A slow rotary dynamic motion of a disc under the influence of torque with the concept of a large parameter

Abstract

A rotary motion is studied, which is used as a paradigm in both mathematics and dynamics. In this motion, the body moves around its axis. Important examples of this motion are the earth, wheels for cars or bicycles, cooling fans, airplanes, radars, and motion of the discs. In this article, we consider one of these motions such as the slow rotational motion of a disc of high natural frequency around a fixed point different from its center of mass. Suppose a constant gyrostatic couple acts on the body about the axis of symmetry. Let us consider the axis of symmetry coincide with one of the main axes of inertia. The controlled nonlinear independent system of equations of motion is obtained in the form of six nonlinear differential equations in the presence of three independent first integrals. This system is reduced to an alternative independent quasi-linear system consisting of two differential equations with only one integral. At first, when the body rotates slowly with a small angular speed about the axis of the symmetry, we achieve a large parameter for the motion which is proportional to the inverse of the angular velocity component. Then, we use the large parameter technique to obtain approximate and high-frequency analytic solutions for this motion. This technique enables us to introduce new conditions of the motion, save a lot of energy required to move the disc, and obtain new approximated solutions in a new domain of the disc parameters. We present a digital example of the problem to clarify the inferences of the motion depending on the parameters of the body. On the other hand, we use the Runge–Kutta algorithmic method to obtain the numerical solutions corresponding to the approximate solutions of the considered independent system. We investigate the inaccuracies and errors of analytic and numerical solutions by comparing them. This problem offers many applications in gyroscope dynamics, industrial engineering, astrophysics, navigation, satellites, and space engineering.