Nonlinear modal interactions of a linear oscillator coupled to a cubic nonlinear oscillator in the gravitational field

Abstract

The work is dedicated to exploring nonlinear modal interactions and mechanisms of energy transfer between a linear oscillator and a nonlinear energy sink in the gravitational field. Nonlinear modal interactions are studied based on the frequency-energy plot. Periodic motions are computed via numerical continuation method. Numerical evidences reveal that a 1:1 in-phase oscillation exists at low energies, and as energy increases, a 1:2 subharmonic tongue emerges from this 1:1 backbone branch. Besides, at high-energy levels, the NES engages in every n:m internal resonance with the LO. Differences on nonlinear modal interactions and energy transfer are compared between the cases of considering and neglecting effects of weights. Distinct nonlinear modal interactions depicted in the form of the frequency-energy plots are found at low-energy levels. For the case of the presence of weights, 1:1 transient resonance capture (TRC) or 1:2 subharmonic TRC governs nonlinear energy transfer. It means that energy transfer is still activated at low input energies, which perhaps breaks the limitation of critical energy threshold. The effects of NES parameters on frequency-energy relations for the underlying undamped system are also discussed via harmonic balance method. These discussions on nonlinear modal interactions and energy transfer provide a guidance for engineering application of the NES.