The tennis racket effect in a three-dimensional rigid body

TL;DR

This paper provides a comprehensive theoretical analysis of the tennis racket effect in three-dimensional rigid body rotation, demonstrating its robustness and generality across different inertia configurations.

Contribution

It offers a complete theoretical description, asymptotic analysis, and an analytical formula for the tennis racket effect, highlighting its geometric nature and broad applicability.

Findings

The tennis racket effect occurs during free rotation around an unstable inertia axis.

The effect is robust to variations in moments of inertia and initial conditions.

An explicit analytical formula estimates the twisting effect in general cases.

Abstract

We propose a complete theoretical description of the tennis racket effect, which occurs in the free rotation of a three-dimensional rigid body. This effect is characterized by a flip ($\pi$- rotation) of the head of the racket when a full ($2\pi$) rotation around the unstable inertia axis is considered. We describe the asymptotics of the phenomenon and conclude about the robustness of this effect with respect to the values of the moments of inertia and the initial conditions of the dynamics. This shows the generality of this geometric property which can be found in a variety of rigid bodies. A simple analytical formula is derived to estimate the twisting effect in the general case. Different examples are discussed.