On the dynamical motion of a gyro in the presence of external forces

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.