Hysteresis elements such as elasto-plastic dampers are important for mechanical structures. They have nonlinearities and thus the vibration characteristics of mechanical structures with hysteresis elements are not clarified yet and need further studies. In this paper, the investigation of the dynamical stability of mechanical structures with a hysteresis element is performed by means of describing function method, which is on the basis of linear approximation of the hysteresis element. The nonlinear vibration system is composed of a linear spring-mass-damper structure system of any order and a hysteresis damper. The describing function of a hysteresis damper is derived and it is shown that the function remains upper half plane in the complex plane. According to the passivity of the linear structure system, this fact leads to the vibration system that is stable for any harmonic excitation. Next, the responses for seismic excitation are examined in simulation by using two typical models of the hysteresis element, a bilinear model and a Ramberg-Osgood model. The effects of the second stiffness of the dampers and the amplitude of the excitation motion are investigated using an actually recorded earthquake motion. The simulation results imply that the determination of the second stiffness requires the trade-off between an oscillation and a drift for designing earthquake-proof mechanical structures with the hysteresis damper.

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