Analysis of Euler’s equations for a symmetric rigid body subject to time-dependent gyrostatic torque

This paper’s main investigation is based on approaching a new solution for the nonlinear Euler’s equations that govern the motion of a rigid body (RB), which is close to Lagrange’s case. The RB is rotating under the influence of two external torques; the gyrostatic torque (GT) and the RB’s constant fixed torques (RBCFTs). The governing equations of motion (EOMs) have been derived and scaled to be decoupled and solved in a complex form. The novel analytical solutions for the angular velocities have been approached and visualized with the help of a mathematical program (Wolfram Mathematica 13.2). The effect of the GT and the RBCFT on these obtained solutions has been presented. In addition, the phase portraits which are a way to visualize the stability and periodicity of the system’s angular velocities have also been explored. These outcomes could help illustrate how GT plays a role in controlling the stability and dynamics of rotating RBs. The importance of this study can significantly affect the stability and control of various mechanical and aerospace systems. Its applications include the design of gyroscopes, spacecraft attitude control, and navigation systems where precise motion control is critical. Understanding this phenomenon helps engineers create more efficient, stable, and reliable systems in fields such as robotics, aviation, and space exploration.