Energy-preserving matrix perturbation theory for coupling dynamic analysis of flexible structures

Abstract

Aiming at the reanalysis problem of time-varying eigenvalues of force-shape coupled systems, this paper proposes an energy-preserving matrix perturbation theory (EPMPT) that can maintain the essential physical properties of the system. The classical matrix perturbation method, which employs interpolated shape functions, fails to evaluate and address solution errors promptly during the continuous perturbation process. This limitation has led to the theoretical issue of “eigenvalue drift”, a flaw that has persisted in the original method since its introduction 40 years ago. In contrast, the presented method uses the dynamic stiffness method to obtain the system eigenvalues and eigenvectors at one time, and provides a perturbation solution to the time-varying eigenvalue problem. Further combined with the J count test technology in the Wittrick-Williams algorithm, an energy-preserving method that can “self-check and self-correct” the solution was developed. The idea that “continuous perturbation should maintain force-shape dynamic self-consistency in the frequency domain” is proposed, and avoiding the energy dispersion and resulting distortion problems caused by long-term numerical simulation. To illustrate the advantage of the EPMPT, a thermally induced vibration of an aerospace structure including force-shape coupling effect, and the vibration of a flexible space solar power arrays including rigid-flexible coupling effect are investigated. Case studied elucidates that EPMPT possesses the capability to notably enhance the computational efficiency associated with generalized eigenvalue and response reanalysis problems. When juxtaposed against conventional step-by-step integration methods, EPMPT has been found to augment computational efficiency by a margin of at least 70 %, and in some instances, up to 90 %.