Perturbation Methods for the Eigencharacteristics of Symmetric and Asymmetric Systems

Abstract

Dynamic analysis for a vibratory system typically begins with an evaluation of its eigencharacteristics. However, when design changes are introduced, the eigensolutions of the system change and thus must be recomputed. In this paper, three different methods based on the eigenvalue perturbation theory are introduced to analyze the effects of modifications without performing a potentially time-consuming and costly reanalysis. They will be referred to as the straightforward perturbation method, the incremental perturbation method, and the triple product method. In the straightforward perturbation method, the eigenvalue perturbation theory is used to formulate a first-order and a second-order approximation of the eigensolutions of symmetric and asymmetric systems. In the incremental perturbation method, the straightforward approach is extended to analyze systems with large perturbations using an iterative scheme. Finally, in the triple product method, the accuracy of the approximate eigenvalues is significantly improved by exploiting the orthogonality conditions of the perturbed eigenvectors. All three methods require only the eigensolutions of the nominal or unperturbed system, and in application, they involve simple matrix multiplications. Numerical experiments show that the proposed methods achieve accurate results for systems with and without damping and for systems with symmetric and asymmetric system matrices.