Solution of eigenvalue problems for non-classically damped systems with multiple frequencies

Abstract

An efficient solution method is presented to solve the eigenvalue problem arising in the dynamic analysis of non-classically damped structural systems with multiple eigenvalues. The proposed method is obtained by applying the modified Newton-Raphson technique and the orthonormal condition of the eigenvectors to the linear eigenproblem through matrix augmentation of the quadratic eigenvalue problem. In the iteration methods, such as the inverse iteration method and the subspace iteration method, singularity may occur during the factorizing process when the shift value is close to an eigenvalue of the system. However, even though the shift value is an eigenvalue of the system, the proposed method provides non-singularity, and that is analytically proved. Since the modified Newton-Raphson technique is adapted to the proposed method, initial values are needed. Because the Lanczos method effectively produces better initial values than other methods, the results of the Lanczos method are taken as the initial values of the proposed method. Two numerical examples are presented to demonstrate the effectiveness of the proposed method and the-results are compared with those of the well-known subspace iteration method and the Lanczos method. (C) 1999 Academic Press.