  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.
