New Model Correcting Method for Quadratic Eigenvalue Problems Using a Symmetric Eigenstructure Assignment

Abstract

Finite element model correction of quadratic eigenvalue problems (QEPs) using a symmetric eigenstructure assignment technique is proposed by Zimmerman and Widengren 1989, which incorporates the measured model data into the finite element model to produce an adjusted finite element model on the damping and stiffness matrices that matches the experimental model data, and minimizes the distance between the analytical and corrected models. In this paper, we mainly develop an efficient algorithm to solve the corresponding optimization problem in a least-squares sense. The resulting matrices obtained by the new method are necessary and sufficient to the optimization problem. Furthermore, the proposed algorithm only needs to solve a linear system and totally requires O(nm 2 ) flops, where n is the size of coefficient matrices of the QEP and m is the number of the measured modes. The numerical results show that the new method is reliable and attractive.

New model updating method for damped structural systems

“…Suppose that M a ∈ R n×n , C 0 , K 0 ∈ R r×r , X ∈ R n×k and Λ ∈ R k×k , where rank(Λ) = k, and Λ is a block diagonal matrix. Let the partitions of the matrices X , M a X Λ 2 , C and K be (7)- (9). Assume that the SVDs of the matrices X 2 , ΣV T 1 ΛV 2 and the QR-decomposition of the X are given by (12), (15) and (20), respectively.…”

“…For damped systems, the theory and computation were proposed by Friswell, Inman and Pilkey [8], Kuo, Lin and Xu [10] recently have proposed a direct method which seems more efficient and reliable. Another line of thought is to update with symmetric low-rank correction of damping and stiffness matrices [9]. However, the system matrices are adjusted globally in these methods.…”

New symmetry preserving method for optimal correction of damping and stiffness matrices using measured modes

“…They considered the mass matrix to be exact and updated the damping and stiffness matrices by using the measured modal data as a reference. Inman and Pilkey [10], Kuo, Lin and Xu [11,12],Yuan and Dai [13], Lee and Eun [14,15] recently have proposed a direct method which seems more efficient and reliable. Another line of thought is to update with symmetric low-rank correction of damping and stiffness matrices [11].…”

A CG-Type Method for Inverse Quadratic Eigenvalue Problems in Model Updating of Structural Dynamics

“…All the existing methods proposed in [11,12,20] aim at updating directly the mass, stiffness, and damping matrices in such a way that the updated model remains symmetric and reproduces the measured data as accurately as possible, but cannot guarantee that the updating with minimal changes. The method proposed in [3,5] have the additional important feature that the eigenvalues and eigenvectors which are not updated remain unchanged by the updating procedure. This guarantees that "no spurious modes appear in the frequency range of interest".…”