Multiple time scale analysis of hysteretic systems subjected to harmonic excitation

Dynamic behavior of smooth hysteretic systems subjected to harmonic excitation is analyzed. Wenʹs differential equation model for hysteresis, which can be applied to a large class of hysteretic systems, is used. A piecewise power series expression for hysteretic restoring force is derived from Wenʹs model assuming that steady state force–displacement curve draws a single loop and that the non-linearity of the restoring force is weak. The method of multiple scales is applied to the equation of motion by using the piecewise power series expression for the cases of primary and secondary resonance to derive the approximate solutions and the differential equations governing the amplitude and phase of the solutions. Phase plane trajectories, resonance curves and stability limit of the solutions are obtained and compared with the results of numerical integration in order to examine the validity of the present analysis.