On the motion of a gyrostat similar to Lagrange's gyroscope under the influence of a gyrostatic moment vector

In this paper, the problem of the motion of a gyrostat fixed at one point under the action of a gyrostatic moment vector (l) over right arrow whose components are (l) over right arrow l(i) (i = 1, 2, 3) about the axes of rotation, similar to a Lagrange gyroscope is investigated. We assume that the center of mass G of this gyrostat is displaced by a small quantity relative to the axis of symmetry, and that quantity is used to obtain the small parameter e (Elfimov in PMM, 42(2): 251-258, 1978). The equations of motion will be studied under certain initial conditions of motion. The Poincare small parameter method (Malkin in USAEC, Technical Information Service, ABC. Tr-3766, 1959; Nayfeh in Perturbation methods, Wiley-Interscience, New York, 1973) is applied to obtain the periodic solutions of motion. The periodic solutions for the case of irrational frequencies ratio are given. The periodic solutions are analyzed geometrically using Euler's angles to describe the orientation of the body at any instant t of time. These solutions are performed by our computer programs to get their graphical representations.