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A rattleback is a rigid, semi-elliptic toy which exhibits unintuitivebehavior; when it is spun in one direction, it soon begins pitching and stopsspinning, then it starts to spin in the opposite direction, but in the otherdirection, it seems to spin just steadily. This puzzling behavior results fromthe slight misalignment between the principal axes for the inertia and thosefor the curvature; the misalignment couples the spinning with the pitching andthe rolling oscillations. It has been shown that under the no-slip conditionand without dissipation the spin can reverse in both directions, and Garcia andHubbard obtained the formula for the time required for the spin reversal $t_r$[Proc. R. Soc. Lond. A 418, 165-197 (1988) ]. In this work, we reformulate therattleback dynamics in a physically transparent way and reduce it to athree-variable dynamics for spinning, pitching, and rolling. We obtain anexpression of the Garcia-Hubbard formula for $t_r$ by a simple product of fourfactors: (1) the misalignment angle, (2) the difference in the inverses ofinertia moment for the two oscillations, (3) that in the radii for the twoprincipal curvatures, and (4) the squared frequency of the oscillation. Weperform extensive numerical simulations to examine validity and limitation ofthe formula, and find that (1) the Garcia-Hubbard formula is good for bothspinning directions in the small spin and small oscillation regime, but (2) inthe fast spin regime especially for the steady direction, the rattleback maynot reverse and shows a rich variety of dynamics including steady spinning,spin wobbling, and chaotic behavior reminiscent of chaos in a dissipativesystem.

Rattleback dynamics and its reversal time of rotation