A Comparative Study on Dynamic Response Analysis Methods for Seismically Isolated Structures with Non-Proportional Damping

The trade-off between accuracy and computational efficiency of dynamic response analysis methods for structures with non-proportional damping is an important topic in engineering research. In this paper, a two-degree-of-freedom seismically isolated structure with non-proportional damping is studied. Analytical solutions of structural displacement responses under seismic excitation are derived using five methods: the real-space exact decoupling method, unidirectionally coupled modal superposition method, forced decoupling method and first-order perturbation correction method in the real domain, together with the complex modal method in the complex domain. Systematic comparisons of the five methods are conducted from two aspects: computational accuracy and computational efficiency. The results of the complex modal method are taken as the benchmark for accuracy evaluation, and the computational time of each method under different numbers of degrees of freedom is measured to evaluate efficiency. The results show that, within the MATLAB computing environment, the complex modal method simultaneously achieves the highest accuracy and efficiency for linear analysis, which challenges the conventional understanding that approximate methods are inherently more efficient than exact methods. The real-space exact decoupling method exhibits excellent efficiency for systems with low and moderate degrees of freedom. However, its numerical stability degrades as the number of degrees of freedom increases, and an efficiency crossover with the forced decoupling method appears at approximately 170 degrees of freedom. The accuracy of the three approximate methods decreases with the rising damping ratio of the isolation layer, and the first-order perturbation correction method maintains the best approximation accuracy across the full damping range. This study reveals an efficiency competition mechanism: computational efficiency is dominated by the number of function calls for small- and medium-scale structures, whereas the order of complexity of floating-point operations becomes the governing factor for large-scale structures. Based on the two-degree-of-freedom linear non-proportionally damped seismically isolated structure examined in this paper and the MATLAB computing environment, this study can provide a reference for the selection of dynamic response analysis methods for seismically isolated structures.