The dynamical motion of a gyrostat for the irrational frequency case

摘要

This work outlines on the three dimensional motion of a rigid body about a fixed point according to Lagrange's case under the action of a gyrostatic moment and a Newtonian force field. It is considered that the center of mass of the body is shifted slightly with respect to the principal axis of dynamic symmetry. Equations of motion are derived using the principal equation of the angular momentum and are solved using the Poincare method of small parameter to achieve the asymptotic solutions for the case of irrational frequencies. Euler's angles characterizing the position of the body at any instant are obtained. The diagrammatic representations of the obtained solutions and Euler's angles are represents through some plots which reflect the good effect of the applied moments on the motion and its impact on the stability of the body. The numerical solutions are obtained using Runge-Kutta algorithms from fourth order. The comparison between the asymptotic solutions and the numerical ones reveal high consistency between them which reveal the good accuracy of the used perturbation method.