 Figue 5: Relative eo of the modal fequencies pedicted by the finite element models with espect to the models numbe of degees of feedom. Figue 6: Finite element model of the expeimental vehicle consisted of 5468 tiangula shell elements and 4636 DOF. 0

(a) (b) Figue 7: Modeshapes pedicted by the finite element model fo the (a) fist mode at 5.39 Hz and (b) fifth mode at 58.70 Hz. Two diffeent paameteizations of the finite element model of the expeimental vehicle ae employed in ode to demonstate the applicability of the poposed finite element model updating methodologies, and point out issues associated with the multi-objective identification. The fist paameteized model consists of thee paametes, while the second paameteized model consist of six paametes. Fo the thee paamete model, shown in Figue 8(a), the fist paamete θ accounts fo the modulus of elasticity of the lowe pat of the expeimental vehicle, the second paamete θ accounts fo the modulus of elasticity of the pats (joints) that connect the lowe pat with the uppe pat (fame) of the expeimental vehicle, while the thid paamete θ 3 accounts fo the modulus of elasticity of the uppe pat of the expeimental vehicle. The nominal finite element model coesponds to values of θ= θ = θ3=. Fo the six paamete model, the fist paamete θ accounts fo the modulus of elasticity of the lowe pat of the expeimental vehicle, the second paamete θ accounts fo the modulus of elasticity of the pats (joints) that connect the lowe pat with the uppe pat of the expeimental vehicle, while the othe fou paametes θ 3, θ 4, θ 5 and θ 6 account fo the modulus of elasticity of the diffeent components of the uppe pat of the expeimental vehicle as shown in Figue 8(b). The nominal finite element model coesponds to values of θ= θ = θ3= θ4= θ5= θ 6 =. The paameteized finite element model classes ae updated using the eight lowest modal fequencies and modeshapes (modes and 3 to 9) obtained fom the modal analysis, excluding the second modal fequency and modeshape, and the two modal goups with modal esiduals given by (). The esults fom the multi-objective identification methodology fo the case of the thee paamete model ae shown in Figue 9. The nomal bounday intesection algoithm was used to estimate the Paeto solutions. Fo each model class and associated stuctual configuation, the Paeto font, giving the Paeto solutions in the two-dimensional objective space, is shown in Figue 9a. The non-zeo size of the Paeto font and the non-zeo distance of the Paeto font fom the oigin ae due to modeling and measuement eos. Specifically, the distance of the Paeto points along the Paeto font fom the oigin is an indication of the size of the oveall measuement and modeling eo. The size of the Paeto font depends on the size of the model eo and the sensitivity of the modal popeties to the paamete values θ []. Figues 9b-d show the coesponding Paeto optimal solutions in the thee-dimensional

paamete space. Specifically, these figues show the pojection of the Paeto solutions in the two-dimensional paamete spaces ( θ, θ), ( θ, θ3) and ( θ, θ3). It should be noted that the equally weighted solution is also computed and is shown in Figue 9. (a) (b) Figue 8: Paameteized finite element model classes of the expeimental vehicle, (a) thee paamete model and (b) six paamete model. It is obseved that a wide vaiety of Paeto optimal solutions ae obtained fo diffeent stuctual configuations that ae consistent with the measued data and the objective functions used. The Paeto optimal solutions ae concentated along a one-dimensional manifold in the thee-dimensional paamete space. Compaing the Paeto optimal solutions, it can be said that thee is no Paeto solution that impoves the fit in both modal goups simultaneously. Thus, all Paeto solutions coespond to acceptable compomise stuctual models tading-off the fit in the modal fequencies involved in the fist modal goup with the fit in the modeshape components involved in the second modal goups. The vaiability in the values of the model paametes ae of the ode of 5%, 7% and 8% fo θ, θ and θ3 espectively. It should be noted that the Paeto solutions 6 to 0 fom a one dimensional solution manifold in the paamete space that coespond to the non-identifiable solutions obtained by minimizing the second objective function. The eason fo such solutions to appea in the Paeto optimal set has been discussed in efeence [3]. Fo the case of the six paamete model, the Paeto font, giving the Paeto solutions in the two-dimensional objective space fom the multi-objective identification methodology ae shown in Figue 0(a). These esults ae compaed with the case of the thee paamete model. The six paamete model classes ae able to fit bette the expeimental esults and this is shown in Figue 0(a) obseving that the size of the Paeto font fo the case of the six paamete model classes is compaatively smalle than the thee paamete model classes and the distance fom the oigin is shote fo the six paamete model classes. The coesponding Paeto optimal solutions fo the six paamete model classes ae shown in Figue 0(b). The vaiability in the values of the model paametes ae of the ode of % fo θ, 3% fo θ, 4% fo θ3, % fo θ4, 9% fo θ5, and 0% fo θ6 espectively. It should be noted that the highest vaiability of 3% is obseved at the stiffness at the connections between the lowe and uppe pat of the vehicle. The lowest vaiability is obseved in the stiffness of the vetical membes located at the ea pat (Figue 8b) of the vehicle model.

