Stability of steady motions of a body with a fixed point in a flow of particles

DOI: 10.34759/trd-2023-129-01

Аuthors

Gadzhiev M. M.^*, Kuleshov A. S.^**

Lomonosov Moscow State University, 1, Leninskie Gory, Moscow, 119991, Russia

*e-mail: maxuta-jr@yandex.ru
**e-mail: kuleshov@mech.math.msu.su

Abstract

The problem of motion of a rigid body with a fixed point in a free molecular flow of particles is considered. Suppose the flow consists of identical non — interacting particles, moving with constant velocity along a fixed direction in a fixed absolute space. Suppose the particles interact absolutely inelastically with the rigid body, i.e. after collision the velocity of a particle with respect to the rigid body is zero. Suppose the surface of the rigid body is strictly convex. Then, under assumption, that the flow velocity considerably exceeds the product of the characteristic value of the angular velocity of the rigid body and the characteristic distance from the rigid body to a fixed point, the explicit expression for the moment, acting on the rigid body with a fixed point from the flow of particles are obtained. It is shown, that equations of motion of the rigid body with a fixed point in a free molecular flow of particles are similar in many aspects to the classical system of equations of motion of a heavy rigid body with a fixed point. The corresponding equations of motion of the rigid body with a fixed point in a free molecular flow of particles have partial solutions for which the rigid body performs permanent rotations with constant angular velocity around the streamlines of the flow. Necessary conditions of stability of these permanent rotations are obtained by analyzing the linearized system. When the rigid body is dynamically symmetric, the necessary and sufficient stability conditions of the corresponding steady motions are obtained by analyzing the effective potential of the system.

Keywords:

Rigid Body with a fixed point, Free molecular flow of particles, Steady motions, Stability

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