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<p>Based on the well-known P.V. Kharlamov equations and the S.L. Sobolev state function, the problem of stabilization of the unstable rotation of two rigid bodies connected by an elastic restoring spherical joint and located in a resisting medium has been considered. One of the rigid bodies has an arbitrary axisymmetric cavity completely filled with an ideal incompressible fluid, and the stabilization can be attained by rotating its rigid part. It is assumed that dissipative and constant moments act on the rigid bodies. Supposing that the centers of mass of the rigid bodies and the fluid are located on the third principal axis of inertia, the equations of perturbed motion have been obtained. In the case of two Lagrangian gyroscopes, these equations are presented as a countable system of ordinary differential equations, and the corresponding transcendental characteristic equation has been written down. Taking into account the fundamental mode of fluid oscillations, a fifth-order characteristic equation has been obtained, and conditions for the asymptotic stability of uniform rotations of the Lagrangian gyroscopes and the fluid have been written in the form of a system of four inequalities. These conditions have been analyzed with respect to the kinetic momentum of the second gyroscope and the joint elasticity coefficient. It has been shown that if the first fluid oscillation mode exceeds unity and the center of mass of the first (with fluid) or the second gyroscope does not coincide with their common point, then, provided that the angular velocity of the second gyroscope or the joint elasticity coefficient is sufficiently large, it will be possible to stabilize the unstable rotation of the Lagrangian gyroscope with the ideal fluid. This means that the ellipsoidal cavity must be squeezed along the rotation axis. It has been shown that if dissipation is absent and the moments are constant, it is always possible to stabilize the unstable free rotation of the Lagrangian gyroscope with fluid by increasing the joint elasticity coefficient and the angular velocity. In this case, the ellipsoidal cavity can be not only squeezed but also stretched. This is a fundamental difference between those two cases. The results obtained are accurate for ellipsoidal and cofocal ellipsoidal cavities. For all other cavities, they are only approximate, so additional oscillation modes of ideal fluid must be taken into account for their refinement.</p>

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ON STABILIZATION OF UNSTABLE ROTATION OF A FREE RIGID BODY WITH FLUID IN A RESISTING MEDIUM BY ROTATING ITS RIGID PART

  • 期刊论文

  • 发表日期：

    2025年8月1日

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  • Yuriy M. Kononov

摘要

Based on the well-known P.V. Kharlamov equations and the S.L. Sobolev state function, the problem of stabilization of the unstable rotation of two rigid bodies connected by an elastic restoring spherical joint and located in a resisting medium has been considered. One of the rigid bodies has an arbitrary axisymmetric cavity completely filled with an ideal incompressible fluid, and the stabilization can be attained by rotating its rigid part. It is assumed that dissipative and constant moments act on the rigid bodies. Supposing that the centers of mass of the rigid bodies and the fluid are located on the third principal axis of inertia, the equations of perturbed motion have been obtained. In the case of two Lagrangian gyroscopes, these equations are presented as a countable system of ordinary differential equations, and the corresponding transcendental characteristic equation has been written down. Taking into account the fundamental mode of fluid oscillations, a fifth-order characteristic equation has been obtained, and conditions for the asymptotic stability of uniform rotations of the Lagrangian gyroscopes and the fluid have been written in the form of a system of four inequalities. These conditions have been analyzed with respect to the kinetic momentum of the second gyroscope and the joint elasticity coefficient. It has been shown that if the first fluid oscillation mode exceeds unity and the center of mass of the first (with fluid) or the second gyroscope does not coincide with their common point, then, provided that the angular velocity of the second gyroscope or the joint elasticity coefficient is sufficiently large, it will be possible to stabilize the unstable rotation of the Lagrangian gyroscope with the ideal fluid. This means that the ellipsoidal cavity must be squeezed along the rotation axis. It has been shown that if dissipation is absent and the moments are constant, it is always possible to stabilize the unstable free rotation of the Lagrangian gyroscope with fluid by increasing the joint elasticity coefficient and the angular velocity. In this case, the ellipsoidal cavity can be not only squeezed but also stretched. This is a fundamental difference between those two cases. The results obtained are accurate for ellipsoidal and cofocal ellipsoidal cavities. For all other cavities, they are only approximate, so additional oscillation modes of ideal fluid must be taken into account for their refinement.