The Diver with a Rotor

TL;DR

This paper introduces a simplified rigid body model with a switchable rotor to analytically study twisting somersaults, providing explicit formulas for dive dynamics and twist-somersault combinations.

Contribution

It presents a novel, analytically solvable model for twisting dives that captures essential shape-changing dynamics using a switchable rotor mechanism.

Findings

Explicit formulas for dive dynamics using elliptic integrals

Analytical solutions for Euler-type equations with rotor on/off

Formulas for achieving specific twist-somersault combinations in given times

Abstract

We present and analyse a simple model for the twisting somersault. The model is a rigid body with a rotor attached which can be switched on and off. This makes it simple enough to devise explicit analytical formulas whilst still maintaining sufficient complexity to preserve the shape-changing dynamics essential for twisting somersaults in springboard and platform diving. With `rotor on' and with `rotor off' the corresponding Euler-type equations can be solved, and the essential quantities characterising the dynamics, such as the periods and rotation numbers, can be computed in terms of complete elliptic integrals. Thus we arrive at explicit formulas for how to achieve a dive with m somersaults and n twists in a given total time. This can be thought of as a special case of a geometric phase formula due to Cabrera 2007.