A new model updating method for damped structural systems

✍ Scribed by Jiashang Jiang; Yongxin Yuan

• Publisher: John Wiley and Sons
• Year: 2009
• Tongue: English
• Weight: 95 KB
• Volume: 32
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.1129

✦ Synopsis

Abstract

In this paper, the following two are considered:

Problem IQEP

Given M~a~∈SR^n×n^, Λ=diag{λ~1~, …, λ~p~}∈C^p×p^, X=[x~1~, …, x~p~]∈C^n×p^, and both Λ and X are closed under complex conjugation in the sense that \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$\lambda_{2j} = \bar{\lambda}_{2j-1} \in {\mathbf{C}}$\end{document}, x~2__j__~=x̄~2__j__−1~∈C^n^ for j = 1,…,l, and λ~k~∈R, x~k~∈R^n^ for k=2__l__+1,…,p, find real‐valued symmetric (2__r__+1)‐diagonal matrices D and K such that ∥M~a~__X__Λ^2^+__DX__Λ+KX∥=min.

Problem II

Given real‐valued symmetric (2__r__+1)‐diagonal matrices D~a~, K~a~∈R^n×n^, find
\documentclass{article}\footskip=0pc\usepackage{amssymb}\usepackage[mathscr]{euscript}\pagestyle{empty}\begin{document}$(\hat{D},\hat{K}) \in {\mathscr{S}}_{DK}$\end{document} such that \documentclass{article}\usepackage{amssymb}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document}$|\hat{D}-D_a |^2+ | \hat{K}-K_a |^2=\rm{inf}_{(D,K) \in {\mathscr{S}}_{DK}}(|D-D_a|^2+|K-K_a|^2)$\end{document}, where 𝒮~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.