Substructure Finite Element Model Updating of a Space Frame Structure by Minimization of Modal Dynamic Residual

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1 1 Sbstrctre Finite Element Model Updating of a Space Frame Strctre by Minimization of Modal Dynamic esidal Dapeng Zh, Xinn Dong, Yang Wang, Member, IEEE Abstract This research investigates a sbstrctre model pdating approach for large-scale strctres. The approach reqires instrmentation in only part of a large strctre (i.e. a sbstrctre), and is capable of pdating the model parameters for the sbstrctre. Prior to pdating, the entire strctral model is divided into the sbstrctre (crrently being instrmented and to be pdated) and the residal strctre. Craig-Bampton transform is adopted to condense the residal strctre sing a limited nmber of dominant mode shapes, while the sbstrctre remains at high resoltion. To pdate the condensed strctral model, physical parameters in the sbstrctre and modal parameters of the residal strctre are chosen as optimization variables; minimization of the modal dynamic residal is chosen as the optimization obective. A space frame strctre, simlating a pedestrian bridge on Georgia Tech camps, is adopted to validate the proposed approach. Abot a qarter of the bridge model is selected as the sbstrctre to be instrmented and pdated. For comparison, a conventional model pdating approach, which minimizes modal property differences, is also adopted to pdate the sbstrctre model. The reslts show that the proposed sbstrctre pdating approach, which minimizes modal dynamic residal, gives better accracy compared to the conventional approach. I. INTODUCTION Thogh significant advances have been achieved in finite element (FE) modeling of engineering strctres, differences sally exist between predictions of an FE model and experimental measrements from the actal strctre. The discrepancies are mainly cased by complexity of the actal strctre and limitations in FE modeling. For higher simlation accracy, it is essential to pdate the finite element model based on experimental measrements on the actal strctre. Nmeros FE model pdating algorithms have been developed and practically applied in the past few decades [1]. Most algorithms can be categorized into two grops, i.e. freqency-domain approaches and time-domain approaches. Freqency-domain approaches pdate an FE model sing freqency-domain strctral characteristics extracted from experimental measrements, sch as vibration modes [, 3]. On the other hand, time-domain approaches directly tilize measred time histories for model pdating [4, 5]. Nevertheless, most of the existing algorithms operate on an entire strctral model with very large amont of degrees of freedom (DOFs), ths they may sffer significant comptational challenges and convergence problem. In order to address the difficlties, some research activities have been devoted to sbstrctre model pdating, which operates on part of a large strctre with relatively small nmber of DOFs. Among freqency-domain approaches, Link adopts Craig-Bampton transform for sbstrctre modeling, and pdates the sbstrctre model by minimizing difference between simlated and experimental modal properties [6, 7]. In [8], interface force vectors are estimated sing mltiple sets of measrement; the difference between mltiple estimations is minimized with genetic algorithms for sbstrctre model pdating. Other researchers adopt freqency spectra for sbstrctre identification, by minimizing difference between simlated and experimental acceleration spectra in certain freqency range [9, 10]. Among time-domain approaches, the sbstrctre model is always bilt by considering interface copling effects as known/nknown external load. For example, some algorithms only pdate the physical parameters inside the sbstrctre by assming interface dynamic force is known or can be pre-estimated [11-13]. ecently, the seqential nonlinear least sqare estimation (SNLSE) method has been explored for sbstrctre model pdating [14]. The nknown interface copling terms are treated as nknown forces, and pdated