The dissertation is dedicated to the study of the asymptotic stability of uniform rotations in the environment with the support of a non-free system of two elastically connected Lagrange gyroscopes and a system of two elastically connected Lagrange gyroscopes with axisymmetric cavities that are completely filled with an ideal incompressible fluid. The rotation of the gyroscopes is supported by constant moments directed along their axes of symmetry and constant moments in the inertial reference system. The equation of disturbed motion is presented in the form of a counting system of ordinary differential equations, and the characteristic equation is reduced to a transcendental equation. Taking into account the main tone of the oscillation of the ideal fluid, a characteristic equation of the sixth degree is obtained, and on the basis of the Lyapunov-Shipar criterion, conditions for the stability of uniform rotations are obtained in the form of a system of five inequalities and their analytical study is carried out in cases of absence of fluid in the first solid body, in the second or in two solid bodies. The possibility of stabilizing unstable uniform rotation in the environment with the support of the Lagrange gyroscope and the Lagrange gyroscope with an ideal fluid is shown by means of a second rotating gyroscope. For example, it is shown that with a sufficiently large angular velocity of rotation of the second gyroscope, stabilization will always be possible by increasing the hinge stiffness coefficients. Based on the obtained conditions of stability of uniform rotations in the environment with the support of two elastically connected Lagrange gyroscopes with an ideal fluid, the problem of stability of uniform rotations of Lagrange gyroscopes and Lagrange gyroscopes with an ideal fluid on a suspension is considered. The stability conditions obtained in the work are compared with similar conditions in the absence of dissipation.
In modern aviation and rocket-space technology, as well as in other sectors of the national economy, structures with elements in the form of elastically connected solid bodies and solid bodies with fluid are widely used. In this regard, the question arises about the influence of the fluid and elastic joints on the stability of oscillations of such mechanical systems. The problem is significantly complicated when it is necessary to take into account the resistance of the environment. One of the most effective approaches to simplifying problems of this class and obtaining analytical solutions is the approach based on the consideration of a system of two elastically connected Lagrange gyroscopes with an ideal fluid. The overview of the current state of the problem under study indicates that the wide possibilities of practical application of research results require analytical determination of the main properties of rotation in the environment with the support of a system of two elastically connected Lagrange gyroscopes and a system of two elastically connected Lagrange gyroscopes with an ideal fluid, since this model well reflects the properties of many real mechanical objects. The reliability of the obtained results is confirmed by comparison with known results in the literature. The analytical studies carried out have shown a sufficiently high efficiency of the obtained stability conditions.
The work is mainly theoretical, but the scientific significance of the results and the practical value of the work lie in the possibility of their application in the work of design bureaus, research institutes, and enterprises engaged in the design and calculation of mechanical objects related to rotation in an environment with the support of systems of solid bodies and solid bodies with fluid in mechanical engineering, aviation, and space technology.
Keywords: nonlinear ordinary differential equations of motion of a system of two solid bodies, elasticity, suspension, environment with support, Lagrange gyroscopes, nonlinear differential equations of motion of an ideal incompressible fluid, arbitrary axisymmetric cavity, uniform rotations, linearization, free oscillations, boundary value problem, eigenfunctions and eigenvalues, spectral problem, counting system of ordinary differential equations, ellipsoidal cavity, first Lyapunov method, asymptotic stability, stabilization.
