# The Motion of a Rigid Body with Irrational Natural Frequency

Authors: A. I. Ismail
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

In this paper, we consider the problem of the rotational motion of a rigid body with an irrational value of the frequencyω. The equations of motion are derived and reduced to a quasilinear autonomous system. Such system is reduced to a generating one. We assume a large parameter μ proportional inversely with a sufficiently small component ro of the angular velocity which is assumed around the major or the minor axis of the ellipsoid of inertia. Then, the large parameter technique is used to construct the periodic solutions for such cases. The geometric interpretation of the motion is obtained to describe the orientation of the body in terms of Euler’s angles. Using the digital fourth-order Runge-Kutta method, we determine the digital solutions of the obtained system. The phase diagram procedure is applied to study the stability of the attained solutions. A comparison between the considered numerical and analytical solutions is introduced to show the validity of the presented techniques and solutions.


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## Body

## 1. Introduction

The rigid body problem of massM that rotates about a fixed point O is classified according to the natural frequency value in either the uniform gravity field of g acceleration or a Newtonian force one. The case of the rational value of the natural frequency is studied in [1] for a rotating heavy solid about a fixed point with a small velocity about one of the axes of the ellipsoid of inertia. The solutions obtained contain singular cases of the natural frequencies named when ω=1,2,3,1/2,1/3,⋯. The singular cases that appeared for the natural frequency values like ω=1 (the disk case) and ω=0.5 are considered in [2, 3], respectively. It remains for us to study four other cases until the solution of the problem is completed up to the third approximation for any natural frequency of movement. Such cases are classified according to the natural frequency values named; the state of irrational values of the natural frequencies which is the subject of this article besides three singular cases will be studied in the future in shaa Allah classified when ω=2,3,1/3.Let the frameOxyzbe fixed in the body and the frameOXYZbe fixed in space. Assume that A, B, and C represent the moments of inertia of the body in the moving coordinate system. Suppose (xo,yo,zo) is the mass center of the body. In the case when ω is irrational, the equations of motion are obtained and reduced to the following system:(1)x¨1+ω2x1=μ−2Fx1,x2,ẋ1,ẋ2,μ−1,x¨2+x2=μ−2Φx1,x2,ẋ1,ẋ2,μ−1,where(2)F=∑k=2∞μ2−kFk,FΦ,ẋi=dxidτ,τ=tro,(3)x1=x1p,γ,x1x2,μ=cγo′′/r0,0≺γo′′≺1,where symbols like (ab) mean cyclic permutations and indicate equations which are omitted; p and γ are the x-components of the angular velocity vector and the fixed unit vector in space, respectively; and c is a constant that depends on the rigid body parameters. The system (1) has the following first integral:(4)x22+ẋ22+2μ−1νx1x2+ν1ẋ1ẋ2+s21+μ−2⋯=L,where ν, ν1, and L are constants that depend on the rigid body parameters, and the function s21=s21τ,μ−1 satisfies the condition s210,μ−1=0 when μ→∞.