in each time step seqentially with state variables and system parameters. Frthermore, Yen and atafygiotis present a sbstrctre identification procedre sing Bayesian theorem, withot reqiring interface measrements or excitation measrements [15]. This research investigates sbstrctre pdating sing freqency domain data. The entire strctral model is divided into the sbstrctre (crrently being instrmented and to be pdated) and the residal strctre. Craig-Bampton transform is adopted to condense the residal strctre sing a limited nmber of dominant mode shapes, while the sbstrctre model remains at high resoltion. To pdate the condensed strctral model, physical parameters in the sbstrctre and modal parameters of the residal strctre are chosen as optimization variables, and minimization of the modal dynamic residal is chosen as the optimization obective. An iterative linearization procedre is adopted for efficiently solving the optimization problem [3, 16, 17]. The approach is previosly validated with a few D strctral models. This research attempts to validate the approach with a more complicated 3D frame strctre. *esearch spported by National Science Fondation (CMMI and CMMI ), the esearch and Innovative Technology Administration of US DOT (#DTT1GUTC1) and Georgia DOT (#P1-1) Dapeng Zh, Xinn Dong, Yang Wang are with the School of Civil and Environmental Engineering, Georgia Institte of Technology, Atlanta, GA 3033 USA (corresponding athor: Yang Wang. yang.wang@ce.gatech.ed; phone: +1(404) ; fax: +1(404) )

2 The rest of the paper is organized as follows. Section II presents the formlation of sbstrctre modeling and model pdating throgh modal dynamic residal approach. Section III describes nmerical validations sing a space frame strctre. The performance of the proposed approach is compared with a conventional pdating procedre that minimizes experimental and simlated modal property differences. Finally, a smmary and discssion are provided. II. SUBSTUCTUE MODELING AND UPDATING This section presents the basic formlation for sbstrctre pdating. Section A describes sbstrctre modeling strategy following Craig-Bampton transform. Section B describes sbstrctre model pdating throgh minimization of modal dynamic residal. A. Sbstrctre modeling Figre 1 illstrates the sbstrctre modeling strategy. Sbscripts s, i, and r are sed to denote DOFs associated with the sbstrctre being analyzed, the interface nodes, and the residal strctre, respectively. The block-bidiagonal strctral stiffness and mass matrices, and M, can be assembled sing original DOFs x x x x T. s i r ss si S S 0 0 is 0 0 ir ri rr (1) M ss Msi MS S M 0 0 Mis M 0 0 M Mir M Mri Mrr Here S and M S denote entries of the stiffness and mass matrices corresponding to the sbstrctre; and M denote entries corresponding to the residal strctre; and M denote the entries at the interface DOFs and contribted by the sbstrctre; and S S M denote entries at the interface DOFs and contribted by the residal strctre. For simplicity, the formlation is provided neglecting damping, althogh the sbstrctre pdating approach can consider damping pon some modification. The dynamic behavior of the residal strctre can be approximated sing Craig-Bampton formlation [6]. The DOFs of the r residal strctre, x n, are approximated by a linear combination sing interface DOFs, x ni, and modal coordinates r n q of the residal strctre, q. r xr Txi Φrq r (3) 1 Here T rr is the Gyan static condensation matrix; ri Φ,..., r φ 1 φ n represents the mode shapes of the residal q strctre with interface DOFs fixed. The eigenvale eqation providing the mode shapes, φ (=1,..., n q ), and modal freqencies, r,, can be written as M φ (4) r, rr rr 0, 1,..., nq Althogh the size of the residal strctre may be large, the nmber of modal coordinates, n q, can be chosen as relatively small to reflect the first few dominant mode shapes only (i.e. n q << n r ). The coordinate transformation is rewritten in vector form as: i () Sbstrctre Sbstrctre DOFs x s Interface DOFs x i esidal DOFs x r Figre 1. Illstration of sbstrctre modeling strategy.

