On rotational stability in an environment with the support of a Lagrange's gyroscope on a suspension taking into account an elastic spherical joint and a Hooke's joint

Based on the well-known equations of P.V. Kharlamov, the problem of the stability of uniform rotations in a medium with the resistance of a Lagrange's gyroscope on a suspension is considered, taking into account the elasticity of the spherical joint, the Hooke's joint, the mass and inertial characteristics of the suspension, and the constant moments. The conditions of asymptotic stability of uniform rotations are obtained, and analytical studies of the influence of the elasticity of the joints, the mass and inertial characteristics of the suspension on the stability conditions are carried out. It is shown that with an increase in the elasticity in the joints, asymptotic stability will always be possible at any angular velocity and at any parameters of the mechanical system. The obtained asymptotic stability conditions are compared with similar conditions in the case of replacing the Hooke's joint with a spherical joint, as well as with stability conditions in the absence of dissipation.