Inelastic Gravitational Billiards

We present an experimental investigation of gravitational billiards where the particle undergoes inelastic collisions with its boundary. The motion is mapped for an inelastic particle contained within parabolic, wedge, and hyperbolic boundaries. For the parabola, stable orbits are found and the wedge demonstrates a characteristic instability for its vertex angle. In the instance of the hyperbola, there are several features of the dynamics similar to the parabola at low driving and the wedge for higher driving. However, the low driving case for a hyperbola can only be completely understood by considering inelasticity effects predicted by a numerical simulation and the observation that the velocity dependent inelasticity allows the particle to sample several nearby trajectories for fixed driving.

Figure 1

(a) A schematic of the boundaries used in the experiment. An image of the particle in motion in the parabolic (b), wedge (c), and hyperbolic (d) boundaries.

Figure 2

Sample trajectories of the particle with respect to the motion of the cell for (a) the parabola driven at a frequency of 5.4 Hz, (b) the wedge at 6.6 Hz, (c) the hyperbola at 4.5 Hz, and (d) the hyperbola at 5.8 Hz.

Figure 3

The mappings of the collision heights (left column) and the times of flight (right column). In (a) and (b) we see the parabola at 5.4 Hz, (c) and (d) show the wedge at 6.6 Hz, (e) and (f) show the hyperbola at 4.5 Hz, and (g) and (h) show the hyperbola at 5.8 Hz.

Figure 4

Plots of $y/y^{\max }$ as a function of $v_t/v_t^{\max }$, where $v_t$ is the tangential component of the velocity, for the case of (a) the parabola driven at 5.4 Hz, (b) the wedge driven at 6.6 Hz, (c) the hyperbola driven at 4.5 Hz, and (d) the hyperbola driven at 5.8 Hz. The data falls within the envelope (gray line) given by $y/y^{\max }$ as a function of $v/v^{\max }$, where $v$ is the total velocity.