to examine the validity and the limitation of the theory. Discussions and conclusion are given in Sec. \[sec:discussion\] and Sec. \[sec:conclusion\], respectively.

Theory {#sec:theory}
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Equations of motion
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![\[fig:notation\](a) A commercially available rattleback made of plastic. (b) Notations of the rattleback. (c) A schematic illustration of the shell-dumbbell model.](1.pdf){width="7cm"}

We consider a rattleback as a rigid body, whose configuration can be represented by the position of the center of mass G and the Euler angles; both of them are obtained by integrating the velocity of the center of mass $\bm{v}$ and the angular velocity $\bm{\omega}$ around it [@Goldstein2002].

We investigate the rattleback motion on a horizontal plane, assuming that it is always in contact with the plane at a single point C without slipping. We ignore dissipation, then all the forces that act on the rattleback are the contact force $\bm{F}$ exerted by the plane at C and the gravitational force $-Mg\bm{u}$, where $\bm{u}$ represents the unit vertical vector pointing upward \[Fig. \[fig:notation\](b)\]. Therefore, the equations of motion are given by $$\begin{aligned}
 \frac{d(M\bm{v})}{dt} &= \bm{F} - Mg\bm{u},\label{eq:em-1}\\
 \frac{d(\hat{I}\bm{\omega})}{dt} &= \bm{r} \times \bm{F}, \label{eq:em-2}\end{aligned}$$ where $M$ and $\hat{I}$ are the mass and the inertia tensor around G, respectively, and $\bm{r}$ 