Efficient Solution Method of Eigenproblems for Damped Structural Systems using the Modified Newton-Raphson Technique

Abstract

An efficient solution method is presented to solve the eigenvalue problem arising in the dynamic analysis of nonproportionally damped structural systems. 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 vector inverse iteration and subspace iteration methods, 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, and this 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 are taken as the initial values of the proposed method. Two numerical examples are presented to demonstrate the effectiveness of the proposed method.