Dynamics of piecewise linear oscillators with van der Pol type damping

Abstract

In the first part of this study, an analysis is presented for a class of strongly non-linear single degree-of-freedom oscillators, excited by harmonic forcing. The damping and restoring forces of these oscillators are symmetric piecewise linear functions of the velocity and displacement, respectively. More specifically, the stiffness is always positive but the damping coefficient assumes negative values for small displacement amplitudes, resembling the characteristics of the classical van der Pol oscillator. The existence, stability and bifurcation of symmetric, single-contact, periodic steady state solutions are first analysed. Then, numerical results are presented for several representative combinations of the system parameters. It is shown that the systems considered exhibit characteristics which are similar to those observed for the classical van der Pol oscillator. Also, for some sets of parameters they undergo bifurcations which lead to loss of stability of periodic or quasiperiodic solutions and the onset of irregular non-periodic response.