Bouncing ball dynamics: simple model of motion of the table and sinusoidal motion

TL;DR

This paper models the nonlinear dynamics of a bouncing ball with a moving table, using polynomial approximations to analyze impact behavior and relate it to sinusoidal motion of the limiter.

Contribution

It introduces a simplified polynomial model of the bouncing ball dynamics to better understand impact sequences and their relation to sinusoidal table motion.

Findings

Derived a Poincaré map for impact analysis.

Showed how polynomial approximation captures key dynamics.

Applied model to standard sinusoidal bouncing ball scenario.

Abstract

Nonlinear dynamics of a bouncing ball moving vertically in a gravitational field and colliding with a moving limiter is considered and the Poincar\'e map, describing evolution from an impact to the next impact, is described. Displacement of the table is approximated in one period by four cubic polynomials. Results obtained for this model are used to elucidate dynamics of the standard model of bouncing ball with sinusoidal motion of the limiter.