Vibrational Effects on the Coefficient of Restitution

TL;DR

This paper investigates how the vibrational properties of plates and added springs influence the coefficient of restitution during ball collisions, revealing conditions for super-elastic collisions where energy is gained.

Contribution

It introduces a theoretical model that explains the emergence of super-elastic collisions and demonstrates how plate thickness and added springs affect restitution behavior.

Findings

Super-elastic collisions occur with $ ext{ε} > 1$ due to energy transfer from plate vibrations.

Adding a spring to the ball increases the probability of super-elastic collisions.

The model successfully simulates the energy transfer leading to super-elastic behavior.

Abstract

A ball dropped from a given height onto a surface, will bounce repeatedly before coming to rest. A ball bouncing on a thick plate will behave very differently than a ball bouncing off the thin lid of a container. For a plate with a fixed thickness, a ball bouncing at the edge of a plate will be very different from the ball bouncing off the middle of the plate. We study the coefficient of restitution $\epsilon$ for a steel ball bouncing steel plates of various thicknesses. We observe how $\epsilon$ changes as the ball repeated bounces and finally comes to rest. Generally, $\epsilon < 1$ due to the dissipation of kinetic energy of the ball into the plate. However this dissipated energy can come back into ball in its later bounces. We see the emergence of super-elastic collisions ($\epsilon > 1$), implying that the ball gained Kinetic Energy due to the collision with the plate. We can increase the probability of such super-elastic collisions (P$_{SE}$) by adding a spring to the ball. We construct a simple theoretical model where the energy lost from previous collisions are transferred back into later ones. This model is able to simulate the occurrence of such super-elastic collisions.