The rattleback revisited

A rattleback is a canoe-shaped body that, when spun on a smooth surface, rotates stably in one direction only; when spun in the reverse direction it oscillates violently (i.e., it “rattles”) and reverses its direction of spin. This behavior can be traced to the misalignment of the principal axes of the body with respect to the symmetry axis of its bottom surface. Although analyses of the phenomenon exist in the literature, there is not a clear, direct presentation of the basic mechanism responsible for the reversal of direction. The goal of this paper is to present, as clearly as possible, a treatment of the phenomenon by focusing on the geometry of the usual rattleback. Two initial conditions are considered: rotation about the vertical axis with no oscillation, and oscillation about a horizontal axis with no rotation. For the first initial state, oscillatory motion about the two horizontal axes is analyzed using a combination of linearization and reasonable assumptions. The reversal is then analyzed using energy considerations. The analysis for the second initial state is more direct and elementary. In combination, these analyses explain the transitions from rotation to oscillation to rotation in the opposite direction. The non-reversal for the rotation opposite initial rotation is also accounted for. We also comment on how the rattleback might be modified allowing it to reverse in both directions and thus to repeatedly reverse its direction of rotation.