Efficient solution methods for free vibration of non-proportionally damped systems

Abstract

Two efficient solution methods are presented to solve the eigenvalue problem arising in the dynamic analysis of non-proportionally damped structural systems. The first 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 vector 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 nonsingularity, if the desired eigenvalue is not multiple, which is analytically proved. Because the modified Newton-Raphson technique is adapted to the proposed method, initial values are needed. The initial values of the proposed method can be obtained by the intermediate results of iteration methods or results of approximate methods. Because the Lanczos method effectively produces better initial values than other methods, the results of the Lanczos method proposed by Chen are taken as the initial values of the proposed method. However, the Lanczos method proposed by Chen is less efficient than the Lanczos method proposed in Chapter 4 because the former method transforms the quadratic eigenproblem of order 2n into the augmented problem of order n. The latter method retains the quadratic eigenproblem of order n without reformulation in the linearized problem with matrices of order 2n. If the latter method is taken as the algorithm to calculate the initial values in the proposed method, the efficiency of the proposed method can be more improved. Two numerical examples are presented to demonstrate the effectiveness of the proposed method. The second solution method is presented to solve the eigenproblem arising in the dynamic analysis of non-proportional damping systems with s...