J 0.058 0.057 0.056 0.055 0.054 0.053 0.05 3 4 5 6 7 8 9 0 Paeto Solutions Equally weighted solution 3 4 5 6 7 8 9 0 0.05 0.065 0.07 0.075 0.08 0.085 0.09 0.095 J θ 0.95 0.9 6 7890 5 4 3 0.85 0.8 0 9 8 0.75 7 6 5 4 0.7 3 0.65 0.85 0.9 0.95.05..5. θ.6.6 θ 3.4...08.06 6 9 0 8 7 3 4 5 6 7 8 5 9 0 4 3 θ 3.4...08.06 3 4 5 6 6 7 8 9 5 0 4 3 9 0 8 7.04 0.85 0.9 0.95.05..5. θ.04 0.65 0.7 0.75 0.8 0.85 0.9 0.95 θ Figue 9: Paeto font and Paeto optimal solutions fo the thee paamete model classes in the (a) objective space and (b-d) paamete space. J 0.06 0.058 0.056 0.054 34 3 4 56 7 89 Paeto Solutions fo 3 paametes Equally weighted fo 3 paametes Paeto Solutions fo 6 paametes Equally weighted fo 6 paametes Nominal model θ value.4. 0.8 θ θ θ 3 θ 4 θ 5 θ 6 θ w= 0.05 5 67890 3 4 5 67890 0 34567890 0.05 0.04 0.06 0.08 0. 0. 0.4 0.6 J 0.6 0.4 5 0 5 0 Numbe of of solution (a) (b) Figue 0: (a) Compaison of Paeto fonts between the thee paamete model classes and the six paamete model classes, (b) Paeto optimal solutions fo the six paamete model classes. The pecentage eo between the expeimental (identified) values of the modal fequencies and the values of the modal fequencies pedicted by the six paametes model fo the nominal values of the paametes, the equally weighted solution and the Paeto optimal solutions, 5, 0, 5 and 0 ae epoted in Table. Table 3 epots the MAC values between the model pedicted and the expeimental modeshapes fo the nominal, the equally weighted and the Paeto optimal models, 5, 0, 5 and 0. It is obseved that fo the modal fequencies the diffeence between the expeimental values and the values pedicted by the Paeto optimal model vay between 0.% and 5.9%. Specifically, fo the Paeto solution that coesponds to the one that minimizes the eos in the modal fequencies (fist objective function), the modal 3

fequency eos vay fom 0.% to.9%. Highest modal fequencies eos ae obseved as one moves towads Paeto solution 0 since such solutions ae based on minimizing a weighted measue of the esiduals in both the modal fequencies and the modeshapes. The eos fom the Paeto solutions ae significantly smalle than the eos obseved fo the nominal model which ae as high as 8.%. The MAC values between the expeimental modeshapes and the modeshapes pedicted by the Paeto optimal model vay between 0.84 and 0.96. Fo the Paeto solution 0, the lowest MAC value is appoximately 0.87. Relative fequency eo (%) Mode Nominal Equally Paeto solution model weighted 5 0 5 0 8.3 6.09.93 3.69 4.66 5.48 5.87 3-5.63-3.67 -.70 -.88-3.0-3.4-3.93 4 0.0 0.78.8.50.3 0.87 0.7 5 0.89 -.6-3.73-3.0 -.56 -.05 -.08 6-3.57-4.57 -. -.4-3.08-3.94-5.8 7.07.39.03..8.9.07 8 3.54 0.9 0.09-0.6-0.3 0.03-0.37 9 -.45-0.77.7.4 0.57-0.7 -.3 Table : Relative eo between expeimental and model pedicted modal fequencies Mode Nominal Equally MAC value Paeto solution model weighted 5 0 5 0 0.94 0.93 0.94 0.95 0.97 0.97 0.93 3 0.89 0.90 0.850 0.86 0.88 0.899 0.9 4 0.9 0.896 0.870 0.877 0.885 0.89 0.895 5 0.88 0.948 0.959 0.960 0.959 0.954 0.947 6 0.909 0.958 0.958 0.960 0.96 0.960 0.958 7 0.907 0.866 0.897 0.885 0.875 0.87 0.866 8 0.88 0.958 0.844 0.90 0.938 0.953 0.958 9 0.909 0.955 0.90 0.99 0.936 0.948 0.956 Table 3: MAC values between expeimental and model pedicted modeshapes The identified vaiability in Paeto optimal solutions has demonstated in [3] to consideably affect the vaiability in the esponse pedictions. Heein, the fequency esponse functions (FRF) pedicted by the Paeto optimal solutions ae compaed in Figue to the fequency esponse function computed diectly fom the measued data at senso locations 7 (see Figue 3) in the fequency ange [0Hz, 90Hz] used fo model updating. Compaed to the initial nominal model, it is obseved that the updated Paeto optimal models tend to consideably impove the fit between the model pedicted and the expeimentally obtained FRF in most fequency egions close to the esonance peaks. Also, it can be clealy seen that a elatively lage vaiability in the pedictions of the fequency esponse functions fom the diffeent Paeto optimal models is obseved which is due to the elatively lage vaiability in the identified Paeto optimal models. This vaiability is impotant to be taken into consideation 4

