Dynamics of the bouncing ball

TL;DR

This paper investigates the trajectories and chaotic behavior of a bouncing ball on a vibrating plate with arbitrary motion, introducing new measurement methods and analyzing correlations and energy in chaotic regimes.

Contribution

It presents an experimental system capable of imposing arbitrary plate motions and reconstructing ball trajectories, advancing understanding of dissipative collisions and chaos in bouncing systems.

Findings

Reconstruction of ball trajectories from collision times.

Introduction of three methods to measure restitution coefficient.

Analysis of correlations and energy in chaotic regimes.

Abstract

We describe an experiment dedicated to the study of the trajectories of a ball bouncing randomly on a vibrating plate. The system was originally used, considering a sinusoidal vibration, to illustrate period doubling and the route to chaos. Our experimental device makes it possible to impose, to the plate, arbitrary trajectories and not only sinusoidal or random, as is generally the case. We show that the entire trajectory of the ball can still be reconstructed from the measurement of the collisions times. First, we make use of the experimental system to introduce the notion of dissipative collisions and to propose three different ways to measure the associated restitution coefficient. Then, we report on correlations in the chaotic regime and discuss theoretically the complex pattern which they exhibit in the case of a sinusoidal vibration. At last, we show that the use of an aperiodic motion makes it possible to get rid of part of the correlations and to discuss theoretically the average energy of the ball in the chaotic regime.