Analysis of the van der Pol System With Coulomb Friction Using the Method of Multiple Scales

Abstract

The effect of Coulomb friction on the nonlinear dynamics of a van der Pol oscillator is presented. A map from the magnitude of a peak to that of the succeeding valley in the time history is analytically described by considering both the exponential growth due to negative viscous damping and the switching condition due to Coulomb friction, which is a function of the sign of the velocity of the system. The steady states and their stability are clarified and the difference from those in the case without Coulomb friction is revealed. The addition of Coulomb friction makes the trivial equilibrium, which is an unstable focus in the system without friction, into a locally asymptotically stable equilibrium set. The branch of stable nontrivial steady states is not bifurcated from the trivial steady state by the effect of Coulomb friction and is different from the branch in the case without Coulomb friction, which is bifurcated from the trivial steady state through Hopf bifurcation. Furthermore, experiments are conducted and the theoretically predicted dynamics due to Coulomb friction is confirmed.