by the model with a set of parameters, and is used in the following simulations for determining the parameter ranges.

Methods {#subsec:method}
-------

        $\gamma$     $f_{sd}$      $\alpha,\ \beta$     $\theta$        $\phi$      $-\xi$ (deg)  
  ---- ---------- --------------- ------------------ -------------- -------------- -------------- --
  GH    $12.28$          —           13.04, 1.522        0.6429         0.0360          1.72      
  SD    $[5,15]$   $[0.05,0.15]$          —           $[0.6,0.95]$   \[0.01,0.1\]      (0,6\]     

![\[fig:omega-dist\](color online) (a) Cumulative distributions of the pitch and the roll frequencies for the parameter set SD in Table \[tab:parameters\]; $\omega_{p} \text{~and~}
\omega_{r}$ of Eq. (\[eq:def-omgpr\]) and their zeroth order approximation $\omega_{p0}$ and $\omega_{r0}$ by Eqs. (\[def:omega\_p0\]) and (\[def:omega\_r0\])](3r.pdf){width="7.5cm"}

are shown. The inset shows the cumulative distribution of $\omega_{p}/\omega_{r}$. The number of samples is $10^{6}$.

![\[fig:k-dist\](color online) (a) Cumulative distributions of the asymmetric torque coefficients $K_{p}$ and $K_{r}$ for SD (Table \[tab:parameters\]). The number of samples is ${10}^{5}$. (b) A 2D color plot for the distribution of ($K_{p}$,$K_{r}$). The color code shown is in the logarithmic scale for the relative frequency $P(K_{p},\,K_{r})$, i.e., $-9 \le \log_{10}P(K_{p},\,K_{r}) \le 0$. The number of samples is $10^{8}$.](4.pdf){width="7.5cm"}

The equations of motion (\[eq:diff-u\]) and (\[eq:diff-omega\]) with the contact point conditions (\[eq:def-z\]) and (\[eq:def-xy\]) are numerically integrated by the fourth-order Runge-Kutta method with an initial condition $\bm{\omega}(0)$ and $\bm{u}(0)$. In the 