Improvement of Eigensolution Method for Nonproportionally Damped Systems by Step Length

Abstract

This study presents an efficient eigensolution method for nonproportionally damped systems. The proposed method is obtained by applying the accelerated Newton–Raphson technique and the orthonormal condition of the eigenvectors to the linearized form of the quadratic eigenproblem. In the Newton–Raphson scheme, a step length and a selective scheme are introduced to increase the convergence of the solution. The step length can be evaluated by minimizing the norm of the residual vector using the least-squares method. While the singularity may occur during the factorizing process in other iteration methods such as the inverse iteration method and the subspace iteration method if the shift value is close to an exact eigenvalue, the proposed method guarantees the nonsingularity by the orthonormal condition of eigenvectors, which can be proved analytically. A numerical example is presented to demonstrate the effectiveness of the proposed method.