On the subject of influence of dissipative and permanent momentums on stability of uniform rotations of two elastically connected free gyroscopes of Lagrange

Abstract

In many works, there are studies of the asymptotic stability of rotation of a free Lagrange gyroscope in a resisting medium. This article generalizes this problem to the case of uniform rotations of two free Lagrange gyroscopes connected by an elastic restoring spherical hinge. The rotation of each gyroscope is maintained by a constant moment in an inertial coordinate system. The characteristic equation of the perturbed motion is presented in the form of an algebraic equation of the fourth degree with complex coefficients. Based on the innor approach, conditions of asymptotic stability are obtained in the form of a system of three inequalities. The left-hand side of these inequalities is represented, respectively, in the form of determinants of the third, fifth, and seventh orders. Up to first-order values of smallness, relative to the reciprocal of the stiffness coefficient, a study is made of the effect of the joint stiffness on stability conditions. From the conditions of positivity of the highest coefficients in three inequalities, it is shown that for a sufficiently large rigidity, the stability conditions are determined by only one inequality. Cases of degeneration of an elastic spherical joint into a spherical inelastic, cylindrical, and universal elastic joint (Hooke's joint) are considered. In the case of an inelastic spherical joint, the system of three inequalities is slightly simplified. The greatest simplification arises in the case of a cylindrical hinge. In this case, the characteristic equation is represented as a quadratic equation with complex coefficients. According to the innoric approach, the conditions of asymptotic stability are written in the form of a single inequality, the left side of which is presented in the form of third-order determinants. It is shown that this inequality coincides with the inequality obtained earlier for the case of a sufficiently large rigidity of the hinge. If the angular velocities of the proper rotations of the gyroscopes coincide, the inequality obtained for the cylindrical hinge coincides with the well-known inequality for one gyroscope. In the case of a universal elastic hinge (Hooke's hinge), the first inequality is represented as a square inequality with respect to the angular velocity of proper rotation.