Applying the Large Parameter Technique for Solving a Slow Rotary Motion of a Disc about a Fixed Point

In this paper, the motion of a disk about a fixed point under the influence of a Newtonian force field and gravity one is considered. We modify the large parameter technique which is achieved by giving the body a sufficiently small angular velocity component r0 about the fixed z-axis of the disk. The periodic solutions of motion are obtained in the neighborhood r0 tends to 0. This case of study is excluded from the previous works because of the appearance of a singular point in the denominator of the obtained solutions. Euler-Poison equations of motion are obtained with their first integrals. These equations are reduced to a quasilinear autonomous system of two degrees of freedom and one first integral. The periodic solutions for this system are obtained under the new initial conditions. Computerizing the obtained periodic solutions through a numerical technique for validation of results is done. Two types of analytical and numerical solutions in the new domain of the angular velocity are obtained. Geometric interpretations of motion are presented to show the orientation of the body at any instant of time t.


Introduction
In [1], the authors considered the limiting case for the motion of a rigid body about a fixed point in the Newtonian force field and gravity one. The small parameter technique is applied to solve this problem. The authors defined this parameter inversely proportional to the third component of the angular velocity which is assumed to be sufficiently high. The periodic solutions and their graphical representations are obtained and illustrated geometrically. In [2], the authors admitted the KBM technique for solving the problem of a rotating heavy solid about a fixed point under the influence of a gyrostatic moment. They assumed a small parameter as in [1] and found the analytical and numerical solutions for the body which moves under its gravity and a gyro moment about the minor or the major axis of the ellipsoid of inertia. In [3], Leshchenko and Ershkov presented a new type of solving procedure for Euler-Poisson equations (rigid body rotation over a fixed point) in the presence of some restricted conditions on the body angular velocity or the applied perturbing torques. The author in [4] gave the regular precession of an asymmetric rigid body acted upon by a uniform gravity field and magnetic one. He obtained the equations of motion of the body and reduced them to a quasilinear autonomous system. He found the solution to the problem and its geometric interpretation of motion. Nayfeh [5] presented some perturbation methods such as Poincare', the KBM, the Multiple scales, and averaging techniques for solving this kind of motion under certain conditions. The authors in [6] studied the rotating symmetric rigid body about a fixed point in the Newtonian force field in a case analogous to Kovalevskaya's problem. They described the motion of the body and derived its equations of motion and find the solution to the problem assuming Kovalevskaya's conditions.
All the previous approximated methods depend on a small parameter achieved inversely proportional to the sufficiently high angular velocity component. This study gives many applications in physics [7], gyros [8,9], astronomy, engineering, aerospace, and other sciences.
In our problem, we assume a slow rotation (weak spin) of the body instead of a fast rotation (high spin). We introduce a large parameter inversely proportional to the weak spin about the z-axis instead of a small parameter inversely proportional to the high spin about that axis. So, we must apply the large parameter technique instead of the small one used in the previous works. The validation of our results will be given.

Formulation of the Problem and Construction of the Periodic Solutions
In this section, we formulate the problem of the motion of the body, deduce the equations of motion, and construct the periodic solutions: Let the moving z-axis make an angle θ 0 ≈ π/2 with the downward fixed Z-axis. This case of a slow spinning rotary body when the natural frequency ω = 1 gives the rotary motion of a disc which is excluded from the previous case γ 0 ″ = cos θ 0 ≈ 0. We achieve a large parameter 0 ≺ ≺μ≺∞ inversely proportional to the sufficiently small value of r 0 . Assume that the disc rotates about the z-axis in the presence of a Newtonian force field [3] and a gravitational one. We reduce the equations of motion for this case to the following autonomous system of two degrees of freedom [4]: where where where the symbols like (ab), (abc), and (abcd) denote omitted equations and p, q, r, and γ, γ′γ″ are the components of the angular velocity vector and the cosines direction of the unit vector Z ∧ .
The system (2) has the following integral [4]: where where the quantities p 20 , γ 20 , _ p 20 , and _ γ 20 are the initial values of the corresponding variables.

Construction of the Periodic Solutions.
In this subsection, we use the large parameter technique [5] to construct the periodic solutions of system (2) when the disc rotates with slow velocity about the minor axis of the ellipsoid of inertia. Assume that Using (8) and the definition of s 21 , we get where E = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi M 2 1 + M 2 2 p , ε = tan −1 ðM 2 /M 1 Þ and M 1 and M 2 are constants.

