Abstract
The small parameter method was applied for solving many rotational motions of heavy solids, rigid bodies, and gyroscopes for different problems which classify them according to certain initial conditions on moments of inertia and initial angular velocity components. For achieving the small parameter method, the authors have assumed that the initial angular velocity is sufficiently large. In this work, it is assumed that the initial angular velocity is sufficiently small to achieve the large parameter instead of the small one. In this manner, a lot of energy used for making the motion initially is saved. The obtained analytical periodic solutions are represented graphically using a computer program to show the geometric periodicity of the obtained solutions in some interval of time. In the end, the geometric interpretation of the stability of a motion is given.
1. Introduction
Consider a heavy solid of mass rotating about a fixed point in presence of a uniform gravity field of force [1]. The fundamental equations of motion and their three first integrals are presented and reduced to a quasilinear autonomous system having one first integral [2]. Consider that the ellipsoid of inertia of the body is arbitrary [3]. The well-known general equations of motion and their first integrals are [4]where
System (1) of equations of motion represents nonlinear differential equations of the considered problem. These equations are of the first order in unknown angular velocity components , , and and geometric angles , and . The quantities , , and represent the moments of inertia of the body and () represent its gravity center. denotes the gravity acceleration. denotes the time of the motion. The aim is to find the solution to this system using the large parameter method [5].
Let the initial value of the angular velocity component about the moving axis be sufficiently small. The following variables are introduced:where are the initial values of the corresponding quantities.
The nonlinear equations of motions and their first integrals (1) are reduced to a quasilinear autonomous system [6]:wheresuch that are the initial values of the corresponding quantities.
The variables , and are obtained as follows:where
Assuming that the velocity is sufficiently small, the parameter is large.
2. Construction of Periodic Solutions, with Zero Basic Amplitudes
In this section, the periodic solutions, with zero basic amplitudes [7], of the autonomous system (4) are achieved and the large parameter method is applied. Without loss of generality of solutions, it is considered that
Consider the generating system , that is, (), of (4) in the form:with a period . There are three possibilities of the values of frequency which are ; where and are primes; equals an irrational number.
Consider the case when , then the solution of the generating system (9) becomeswhere and are the constants to be determined. The autonomous system (4) has periodic solutions with a period , where is a function of such that . These solutions are reduced to the generating ones (10) when and written in the form:
With initial conditions:where and when .
From the first integral (4) and initial conditions (12), one has the following:
Let , , and change with time according to
The following derivatives are obtained:
From (5), (7), (11), and (17), it is obtained thatwhere are the initial values of the corresponding quantities.
Using (5), (11), and (17), the following is obtained:
Substituting (11), (17), and (19) into system (4) and equating coefficients of similar power terms of , the following is obtained:
Canceling the singular terms [8] from (20), one gets
Substituting (21) into (14), (15), and (16) and integrating, it is obtained that
From the previous results, the following is obtained:
Making use of (21), (22), (11), and (13), the periodic solutions of the autonomous system are deduced. Using (6), (18), (22), and (23), the following periodic solutions, with zero basic amplitudes, are obtained:where the correction of the period becomes
3. Conclusion
It is concluded that the method of the small parameter failed to solve this problem under the studied condition which is sufficiently small because achieving the solutions by this method depends on assuming sufficiently large angular velocity to define the small parameter () proportional to (). With the sufficiently small assumption, the choosing of the small parameter () is impossible, and so the author had to look for another technique.
The large parameter technique is the only one that solves this problem under the studied condition. The advantage of this method is that you save an enormous amount of energy given to the body at the start of the motion. The presented method proves the ability to solve this problem when the component of the angular velocity about the moving z-axis is sufficiently small. Under this technique, gyroscopic motions are obtained under low energy initially instead of high energy in using the small parameter technique. It is clear about the periodicity of the solutions and from Figures 1 and 2 in a defined interval of time. The simple smooth closed curves with different amplitudes of the solution against show the stability [9] of the motions, see Figure 3.
Data Availability
No datasets were generated or analyzed during the current study.
Conflicts of Interest
The author declares that there are no conflicts of interest.