Eigenvalue Problem of MDOF Non-Proportional Complex Damping System and Transfer Function by Mode Superposition

Abstract

First, we show how to obtain the eigenvalues and eigenvectors of a multi-degree of freedom system with non-proportional complex damping. These eigenvalues and eigenvectors are complex and consist of pairs of forward waves and their conjugate backward waves for numbers of masses. From the orthogonality of the eigenvectors, it follows that the seismic transfer function for each mass is a superposition of the transfer functions of the complex modes of the forward wave and is conjugate symmetric. Similarly, using the orthogonality and approximate orthogonality of the real eigenvectors, an approximate transfer function is derived by superposition of the transfer functions of the real modes. Finally, numerical calculations using a simple model are performed to demonstrate the validity of the above complex eigenvalue analysis method and the transfer function based on the superposition of complex modes, and the reliability of the transfer function based on the superposition of real modes is discussed.