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∈Rn×p, a diagonal matrix Λ∈Rp×p and matrices K0∈Rr×r,M0∈Rr×r, find n×n matrices K,M such that ∥KX-MXΛ∥=min, s.t. K([1,r])=K0,M([1,r])=M0, where K([1,r]) and M([1,r]) are, respectively, the r×r leading principal submatrices of K and M. We then consider a best approximation problem: Given n×n matrices Ka,Ma with Ka([1,r])=K0,Ma([1,r])=M0, find (K^,M^)∈SE such that ∥Ka-K^∥2+∥Ma-M^∥2=inf(K,M)∈SE(∥Ka-K∥2+∥Ma-M∥2), where SE is the solution set of LSP. We show that the best approximation solution (K^,M^) is unique and derive an explicit formula for it.