Effect of 1:3 resonance on the steady-state dynamics of a forced strongly nonlinear oscillator with a linear light attachment

We study the 1:3 resonant dynamics of a two degree-of-freedom (DOF) dissipative forced strongly nonlinear system by first examining the periodic steady-state solutions of the underlying Hamiltonian system and then the forced and damped configuration. Specifically, we analyze the steady periodic responses of the two DOF system consisting of a grounded strongly nonlinear oscillator with harmonic excitation coupled to a light linear attachment under condition of 1:3 resonance. This system is particularly interesting since it possesses two basic linearized eigenfrequencies in the ratio 3:1, which, under condition of resonance, causes the localization of the fundamental and third-harmonic components of the responses of the grounded nonlinear oscillator and the light linear attachment, respectively. We examine in detail the topological structure of the periodic responses in the frequency-energy domain by computing forced frequency-energy plots (FEPs) in order to deduce the effects of the 1:3 resonance. We perform complexification/averaging analysis and develop analytical approximations for strongly nonlinear steady-state responses, which agree well with direct numerical simulations. In addition, we investigate the effect of the forcing on the 1:3 resonance phenomena and conclude our study with the stability analysis of the steady-state solutions around 1:3 internal resonance, and a discussion of the practical applications of our findings in the area of nonlinear targeted energy transfer.