Nonlinear Vibratory Energy Exchanges between a Two-Degree-of-Freedom Pendulum and a Nonlinear Absorber

The multi-time-scale responses of a two-degree-of-freedom pendulum coupled with a nonlinear absorber were studied. The absorber was positioned in an arbitrary direction with respect to those of the pendulum oscillations. The phase-dependent slow invariant manifold of the system and its stable zones were traced at a fast time scale while system responses were studied at a slow time scale around the slow, invariant manifold, leading to detection of equilibrium and singular points. Moreover, the amplitude-frequency curves of the system were detected, showing the possibility of the existence of isolated branches, which could correspond to high energy levels for pendulum oscillations. All analytic developments were confronted with numerical results collected from direct numerical integration of system equations. Depending on the characteristics of external excitations, the system can face periodic or modulated regimes.