Many dynamical systems involve intervals of smooth approximations and sharp transitions, resulting in discontinuities. The surface where the system loses continuity is called the switching surface. Ensuring uniqueness in the presence of discontinuities is somewhat difficult. Rigorous ideas from piecewise-smooth dynamics and singular perturbation theory shed light on the problem. In mechanical systems, these discontinuities are of interest to stick-slip vibrations, impact, or a combination of these phenomena. Some examples are squeaking doors, bearing systems, rotating drilling systems, robotic manipulators and passive walkers. One special case that occurs in impacts with friction is the Painleve paradox.

Here, we investigate the nonlinear dynamics of a self-excited smooth discontinuous (SD) oscillator with geometric nonlinearity at the switching surface by using hidden dynamics, where a nonlinear term is added to the Filippov convex method. The belt friction involved in the SD oscillator is modelled as the coulomb friction. The discontinuity in the friction model is a switch in the direction of the contact force in the transition between the left and rightward slipping motion. The switching surface is blow up into the switching layer to study the dynamics. There exist multiple sliding regions depending upon the parametric values of the system. These sliding regions are calculated from the theory and verified using numerical simulations. The system exhibits multiple sliding modes, which are not part of the standard theory of Filippov. We compare the dynamics of the present system when simulated using the Filippov and hidden dynamics methods.