A model updating method for undamped structural systems

摘要

In this paper, we first give the representation of the general solution of the following least-squares problem (LSP): Given a full column rank matrix X epsilon R-n x p, a diagonal matrix Lambda epsilon R-p x p and matrices K-0 epsilon R-r x r, M-0 epsilon R-r x r, find n x n matrices K . M such that parallel to K X - M X Lambda parallel to = min, s.t. K(vertical bar l, r vertical bar) = K-0. M(vertical bar l. r vertical bar) = M-0, where K(vertical bar l, r vertical bar) and M(vertical bar l, r vertical bar) are, respectively, the r x r leading principal submatrices of K and M. We then consider a best approximation problem: Given n x n matrices Ka. M-a with K-a(vertical bar l, r vertical bar) = K-0. M-a(vertical bar l, r vertical bar) = M-0, find ((K) over cap, (M) over cap) epsilon S-E such that parallel to K-a - (K) over cap parallel to(2) + parallel to M-a - (M) over cap parallel to(2) = inf ((K . M)epsilon SE) (parallel to K-a - K parallel to(2) + parallel to M-a - M parallel to(2)), where S-E is the solution set of LSP. We show that the best approxiniation solution ((K) over cap, (M) over cap) is unique and derive an explicit formula for it. (C) 2007 Elsevier B.V. All rights reserved.