On a motion of an equilibrated gyroscope in the newtonian force field

Abstract

This article sheds light on the rotatory motion of a symmetric gyrostat around a fixed point under the effectiveness of both a magnetic field and a Newtonian one besides the action of a gyrostatic moment around the principal inertia’s axes. The gyrostat is assumed to have initially high angular velocity around the principal axis of dynamic symmetry. The controlling system of motion and its first integrals are significantly decreased to a new autonomous system and only one first integral. Poincaré’s method of a small parameter is utilized to achieve the asymptotic solutions of the controlling system. Euler’s angles are evaluated to perceive the motion at every blink. The achieved solutions are graphically portrayed through certain plots for various values of the charge causing the magnetic field and gyrostatic projection's values to estimate the output of these values on the gyrostatic motion. Moreover, the graphs of phase planes for these solutions are sketched to provide a complete explanation of the stability of considered motion. Many applications such as submarines, aircraft, and satellites, are of great interest to the topic concerned.