On Dynamics of Jellet's Egg. Asymptotic Solutions Revisited

We study here the asymptotic condition $\dot E=-\mu g_n\boldsymbol{v}_A^2=0$ for an eccentric rolling and sliding ellipsoid with axes of principal moments of inertia directed along geometric axes of the ellipsoid, a rigid body which we call here Jellett's egg (JE). It is shown by using dynamic equations expressed in terms of Euler angles that the asymptotic condition is satisfied by stationary solutions. There are 4 types of stationary solutions: tumbling, spinning, inclined rolling and rotating on the side solutions. In the generic situation of tumbling solutions concise explicit formulas for stationary angular velocities $\dot\varphi_{\mathrm{JE}}(\cos\theta)$, $\omega_{3\mathrm{JE}}(\cos\theta)$ as functions of JE parameters $\widetilde{\alpha},\alpha,\gamma$ are given. We distinguish the case $1-\widetilde{\alpha}<\alpha^2<1+\widetilde{\alpha}$, $1-\widetilde{\alpha}<\alpha^2\gamma<1+\widetilde{\alpha}$ when velocities $\dot\varphi_{\mathrm{JE}}$, $\omega_{3\mathrm{JE}}$ are defined for the whole range of inclination angles $\theta\in(0,\pi)$. Numerical simulations illustrate how, for a JE launched almost vertically with $\theta(0)=\tfrac{1}{100},\tfrac{1}{10}$, the inversion of the JE depends on relations between parameters.