The possibility of stabilizing an unstable uniform rotation in a resisting medium of a "sleeping" Lagrange gyroscope using a rotating second gyroscope and elastic spherical hinges is considered. The "sleeping" gyroscope rotates around a fixed point with an elastic recovery spherical hinge, and the second gyroscope is located above it. The gyroscopes are also connected by an elastic spherical restorative hinge and their rotation is supported by constant moments directed along their axes of rotation. It is shown that stabilization will be impossible in the absence of elasticity in the common joint and the coincidence of the center of mass of the second gyroscope with its center. With the help of the kinetic moment of the second gyroscope and the elasticity coefficients of the hinges, on the basis of an alternative approach, the stabilization conditions obtained in the form of a system of three inequalities and the conditions found on the elasticity coefficients at which the leading coefficients of these inequalities are positive. It is shown that stabilization will always be possible at a sufficiently large angular velocity of rotation of the second gyroscope under the assumption that the center of mass of the second gyroscope and the mechanical system are below the fixed point. The possibility of stabilizing the unstable uniform rotation of the "sleeping" Lagrange gyroscope using the second gyroscope and elastic spherical joints in the absence of dissipation is also considered. The "sleeping" gyroscope rotates at an angular velocity that does not meet the Mayevsky criterion. It is shown that stabilization will be impossible in the absence of elasticity in the common joint and the coincidence of the center of mass of the second gyroscope with its center. On the basis of the innovation approach, stabilization conditions were obtained in the form of a system of three irregularities using the kinetic moment of the second gyroscope and the elastic coefficients of the hinges. The condition for the angular momentum of the first gyroscope and the elastic coefficients at which the leading coefficients of these inequalities are positive are found. It is shown that if the condition for the angular momentum of the first gyroscope is fulfilled, stabilization will always be possible at a sufficiently large angular velocity of rotation of the second gyroscope, and in this case the center of mass of the second gyroscope can be located above the fixed point.
ON STABILIZATION OF UNSTABLE ROTATION IN THE RESISTING MEDIUM OF THE LAGRANGE GYROSCOPE USING THE SECOND GYROSCOPE AND ELASTIC HINGES
