光滑平面上对称充液腔体定常转动态的稳定性、分叉和突变

本文对形体旋转对称,重心位于对称轴上,并充满粘性均匀液体的对称陀螺在光滑水平面上的运动,给出其定态转动(特别着重斜转态)在各种参数条件下的数目,取向角及Румянцев-Movchan意义下的稳定性。将角动量竖直分量的平方M~2取为控制参数,对定态转动得到三种分叉类型和相应的突变方式。考虑到不可避免地存在微弱摩擦力矩会导致M~2的极缓慢衰减,本文根据分叉突变分析,避开了繁琐的动力学论证,对于初始处于零章动角的稳定准竖立正转定态的陀螺,证明只存在两种不同的倾倒方式。这是突变理论在充液腔体旋转运动问题上的有理论和实际意义的应用。此外,得到的q_4支对陀螺的稳定性控制具有实际意义。

STABILITY, BIFURCATIONS AND CATASTROPHE OF STEADY ROTATION OF SYMMETRIC GYROSCOPE WITH VISCOUS-LIQUID-FILLED CAVITY ON PLANE

The number, the angles of orientation and the stability in Rumyantsev-Movchan's sense of steady rotations of a symmetric gyroscope with a cavity completely filled with a uniform viscous liquid on a plane are given for various values of the parameters. By taking the square of the upright component of the angular momentum M2 as a control parameter, three types of bifurcation diagrams of the steady rotation with two types of jumps are obtained. By taking account of the M2-damping owing to the moment of unavoid...