Quaternion Solution for the Rock'n'roller: Box Orbits, Loop Orbits and Recession

TL;DR

This paper analyzes box and loop trajectories in mechanical systems, focusing on a quaternionic approach to the rock'n'roller, revealing orbit types and the phenomenon of recession in small-amplitude motions.

Contribution

It provides a complete analytical quaternionic solution for the rock'n'roller's motion, elucidating orbit types and recession phenomena in a unified framework.

Findings

Identifies conditions for box and loop orbits.

Derives explicit solutions for symmetric cases.

Links recession to box orbit dynamics.

Abstract

We consider two types of trajectories found in a wide range of mechanical systems, viz. box orbits and loop orbits. We elucidate the dynamics of these orbits in the simple context of a perturbed harmonic oscillator in two dimensions. We then examine the small-amplitude motion of a rigid body, the rock'n'roller, a sphere with eccentric distribution of mass. The equations of motion are expressed in quaternionic form and a complete analytical solution is obtained. Both types of orbit, boxes and loops, are found, the particular form depending on the initial conditions. We interpret the motion in terms of epi-elliptic orbits. The phenomenon of recession, or reversal of precession, is associated with box orbits. The small-amplitude solutions for the symmetric case, or Routh sphere, are expressed explicitly in terms of epicycles; there is no recession in this case.