## 2. The Periodic Solutions

In this section, we obtain the generating system of (1), and then, we will solve it under a new condition of motion. We adapt a large parameter method [1] to solve the system (1) in the presence of the first integral (4). By putting μ→∞ into the system (1), we get the generating system as in [4] which has generating periodic solutions of a period T0=2π.In this case, from the system (1), the required periodic solutions with a period T0+α are assumed in the following forms [5]:(5)x1τ,μ−1=η1β3,μ−1cosωτ+η2β3,μ−1sinωτ+∑k=2∞μ−kGkτ,x2τ,μ−1=M3+β3cosτ+∑k=2∞μ−kHkτ,where M3 is an arbitrary constant and β3=β3μ−1 and ηiβ3,μ−1i=1,2 are analytic functions of μ−1which vanish when μ→∞. The function α is an analytic function of μ−1 which vanishes when μ→∞. The functions Gkτ and Hkτ are obtained by substituting (5) into (1) and equating coefficients of like powers of μ−1.The functionsηiβ3,μ−1 can be written in an expansion series as follows [6]:(6)ηiβ3,μ−1=∑k=1∞Qki+∂Qki∂M3β3+12∂2Qki∂M32β32+⋯μ−k.Using Equation (3), the periodicity conditions are obtained as follows [7]:(7)x1T0,μ−1−η1β3,μ−1+αωη2β3,μ−1=0,ẋ1T0,μ−1−ωη2β3,μ−1−αω2η1β3,μ−1=0,ẋ2T0,μ−1−αM3+β3=0.Under condition (7) and series (6), it produces an endless system of equations that designates the coefficients Qki. The functions Gkτ and Hkτ are expressed as in [8].Now, we aim to find the periodic solutions of the generalized system. According to [9], series (6) begins from a term of order not lower than μ−2. It follows that the expansion of x1τ,μ−1 represents a power series that begins from a term of order not lower than μ−2. In this case, the quantities H2T0and Ḣ2T0 are obtained.Introduce the following variables [10]:(8)x1=pc−1γ0″−μ−1e+e1x2,x2=γγ0″−1−μ−1νx1,qc−1γ0″=−A1−1ẋ1+μ−1A1−1y0′a−1−e2ẋ2+⋯,rr0−1=1+0.5μ−2s11+⋯,γ′γ0″−1=ẋ2+μ−1ν1ẋ1+⋯,γ″γ0″−1=1+μ−1s21+μ−2s22−0.5s11+⋯,where e1=z0′/1−ω2A1b−1−a−1,e2−e1=z0′a−1, and yo′,zo′,a,b,A1 are constants that depend on the rigid body parameters. The vectors p,q,r and γ,γ′,γ″ represent the components of the angular velocity and the unit vector of the downward fixed axis in space, respectively, and sii=siiτ,μ−1,i=1,2, satisfies the condition sii0,μ−1=0 whenμ→∞.Letr0 be sufficiently small; we define a large parameterμ and apply the large parameter method [3] to obtain the analytical solutions in the form of power series expansions in terms of μ−1 as follows:(9)pc−1γ0″=μ−1x0′1−aγ0″+e1M3+β3cosτ+⋯,qc−1γ0″=μ−1μ−1y0′1−bγ0″+e2A1−1M3+β3sinτ+⋯,rr0−1=1−μ−2M3+β3x0′1−cosτ+y0′sinτ+⋯,γ=M3+β3cosτ−0.5μ−2Γ2cosτ+⋯,γ′=−M3+β3sinτ+0.5μ−2Γ2sinτ+⋯,γ″=γ0″+μ−2M3+β3x0′1−aγ0″1−cosτ+y0′1−bγ0″sinτ−0.5M3+β3C1z0′a+b−11−cos2τ+⋯,where C1,x0′ are constants that depend on the rigid body parameters, and(10)0<Γ<1.The correction of the periodα is obtained in the form(11)α=2μ−2πnM3+β3x0′−z0′γ0″+⋯.


## 3. The Geometric Interpretation of Motion

In this section, we discuss the problem geometrically to show the orientation of the body at any instant in time. Substituting Equation (9) into Euler’s angles ψ,θ, and φ in which t has been replaced by t+h, using M3=tanθo, we obtain the following expressions for the angles [11]:(12)θ=θo−μ−2θ1t+h−θ1h+⋯,ψ=ψo−MgCro−1zot+μ−1cosecθocosθoψ1t+h−ψ1h+⋯,φ=φo+ro+MgCro−1zocosθo−xosinθot+⋯,where(13)θ1t=yo′a−1A1−1sinro−1t+xo′b−1B1−1cosro−1t−0.5zo′tanθoA1−1A1+1−1cos2ro−1t,ψ1t=yo′a−1A1−1cosro−1t−xo′b−1B1−1sinro−1t+0.25tanθoe1+e2/A1sin2ro−1t,where B1 is a constant depending on the moments of inertia.


## 4. Numerical Considerations

In this section, we study the analytical and numerical solutions for our problem as follows.


