Abstract
The completeness of the complex response spectrum method for both formally underdamped and overdamped modes is theoretically proved, and the physical meanings of the decoupled modes as well as involved parameters are recognized and clarified in this paper. For the system with relatively large non-classical damping, the eigenvalue pairs generated by the complex mode decomposition method are real and the so-called modal damping ratios are larger than unity. In this paper, we firstly clarified that the decoupled modes are virtual and the so-called modal frequency and damping ratio are mathematical parameters that have no physical meaning. Then, the completeness of the complex response spectrum method for both formally underdamped and overdamped modes is rigorously proved by allowing the “damping frequency” to be an imaginary number. For the virtually overdamped modes, Duhamel integral involved in the calculation for formally underdamped modes automatically convert to hyperbolic Duhamel integral. A numerical example taken from the published literature is given to verify this method. Structural responses for the system with coupled damping under multi-support seismic excitations are further analyzed and numerical results indicate the accuracy of complex response spectrum method.
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