MOTION OF THE ECCENTRIC ELLIPSOID: GYROSCOPIC BALANCE CONDITION AND STABILITY

This paper studies the influence of the friction coefficient\(\mu\) and the eccentric distance \(\rho\) on the gyroscopic balance in the rotating process of the eccentric ellipsoid. The time evolution of the angle \(\theta=\theta(t)\) and two kinds of stationary solutions are obtained. Our numerical results show that the frictional force is an essential condition for spinning vertically, and can enhance it. That is, when \(\rho\) is constant, the time required for the ellipsoid to standing upright decreases with the increase of \(\mu\). The eccentric distance will delay the process of the standing upright of the rotator, and with the increase of \(\rho\), for a given frictional coefficient \(\mu\),it takes more time to reach the gyroscopic balance. When \(\rho\) keeps increasing, the ellipsoid will eventually take a position of a stabilized angle \(\theta_{\rm f}\) rather than an upright one, and \(\theta_{\rm f}\) is proportional to \(\rho\). The equivalent section curves are obtained corresponding to these values of eccentricity. When \(\rho\) continues to increase, a point will be reached where there will be no gyroscopic balance anymore.