On the construction of periodic solutions of quasi-linear autonomous systems with several degrees of freedom

Abstract

This paper attempts to explore the general rotatory 3D motion of a magnetic heavy solid body around one fixed point under the impact of a gyrostatic moment when the body awarded initially high angular velocity about one of its inertia principal axes. The asymptotic technique of Krylov-Bogoliubov-Mitropolski and its adjustment are applied to achieve new asymptotic periodic solutions of the controlling equations of motion. Discussion of these solutions is presented taking into consideration their graphs representations. The graphical depictions of such solutions and their corresponding phase planes are plotted to give an induction about its behaviour during the interval time of motion and to reveal the good effect of the different applied forces and moments on the body's motion. The application of the achieved results of the present work can be widely found in gyroscopic devices especially that used inertia reference systems like in satellites, airplanes, and missiles. Moreover, for devices that are responsible for the stability of motion for these applications.