Effect of dry friction on a parametric nonlinear oscillator

Abstract

Parametrically excited oscillators are used in several domains, in particular to improve the dynamical behaviour of systems like in the case of the parametric amplification or parametric energy harvesting. Although dry friction is often omitted during system modelling due to the complexity of its nonsmooth nature, it is sometimes necessary to account for this kind of damping to adequately represent the system motion. In this paper, it is proposed to investigate the effect of dry friction on the dynamical behaviour of a nonlinear parametric oscillator. Using the pendulum case as example, the problem is formulated according to a Mathieu-Duffing equation. Semi-analytical developments using the harmonic balance method and the method of varying amplitudes are used to find the solutions of this equation and their stability. These results are validated thanks to a comparison with time integration simulations. Effects of initial conditions on the basins of attractions of the solutions are also studied using these simulations. It is found that trivial and non-trivial solutions of the oscillator including dry friction are not connected, giving birth to isolated periodic solutions branches.Thus, both initial displacement and phase between the excitation and the oscillator displacement must be carefully chosen to reach periodic solutions. Finally, a method based on the energy principle is used to find the critical forcing amplitude and frequency needed to obtain the birth of nontrivial solutions for the nonlinear parametric oscillator including dry friction.