3 xi xi Γ xr q, where I Γ r T Φr (5) Sppose and M denote the new stiffness and mass matrices of the residal strctre after transformation: T Γ Γ T M Γ MΓ (6) Link described a model pdating method for both the sbstrctre and the residal strctre [7]. The sbstrctre model is pdated as n S S 0 S 0, 1 MS MS 0 M S 0, (7) where S 0 and M S 0 are the stiffness and mass matrices of the sbstrctre and sed as initial starting point in the model pdating; and correspond to physical system parameters to be pdated, sch as elastic modls and density of each sbstrctre element; n and n represent the total nmber of corresponding parameters to be pdated; and S0, M are S0, constant matrices determined by the type and location of these parameters. Sbscript 0 will be sed hereinafter to denote variables associated with the initial strctral model, which serves as the starting point for model pdating. The matrices of the condensed residal strctral model, and n 1 M in Eq.(6), each contains (n i + n q ) (n i + n q ) nmber of entries. Assming that physical changes in the residal strctre do not significantly alter the generalized eigenvectors of and M, only (n i + n q ) nmber of modal parameters are selected as pdating parameters for each condensed matrix of the residal strctral model : where and ni nq i q M M0 M 0, (8) 0 0, 1 are the modal parameters to be pdated; 0 n n 1 and M are the initial stiffness and mass matrices of the 0 condensed residal strctre model; 0, and M 0, represent the constant correction matrices formlated sing modal back-transform: where and 0, φ0, φ φ l l,t 0, 0, 0, 0, M φ φ (9) l l,t 0, 0, 0, l l T φ0,1 φ 0, ni n Φ q 0 φ0,1 φ 0, ni n (10) q are the -th generalized eigenvale and eigenvector of the initial transformed residal strctral matrices: Φ M Φ I T Φ0 0Φ 0 diag( 0,1,, 0,( n n ) ) (11) T Using all model matrices to be pdated, i.e. Eq. (7) for sbstrctre and Eq. (8) for residal strctre, the condensed entire strctral model with redced DOFs, x x q, can be pdated with variables α, β, ζ and η. For brevity, these variables T s i r n will be referred to in vector form as α, β, n ζ ni nq and ni nq η. T q i S S n ni nq n ni n q S0, ,, 1 1 0, M M M M S S n ni nq n ni n q S0, ,, 1 1 M0, 1 1 (1) (13) where S,, S,, S, and S, represent the constant sensitivity matrices corresponding to variables α, β, ζ and η, respectively.

4 B. Sbstrctre model pdating throgh minimization of modal dynamic residal To pdate the sbstrctre model, a modal dynamic residal approach is proposed in this stdy. The model pdating approach attempts to minimize modal dynamic residal of the generalized eigenvale eqation. nm ψ m, minimize α, ζ M β, η α, β, ζ, η, ψ 1 ψ, sbect to α α α ; β β β ; ζ ζ ζ ; η η η L U L U L U L U (14) where denotes any vector norm; n m denotes the nmber of measred modes from experiments; denotes the -th modal freqency extracted from experimental data; ψ denotes the entries in the -th mode shape that correspond to measred m, (instrmented) DOFs; ψ corresponds to nmeasred DOFs; α, β, ζ and η are the system parameters to be pdated (see Eq., (1) and (13)); α L, β L, ζ L and η L denote the lower bonds for vectors α, β, ζ and η, respectively; α U, β U, ζ and U η U denote the pper bonds for vectors α, β, ζ and η, respectively. Note that the sign in Eq. (14) is overloaded to represent element-wise ineqality. In smmary, and DOFs at high density, and ths, are constant in the optimization problem. The optimization variables are α, β, ζ, η and ψ are extracted sing experimental data from the sensors deployed on the sbstrctre and interface m, ψ. Eq. (14) leads to a complex nonlinear optimization problem that is generally difficlt to solve. However, if mode shapes at nmeasred DOFs, ψ, were known, Eq. (14) becomes a convex optimization problem[17, 18]. The optimization variables are system parameters (α, β, ζ and η), and the problem can be efficiently solved. Likewise, if system parameters (α, β, ζ and η) were known, Eq. (14) also becomes a convex optimization problem with variable ψ. Therefore, an iterative linearization procedre for efficiently solving the optimization problem is adopted in this stdy, similar to [3]. Figre shows the psedo code of the procedre. Each iteration step involves two operations, modal expansion and parameter pdating. At each iteration, the first operation is essentially modal expansion for nmeasred DOFs ψ, where system parameters (α, β, ζ and η) are from initial gess (starting vales) for the first iteration step, or for later steps are from the pdating reslts in the previos iteration. These parameters are ths constant in the first operation. The optimization problem over variables ψ can be conveniently coded and efficiently solved sing off-the-shelf solvers sch as CVX [19]. The second operation at each iteration is the pdating of model parameters (α, β, ζ and η) sing the expanded complete mode shapes from the first operation. Ths, ψ is held as constant in the second operation. Again, off-the-shelf solvers sch as CVX can be adopted to efficiently solve the optimization problem over variables (α, β, ζ and η). Note that when Eclidean norm (-norm) is adopted, the optimization problem withot constraints is eqivalent to a least sqare form for each operation and the detailed formlation can be fond in [0]. III. VALIDATION BY NUMEICAL SIMULATION OF A SPACE FAME BIDGE To validate the proposed approach for sbstrctre model pdating, nmerical model of a space frame bridge, which simlates a pedestrian bridge on Georgia Tech camps, is constrcted. Figre 3 shows the space frame model containing 46 nodes, each node with six DOFs. Althogh mainly a frame strctre, the segment cross bracings in top plane and two side planes are trss members. Transverse and vertical springs (k y and k z ) are allocated at both ends of the bridge to simlate non-ideal bondary conditions. In this stdy, it is assmed to have accrate information on strctral mass; strctral stiffness parameters are to be pdated. TABLE I smmarizes the strctral stiffness parameters of the model. The parameters are divided into three categories. The first category contains six parameters (starting from top in the table), which are elastic modli of the frame and trss (diagonal bracings in top plane) members along the entire length of the bridge. The second category contains ten parameters, which are the elastic modli of trss members (diagonal bracings in two side planes) for different segments. The start with α, β, ζ and η = 0 (meaning M and start with M 0 and 0 ) ; EPEAT { (i) hold α, β, ζ and η as constant and minimize over variable ψ ; () hold ψ as constant and minimize over variables α, β, ζ and η ; } UNTIL convergence ; Figre. Psedo code of the iterative linearization procedre.

5 Figre 3. Sbstrctre modeling of a space frame bridge. TABLE I. STUCTUAL STIFFNESS PAAMETES Elastic modli of members along the bridge (kips/in ) Elastic modli of side-plane diagonal bracings (trss members) for each segment (kips/in ) Spport springs (kips/in) Updating parameters Initial vale Actal vale Change (%) E 1 Longitdinal top chord 9,000 30,450 5 E Longitdinal bottom chord 9,000 30,450 5 Frame members E 3 Vertical members 9,000 7,550-5 E 4 Transverse top chord 9,000 30,450 5 E 5 Transverse & diagonal bottom chord 9,000 30,450 5 Trss members E 6 Diagonal bracings in top plane 9,000 7,550-5 E S nd segment 9,000 6, E S3 3 rd segment 9,000 6, E S4 4 th segment 9,000 6, E S5 5 th segment 9,000 7,550-5 E S6 6 th segment 9,000 7,550-5 E S7 7 th segment 9,000 7,550-5 E S8 8 th segment 9,000 4, E S9 9 th segment 9,000 6, E S10 10 th segment 9,000 7,550-5 E S11 11 th segment 9,000 7,550-5 k y1 Left transverse k z1 Left vertical k y ight transverse k z ight vertical third category contains stiffness parameters of the for types of spport springs. TABLE I provides initial (nominal) vales for all parameters, as starting point for model pdating. The table also lists actal vales, which ideally are to be identified. The relative changes from initial to actal vales, to be identified, are also listed. A sbstrctre containing first three segments from left is selected for model pdating (Figre 3). The selected sbstrctre contains 10 sbstrctre nodes and 4 interface nodes. Since each node has six DOFs and the longitdinal DOFs of the two spport nodes are constrained, the sbstrctre DOF vector is x s and the interface DOF vector is x i. For practicality, it is assmed only translational DOFs of the sbstrctre and interface nodes are instrmented with accelerometers for captring sbstrctre vibration modes; rotational DOFs are not measred. As described by formlation in Section II, no measrement is reqired on the residal strctre. For model condensation, dynamic response of the residal strctre is approximated sing twenty modal coordinates, i.e. n q = 0 in Eq.