Geometric Interpretation of the Motion
In this section, we search the description of the motion of the disk at any instant of the time using Euler's angles θ, ψ, ϕ [11,12]: where θ 2 t ð Þ = −b −1 x 0 ′ cos r 0 t + a −1 y 0 ′ sin r 0 t + 0:25k cos 2r 0 t, These formulas depend on four arbitrary constants θ 0 , ψ 0 , ϕ 0 , and r 0 , where r 0 is sufficiently small. The obtained analytical periodic solutions are considered a generalization of the ones obtained in [13,14].

Numerical Solutions
In this section, we use a smooth and suitable numerical method for obtaining the approximated numerical solutions for the autonomous system (2). Such a method is named the fourth-order Runge-Kutta method [15] which is used through a computerized program to find the numerical solutions of the considered problem. In the other side, we computerize the obtained analytical solutions and compare them with the numerical ones aiming to find the errors between them: 4.1. The Analytical Solutions. Rewriting the resulted analytical solutions in the form: Let the disk parameters are We obtain the following parameters of the motion: Making use of (30), (31), and (32) through a computer program, we obtain the values of the analytical solutions (see Table 1).

4.2.
The Numerical Solutions. The system (30) is rewritten in the form: where f 1 = F 2 , g 1 = Φ 2 . Using (33), (31), (32), the fourth-order Runge-Kutta method, and the same initial values in the Table 1, we find the results of the numerical solutions in Table 2. Table 1 is in good agreement with Table 2; that is, the analytical solutions are in full approximation with the numerical ones.

Conclusions
In this paper, we studied the singular value of the natural frequency problem ω = 1 in a limiting case when γ o ′ ′ = cos θ o ≈ 0. In this case, the body rotates about the z-axis which should coincide with the minor axis of the ellipsoid of inertia. This problem corresponds to the disc motion The motion takes a slow spin rotation about the symmetric z-axis of the disc. The equations of motion and their first integrals are derived and reduced to two quasilinear differential equations of the second order and one first integral [16][17][18][19]. The periodic solutions for this problem are constructed applying the periodicity conditions and assuming a large parameter [20] proportional to 1/r 0 . We used here the large parameter technique instead of the small one well-known in [21][22][23][24][25][26]. The advantage of this technique comes from the saving of the high initial energy which is given for the body to start the motion, and the solving of the problem in a new domain of the motion F(t, μ ⟶ ∞, r o ⟶ 0) and under new considerations. A geometric interpretation using Euler's angles of motion of the body as a function of time is presented. This interpretation shows the orientation of the disc at any instant of time t. There are numerical considerations of the problem and the solutions using the fourth-order Runge-Kutta method which is a 5 International Journal of Aerospace Engineering specialized and smooth method for finding the approximated periodic solutions of the nonlinear differential equations. Computerized programs are given as a validation of the technique and the results. These programs are carried in a closed interval of time t ∈ ½0, 300 and showed that a full agreement between the analytical solutions and the numerical ones. The agreement of the numerical results with the analytical ones proves that the accuracy (validation) of the resulted solutions and studied techniques (see Tables 1 and 2). From the above tables, we find that the errors between the analytical and the numerical solutions are very small and can be neglected. To study the behavior of the body, we investigate a geometric interpretation of the motion using Euler's angles. We obtain an arbitrary initial angle of nutation θ o , precession ψ o , and pure rotation φ o . We note also that the expressions for the Euler's angles depend on four arbitrary constants θ 0 , ψ 0 , ϕ 0 , and r 0 (where r 0 is sufficiently small). Moreover, we note that the disc spins slowly about the minor axis of the ellipsoid of inertia (that is a case of weak oscillations is obtained). In the first approximation, the case of a pseudoregular precession about the vertical axis is attained. As an example, the case of a regular Precession of a slowly spinning Lagrange gyroscope (A = B, x 0 = y o = 0) is obtained as a special case of this motion. There are many applications of these results in both military and civil life. This study is important for the satellite motion which has the correspondence of inertia moments, the antennas, the navigations, the solar collectors, and aerospace dynamics.

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

Conflicts of Interest
The author declares that he has no competing interests.