The periodic solutions of a symmetric charged gyrostat for a slightly relocated center of mass

Abstract

This paper aims to investigate the rotatory dynamical motion of a charged symmetric gyrostat when its center of mass is slightly displaced from the axis of dynamic symmetry as a novel case. A uniform electromagnetic field, due to the presence of point charge located on the axis of symmetry, and a gyrostatic moment are acted on the gyrostatic motion. Equations of motion are derived and solved analytically using the small parameter method of Poincaré for irrational frequencies’ case of the examined motion of the gyrostat. The approximate formulas of the rotating Euler’s angles of the gyrostat are obtained. The time histories of the achieved results and the Euler’s ones are graphed to display the excellent influence of the chosen parameters of the gyrostat on the motion. Moreover, the phase plane diagrams of these results are graphed also to reveal the stability of the gyrostatic motion at any time. The attained results can be used to improve the dynamical behavior of several engineering applications, particularly those that employ the gyroscopic theory such as satellites, submarines, compasses, and automatic pilots on aircraft and ships.