Dynamics of an Inelastic Gravitational Billiard with Rotation

TL;DR

This paper develops a comprehensive mathematical model for inelastic gravitational billiards with rotation, capturing complex dynamics across various boundary shapes and comparing predictions with experimental data.

Contribution

It introduces a new model that includes rotational effects and energy dissipation for arbitrary billiard boundaries, aligning well with experimental results.

Findings

Parabolic billiard exhibits stable, periodic motion.

Wedge and hyperbolic billiards show chaotic behavior at high frequencies.

Model predictions agree with experimental data using simple parameters.

Abstract

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 value 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.