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arXiv · 1207.6022

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

Peter Lynch·Miguel D Bustamante
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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.

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Peter Lynch, Miguel D Bustamante. 2012-07-25. Quaternion Solution for the Rock'n'roller: Box Orbits, Loop Orbits and Recession. https://doi.org/10.1134/s1560354713010127

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