Abstract
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 . This 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. The geometric interpretation of the considered motion will be given in terms of Euler’s angles. The numerical solutions for the system of equations of motion are obtained by one of the well-known numerical methods. The 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. The stability of the motion is considered by the phase diagram procedures.
1. 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 be a fixed system in space, and the frame 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 for the surface of the ellipsoid of inertia that makes an angle with the fixed axis in space. Let the body spins with small speed angular velocity about the axis . Suppose that represent the components of the angular velocity vector of the body about the main axes of the ellipsoid of the inertia surface; are the directional cosines vector of the axis ; is the acceleration of gravity; are the principal moments of inertia. The point () is the center of mass in the moving coordinate system; is the position vector of the center of attraction on the fixed downward coordinate axis, and is the position vector of the element . Let , and be the unit vectors in the shown directions (Figure 1). Consider is the attraction force element due to the attracting center and acted on the element at the point .
2. Formulation of the Problem
Without a loss of generality, we choose the positive direction of both the axis and the axis that do not make an obtuse angle with the direction of axis . Under the restriction on and the choice of the coordinate system, we get [2]
The differential equations of motion can be reduced to an autonomous system of two degrees of freedom and one first integral as follows [3]:where
The symbols like ABC are abbreviated equations.
3. 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]
The generating system for (2) is obtained when as follows:
The solutions for system (10) with a period arewhere are constants.
Let system (2) has periodic solutions with a period in the form [5]
For these solutions, we let the initial conditions
Here, at . Considering first integral (3) with conditions (13), we get
Let , , and are changed with time according to
The following derivatives are obtained:
Using equations (7), (12), and (18), we getwhere and are the initial values of the corresponding functions.
Using (4), (12), (18), and (19), we obtain
Substituting from (12), (18), and (20) into (2) and equating coefficients of in both sides, we get
Canceling singular terms from (21) as in [6], we get
Substituting from (22) into (15)–(17) and integrating, we obtain
From the previous results, we get
From (13) and (23), we obtain from the order greater than .
The periodic solutions are obtained by substituting (22) and (23) into (21) and using (12) and (14). Finally, the periodic solutions are obtained from (5), (19), (23), and (24).
4. 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 as follows:where , and are constants.
Let system (2) has periodic solutions with a period that reduces to generating solutions (21) when , where is a function of such that . Consider the following initial conditions:
The notation denotes the following substitution:where , and represent the deviations of the initial values of the required solutions from their values of the generating ones , and , respectively. These deviations are functions of and vanish when . Now, we construct the required solutions in the following forms [8]:where and are periodic functions in and , respectively. The quantity is determined from the first integral (3). Let , and are changed with time according to
Substituting initial conditions (26) into integral (3), when , we deduce that
The derivatives become
Using equations (7), (28), and (33), we get
Using (4), (28), (33), and (34), we obtain
Substituting from (28), (33), and (35) into initial system (2) and equating coefficients of and in both sides, we obtain the following:
Coefficients of :
We neglect the singular terms [4] to getsuch that determinant (37) becomes
For this case, the solution of (37) becomes
The particular solutions for (36) become
Coefficients of :
Neglecting singular terms from (42) and (43) yields [4]
Substituting from (38), (40), and (44) into (29) and (30) and integrating, we get
Substituting (44) into (42) and (43) and solving the resulted equations, we get . The periodic solutions are constructed using (28), (32), (41), and (45). Using (5) and (34), we get the first terms of the required solutions as follows:
The correction of the period is
5. Geometric Interpretation of Motion
In this section, we describe the body motion using Euler’s angles which come from the obtained solutions (Figure 2). Replacing the time by where is an arbitrary interval, the periodic solutions remain periodic since the initial system is autonomous [9]. For this case, we obtain from (32),where are arbitrary initial angles.
Making use of (46) and (49) when , we find Euler’s angles as follows:where
6. 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. The comparison between the obtained numerical solutions and analytical ones is presented to know the difference between them. The numerical and analytical solutions are in good agreement with others which proves the accuracy of used methods and obtained results.
7. Conclusion
The 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). The 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 [10–12]. 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 , such that is sufficiently small. The obtained solutions give special cases of motions when and when , or . 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.
The analytical solutions (46) are represented indefinite intervals of time through computer programs (Table 1). The 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. These results show that the analytical solutions are in full agreement with the numerical ones which proves the accuracy of the considered techniques and results. This case of study is considered as a general case of such ones studied in [5]. The stability phase diagrams of the solutions and 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. This gives the validity of the obtained solutions and the considered procedures. The 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
The authors declare that they have no conflicts of interest.