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

Abstract

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 2 j = λ ¯ 2 j - 1 ∈ C , x 2 j = x ¯ 2 j - 1 ∈ C n for j = 1 , … , l , and λ k ∈ R , x k ∈ R n for k = 2 l + 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 a X Λ 2 + ( D + G ) X Λ + 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 ‖ 2 + ‖ 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.