Complex Mode Decomposition Response Spectrum Method of Tuned Mass Damping Structures

To address the issue of low computational accuracy caused by the forced decoupling of tuned mass damper (TMD) damping structures with non-proportional damping matrices in real modal decomposition response spectrum methods, this paper introduces the complex modal theory. Utilizing the orthogonality and superposition of complex modes in the state space, and based on the complex complete quadratic combination (CCQC) method and response spectra, a complex modal decomposition response spectrum method suitable for shear-type TMD damping structures is proposed. To align with seismic design codes, this paper derives a real-number form of the standard seismic action expression based on the complex conjugate characteristics of complex modes and the relationship between the parameter matrix and the state equation coefficient matrix. Numerical analysis compares errors of real modal and complex modal response spectrum methods, showing that the error of the real modal method exceeds 10% , while the error of the complex modal method is within 5% . The study indicates that the complex modal method, which considers the effects of non-proportional damping and modal velocity, has higher computational accuracy and stability than the real modal method. The standard seismic action expression matching the design code form is simple and compatible with the design response spectrum.