Bifurcations of Nonlinear Normal Modes of Linear Oscillator with Strongly Nonlinear Damped Attachment

Abstract

Linear oscillator coupled to damped strongly nonlinear attachment with small mass is considered as a model design for nonlinear energy sink (NES). Damped nonlinear normal modes of the system are considered for the case of 1:1 resonance by combining the invariant manifold approach and multiple scales expansion. Special asymptotical structure of the model allows a clear distinction between three time scales. These time scales correspond to fast vibrations, evolution of the system toward the nonlinear normal mode and time evolution of the invariant manifold, respectively. Time evolution of the invariant manifold may be accompanied by bifurcations, depending on the exact potential of the nonlinear spring and value of the damping coefficient. Passage of the invariant manifold through bifurcations may bring about destruction of the resonance regime and essential gain in the energy dissipation rate.