Thesis for a candidate’s degree (physical and mathematical sciences) by speciality 01.02.01 theoretical mechanics. Institute of Applied Mathematics and Mechanics of NAS of Ukraine, Donetsk, 2004.
The dissertation is devoted to the study of rotational motions of a dynamically symmetric rigid body about a fixed point under the action of restoring and perturbation torques. Restoring torques are assumed to be depending on: a) slow time, b) nutation angle and joint action of these factors. Perturbation torques are slowly varying in time and caused by influence of: a) resistant medium, b) time optimal suppression of the equatorial component of the vector of angular velocity.
Investigations are carried on at various collections and assumptions relative to the order of smallness and values of restoring torques. The averaging method is applied for the analysis of a nonlinear system of equations of motion.
Perturbed motions of Lagrange’s top were considered in case where the angular velocity of the body on the symmetry axis is large enough and two projections of the perturbation torque on the principal axes of inertia of the body are small as compared with the restoring torque, while the third projection coincides with the restoring torque in order of magnitude. Corresponding standard system is a two-frequency system where the frequency ratio is constant. In this case the averaging of the nonlinear system is equivalent to the averaging of a quasi-linear system with constant frequencies. The averaging was carried on in non-resonant and resonant cases. Averaged first-approximation systems of equations of motion for slow variables characterizing evolution of rotations of a rigid body were obtained. The influence of perturbation torque of symmetric linear dissipation by the environment and small control moments is investigated.
Perturbed rotational motions of a rigid body similar to the Lagrange case were studied. It was assumed that the angular velocity of the body is large and the perturbation torques are small as compared with the restoring torques. The averaging method was used. Averaged systems of equations of motion were obtained in the first and second approximations. The terms of the second approximation supplement the expression for the angular precession velocity known from approximate gyroscope theory. A number of specific problems of dynamics and control of rotation of a rigid body has been solved. The obtained solutions have an independent value for applications.
Key words: rigid body, restoring and perturbation torques, averaging method, Lagrange case, control.