A direct method for the simultaneous updating of finite element mass, damping and stiffness matrices.

This paper is mainly concerned with a model updating problem for damped structural systems, which can be described as the following inverse quadratic eigenvalue problem (IQEP) and an optimal approximate problem (OAP). Problem IQEP: Given matrices Λ = diag { λ 1 , ... , λ p } ∈ C p × p and X = [ x 1 , ... , x p ] ∈ C n × p with p ≤ n , λ i ≠ λ j for i ≠ j , i , j = 1 , ... , p , rank (X) = p and both Λ and X being closed under complex conjugation, find real symmetric matrices M, D and K such that M X Λ 2 + D X Λ + K X = 0. Problem OAP: Given real symmetric matrices M a , D a and K a , find (M ^ , D ^ , K ^) ∈ S E such that ‖ M ^ − M a ‖ N 2 + ‖ D ^ − D a ‖ N 2 + ‖ K ^ − K a ‖ N 2 = min (M , D , K) ∈ S E (‖ M − M a ‖ N 2 + ‖ D − D a ‖ N 2 + ‖ K − K a ‖ N 2) , where S E is the solution set of Problem IQEP and ‖ ⋅ ‖ N is a weighted Frobenius norm. The representation of the general solution to Problem IQEP and the explicit formula for the optimal approximate solution of Problem OAP are given by applying the singular value decomposition. A given numerical example shows that the updated model can be in good agreement with the modal test data. [ABSTRACT FROM AUTHOR]