粗糙平面上陀螺的稳定性

本文用摄动法讨论微粗糙平面上轴对称陀螺绕直立的极轴或赤道轴永久转动的稳定性。作为摄动因素的摩擦力是滑动速度的任意非线性函数。导出的解析形式稳定性判据与Contensou-Magnus线性理论一致,但摒弃了线性摩擦假定,指出刚体在接触点处滑动速度的方向是影响陀螺旋转稳定性的主要因素,并给出明确的物理解释。

THE STABILITY OF A TOP ROTATING ON A ROUGH HORIZONTAL PLANE

The Stability of a symmetrical rigid body rotating on a slightely rough horizontal plane is considered in present paper. The sliding friction is an arbitrary nonlinear function of the sliding speed. The perturbation method is used, selecting the friction coefficient as a small parameter. The obtained stability criterium of the permanent rotation about the polar axis is identical with the linear theory of Contensou-Magnus, but the hypothesis of the linearity of the sliding friction can be omitted. It is pointed out that the direction of the sliding velocity of the top is a decisive factor to determine the stability behaviour and an explicit physical explanation is given. The stability of rotation about the transverse axis of the top with a non-spherical surface is discussed using the same method.