Energy analysis of Duffing oscillators with quadratic damping: exact solutions

Abstract

Oscillators with a Duffing-type restoring force and quadratic damping are dealt with in this paper. Four characteristic cases of this restoring force are analysed: hardening, softening, bistable and a pure cubic one. Their energy-displacement relationships are considered, and the corresponding closed-form exact solutions are obtained in terms of the incomplete Gamma function, which represent new results. Such results provide insight into damped dynamics of the class of system, including finding the phase trajectories as well as the comparison between these cases from the viewpoint of the energy loss per cycle.