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

An efficient solution method is presented to solve the eigenvalue problem a rising 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 eig envectors to the linear eigenproblem through matrix augmentation of the qua dratic eigenvalue problem. In the iteration methods, such as the inverse it eration method and the subspace iteration method, singularity may occur dur ing 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 analytic ally proved. Since the modified Newton-Raphson technique is adapted to the proposed method, initial values are needed. Because the Lanczos method effe ctively produces better initial values than other methods, the results of t he Lanczos method are taken as the initial values of the proposed method. T wo 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.