Application of a Large-Parameter Technique for Solving a Singular Case of a Rigid Body

A I Ismail,Eugen Radu

doi:10.1155/2021/8842700

Abstract

In this paper, the motion of a rigid body in a singular case of the natural frequency ( ω = 1 / 3 ) is considered. This case of singularity appears in the previous works due to the existence of the term ω 2 − 1 / 9 in the denominator of the obtained solutions. For this reason, we solve the problem from the beginning. We assume that the body rotates about its fixed point in a Newtonian force field and construct the equations of the motion for this case when ω = 1 / 3 . We use a new procedure for solving this problem from the beginning using a large parameter ε that depends on a sufficiently small angular velocity component r o . Applying this procedure, we derive the periodic solutions of the problem and investigate the geometric interpretation of motion. The obtained analytical solutions graphically are presented using programmed data. Using the fourth-order Runge-Kutta method, we find the numerical solutions for this case aimed at determining the errors between both obtained solutions.

Highlights

In [1], the problem of the motion of a fast coherent body around a fixed point under the influence of a Newtonian field of attraction at the value ω = 1/3 of the natural frequency was studied
Achieving a large parameter depends on the properties of the motion, and the periodic solutions are obtained in a new domain using the large-parameter technique
We conclude that the equations of motion for a singular case excluded from the previous works [14] are obtained and reduced to a semilinear system of the second order of two variables

Summary

Introduction

In [1], the problem of the motion of a fast coherent body around a fixed point under the influence of a Newtonian field of attraction at the value ω = 1/3 of the natural frequency was studied. This anomaly appeared in [2] and is specialized in various bodies classified according to the inertia. Achieving a large parameter depends on the properties of the motion, and the periodic solutions are obtained in a new domain using the large-parameter technique. The geometric interpretation of the motion is illustrated to describe Euler’s angles in a new domain depending on the time, the angular velocity, and the large parameter.

The Considered Problem

Geometric Interpretation of Motion

Result and Discussion

Conclusions