New cases of regular precession of an asymmetric liquid-filled rigid body

Abstract

To describe the rotation of a rigid body with an ellipsoidal cavity filled with an ideal vorticated liquid, the Poincare–Hough–Zhukovsky equations are used. It is obtained constraints (hereinafter referred to as configuration conditions) on the mass distribution and cavity dimensions of an asymmetric liquid-filled rigid body under which the rigid body can perform the regular precession. Two possible nontrivial cases are indicated when one or two components of the direct vector of the axis of proper rotation are equal to zero. It is shown that if the axis of proper rotation coincides with one of the principal axes of inertia of the system, it suffices to fulfill one configuration condition. The ratio between the periods of proper rotation and precession is found. The regular precession of a system in which the principal moments of inertia are close to each other and the cavity is close to spherical is considered. For the case when the difference between the two equatorial moments of inertia is an order of magnitude smaller than the difference between the equatorial and polar moments, the main part of the ratio between the periods coincides with the Euler period, and the correction is due to the inequality between the equatorial moments of inertia. In the case when the axis of proper rotation is orthogonal to a principal axis and is not coincident with any other principal axis, the number of configuration conditions is two. Expressions for the rates of precession and proper rotation are obtained, and the position in the body of the axis of proper rotation is indicated. Special cases are given that allow simplification of the configuration conditions.