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.
Highlights
In order to analyze the vibration of any structural system, one usually first calculates the system’s eigensolutions
Eigenvalue perturbation theory can be applied to obtain the approximate eigencharacteristics of the system without resolving an entirely new eigenvalue problem. e expressions for the approximate eigensolutions of the perturbed system consist solely of matrix multiplications, which can be efficiently performed by any computer
Three methods are proposed based on the eigenvalue perturbation theory to approximate the log10(δ)
Summary
In order to analyze the vibration of any structural system, one usually first calculates the system’s eigensolutions. Simple methods are introduced that can be used to obtain the approximate eigenvalues and eigenvectors of various systems, including those with asymmetric mass, stiffness, or damping matrices and those with varying magnitudes of perturbations.
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