The seminal physical model for investigating formulations of nonlinear dynamics is the billiard. Gravitational billiards provide an experimentally accessible arena for their investigation. We present a mathematical model that captures the essential dynamics required 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 good agreement with the experimental data for all boundary shapes investigated. It is shown that the data can be successfully modeled with a simple set of parameters without an assumption of exotic energy dependence.
Gravitational billiards provide an experimentally accessible arena for testing formulations of nonlinear dynamics. We present a mathematical model that captures the essential dynamics required for describing the motion of a realistic billiard for arbitrary boundaries. Simulations of the model are applied to parabolic, wedge and hyperbolic billiards that are driven sinusoidally. Direct comparisons are made between the model's predictions and previously published experimental data.It is shown that the data can be successfully modeled with a simple set of parameters without an assumption of exotic energy dependence.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.