Non-Regular Precession of a Gyrostat in Three Uniform Fields

This article presents a solution to the problem of the conditions of gyrostat non-regular precession in three homogeneous fields, in which the ratio of precession and proper rotation velocities is constant. The case of a gyrostat with axial dynamic symmetry, the axis of its proper rotation coinciding with the axis of symmetry of the gyrostat, is highlighted. It is shown that the precession is possible only at a precession rate twice the rate of its proper rotation, and the gyrostatic moment deflected from the axis of symmetry by some angle ε. An expression for each of the rates is obtained through elementary functions of time. At 0 < ε < ε_*, the motion is periodic, at ε ≥ ε_*, the velocity tends to zero and the solid makes no more than one revolution around the axis of its proper rotation, the angle ε_* is expressed through the constant nutation angle θ. A relationship has been found between the nutation angle and the ratio of the axial and equatorial moments of inertia, under spherical symmetry cosθ = 1/4. The set of permissible positions of the centers of force at arbitrary given angles between the lines of force of homogeneous fields and for the special case of orthogonal fields is indicated.