| Abstrakt: |
On the basis of the known equations of motion of the system of coupled gyrostats by P.V. Kharlamov and the functions of state by S.L. Sobolev, the equations of rotation in a medium with resistance of a free system of two elastically connected rigid bodies with cavities completely filled with an ideal incompressible fluid were derived. Rigid bodies are connected by an elastic restoring spherical joint. Assuming that the center of mass of the rigid bodies is located on the third main axis of inertia and the fluid is ideal, the equation of disturbed motion of the considered mechanical system is obtained in the form of a countable system of ordinary differential equations. In the case of two Lagrangian gyroscopes with arbitrary axisymmetric cavities filled with an ideal fluid, a transcendental characteristic equation has been derived. Taking into account the fundamental tone of liquid oscillations, a characteristic equation of the sixth order was obtained, and on the basis of the Lenard–Schipar criterion, it was written in the innor form, and the conditions for the asymptotic stability of uniform rotation of Lagrange gyroscopes with a liquid were written out in the form of a system of five inequalities. These inequalities are presented in the form of the first, third, sixth, and eighth powers with respect to the coefficient of the spherical joint elasticity. It was proved that if the first tones of liquid oscillations in two cavities are greater than one and do not coincide, then this is sufficient for the higher inequality coefficients to be positive. It was shown that if the first oscillation tones coincide, only the degree of the last inequality decreases, while the higher inequality coefficients remain positive; therefore, internal resonance is impossible. Thus, when the first tones of fluid oscillations are greater than one, the asymptotic stability will always be possible with the increase in the elasticity coefficient. For ellipsoidal cavities, this means that they must be oblate along the axis of rotation. It was shown that in the absence of the spherical joint elasticity, the characteristic equation has a zero root, and the conditions of stability are already presented in the form of a system of four inequalities, which are only necessary. [ABSTRACT FROM AUTHOR] |
Full text from SpringerLink