### 4.1. The Analytical Solutions

We rewrite the analytical solutionsx1,x2 in the following form:(14)x11−μ−2e1ν=−μ−1e+xo′γo′′1−a+e1M3+β3cosro−1t−e1γo′′−1M3+β3cosro−1t−0.5μ−2Γ2cosro−1t+⋯,ẋ11−μ−2e1ν=−μ−1−e1M3+β3sinro−1t+e1γo′′−1M3+β3sinro−1t−0.5μ−2Γ2sinro−1t+⋯,(15)x21−μ−2e1ν=γo′′−1M3+β3cosro−1t−0.5μ−2Γ2cosro−1t+μ−2eν−μ−2νxo′γo′′1−a+e1M3+β3cosro−1t+⋯,ẋ21−μ−2e1ν=γo′′−1−M3+β3sinro−1t+0.5μ−2Γ2sinro−1t+μ−2νe1M3+β3sinro−1t+.⋯Let the stepi=0:300and t=iT/300, where T is the maximum value of the variable t. Let us introduce the following data:(16)A=10,B=15,C=20,x0=4,y0=7,z0=8,M=600,γ0″=0.99,r0=0.000001,T=21.76559.Let the body parameters be(17)A1=0.5,B1=‐0.6666667,C1=0.25,ω=3/3,ℓ=11.357820,x0′=0.3521804,y0′=0.6163157,z0′=0.7043607,g=1.020000E‐04,c=0.1864267395,ε=5.272944E05,a=0.5,b=0.75,ν=2,ν1=6,e=‐4,e1=‐6.4,e2=‐2.4.Introduce the following computerized symbols:(18)dx2dt=y,x2=J.Using the above data and a computer program, we obtain the analytical solutions in the graphs (see Figures1 and 2).Figure 1
The analytical solutionx1 against the time t.Figure 2
The analytical solutionJ=x2 against the time t.


### 4.2. The Numerical Solutions

Using (18), we rewrite the system (1) in the form(19)x¨1=−ω2x1+μ−2Fx1,J,ẋ1,y,μ−1,ẏ=−J+μ−2Φx1,J,ẋ1,y,μ−1.Using the fourth-order Runge-Kutta method [12] through a computer program and the data (16) and (17) with the same initial values of the analytical solutions, we obtain the numerical solutions in the graphs (see Figures 3 and 4). For checking the accuracy of both solutions, we draw the graphs (see Figures 5 and 6). We find agreement between the analytical and numerical solutions which satisfied the excellent results for the analytical and numerical techniques. The smooth simple curves obtained (see Figures 7–12) show that the obtained solutions are stable [13, 14].Figure 3
The numerical solutionx1 against the time t.Figure 4
The numerical solutionJ=x2 against the time t.Figure 5
The analytical and numerical solutionsx1 against the time t.Figure 6
The analytical and numerical solutionsJ=x2 against the time t.Figure 7
The stability of the analytical solutionẋ1≡x1.Figure 8
The stability of the numerical solutionẋ1≡x1.Figure 9
The stability of the analytical and numerical solutionsẋ1≡x1.Figure 10
The stability of the analytical solutionsy=ẋ2 and J=x2.Figure 11
The stability of the numerical solutionsy=ẋ2 and J=x2.Figure 12
The stability of the analytical and numerical solutionsy=ẋ2 and J=x2.


## 4.1. The Analytical Solutions

We rewrite the analytical solutionsx1,x2 in the following form:(14)x11−μ−2e1ν=−μ−1e+xo′γo′′1−a+e1M3+β3cosro−1t−e1γo′′−1M3+β3cosro−1t−0.5μ−2Γ2cosro−1t+⋯,ẋ11−μ−2e1ν=−μ−1−e1M3+β3sinro−1t+e1γo′′−1M3+β3sinro−1t−0.5μ−2Γ2sinro−1t+⋯,(15)x21−μ−2e1ν=γo′′−1M3+β3cosro−1t−0.5μ−2Γ2cosro−1t+μ−2eν−μ−2νxo′γo′′1−a+e1M3+β3cosro−1t+⋯,ẋ21−μ−2e1ν=γo′′−1−M3+β3sinro−1t+0.5μ−2Γ2sinro−1t+μ−2νe1M3+β3sinro−1t+.⋯Let the stepi=0:300and t=iT/300, where T is the maximum value of the variable t. Let us introduce the following data:(16)A=10,B=15,C=20,x0=4,y0=7,z0=8,M=600,γ0″=0.99,r0=0.000001,T=21.76559.Let the body parameters be(17)A1=0.5,B1=‐0.6666667,C1=0.25,ω=3/3,ℓ=11.357820,x0′=0.3521804,y0′=0.6163157,z0′=0.7043607,g=1.020000E‐04,c=0.1864267395,ε=5.272944E05,a=0.5,b=0.75,ν=2,ν1=6,e=‐4,e1=‐6.4,e2=‐2.4.Introduce the following computerized symbols:(18)dx2dt=y,x2=J.Using the above data and a computer program, we obtain the analytical solutions in the graphs (see Figures1 and 2).Figure 1
The analytical solutionx1 against the time t.Figure 2
The analytical solutionJ=x2 against the time t.


