Permanent rotations of a heavy gyrostat having a stationary point

Abstract

Kharlamov [1] and Tsodokova [2] have specified the cone of permanent axes ot a gyrostat having a single stationary point. The present paper is an investigation of this cone and the domain of stability of the permanent rotations. It is appropriate at this point to note that Kharlamov's remarks in [1] concerning the present author's papers [3] and [4] are quite valid. Our notation is as follows: Ox 1 x 2 x 3 is a coordinate system invariably connected with the solid portion of the gyrostat. The origin of the system lies at the point O of the gyrostat and its axes coincide with the principal axes of inertia. a 1, a 2, A 3 are the principal moments of inertia: I is the inertia tensor of the gyrostat for the stationary point O: k( k 1, k 2, k 3) is the gyrostatic moment : r( r 1, r 2 r 3) is the radius vector of the center of mass : ω ( p 1, p 2, p 3) is the angular velocity vector of the gyrostat ; γ (γ 1,γ 2,γ 3) is the unit vector of the stationary axis (vertical) directed vertically upward : e( e 1, e 2, e 3) is the unit vector of the permanent axis : ω is the projection of the angular velocity vector on the vertical ; P is the weight of the gyrostat.