On the stability of the rotation of the free system of two elastic Lagrange gyroscopes, one of which has an ideal liquid

On the basis of the known equations of motion of the system of coupled gyrostats by P. V. Kharlamov and the state function by S. L. Sobolev, the equations of rotation of a free system of two elastically coupled Lagrange gyroscopes were derived, one of which has an arbitrary axisymmetric cavity completely filled with an ideal incompressible fluid. The rigid bodies are connected by an elastic restoring spherical hinge. A transcendental characteristic equation of the perturbed uniform rotation of the mechanical system under consideration is derived. Taking into account the fundamental tone of the fluid oscillation, a characteristic equation of the fifth order is obtained. On the basis of the Liénard – Chipart criterion written in the inor form, the necessary conditions for the stability of the uniform rotation of the Lagrange gyroscopes and the fluid are written out as a system of four inequalities. With respect to the elasticity coefficient, these inequalities have degrees 1, 3, 6 and 8, respectively. Analytical studies of the leading coefficients of these stability conditions are carried out. It is shown that when the center of mass of the first solid body with liquid or the second does not coincide with the common point of these bodies, then at sufficiently large values of the elasticity coefficient the necessary stability conditions will always be satisfied. In the absence of elasticity in the hinge, the characteristic equation has a double zero root and in this case the stability conditions require additional studies. The obtained stability conditions are exact for an ellipsoidal cavity and approximate for other axisymmetric cavities. To clarify the obtained glass conditions for these voids, it is necessary to take into account additional tones of oscillation of an ideal liquid.