in the pedictions fom updated models in model updating techniques. It should be noted that besides fequency esponse functions, othe moe impotant esponse quantities of inteest ae the eliability of the stuctue against vaious modes of failue, as well as the fatigue accumulation and lifetime of the stuctue subjected to stochastic loads aising fom the vaiability in oad pofiles. 0 0 Expeimental Nominal model Paeto model Paeto model 5 Paeto model 0 Paeto model 5 Paeto model 0 0 0 FRF 0-0 - 0-3 0-4 0 30 40 50 60 70 80 90 Fequency (Hz) Figue : Compaison between measued and the pedicted FRF fom the Paeto models, 5, 0, 5, 0. The discepancies between the expeimental and the model pedicted modal fequencies as well as the deviations of the MAC values fom unity ae due to (a) the model eo, (b) the paameteization employed, and (c) the measuement eos. Specifically, the model eo aises fom the assumptions used to constuct the mathematical model of the stuctue. Fo the laboatoy vehicle model one should emphasize that the souces of model eo ae due to the assumptions used to build up the connections between the vaious pats compising the stuctue, as well as the use of shell elements to epesent the membes of the stuctue and the connections between the lowe and the uppe pat of the model. Also, elative small eos esults fom the size of the finite elements employed in the discetization scheme. Anothe souce that affects the model updating esults and the eos between the model pedictions and the measuements is the paameteization employed. An exhaustive seach fo the optimal paameteization scheme (numbe and type of paametes) has not been exploed in this wok. Howeve, intoducing moe paametes to be updated will impove the fit and educe the eos between the pedictions and the expeiment. Howeve, these eos cannot be eliminated and the emaining eos could be attibuted mainly to the model eo. The esulting eos povide guidance fo modifying the assumptions made to build the model in an effot to futhe impove modeling and obtain highe fidelity models able to adequately epesent the behavio of the expeimental vehicle stuctue in the fequency ange of inteest. 5 CONCLUSIONS Methods fo modal identification and stuctual model updating wee used to develop high fidelity finite element models of an expeimental vehicle model using acceleation measuements. A multi-objective stuctual identification method was used fo estimating the paametes of the finite element stuctual models based on minimizing two goups of modal esiduals, one associated with the modal fequencies and the othe with the modeshapes. The 5

constuction of high fidelity models consistent with the data depends on the assumptions made to build the mathematical model, the finite elements selected to model the diffeent pats of the stuctue, the dicetization scheme contolling the size of the finite elements, as well as the paameteization scheme used to define the numbe and type of paametes to be updated by the methodology. The multi-objective identification method esulted in multiple Paeto optimal stuctual models that ae consistent with the measued (identified) modal data and the two goups of modal esiduals used to measue the discepancies between the measued modal values and the modal values pedicted by the model. A wide vaiety of Paeto optimal stuctual models was obtained that tade off the fit in vaious measued modal quantities. These Paeto optimal models ae due to uncetainties aising fom model and measuement eos. The size of obseved vaiations in the Paeto optimal solutions depends on the infomation contained in the measued data, as well as the size of model and measuement eos. The vaiability in the Paeto optimal vehicle models esults in consideable vaiability in the pedictions of the esponse and eliability fom these stuctual models. Such vaiability should be taken into consideation when using the updated models fo pedictions. ACKNOWLEDGEMENTS This eseach was co-funded 75% fom the Euopean Union (Euopean Social Fund), 5% fom the Geek Ministy of Development (Geneal Secetaiat of Reseach and Technology) and fom the pivate secto, in the context of measue 8.3 of the Opeational Pogam Competitiveness (3 d Community Suppot Famewok Pogam) unde gant 03-ΕΔ-54 (PENED 003). This suppot is gatefully acknowledged. REFERENCES [] C. Papadimitiou, J.L. Beck, L.S. Katafygiotis, Updating obust eliability using stuctual test data. Pobabilistic Engineeing Mechanics, 6, 03-3, 00. [] C.P. Fitzen, D. Jennewein, T. Kiefe, Damage detection based on model updating methods. Mechanical Systems and Signal Pocessing, (), 63-86, 998. [3] A. Teughels, G. De Roeck, Damage detection and paamete identification by finite element model updating. Achives of Computational Methods in Engineeing, (), 3-64, 005. [4] M.W. Vanik, J.L. Beck, S.K. Au, Bayesian pobabilistic appoach to stuctual health monitoing. Jounal of Engineeing Mechanics (ASCE), 6, 738-745, 000. [5] E. Ntotsios, C. Papadimitiou, P. Panetsos, G. Kaaiskos, K. Peos, Ph. Pedikais, Bidge health monitoing system based on vibation measuements. Bulletin of Eathquake Engineeing, doi: 0.007/s058-008-9067-4, 008. [6] P. Metallidis, G. Veos, S. Natsiavas, C. Papadimitiou, Fault detection and optimal senso location in vehicle suspensions. Jounal of Vibation and Contol, 9, 337-359, 003. [7] P. Metallidis, I. Stavakis, S. Natsiavas, Paametic identification and health monitoing of complex gound vehicle models. Jounal of Vibation and Contol, 4, 0-036, 008. 6

