Responses of a two degree-of-freedom system coupled to a nonlinear damper under multi-forcing frequencies

In this paper, forced responses are investigated in a two degree-of-freedom linear system with a linear coupling to a Non-linear Energy Sink (NES) subjected to quasi-periodic excitation. The quasi-periodic regimes associated to quasi-periodic forcing in the regime of 1:1-1:1 are studied analytically using the complexification method combined to the averaging method in terms of multi-time parameter. Local bifurcations of the quasi-periodic regimes are also analyzed using the excitation frequencies as control parameters. The nonlinear differential system is also solved numerically in time domain and the responses are analyzed in view of the analytical results. Stable and unstable quasi-periodic responses are found in good agreement with the analytical study, and strongly modulated responses are noticed. We observe that a single NES can be efficient for the reduction of two resonance peaks even if they are well separated, incommensurable, and excited simultaneously.