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 2j = -λ2j-1 ∈ C, x2j = -x2j-1 ∈ Cn for j = 1, · · ·, l, and λk ∈ R, xk ∈ Rn for k = 2l+1, · · ·, p, find real-valued symmetric matrices D,K and a real-valued skew-symmetric matrix G (that is, GT = -G) such that MaXΛ2 +(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 such that, where SE is the solution set of Problem I and {double pipe} · {double pipe} 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 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.