[8] K.V. Yuen, J.L. Beck, Reliability-based obust contol fo uncetain dynamical systems using feedback of incomplete noisy esponse measuements. Eathquake Engineeing and Stuctual Dynamics, 3 (5), 75-770, 003. [9] J.L. Beck, L.S. Katafygiotis, Updating models and thei uncetainties- I: Bayesian statistical famewok. Jounal of Engineeing Mechanics (ASCE), 4 (4), 455-46, 998. [0] D. Giagopoulos, C. Salpistis, S. Natsiavas, Effect of nonlineaities in the identification and fault detection of gea-pai systems. Intenational Jounal of Non-Linea Mechanics, 4, 3-30, 006. [] Y. Haalampidis, C. Papadimitiou, M. Pavlidou, Multi-objective famewok fo stuctual model identification. Eathquake Engineeing and Stuctual Dynamics, 34 (6), 665-685, 005. [] K. Chistodoulou, C. Papadimitiou, Stuctual identification based on optimally weighted modal esiduals. Mechanical Systems and Signal Pocessing,, 4-3, 007. [3] K. Chistodoulou, E. Ntotsios, C. Papadimitiou, P. Panetsos, Stuctual model updating and pediction vaiability using Paeto optimal models. Compute Methods in Applied Mechanics and Engineeing, 98 (), 38-49, 008. [4] E. Ntotsios, C. Papadimitiou, Multi-objective optimization algoithms fo finite element model updating. Intenational Confeence on Noise and Vibation Engineeing (ISMA008), Katholieke Univesiteit Leuven, Leuven, Belgium, Septembe 5-7, 008. [5] R.B. Nelson, Simplified calculation of eigenvecto deivatives. AIAA Jounal, 4 (9), 0-05, 976. [6] A. Teughels, G. De Roeck, J.A.K. Suykens, Global optimization by coupled local minimizes and its application to FE model updating. Computes and Stuctues, 8 (4-5), 337-35, 003. [7] H. G. Beye, The theoy of evolution stategies, Belin, Spinge-Velag, 00. [8] E. Zitzle, L. Thiele, Multi-objective evolutionay algoithms: A compaative case study and the stength Paeto appoach. IEEE Tansactions on Evolutionay Computation, 3, 57-7, 999. [9] I. Das, J.E. J. Dennis, Nomal-Bounday Intesection: A new method fo geneating the Paeto suface in nonlinea multi-citeia optimization poblems. SIAM Jounal of Optimization, 8, 63-657, 998. [0] E. Ntotsios, C. Papadimitiou, Multi-objective optimization famewok fo finite element model updating and esponse pediction vaiability. Inaugual Intenational Confeence of the Engineeing Mechanics Institute (EM08), Depatment of Civil Engineeing Univesity of Minnesota, Minneapolis, Minnesota, May 8-, 008. [] D. Giagopoulos, S. Natsiavas, Hybid (numeical-expeimental) modeling of complex stuctues with linea and nonlinea components. Nonlinea Dynamics, 47, 93-7, 007. [] E. Ntotsios, Expeimental modal analysis using ambient and eathquake vibations: Theoy, Softwae and Applications. MS Thesis Repot No. SDL-09-, Depatment of Mechanical and Industial Engineeing, Univesity of Thessaly, Volos, 008. [3] COMSOL AB, COMSOL Multiphysics use s guide, 005, [http://www.comsol.com/]. 7