A new model updating method for damped structural systems

In this paper, the following two are considered: Problem IQEP Given M_a∈ SR ^n×n, Λ=diag{λ₁, …, λ_p}∈ C ^p×p, X=x₁, …, x_p]∈ C ^n×p, and both Λ and X are closed under complex conjugation in the sense that $\lambda_{2j} = \bar{\lambda}_{2j-1} \in {\mathbf{C}}In this paper, the following two are considered: Problem IQEP Given M_a∈ SR ^n×n, Λ=diag{λ₁, …, λ_p}∈ C ^p×p, X=x₁, …, x_p]∈ C ^n×p, and both Λ and X are closed under complex conjugation in the sense that $\lambda_{2j} = \bar{\lambda}_{2j-1} \in {\mathbf{C}}$, x_2j=x?_2j?1∈ C ⁿ for j = 1,…,l, and λ_k∈ R , x_k∈ R ⁿ for k=2l+1,…,p, find real‐valued symmetric (2r+1)‐diagonal matrices D and K such that ∥M_aXΛ²+DXΛ+KX∥=min. Problem II Given real‐valued symmetric (2r+1)‐diagonal matrices D_a, K_a∈ R ^n×n, find $(\hat{D},\hat{K}) \in {\mathscr{S}}_{DK}$ such that $\|\hat{D}-D_a \|^2+ \| \hat{K}-K_a \|^2=\rm{inf}_{(D,K) \in {\mathscr{S}}_{DK}}(\|D-D_a\|^2+\|K-K_a\|^2)$, where ??_DK is the solution set of IQEP. By applying the Kronecker product and the stretching function of matrices, the general form of the solution of Problem IQEP is presented. The expression of the unique solution of Problem II is derived. A numerical algorithm for solving Problem II is provided. Copyright © 2009 John Wiley & Sons, Ltd.