Evolution of rotations of a spheroid with cavity containing a viscous fluid in a resistive medium

Abstract

The problems of the dynamics of rigid bodies with the fluid containing cavities are classical problems of mechanics. The interest to the problems of rotations of rigid bodies with the fluid-containing cavities increased with the development of the rocket and space technology. There is a necessity to study the problems of the dynamics of bodies with cavities containing a viscous fluid, to calculate the motion of spacecraft about the its center of mass, as well as their orientation and stabilization. The torques of forces of viscous fluid in cavity, acting on the body, are often relatively small and can be considered as perturbations. It is natural to use the methods of small parameter to analyze the dynamics of rigid body under the action of applied torques. The method applied in this paper is the Krylov--Bogoliubov asymptotic averaging method. The paper develops on approximate solution by means of averaging method to the system of Euler's equations with additional perturbation terms for a spheroid filled with a viscous fluid in a resistive medium. The numerical integration of the averaged system of equations is conducted for the body motion. The graphical presentations of the solutions are represented and discussed. The main objective of this paper is to extend the previous results for the problem of motion about a center of mass of a spheroid with a cavity filled with a fluid of high viscosity (in the absence of resistive medium). Evolution of perturbed Euler--Poinsot motion under the influence of small internal and external torques is studied. We present new qualitative and quantitative results of investigations of motion in a resistive medium of a nearly dynamically spherical rigid body with a cavity containing viscous fluid with small Reynolds number. We received the system of motion equations in standard form, which refined in square approximation by small parameter. The averaging method was applied to the nonlinear system equation of rotational motion. In this paper we present the input equations of motion, conduct an averaging procedure and receive the averaged equations which, being simpler than the original ones, describe the motion over the large time interval. We establish qualitative and quantitative properties of motion and provide a description of the evolution of the body motion. Results summed up in this paper make it possible to analyze motions of artificial satellites and celestial bodies under the action of small perturbation torques.