Limiting phase trajectories and non-stationary resonance oscillations of the Duffing oscillator. Part 1. A non-dissipative oscillator

This paper demonstrates an analytical description of the non-stationary resonance response of the Duffing system subject to 1:1 resonance. The approximate procedure is based on a recently introduced concept of the limiting phase trajectories (LPTs). The LPT is understood as a trajectory of the system with the initial rest state, and thus corresponds to the most intensive energy transfer from an external source of harmonic excitation to the oscillator. It is shown that the LPT of the Duffing system is similar to the trajectory of a free particle moving between two motion-limiters. The use of special non-smooth transformations gives an explicit asymptotic expression of the LPT of the resonance Duffing oscillators. The theoretical results are confirmed by numerical simulations.