The role of the hysteretic restoring force on modal interactions in nonlinear dynamics

Hysteretic behaviour characterizes elements of a wide class of mechanical systems: the dependence of their restoring force on the deformation history has a great influence on the dynamic response. For this reason, an increasing interest has been registered in the study of typical phenomena of nonlinear dynamics, also because hysteresis belongs to the class of strong nonlinearities. The paper is devoted to the analysis of the modification of the response of two degrees-of-freedom chain systems due to the characteristics of the restoring force of a hysteretic element; attention is given to its dissipation characteristic, comparing the cases of full and reduced hysteresis. With increasing excitation amplitude, the strong hysteretic nonlinearity modifies the nonlinear frequencies and in turn their ratio, easily leading to internal resonance conditions. For a system close to a 3:1 resonance condition, the modification of the frequency response curves (frcs) for increasing excitation intensity is illustrated and compared with the response of similar systems not in internal resonance. The general trend of the phenomena is slightly qualitatively influenced by the dissipation property, whereas the quantitative differences are notable. The only evident difference is the presence of a frequency range of coexisting solutions in the case of reduced hysteresis. In both cases, after a bifurcation a novel mode arises around the first resonance, with similar frequency and different shape. The case of reduced hysteresis makes it possible to better investigate the evolution of nonlinear modes of a system, close to the Hamiltonian system embedded by the actual dissipative one. The occurrence of this novel mode, peculiar of systems with strong nonlinearities, is responsible of a substantial transfer of energy between the two modes in internal resonance conditions. Finally, an example of quasiperiodic oscillations is presented to show one of the other possible scenarios which arise even in the presence of significant dissipation.