New Treatment of the Rotary Motion of a Rigid Body with Estimated Natural Frequency

In this paper, the problem of the motion of a rigid body about a fixed point under the action of a Newtonian force field is studied when the natural frequency ω � 0.5. (is case of singularity appears in the previous works and deals with different bodies which are classified according to the moments of inertia. Using the large parameter method, the periodic solutions for the equations of motion of this problem are obtained in terms of a large parameter, which will be defined later. (e geometric interpretation of the considered motion will be given in terms of Euler’s angles. (e numerical solutions for the system of equations of motion are obtained by one of the well-known numerical methods. (e comparison between the obtained numerical solutions and analytical ones is carried out to show the errors between them and to prove the accuracy of both used techniques. In the end, we obtain the case of the regular precession type as a special case. (e stability of the motion is considered by the phase diagram procedures.


Introduction
Consider a rigid body of mass M moves in an asymmetric field around a fixed point O [1]. Let us assume that the surface of its ellipsoid of inertia is optional, as well as the mass center. Let the frame OX, OY, and OZ be a fixed system in space, and the frame Ox, Oy, and Oz is the main axes frame for the surface of the ellipsoid of inertia of the body which moves with the it. Initially, we consider the main axis z for the surface of the ellipsoid of inertia that makes an angle ξ 0 ≠ (kπ/2); k � 0, 1, and 2 with the fixed axis Z in space. Let the body spins with small speed angular velocity r 0 about the axis z. Suppose that p, q, and r represent the components of the angular velocity vector of the body about the main axes of the ellipsoid of the inertia surface; c, c ′ , and c ″ are the directional cosines vector of the axis Z; g is the acceleration of gravity; A, B, and C are the principal moments of inertia. e point (x 0 , y 0 , z 0 ) is the center of mass in the moving coordinate system; R is the position vector of the center of attraction 0 1 on the fixed downward coordinate Z axis, and ρ is the position vector of the element dm. Let i , j , k , and Z be the unit vectors in the shown directions ( Figure 1). Consider d F is the attraction force element due to the attracting center and acted on the element dm at the point p(x, y, z).

Formulation of the Problem
Without a loss of generality, we choose the positive direction of both the axis z and the axis x that do not make an obtuse angle ξ 0 with the direction of axis Z. Under the restriction on ξ 0 and the choice of the coordinate system, we get [2] c 0 ≥ 0, 0 < c 0 ″ < 1.
(1) e differential equations of motion can be reduced to an autonomous system of two degrees of freedom and one first integral as follows [3]: where , , e symbols like ABC are abbreviated equations.

Construction of Periodic Solutions with Zeros Basic Amplitudes
In this section, we use the suggested method for constructing the aimed solutions for the autonomous system (2). Consider the condition [4] e generating system for (2) is obtained when ε ⟶ ∞ as follows: e solutions for system (10) with a period T 0 � 4π are where a * 0 and b * 0 are constants. Let system (2) has periodic solutions with a period T 0 + α in the form [5] For these solutions, we let the initial conditions Here, a * (ε), b * (ε) ⟶ 0 at ε ⟶ ∞. Considering first integral (3) with conditions (13), we get Let a * , ψ, and ϕ are changed with time according to e following derivatives are obtained: Using equations (7), (12), and (18), we get where ψ 0 and ϕ 0 are the initial values of the corresponding functions.

Construction of Periodic Solutions with Nonzeros Basic Amplitudes
We use the large parameter method [7] for constructing the periodic solutions with nonzeros basic amplitudes for system (2) when A<B<C or A>B>C. Consider generating system (10) has periodic solutions with a period T 0 � 2πn as follows: where E � , and M 1 , M 2 , and M 3 are constants.
Let system (2) has periodic solutions with a period T 0 + α that reduces to generating solutions (21) when ε ⟶ ∞, where α is a function of ε such that α(∞) � 0. Consider the following initial conditions: e notation ∼ denotes the following substitution: where β 1 , 0.5β 2 , and β 3 represent the deviations of the initial values of the required solutions from their values of the generating ones M 1 , M 2 , and M 3 , respectively. ese deviations are functions of ε and vanish when ε ⟶ ∞. Now, we construct the required solutions in the following forms [8]: where p * n and c * n are periodic functions in ψ and ϕ, respectively. e quantity M 3 is determined from the first integral (3). Let E, ψ, and ϕ are changed with time according to Advances in Astronomy Substituting initial conditions (26) into integral (3), when τ � 0, we deduce that (32) (33) Using equations (7), (28), and (33), we get Using (4), (28), (33), and (34), we obtain

The Numerical Solutions
In this section, we assume numerical values data for the parameters of a rigid body, and we achieve a computer program to solve the quasilinear system using the fourth order Runge-Kutta method [7]. We make another program to represent the analytical solutions numerically in a period t between 0 and 300 (Table 1). We use the initial values from Table 1 for obtaining the numerical solutions represented in Table 2. e comparison between the obtained numerical solutions and analytical ones is presented to know the difference between them. e numerical and analytical solutions are in good agreement with others which proves the accuracy of used methods and obtained results.

Conclusion
e solutions (46) and the correction of the period (47) are obtained using the large parameter method, which had never been used for solving this kind of problem in the presence of the new assumptions for motion (the weak oscillations of the body about the minor or the major axis of the ellipsoid of inertia instead of the strong oscillations in the previous works). e advantage of this method is that the energy motion of the body is assumed to be sufficiently small instead of sufficiently large with other techniques [10][11][12]. Also, the obtained solutions treat a singular situation for the natural frequency which was excluded from previous works [13,14].
Equations (50) and (51) describe the rotation of the body at any time and show that this motion depends on four arbitrary constants ξ 0 , ζ 0 , η 0 , andr 0 , such that r 0 is sufficiently small. e obtained solutions give special cases of motions when (M 1 � M 2 � 0) and when M 1 � 0, M 2 ≠ 0, or M 2 � 0, M 1 ≠ 0. Also, the obtained solutions give many gyroscopic motions, which depend on the values of the moments of inertia and the initial position of the body center of gravity. In the end, we obtain the case of regular precession [10] as a special case. e analytical solutions (46) are represented indefinite intervals of time through computer programs (Table 1). e numerical solutions are obtained using the fourth order Runge-Kutta method in terms of another program ( Table 2). Tables 1 and 2 give in detail the obtained results of both the analytical solutions and numerical ones. ese results show that the analytical solutions are in full agreement with the numerical ones which proves the accuracy of the considered techniques and results. is case of study is considered as a general case of such ones studied in [5]. e stability phase diagrams of the solutions p 2 and c 2 are given (Figures 3 and  4). From these diagrams, we note that the stability for both the analytical and the numerical solutions in full agreement.
is gives the validity of the obtained solutions and the considered procedures. e considered procedures and results are very useful for the general reader's concern with the new applications dealing with the use of functionally graded materials in such structures based on the recent works [15].

Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.