On Stability of Rotation in Resisting Medium of Free System of Three Elastically Connected Rigid Bodies

The equation of rotation in a resisting medium of a free system of three rigid bodies connected in series by elastic spherical hinges is derived. In the case of Lagrange gyroscopes, the characteristic equation of uniform rotations of these gyroscopes under the constant moments directed along their axes of symmetry is obtained. On the basis of the Liénard–Chipart criterion, written in the inner form, the conditions of asymptotic stability are obtained in the form of a system of five inequalities for kinetic moments or the elasticity coefficients of the hinges. The conditions for the kinetic moments and elasticity coefficients under which instability is possible are written out. The algorithm for writing down the stability conditions in the absence of elasticity in one or two hinges is presented. It is shown that when the side bodies are identical, rotate with the same angular velocity, and have the same dissipation and elasticity coefficients, the characteristic equation decomposes into two equations. The first equation describes the asymptotic steady vibrations of a single free Lagrange gyroscope subjected to the restoring moment. The stability conditions in its absence are written out to assess the impact of dissipation on the stability conditions.