Solving a Problem of Rotary Motion for a Heavy Solid Using the Large Parameter Method

(e 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. (e 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.


Introduction
Consider a heavy solid of mass M rotating about a fixed point O in presence of a uniform gravity field of force [1]. e 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]. e well-known general equations of motion and their first integrals are [4] dp dt System (1) of equations of motion represents nonlinear differential equations of the considered problem. ese equations are of the first order in unknown angular velocity components p, q, and r and geometric angles c, c ′ , and c ″ . e quantities A, B, and C represent the moments of inertia of the body and (x 0 , y 0 , z 0 ) represent its gravity center. g denotes the gravity acceleration. t denotes the time of the motion. e aim is to find the solution to this system using the large parameter method [5].
Let the initial value of the angular velocity component r � r o about the moving z axis be sufficiently small. e following variables are introduced: . ≡ d dτ , where r 0 and c 0 ″ are the initial values of the corresponding quantities. e nonlinear equations of motions and their first integrals (1) are reduced to a quasilinear autonomous system [6]: where such that p 20 and c 20 are the initial values of the corresponding quantities. e variables q 1 , r 1 , c 1 ′ , and c 1 ″ are obtained as follows: where Assuming that the velocity r o is sufficiently small, the parameter λ is large.

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 ((1/λ) � 0), that is, (λ ⟶ ∞), of (4) in the form: with a period T 0 � 2πn. ere are three possibilities of the values of frequency ω which are 1 − ω � 1; 2 − ω � m/n where m and n are primes; 3 − ω equals an irrational number.
Consider the case when ω � m/n, then the solution of the generating system (9) becomes where a * 0 and b * 0 are the constants to be determined. e autonomous system (4) has periodic solutions with a period T 0 + α, where α is a function of 1/λ such that α(0) � 0. ese solutions are reduced to the generating ones (10) when (1/λ) � 0 (λ ⟶ ∞) and written in the form: With initial conditions: From the first integral (4) and initial conditions (12), one has the following: Let a * , ψ, and ϕ change with time according to e following derivatives are obtained: From (5), (7), (11), and (17), it is obtained that where ψ 0 and ϕ 0 are the initial values of the corresponding quantities.

Conclusion
It is concluded that the method of the small parameter failed to solve this problem under the studied condition r 0 which is sufficiently small because achieving the solutions by this method depends on assuming sufficiently large angular velocity r 0 to define the small parameter (ε) proportional to (1/r o ). With the sufficiently small assumption, the choosing of the small parameter (ε) is impossible, and so the author had to look for another technique. e large parameter technique is the only one that solves this problem under the studied condition. e advantage of this method is that you save an enormous amount of energy given to the body at the start of the motion. e presented method proves the ability to solve this problem when the

Data Availability
No datasets were generated or analyzed during the current study.  The periodic motions of the body at different times   Advances in Astronomy 7