Nonlinear Dynamics and Phenomena in Oscillators with Hysteresis

Many mechanical systems are characterized by hysteretic behaviors with a restoring force dependent on their deformation history. Several materials and elements base their capacity to dissipate vibratory energy on hysteresis. The wide variety of engineering applications explains the high number of studies devoted to the dynamic hysteretic response of structural systems and the hysteretic models proposed, with different levels of complexity. The Bouc–Wen model has been adopted here, because it is simple, yet, at the same time, is able to represent diverse types of hysteretic behaviors. Hysteresis can be classified among material nonlinearities and is recognized as a strong nonlinearity due to the high variation of stiffness and damping with deformation. The main characteristics of the dynamic response are first illustrated by means of frequency response curves of a hysteretic oscillator, highlighting the dependence of the response on the oscillation amplitude. The aim of this chapter is to investigate nonlinear modal interactions in the dynamic response of a two degree-of-freedom system (2DOF). These phenomena are notably important in internal resonance conditions; since when increasing excitation intensity frequencies of hysteretic system change and in turn their ratio changes, several internal resonance conditions occur, where the interaction phenomena between the two modes produce strong modifications of the response with possible beneficial effects. Two configurations are investigated: the hysteretic element at the top and the hysteretic element at the base. Qualitative similar results are obtained, characterized by a transfer of energy between the two modes, which greatly influences the evolution of the response amplitude with the excitation intensity and can be exploited in the vibration mitigation of the forced response around the first mode.