Dynamics of a dissipative, inelastic gravitational billiard

The seminal physical model for investigating formulations of nonlinear dynamics is the billiard. This article expands on our previously published work concerning a real-world billiard. Here we provide a detailed mathematical model for describing the motion of a realistic billiard for arbitrary boundaries, where we include rotational effects and additional forms of energy dissipation. Simulations of the model are applied to parabolic, wedge, and hyperbolic billiards that are driven sinusoidally. The simulations demonstrate that the parabola has stable, periodic motion, while the wedge and hyperbola (at high driving frequencies) appear chaotic. The hyperbola, at low driving frequencies, behaves similarly to the parabola, i.e., has regular motion. Direct comparisons are made between the model's predictions and previously published experimental data. The representation of the coefficient of restitution employed in the model resulted in approximate agreement with the experimental data for all boundary shapes investigated. We show how the coefficient of restitution varies under different model assumptions. It is shown that the data can be successfully modeled with a simple set of parameters. DOI: 10.1103/PhysRevE.87.032901