respect to the $x$-$z$ plane.

At this stage, we introduce the dimensionless inertial parameters $\alpha$, $\beta$, and $\gamma$ for later use after Bondi [@Bondi1986] as $$\alpha \equiv \frac{I_{xx}}{Ma^{2}}+1, \ \beta \equiv \frac{I_{yy}}{Ma^{2}}+1, \ \gamma \equiv \frac{I_{zz}}{Ma^2},\label{eq:def-abg}$$ which are dimensionless inertial moments around the contact point C. Note that $$\alpha > \beta > 1,$$ because we have assumed $I_{xx} > I_{yy}$.

Small amplitude approximation of oscillations under $\omega_{z}=0$ {#subsec:linearization}
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We consider the oscillation modes in the case of no spinning $\omega_{z} = 0$ in the small amplitude approximation, namely, in the linear approximation in $|\omega_{x}|,\,|\omega_{y}|\ll\sqrt{g/a}$, which leads to $|x|,\,|y| \ll a$, $|u_{x}|,\, |u_{y}| \ll 1 $, and $u_{z} \approx -1$. In this regime, the $x$ and $y$ components of Eq. (\[eq:diff-u\]) can be linearized as $$\dot{\bm{u}}_{\perp} \approx \hat{\varepsilon}\,\bm{\omega}_{\perp}, \quad
\hat{\varepsilon} \equiv
\begin{pmatrix}
0, & 1 \\
-1, & 0 \\
\end{pmatrix} = \hat{R}(-\pi/2).\label{eq:lin-u}$$ By using Eq. (\[eq:def-xy\]) with $u_{z} \approx -1$, Eq. (\[eq:diff-omega\]) can be linearized as $$\begin{aligned}
 \hat{J}\,\dot{\bm{\omega}}_{\perp} &\approx \frac{g}{a^2} (\bm{r}\times\bm{u})_{\perp}\notag\\
 &= -\frac{g}{a}\hat{\varepsilon}\,[-\hat{R}(\xi)\hat{\Theta}^{-1}\hat{R}^{-1}(\xi) +1 ] \bm{u}_{\perp},\label{eq:lin-omg1}\end{aligned}$$ with the inertial matrix $$\hat{J} \equiv 
\begin{pmatrix}
\alpha, & 0 \\
0, & \beta \\
\end{pmatrix}.$$ From the linearized equations (\[eq:lin-u\]) and (\[eq:lin-omg1\]), we obtain $$\hat{J}\ddot{\bm{\omega}}_{\perp}= - \frac{g}{a}(\hat{\Gamma} -1) \bm{\omega}_{\perp},\label{eq:lin-omg2}$$ where $$\hat{\Gamma} \equiv 