Quasi-Periodic Response Regimes of Linear Oscillator Coupled to Nonlinear Energy Sink Under Periodic Forcing

Quasi-periodic response of a linear oscillator attached to nonlinear energy sink with relatively small mass under external sinusoidal forcing in a vicinity of main (1:1) resonance is studied analytically and numerically. It is shown that the quasi-periodic response is exhibited in well-defined amplitude-frequency range of the external force. Two qualitatively different regimes of the quasi-periodic response are revealed. The first appears as a result of linear instability of the steady-state regime of the oscillations. The second one occurs due to interaction of the dynamical flow with invariant manifold of damped-forced nonlinear normal mode of the system, resulting in hysteretic motion of the flow in the vicinity of this mode. Parameters of external forcing giving rise to the quasi-periodic response are predicted by means of simplified analytic model. The model also allows predicting that the stable quasi-periodic regimes appear for certain range of damping coefficient. All findings of the simplified analytic model are verified numerically and considerable agreement is observed.