An analytical study of piecewise linear oscillator with Coulomb friction

This relatively short paper deals with the analytical study of piecewise linear (PWL) oscillators subjected to Coulomb friction. Such PWL oscillators are effective reduced order models for cracked beam structures. The Coulomb friction is introduced to model the relative motion between the cracked surfaces upon crack closure and the resulting energy dissipation. This class of dynamical systems poses challenge in terms of computing time instants of transition from one linear state to another. Whereas, this difficulty can be overcome by invoking the method of averaging. In this study we introduce PWL basis functions which are non-analytic and enable direct application of the method of averaging to the resonantly forced, essentially nonlinear PWL oscillators (PWLOs). The Coulomb friction is introduced in a piecewise form as well such that it is active only during the phase of crack closure. Three different forms of Coulomb friction are considered. In the case of the simplest form of Coulomb friction, the amplitude of harmonic excitation is required to be above a certain threshold to induce steady state oscillations. The analytical model quite well predicts this amplitude threshold. The presented results from the method of averaging matches very well with the numerical simulations within the limits of asymptotic validity.