偏心椭球体回转平衡及其稳定性的研究

本文研究了摩擦系数 和偏心距离对偏心椭球体回转平衡的影响,得到了椭球极轴与地面法向夹角 的运动方程,导出了两类稳定解.结果表明摩擦力是椭球能否旋转至直立的必要条件,当 偏心距一定, 摩擦系数 越大其运动至直立所需的时间越短.偏心距阻碍直立现象的出现,摩擦系数 一定,偏心距 越大,其运动至直立所需的时间越长.当偏心距值增大到一定程度,其将无法直立而是至一稳定的角度,并且稳定角度正比于偏心距,偏心距继续增大,将不会出现回转平衡.

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.