Abstract

The equations of motion of the Goryachev–Sretensky gyrostat are studied. It has been established that one component of the angular velocity vector oscillates with zero average value on all solutions outside the set of zero level of the area integral. All stationary solutions are found, including two states of rest and two parametric families of permanent rotations. Using Chetaev’s method of integral connectives, sufficient conditions for the stability of stationary solutions are obtained. Based on the analysis of the roots of the characteristic equation, instability conditions were obtained. The possibility of gyroscopic stabilization due to the moment of circular-gyroscopic forces under certain conditions has been established.

中文翻译：

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论Goryachev-Sretensky陀螺仪运动方程稳态解的稳定性

摘要

研究了 Goryachev-Sretensky 陀螺仪的运动方程。已经确定，在面积积分的零水平集合之外的所有解上，角速度矢量的一个分量以零平均值振荡。找到所有静止解，包括两种静止状态和两个永久旋转参数族。利用Chetaev积分联结法，得到了平稳解稳定的充分条件。通过对特征方程根的分析，得到了不稳定条件。已经确定了在某些条件下由于圆形陀螺力的力矩而实现陀螺稳定的可能性。