Abstract
This work shed light on the motion of a symmetric rigid body (gyro) about one of its principal axes in the presence of a Newtonian force field besides a gyro moment in which its second component equals null. It is assumed that the body center of mass is shifted slightly relative to the dynamic symmetry axis. The governing equations of motion are investigated taking into account some initial conditions. The desired solutions of these equations are achieved in framework of the small parameter method. The periodic solutions for the case of irrational frequencies are investigated. Euler’s angles have been used to interpret the motion at any time. The geometrical representations of the obtained solutions and the phase plane schemas of these solutions are announced during several plots. Discussion of the results is presented to reinforce the importance of the considered gyro moment and the Newtonian force field. The significance of this problem is due to the framework of its several applications in different industries such as airplanes, submarines, compasses, spaceships, and guided missiles.
Highlights
The search for the analytical solutions of the rigid body problem about one of its fixed points is an important task in analytical mechanics
The task of this work is to investigate the rotational motion of a symmetric rigid body under the effectiveness of a Newtonian field of force and undergo the gyrostatic moment vector in which its second component equals zero
A conclusion that may be constructed here is that the small parameter method of Poincareis applied to obtain the approximate analytic periodic solutions for the mentioned problem up to the fraction order 3/2 of the small parameter e
Summary
The search for the analytical solutions of the rigid body problem about one of its fixed points is an important task in analytical mechanics. The periodic solutions for the rotational motion of a rigid body under the influence of a Newtonian field of force were obtained in Ismail[9] using Krylov- The rotational motion of a body undergoes a constant torque which was investigated by Livneh and Wie.[17] The solutions of this problem close to the equilibrium points undergo a Newtonian field of force are achieved in El-Sabaa.[20]
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