Shell-dumbbell model
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To consider a rattleback whose inertial and geometrical parameters can be set separately, we construct a simple model of the rattleback, or the *shell-dumbbell model*, which consists of a light shell and two dumbbells: the light shell defines the shape of the lower part of the rattleback and the dumbbells represent the masses and the moments of inertia. The shell is a paraboloid given by Eq. (\[eq:def-z\]). The dumbbells consist of couples of weights, $m_{x}/2$ and $m_{y}/2$, fixed at $(\pm r_{x},0,0)$ and $(0,\pm r_{y},0)$ in the body-fixed co-ordinate, respectively \[Fig. \[fig:notation\](c)\]. Then the total mass is $$M = m_{x} + m_{y}$$ and the inertia tensor is diagonal with its principal moments $$\begin{aligned}
    I_{xx} &= m_{y}r_{y}^{2},\quad I_{yy} = m_{x}r_{x}^{2}, \\
    I_{zz} &= m_{y}r_{y}^{2} + m_{x}r_{x}^{2}.\end{aligned}$$ Note that the simple relation $$I_{zz} = I_{xx} + I_{yy}$$ holds for the shell-dumbbell model. We define $$f_{sd} \equiv I_{yy}/I_{zz},$$ then the dimensionless parameters $\alpha,\,\beta,$ and $\gamma$ defined by Eq. (\[eq:def-abg\]) are given by, $$\gamma = I_{zz}/Ma^{2}, \, \alpha = (1-f_{sd})\gamma + 1, \, \beta = f_{sd}\gamma + 1.$$ The parameter $f_{sd}$ satisfies $0<f_{sd}<0.5$, since we have assumed $\alpha > \beta$. The shell-dumbbell model makes it easier to visualize an actual object represented 