(3). As a reslt, the entire strctral model is condensed to n s +n i +n q =10 DOFs (redced from 74 DOFs in the original strctre). Since accrate strctral mass matrix is assmed to be known, mass parameters β (Eq.(7)) is not among the pdating parameters. Figre 4 shows the detailed view of the sbstrctre containing the first three segments. The sbstrctre stiffness parameters α (being pdated) inclde the five elastic modli of the frame members (E 1 ~E 5 ), the elastic modli of top bracing trss members (E 6 ), the elastic modli of side-bracing trss members at the nd and 3 rd segments (E S and E S3 ), and the spring stiffness vales at the left spport (k y1 and k z1 ). On the other hand, the residal strctre is pdated throgh modal parameters of the residal strctre with free interface (ζ, ζ 3,..., ζ 44 and η 1, η,..., η 44 ). Note that n i +n q = 44 and that modal parameter ζ 1 is not inclded. The reason is the first resonance freqency of the residal strctre with free interface is zero (corresponding to free-body movement). As a reslt, the first modal correction matrix 0,1 in Eq.(9) is a zero matrix, and so is the corresponding sensitivity matrix S,1. Using modal freqencies and sbstrctre mode shapes ( and ψ ) as "experimental data", the m, proposed modal dynamic residal approach is applied for sbstrctre model pdating. For each approach, the pdating is

6 elative error (%) Figre 4. Detailed view of the sbstrctre showing stiffness parameters to be pdated. performed assming different nmbers of measred modes are available (i.e. modes corresponding to the lowest three to six natral freqencies). TABLE II smmarizes the pdating reslts sing the proposed modal dynamic residal approach for sbstrctre model pdating. The reslts are presented in terms of relative change percentages from initial vales. For every available nmber of modes, most of the pdated parameter changes are close to the ideal percentages listed in TABLE I. The pdating reslts for E 4, the elastic modli of the transverse frame members in top plane are between -4.03% (with 6 modes) and -5.51% (with 3 modes). These reslts are most different from the actal/ideal change of +5%. The sspected reason is this parameter is less sensitive to translational DOFs. Another stdy shows if any rational DOF is measred, this parameter can be sccessflly pdated. For clear demonstration of pdating accracy, Figre 5 plots the relative errors of the pdating reslts, i.e. relative difference of pdated vales from the actal parameter vales, for different nmber of available modes. The figre shows that except for E 4, the pdating reslts accrately identify all other sbstrctre stiffness parameters. In addition, the pdating accracy generally improves when more measred modes are available. Althogh mostly small, pdating errors do exist in this nmerical example. The errors are mainly cased by the TABLE II. UPDATED PAAMETE CHANGES (%) FO SUBSTUCTUE ELEMENTS BY MINIMIZATION OF MODAL DYNAMIC ESIDUAL Nmber of available modes Longitdina l top (E 1) Frame member Trss member Spring Longitdinal Vertical Transverse Transverse Top Side in nd Side in 3 rd ky1 bottom (E ) (E 3) top (E 4) bottom (E 5) (E 6) segment (E S) segment (E S3) 3 modes modes modes modes kz E 1 E E 3 E 4 E 5 E 6 E S E S3 k y1 k z1 3 modes 4 modes 5 modes 6 modes Figre 5. elative errors of the pdated parameters by minimization of modal dynamic residal

7 elative error (%) approximations made in the formlation for sbstrctre model pdating. First, the Craig-Bampton transform sed in model condensation (Eq. (3)) adopts the static condensation matrix as the transformation matrix from interface DOFs to residal DOFs, which neglects interface dynamic contribtion. Second, the Craig-Bampton transform ses only a few dominant modes describing dynamic behavior of the residal strctre; higher-freqency modes are neglected. Third, while pdating modal parameters for the residal strctre, it is assmed that potential physical parameter changes in the residal strctre do not significantly alter the generalized eigenvectors of the residal strctral matrices (Eq.