An iterative updating method for damped gyroscopic systems

The problem of updating damped gyroscopic systems using measured modal data can be mathematically formulated as following two problems. Problem I: Given Ma ∈ Rn×n,Λ = diag{λ1, · · · , λp} ∈ Cp×p, X = [x1, · · · , xp] ∈ Cn×p, where p < n and both Λ and X are closed under complex conjugation in the sense that λ2j = λ̄2j−1 ∈ C, x2j = x̄2j−1 ∈ C for j = 1, · · · , l, and λk ∈ R, xk ∈ R for k = 2l+1, · · · , p, find real-valued symmetric matrices D,K and a real-valued skew-symmetric matrix G (that is, G = −G) such that MaXΛ+(D+G)XΛ+KX = 0. Problem II: Given real-valued symmetric matrices Da,Ka ∈ Rn×n and a real-valued skew-symmetric matrix Ga, find (D̂, Ĝ, K̂) ∈ SE such that ‖D̂−Da‖+‖Ĝ−Ga‖+‖K̂−Ka‖ = min(D,G,K)∈SE(‖D− Da‖ + ‖G − Ga‖ + ‖K − Ka‖), where SE is the solution set of Problem I and ‖ · ‖ is the Frobenius norm. This paper presents an iterative algorithm to solve Problem I and Problem II. By using the proposed iterative method, a solution of Problem I can be obtained within finite iteration steps in the absence of roundoff errors, and the minimum Frobenius norm solution of Problem I can be obtained by choosing a special kind of initial matrices. Moreover, the optimal approximation solution (D̂, Ĝ, K̂) of Problem II can be obtained by finding the minimum Frobenius norm solution of a changed Problem I. A numerical example shows that the introduced iterative algorithm is quite efficient. Keywords—model updating, iterative algorithm, gyroscopic system, partially prescribed spectral data, optimal approximation.