perform extensive numerical simulations to examine validity and limitation of the formula, and find that (1) the Garcia-Hubbard formula is good for both spinning directions in the small spin and small oscillation regime, but (2) in the fast spin regime especially for the steady direction, the rattleback may not reverse and shows a rich variety of dynamics including steady spinning, spin wobbling, and chaotic behavior reminiscent of chaos in a dissipative system.'
author:
- Yoichiro
- Hiizu
title: Rattleback dynamics and its reversal time of rotation
---

Introduction {#sec:introduction}
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Spinning motions of rigid bodies have been studied for centuries and still are drawing interest in recent years, including the motions of Euler’s disks [@Moffatt2000], spinning eggs [@Moffatt2002], and rolling rings [@Jalali2015], to mention just a few. Also, macroscopic systems which convert vibrations to rotations have been studied in various context such as a circular granular ratchet [@Heckel2012], and bouncing dumbbells, which show a cascade of bifurcations [@Kubo2015]. Another interesting example of rigid body dynamics which involves such oscillation-rotation coupling is a rattleback, also called as a celt or wobble stone, which is a semi-elliptic spinning toy \[Fig. \[fig:notation\](a)\]. It spins smoothly when spun in one direction; however, when spun 