## Comparison of four numerical methods for calculating seismic response of SDOF system

**JIN Xing, MA Qiang, LI Shanyou(Institute of Engineering Mechanics, China Seismological Bureau, Harbin 150080, China)**

The main characteristics of numerical methods for calculating dynamic response of singledegree offreedom (SDOF) system in frequencydomain and timedomain are briefly reviewed in this paper. Based on the uniform recursive formula, four methods (central difference method, Newmark′s method, Z-transform method and Duhamel′s step integral method) are systematically studied. The advantages, disadvantages, relative precision and usage range of the four methods are pointed out through analyzing the invariance of recursive parameters b1 and b2 which are related to the poles of the transfer function of system, the constraint condition of the system transform function at low frequency and the phase properties of the digital filter in theory and numerical calculation have been presented. Based on the invariance of operators in dynamic equation common characteristics of the four methods and the interrelation among recursive equation of relative displacement, relative velocity and relative acceleration are also obtained. This means that if the recursive algorithm of relative displacement is known, the corresponding recursive algorithm of relative velocity or acceleration will be obtained easily. Finally, a set of new algorithms for calculating displacement, velocity and acceleration response of SDOF system to arbitrary ground motion are suggested.

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