Rotation around a fixed point of a solid body with an ellipsoidal cavity completely filled with an inhomogeneous fluid

DOI: 10.34759/trd-2023-128-06

Аuthors

Temnov A. N.*, Yan N. O.**

Bauman Moscow State Technical University, MSTU, 5, bldg. 1, 2-nd Baumanskaya str., Moscow, 105005, Russia

*e-mail: antt45@mail.ru
**e-mail: yno64528@gmail.com

Abstract

In this paper, the equations of spherical motion of a solid body with a rotating inhomogeneous incompressible fluid filling a completely ellipsoidal cavity are obtained and investigated. The stability of rotation of a solid with an inhomogeneous fluid having a linear density distribution is considered. The purpose of this article is to study the effect of an inhomogeneous fluid on the stability of rotation of a solid with a fluid around the axis of dynamic symmetry. In the formulation of the problem, a solid body with an ellipsoidal cavity rotates at an angular velocity around a fixed point that coincides with the geometric center of the cavity, and an inhomogeneous ideal fluid completely filling this cavity performs a homogeneous vortex motion in it with the angular velocity of the fluid. To derive the equation of motion of the system under consideration, we use the Euler-Lagrange equations, which, under the action of potential forces. To solve the problem of the stability of the system’s motion, we use the second Lyapunov method, and construct the Lyapunov function using the Chetaev method. Sufficient conditions for the stability of the rotation of a solid body with a fluid around the vertical axis of dynamic symmetry are derived. The condition is obtained in the form of inequalities of the roots of quadratic forms corresponding to the perturbed motion of a body with a fluid. The obtained equations of motion make it possible to study the stability of stationary motions of the system in question in order to assess the effect of fluid stratification on the dynamics of the body.

Keywords:

inhomogeneous liquid, solid body, ellipsoidal cavity, uniform vortex motion, stability

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