Nonlinear resonances of hysteretic oscillators

The response of single and two-degree-of-freedom mechanical systems with elements exhibiting a hysteretic restoring force is studied under harmonic imposed motion to characterize the nonlinear dynamic properties of the system. The hysteretic behavior of the element is described by the Bouc–Wen model, which is simple but able to represent different hysteretic behaviors. Since for a given restoring force the nonlinear response is affected by the oscillation amplitude, frequency-response curves for various excitation levels are constructed. The curve of periodic solutions for increasing excitation amplitude has been investigated analyzing the evolution of the resonance frequencies. Furthermore, in addition to known behaviors, to some extent already observed for single-degree-of-freedom hysteretic oscillators, a richer class of solutions and bifurcations is found for a 2-DOF system. New branches of stable periodic oscillations are bifurcated where conditions of internal resonance are encountered by varying the excitation amplitude; these are characterized by the onset of novel modes with an oscillation shape significantly different than that of modes on the fundamental branch. Finally, comparison between systems close to and far from internal resonance allowed to enlighten the role of the nonlinear coupling in the vibration mitigation.