An algorithm for solving frequency responses of a system with Rayleigh damping

This paper proposes a fast algorithm for solving frequency responses of a system with Rayleigh damping. The algorithm combines modal superposition with a model order reduction. Only the modes corresponding to a frequency range which is a little bigger than that of interest are used for modal superposition. Complementary part of contribution of computed modes for frequency response is calculated by a model order reduction method. The Ritz vectors resulted from a shifted undamped system is used in the reduction. Their numbers can adaptively be determined by monitoring the Euclidean norms of the relative residuals associated with the function to be evaluated only at the two endpoints of the frequency band of interest. Eigenproblem of the reduced system is then solved to yield the set of orthogonalized Ritz vectors. The final approximate solution can be represented as superposition of computed modes and orthogonalized Ritz vectors. The accuracy, robustness and computational efficiency of the proposed method are highlighted with the solutions of frequency response problems associated with two large finite element models. It is shown that the present method can remarkably reduce the CPU time required by the direct method to frequency response analysis.