On the subject of influence of dissipative and constant of moments on the stability of uniform rotations of non-free two elastically connected gyroscopes of Lagrange
The conditions for asymptotic stability of uniform rotations in a resisting medium of two heavy Lagrange gyroscopes connected by an elastic spherical hinge are obtained in the form of a system of three inequalities. The bottom gyroscope has a fixed point. The rotation of the gyroscopes is maintained by constant moments in the inertial coordinate system. The influence of the elasticity of the hinge on the stability conditions is estimated. It is shown that for a sufficiently high rigidity of the hinge, the asymptotic stability conditions are determined by only one inequality, which coincides with the inequality obtained for the case of a cylindrical hinge. When the angular velocities of the gyroscopes' own rotations coincide, this inequality coincides with the well--known condition for one gyroscope. Cases of degeneration of an elastic spherical hinge into a spherical inelastic, cylindrical and universal elastic hinge (Hooke's hinge) are considered. For the Hooke hinge, it is shown that there is no asymptotic stability at a sufficiently high angular velocity of gyroscopes rotation.