The Direct Updating Methods For Structural Dynamic Models

Model updating is an important research topic in the structural dynamics.This dissertation studies some model updating problems on damped structural systems and undamped acoustic-structural systems by means of the theory of matrix decomposition.For the damped structural systems,a direct method for the simultaneous updating of finite element mass,damping and stiffness matrices using complex modal measured data is proposed.The model updating problem is equivalent to solving the following inverse quadratic eigenvalue problem(IQEP):Given matrices Λ=diag{λ1,…,λp}∈Cp×p and X=[x1,…,xp]∈Cn×p with p≤n,λi≠λj for i≠j,i,j=1,…,p,rank(X)=p and both A and X being closed under complex conjugation,find real symmetric matrices M,D and K such that MXΛ2+DXΛ+KX=0;and an optimal approximate problem(OAP):Let Ma,Da and Ka be the mass,damping and stiffness matrix on dampd structure,find(M,D,K)∈SE such that ‖M-Ma‖N2+‖DDa‖N2+‖K-Ka‖N2=min(M,D,K)∈SE(‖M-Ma‖N2+‖D-Da‖N2+‖K-Ka‖N2),where SE is the solution set of Problem IQEP and‖·‖N is a weighted Frobenius norm.The representation of the general solution of Problem IQEP and the explicit formula for the optimal approximate solution of Problem OAP are given by applying the singular value decomposition and matrix derivatives.A given numerical example shows that the measured modal data are well integrated into the updated model.For the undamped acoustic-structural systems,a direct method for the simultaneous updating of finite element mass and stiffness matrices is proposed.The desired coefficient matrix properties,including symmetry,characteristic equation and orthogonality,are imposed as constraints to form the matrix minimization problem.The general solution of constrained matrix equations are obtained by means of the generalized singular value decomposition.Furthermore,the optimal corrected mass and stiffness matrices are obtained by using Kronecker product and matrix derivatives.A numerical example show that the proposed method is effective.