is set to zero where $\bm{u}(0) = (0,0,-1)^{t}$.

The parameters used in the simulations are listed in Table \[tab:parameters\]. For the parameter set SD, the ranges are shown. When numerical results are plotted against $K_{p}$ or $K_{r}$, given by Eqs. (\[eq:Kp\]) or (\[eq:Kr\]), respectively, sets of parameters are chosen randomly from the ranges until resulting $K_{p}$ or $K_{r}$ falls within the range of $\pm0.1\%$ of a target value. The ranges of SD are chosen to meet the following two conditions: (1) $0<\phi \ll \theta <1$, $\beta < \alpha$, and $|\xi| \ll 1$ and (2) the pitch frequency should be higher than the roll frequency. As argued in Sec. \[subsec:GHtheory\], usual rattlebacks such as one in Fig. \[fig:notation\](a) satisfy these two conditions. Figure \[fig:omega-dist\] shows the cumulative distributions for the eigenfrequencies $\omega_{p}$ and $\omega_{r}$, and their approximate expressions $\omega_{p0}$ and $\omega_{r0}$ for the parameter set SD; it shows $(\omega_{p}/\omega_{r}) > 1.3$ in accordance with the condition (2).

The parameter set GH gives $K_{p} = 0.553$ and $K_{r}=0.0967$, and the distributions of $K_{p}$ and $K_{r}$ for SD are shown in Fig. \[fig:k-dist\], where one can see $K_{p}\gg K_{r}$. From Eq. (\[eq:k-rat\]), this corresponds to $\omega_{p}^2 \gg
\omega_{r}^2$, i.e., the pitch frequency is 