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

摘要

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 ∈ R~(n×n), Λ = diag{λ_1,…,λ_p} ∈C~(p×p), X = [x_1,…,x_p] ∈C~(n×p), where p < n and both Λ and X are closed under complex conjugation in the sense that λ_(2j) = λ_(2j-1) ∈ C, x_(2j) = x_(2j-1) ∈ C~n for j = 1,…,l, and λ_k ∈ R, x_k ∈ R~n for k = 21 + 1,… ,p, find real-valued symmetric matrices D, K and a real-valued skew-symmetric matrix G (that is, G~T = -G) such that M_aXΛ~2 + (D + G)XΛ + KX = 0. Problem 2: Given real-valued symmetric matrices D_a.K_a ∈ R~(n×n) and a real-valued skew-symmetric matrix G_a, find (D,G,K) ∈S_E such that ||D-D_a||~24 ||G-G_a||~2 + ||K-K_a||~2 = min_((D,G,K)∈S_E)(||D-D_a||~2 + ||G-G_a||~2 + ||K-K_a||~2), where S_E is the solution set of Problem 1 and || ? || is the Frobenius norm. We provide the representation of the general solution of Problem 1 and show that the optimal approximation solution (D, G, K) is unique and derive an explicit formula for it.