Solutions of Euler's Dynamic Equations for the Motion of a Rigid Body

The aim of this work is to investigate the analytical solutions for the equations of motion of a rigid body about a fixed point through the process of decoupling Euler's dynamic equations. This body is acted upon by a gyrostatic torque l = (l(1); l(2); l(3)) about the axes of rotation, and in the presence of a moment about the same axes, it depends on an external loading in which its components have been expressed as a harmonic function of time. The achieved analytical solutions for the equations of motion are obtained under conditions consistent with the physical nature of the body, and the uniqueness of the solution is proved. Some new theoretical applications are presented when the body is symmetric about one of its principal axes and when the body is in complete symmetry. The graphical representations for the motion of the body are represented to show the effectiveness of the physical parameters of the body. Moreover, the phase plane plots are given to ensure that the considered motion is free of chaos. (C) 2017 American Society of Civil Engineers.