by the pitching and the rolling always have opposite signs to each other.

### Typical rattleback parameters

Typical rattleback parameters fall in the region that satisfies the following two conditions: (1) the skew angle is small, $$|\xi| \ll 1,$$ and (2) the pitch frequency is higher than the roll frequency. Under these conditions, the modes $p$ and $r$ of Eq. (\[eq:def-omgpr\]) correspond to the pitching and the rolling oscillations respectively, and $$\omega_{p}^2 \approx  \omega_{p0}^2,\qquad
\omega_{r}^2 \approx  \omega_{r0}^2
\label{eq:omg_pr-approx-omgpr0} 
$$in accord with the inequality (\[ineq:omega\_pr\]) [@Note1]. From Eqs. (\[eq:def-kpr\]), (\[eq:Kp\]), and (\[eq:Kr\]), the signs of the asymmetric torque coefficients and the averaged torques for typical rattlebacks are given by $$K_{p}>0 \quad \text{and}\quad K_{r}>0, \label{eq:sign-typ-k}$$ and $$\overline{T}_{p}<0 \quad \text{and}\quad \overline{T}_{r}>0,$$ by noting $\xi<0$, $\alpha > \beta$, $\theta >\phi$. The fact that $\omega_{p0}>\omega_{r0}$ for a typical rattleback means that the shape factor, $1/\phi-1$ or $1/\theta-1$, contributes much more than the inertial factor, $1/\alpha$ or $1/\beta$, in Eqs. (\[def:omega\_p0\]) and (\[def:omega\_r0\]) although these two factors compete, i.e. $1/\phi-1>1/\theta-1$ and $1/\alpha<1/\beta$. This is a typical situation because the two curvatures of usual rattlebacks are markedly different, i.e., $\phi \ll \theta < 1$ as can be seen in Fig. \[fig:notation\](c). Moreover, we can show that the 