Про стійкість обертання вільної системи двох пружно зв’язаних твердих тіл

Abstract

The equations of rotation of the free system of two rigid bodies connected by an elastic spherical joint or a Hook joint are derived. Assuming that the center of mass of the rigid bodies is located on the third main axis of inertia, the equations of the disturbed motion of the mechanical system under consideration are written in the form of eight ordinary differential equations with periodic coefficients. In the case of two Lagrangian gyroscopes, a characteristic equation of the fourth order is obtained. On the basis of the Lénar-Schipar criterion, written in innor form, the necessary conditions for the stability of uniform rotations of Lagrange gyroscopes in the form of a system of three inequalities are obtained. Analytical studies of these stability conditions were carried out. It is proved that the first inequality is always satisfied. It follows from the third inequality that when gyroscopes have equal axial moments of inertia and rotate with the same angular velocities in different directions or there is no elasticity in the hinge, then the characteristic equation has multiple roots and the question of stability requires additional research. The conditions of stability with respect to kinetic moments are written and it is shown that the older coefficients of these two inequalities are positive, from which it follows that stability will always be possible with sufficiently large values of one of the two kinetic moments. Similar conclusions were obtained in the case of Hook's hinge, and it was also shown that when the gyroscopes are the same, the characteristic equation breaks down into two equations. The first and second equations describe the steady free rotation of one gyroscope on which the restoring moment acts, only in one of the equations it is necessary to add the mass moment of the second gyroscope to the equatorial moment. In the absence of elasticity in the joint, multiple zero roots appear and the question of stability requires additional research.