Non-smooth approximations of the limiting phase trajectories for the Duffing oscillator near 1:1 resonance

This paper demonstrates an analytical description of the non-stationary response of the forced Duffing oscillator near the 1:1 resonance. Our approach exploits the concept of the limiting phase trajectories (LPTs). The LPT is understood as a trajectory of the system being initially at rest, and motion over the LPT corresponds to a maximum possible energy transfer from an external source of harmonic excitation to the oscillator. It is shown that for a wide range of parameters the behavior of the Duffing oscillator moving over the LPT is similar to the dynamics of a free particle between two motion-limiters. The vibro-impact hypothesis suggests a perturbation technique based on non-smooth generating functions, which yields an explicit asymptotic expression for the LPT. Different types of the dynamical behavior of the oscillator subjected to harmonic or narrow-band perturbations are studied.