(8)). Nevertheless, the overall sbstrctre pdating performance throgh minimization of modal dynamic residal is reasonably accrate. For comparison, sbstrctre model pdating is also performed sing a conventional approach that minimizes experimental and simlated modal property differences [7]. The conventional model pdating formlation aims to minimize the difference between experimental and simlated natral freqencies, as well as the difference between experimental and simlated mode shapes of the sbstrctre. minimize α, β, ζ, η nm FE 1 MAC MAC 1 (15) FE where and represent the -th simlated (from the condensed model in Eq. (9) and (10)) and experimentally extracted freqencies, respectively; MAC represents the modal assrance criterion evalating the difference between the -th simlated and experimental mode shapes. Note that mode shape entries only corresponding to measred DOFs are compared (i.e. between FE ψ and ψ m, m, ). A nonlinear least-sqare optimization solver, lsqnonlin in MATLAB toolbox [1], is adopted to nmerically solve the optimization problem minimizing modal property differences. The optimization solver seeks a minimm throgh Levenberg-Marqardt algorithm, which adopts a search direction interpolated between the Gass-Newton direction and the steepest descent direction []. TABLE III smmarizes the pdating reslts sing the conventional approach minimizing modal property differences. Many of the pdated/identified parameter changes are apparently different from the correct/ideal vales listed in TABLE I. For clear demonstration of pdating accracy, Figre 6 plots the relative errors of the pdating reslts. The figre shows that the pdating reslts from conventional approach have mch larger errors than the reslts from the proposed modal dynamic residal approach (Figre 5), particlarly for spport spring stiffness k y1 and k z1. The conventional approach minimizing modal property differences, when sed for sbstrctre model pdating, cannot achieve a reasonable accracy in this example. The main reason is that the obective fnction sing modal property differences is not sensitive to minor changes in strctral parameters. Moreover, the obective fnction sing modal property differences is highly non-convex to strctral parameters, so the optimization algorithm easily gets stck at a local minimm. TABLE III. UPDATED PAAMETE CHANGES (%) FO SUBSTUCTUE ELEMENTS BY MINIMIZATION OF MODAL POPETY DIFFEENCES Nmber of available modes Longitdina l top (E 1) Frame member Trss member Spring Longitdinal Vertical Transverse Transverse Top Side in nd Side in 3 rd ky1 bottom (E ) (E 3) top (E 4) bottom (E 5) (E 6) segment (E S) segment (E S3) 3 modes modes modes modes kz modes 4 modes 5 modes 6 modes E 1 E E 3 E 4 E 5 E 6 E S E S3 k y1 k z1 Figre 6. elative errors of the pdated parameters by minimization of modal property differences

8 IV. SUMMAY & CONCLUSION This paper stdies sbstrctre model pdating throgh minimization of modal dynamic residal. The entire strctral model is divided into the sbstrctre (crrently being instrmented and to be pdated) and the residal strctre. Craig-Bampton transform is adopted to condense the residal strctre sing a limited nmber of dominant mode shapes, while the sbstrctre remains at high resoltion. To pdate the condensed strctral model, physical parameters in the sbstrctre and modal parameters of the residal strctre are chosen as optimization variables; minimization of the modal dynamic residal is determined as the optimization obective. An iterative linearization procedre is adopted for efficiently solving the optimization problem. The presented sbstrctre pdating method is validated throgh a space frame bridge example. Abot a qarter of the bridge is selected as the sbstrctre being instrmented and pdated. The proposed approach accrately identifies most of the parameters for the selected sbstrctre. For comparison, a conventional approach minimizing modal property differences is also applied, bt cannot achieve reasonable pdating reslts. Ftre research will contine to investigate the sbstrctre model pdating approach on more complicated strctral models, throgh both simlations and experiments. ACNOWLEDGMENT This research is sponsored by the National Science Fondation (#CMMI and #CMMI ), the esearch and Innovative Technology Administration of US DOT (#DTT1GUTC1) and Georgia DOT (#P1-1). Any opinions, findings, and conclsions or recommendations expressed in this pblication are those of the athors and do not necessarily reflect the view of the sponsors. EFEENCES [1] M. I. Friswell and J. E. Mottershead, Finite element model pdating in strctral dynamics. Dordrecht, Boston: lwer Academic Pblishers, [] B. Jaishi and W. 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