The Motion of a Rigid Body with Irrational Natural Frequency

A I Ismail

doi:10.1155/2020/8898733

Abstract

In this paper, we consider the problem of the rotational motion of a rigid body with an irrational value of the frequency ω . The equations of motion are derived and reduced to a quasilinear autonomous system. Such system is reduced to a generating one. We assume a large parameter μ proportional inversely with a sufficiently small component r o of the angular velocity which is assumed around the major or the minor axis of the ellipsoid of inertia. Then, the large parameter technique is used to construct the periodic solutions for such cases. The geometric interpretation of the motion is obtained to describe the orientation of the body in terms of Euler’s angles. Using the digital fourth-order Runge-Kutta method, we determine the digital solutions of the obtained system. The phase diagram procedure is applied to study the stability of the attained solutions. A comparison between the considered numerical and analytical solutions is introduced to show the validity of the presented techniques and solutions.

Highlights

The rigid body problem of mass M that rotates about a fixed point O is classified according to the natural frequency value in either the uniform gravity field of g acceleration or a Newtonian force one
It remains for us to study four other cases until the solution of the problem is completed up to the third approximation for any natural frequency of movement. Such cases are classified according to the natural frequency values named; the state of irrational values of the natural frequencies which is the subject of this article besides three singular cases will be studied in the future in shaa Allah classified when ω = 2, 3, 1/3
We conclude that the problem of the motion of a rigid body in a uniform gravity field with an irrational value of the natural frequency which is excluded from the previous works [1,2,3] is considered

Summary

Introduction

The rigid body problem of mass M that rotates about a fixed point O is classified according to the natural frequency value in either the uniform gravity field of g acceleration or a Newtonian force one. It remains for us to study four other cases until the solution of the problem is completed up to the third approximation for any natural frequency of movement. Such cases are classified according to the natural frequency values named; the state of irrational values of the natural frequencies which is the subject of this article besides three singular cases will be studied in the future in shaa Allah classified when ω = 2, 3, 1/3.

The Periodic Solutions

The Geometric Interpretation of Motion

Numerical Considerations

Conclusions

Conflicts of Interest