Applying the Large Parameter Technique for Solving a Slow Rotary Motion of a Disc about a Fixed Point

A I Ismail

doi:10.1155/2020/8854136

Abstract

In this paper, the motion of a disk about a fixed point under the influence of a Newtonian force field and gravity one is considered. We modify the large parameter technique which is achieved by giving the body a sufficiently small angular velocity component r0 about the fixed z-axis of the disk. The periodic solutions of motion are obtained in the neighborhood r0 tends to 0. This case of study is excluded from the previous works because of the appearance of a singular point in the denominator of the obtained solutions. Euler-Poison equations of motion are obtained with their first integrals. These equations are reduced to a quasilinear autonomous system of two degrees of freedom and one first integral. The periodic solutions for this system are obtained under the new initial conditions. Computerizing the obtained periodic solutions through a numerical technique for validation of results is done. Two types of analytical and numerical solutions in the new domain of the angular velocity are obtained. Geometric interpretations of motion are presented to show the orientation of the body at any instant of time t.

Highlights

In [1], the authors considered the limiting case for the motion of a rigid body about a fixed point in the Newtonian force field and gravity one
In [3], Leshchenko and Ershkov presented a new type of solving procedure for Euler-Poisson equations in the presence of some restricted conditions on the body angular velocity or the applied perturbing torques
The author in [4] gave the regular precession of an asymmetric rigid body acted upon by a uniform gravity field and magnetic one. He obtained the equations of motion of the body and reduced them to a quasilinear autonomous system. He found the solution to the problem and its geometric interpretation of motion

Summary

Introduction

In [1], the authors considered the limiting case for the motion of a rigid body about a fixed point in the Newtonian force field and gravity one. In [2], the authors admitted the KBM technique for solving the problem of a rotating heavy solid about a fixed point under the influence of a gyrostatic moment They assumed a small parameter as in [1] and found the analytical and numerical solutions for the body which moves under its gravity and a gyro moment about the minor or the major axis of the ellipsoid of inertia. The authors in [6] studied the rotating symmetric rigid body about a fixed point in the Newtonian force field in a case analogous to Kovalevskaya’s problem They described the motion of the body and derived its equations of motion and find the solution to the problem assuming Kovalevskaya’s conditions.

Formulation of the Problem and Construction of the Periodic Solutions

Numerical Solutions

Conclusions