An exact solution method for perturbation analysis of standard eigenvalue problems

Abstract

In the presence of perturbation, the accurate method of calculating the perturbed values of eigenvalues is an important issue to ensure the safety in engineering structures. Based on a power series expansion for the eigenvalues and eigenvectors, all the present perturbation methods neglect the higher-order terms in the calculating process, which may result in the deterioration of accuracy. However, the present methods for improving the accuracy usually result in amazing amounts of computation. The contradiction between accuracy and efficiency has existed for a long time, which made the research stagnant in recent years. In this paper, an exact solution method for perturbation analysis of standard eigenvalue problem is proposed to calculate the exact perturbed values of eigenvalues, overcoming the drawbacks of lack of accuracy due to neglecting the higher-order terms in matrix perturbation series expansion method. In the presented method, the perturbation solution equation of the standard eigenvalue problem is derived without any approximations in the derivation process. By substituting the normal eigenvalues obtained by solving the standard eigenvalue problem equation of the normal matrix into the derived perturbation solution equation, the perturbed values of eigenvalues can be calculated efficiently, which can satisfy the requirement of high accuracy and high efficiency for standard eigenvalue analysis with perturbed parameters. In three numerical examples with perturbed parameters of the undamped spring-mass system, the Bernoulli-Euler cantilever beam and the plane frame, the accuracy and efficiency of the proposed method are verified by comparing numerical results with those obtained by matrix perturbation series expansion method. The results show: (1) The relative errors of the perturbed values of eigenvalues yielded by the proposed method and the lower-order matrix perturbation series expansion method are much bigger and can be reduced reduce remarkably with the increase of the order. The results obtained by higher-order matrix perturbation series expansion method are quite close to those obtained by the proposed method. It is known that as the order increases, the results obtained by the lower-order matrix perturbation series expansion method become much closer to the exact values. It is indicated that the accuracy of the proposed method is much higher. (2) Although the results obtained by higher-order matrix perturbation series expansion method are quite close to those obtained by the proposed method, the execution time of the proposed method is much shorter than that of higher-order matrix perturbation series expansion method. It is indicated that the proposed method is much more efficient, especially for large and complex structures. To sum up, the accuracy and efficiency of the proposed method are both remarkably higher than those of the matrix perturbation series expansion method for calculating the exact perturbed values of real eigenvalues. The reason for the proposed method of high accuracy is that the perturbed values of eigenvalues are exact without approximation while the higher-order terms in the matrix perturbation series expansion method are neglected. Predictably, this method will become a powerful tool for standard eigenvalue analysis with perturbed parameters in the future. Moreover, it can be highly applicable to the fields of engineering practice and will make great contributions.