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 Ma∈SRn×n and Ka∈SRn×n be the analytical mass and stiffness matrices and Λ=diag{λ1,…,λp}∈Rp×p and X=[x1,…,xp]∈Rn×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.