Abstract：The resistance-displacement relationship of dynamic foundation systems subjected to cyclic loading exhibits two significant characteristics—nonlinearity and hysteresis，due to soil-structure interaction effects. In this study，foundation-basement is abstracted as fixed parameter lumped parameter system for simplicity. A nonlinear dynamic model possessing viscous damping is set up with the third power polynomial function，which is used to represent the nonlinear behavior of force-displacement interacting between soil and foundation. And a new hyperbolic hysteresis loop model of dynamic foundations is presented exteriorly adopting Masing rules extend the exponential backbone curve into the loop which simulates nonlinear and hysteresis characteristics of the system. The nonlinear mathematical model developed includes the parameters such as soil stiffness，nonlinearity of passive earth pressure，displacement scale and velocity direction. It¢s difficult to solve the constitutive function intactly by using general analytical method for its complexity，so the paper makes use of Fourier progression approach in study. The paper investigates periodic solution of the dynamic system with the hyperbolic hysteresis loop model and obtains the time-history and frequency response characteristics of the foundations under harmonic excitation. The studies show that the Fourier method，which is used to analyze the systems of dynamic foundations with the hyperbolic hysteresis model，is precise and convergent，and the results of emulation agree with the specific experimental data.
