A direct updating method for damped gyroscopic systems using measured modal data

The matrix model updating problem (MMUP), considered in this paper, concerns updating a symmetric second-order finite element model so that the updated model can reproduce a given set of measured eigenvalues and eigenvectors by replacing the corresponding ones from the original model, and preserves the symmetry of the original model. The MMUP can be mathematically formulated as following two problems. Problem 1: Given M(a) is an element of R(nxn), Lambda = diag{lambda(1),...,lambda(p)} is an element of- C(pxp), X = [x(1),..., x(p)] c- C(nxp), where p < n and both Lambda and X are closed under complex conjugation in the sense that lambda(2j) = <(lambda)over bar>(2j-1) is an element of C, x(2j) = (x) over bar (2j-1) is an element of C(n) for j = 1,..., l, and lambda(k) is an element of R, x(k) is an element of R(n) for k = 2l + 1,...,p, find real-valued symmetric matrices D, K and a real-valued skew-symmetric matrix G (that is, G(T) = -G) such that MaX Lambda(2)+ (D + G)X Lambda + KX = 0. Problem 2: Given real-valued symmetric matrices D(a), K(a) is an element of R(nxn) and a real-valued skew-symmetric matrix G(a), find ((D) over cap,(G) over cap,(K) over cap) is an element of S(E) such that parallel to(D) over cap - D(a)parallel to(2)+ parallel to(G) over cap - G(a)parallel to(2) + parallel to(K) over cap - K(a)parallel to(2) = min((D,G,K)is an element of SE) (parallel to D - D(a)parallel to(2) + parallel to G - Ga parallel to(2) + parallel to K - K(a)parallel to(2)), where S(E) is the solution set of Problem 1 and parallel to . parallel to is the Frobenius norm. We provide the representation of the general solution of Problem 1 and show that the optimal approximation solution ((D) over cap,(G) over cap,(K) over cap) is unique and derive an explicit formula for it. (C) 2009 Elsevier Inc. All rights reserved.