Stability of the uniform rotation of a gyrostat round the vertical main axis on an absolutely smooth horizontal plane

Abstract

The motion of a gyrostat on an absolutely smooth plane is discussed. A Hamilton function which gives the canonical equations of motion is obtained. This admits of particular solutions, namely uniform rotations round a vertical axis which are identical with that of the uniform rotations of the rotor. A transition to a system with two degrees of freedom is realized, and the expansion of the Hamiltonian in the vicinity of the corresponding position of equilibrium, with an accuracy to within fourthorder terms, is obtained. In the region of admissible values of the parameters the domain of the necessary stability conditions, and the domains where the Hamiltonian functions are of fixed sign and alternating, are examined. In those cases where the Hamiltonian is not fixed sign, its normalization is performed, both a non-resonance situation and resonances of the first, second and fourth order being considered. The sufficient conditions for stability of uniform gyrostat rotation in terms of constraints on the coefficients of normal forms are obtained. For a clear interpretation of the results, special cases where the values of all the parameters except two are fixed, are given. The plane domain of the necessary stability conditions and resonance curves are constructed, and using computer results stability on the curves is discussed. The stability of uniform rotations of a heavy solid around the vertical principal and minor axes on an absolutely smooth, and on an absolutely rough horizontal plane, and also on a plane with viscous friction is discussed in /1–4/. The stability of uniform rotations of a gyrostat round the vertical principal axis on absolutely smooth and absolutely rough horizontal planes was considered in /5, 6/. Investigations of the motion of a solid on an absolutely rough plane, the body being perturbed with respect to rotation round the principal axis (in particular with respect to the steady position of equilibrium), are described in /7, 8/. The stability of two types of rotation of a homogeneous ellipsoid on an absolutely smooth horizontal plane, in particular the stability of the uniform rotations of an ellipsoid round the vertical principal axis is discussed in /9/.