An iterative updating method for undamped structural systems

Finite element model updating is a procedure to minimize the differences between analytical and experimental results and can be mathematically reduced to solving the following problem. Problem P: Let M _a ∈SR ^n×n and K _a ∈SR ^n×n be the analytical mass and stiffness matrices and Λ=diag{λ ₁,…,λ _p }∈R ^p×p and X=[x ₁,…,x _p ]∈R ^n×p be the measured eigenvalue and eigenvector matrices, respectively. Find ((hat{M}, hat{K}) in mathcal{S}_{MK}) such that (| hat{M}-M_{a} |^{2}+| hat{K}-K_{a}|^{2}= min_{(M,K) in {mathcal{S}}_{MK}} (| M-M_{a} |^{2}+|K-K_{a}|^{2})), where (mathcal{S}_{MK}={(M,K)| X^{T}MX=I_{p}, MX varLambda=K X }) and ∥?∥ is the Frobenius norm. This paper presents an iterative method to solve Problem P. By the method, the optimal approximation solution ((hat{M}, hat{K})) of Problem P can be obtained within finite iteration steps in the absence of roundoff errors by choosing a special kind of initial matrix pair. A numerical example shows that the introduced iterative algorithm is quite efficient.