## 4.2. The Numerical Solutions

Using (18), we rewrite the system (1) in the form(19)x¨1=−ω2x1+μ−2Fx1,J,ẋ1,y,μ−1,ẏ=−J+μ−2Φx1,J,ẋ1,y,μ−1.Using the fourth-order Runge-Kutta method [12] through a computer program and the data (16) and (17) with the same initial values of the analytical solutions, we obtain the numerical solutions in the graphs (see Figures 3 and 4). For checking the accuracy of both solutions, we draw the graphs (see Figures 5 and 6). We find agreement between the analytical and numerical solutions which satisfied the excellent results for the analytical and numerical techniques. The smooth simple curves obtained (see Figures 7–12) show that the obtained solutions are stable [13, 14].Figure 3
The numerical solutionx1 against the time t.Figure 4
The numerical solutionJ=x2 against the time t.Figure 5
The analytical and numerical solutionsx1 against the time t.Figure 6
The analytical and numerical solutionsJ=x2 against the time t.Figure 7
The stability of the analytical solutionẋ1≡x1.Figure 8
The stability of the numerical solutionẋ1≡x1.Figure 9
The stability of the analytical and numerical solutionsẋ1≡x1.Figure 10
The stability of the analytical solutionsy=ẋ2 and J=x2.Figure 11
The stability of the numerical solutionsy=ẋ2 and J=x2.Figure 12
The stability of the analytical and numerical solutionsy=ẋ2 and J=x2.


## 5. Conclusions

In this section, we conclude that the problem of the motion of a rigid body in a uniform gravity field with an irrational value of the natural frequency which is excluded from the previous works [1–3] is considered. The equations of motion and their first integrals in the presence of new conditions of motion are obtained and reduced to a semilinear autonomous system of two degrees of freedom and one first integral. We assumed a large parameter μ that is achieved to be inversely proportional to the angular velocity component ro which is supposed to be sufficiently small. Under this assumption, the well-known Poincare method [15] cannot solve this problem because we cannot achieve the small parameter which must be proportional to a sufficiently high angular velocity component ro→∞. Therefore, we solve the problem with the large parameter technique. The advantages of this method are as follows: using small energy at the initial moment instead of high energy, obtaining a slow gyroscopic motion instead of fast gyroscopic motion, and giving analytical and numerical solutions in a new domain of the motion fro→0,μ→∞,t. Also, the correction of the period for these solutions is obtained in the new domain. The geometric interpretation of motion is attained. When μ→∞, we obtain from Equation (17) the cases of regular permutation and pure rotation. Applying the numerical fourth-order Runge-Kutta method [12] through a computer program, we find the numerical solutions for this problem. Another computer program is carried out on the analytical obtained solutions using the large parameter technique and assuming the same initial values and data (16) and (17). We get the numerical results for both solutions and their graphical representations. The agreement of results through the graphical representations shows the advantage of both techniques for obtaining the high accuracy of the obtained solutions. This problem has many applications in aerospace sciences and technologies because of the wide use of gyros in these fields [16]. The procedures used here are useful for solving sophisticated problems such as [17] in a new domain of the considered parameters. This can be done by reflecting the problem parameters. In shaa Allah in next papers, we study the remaining singular cases of the natural frequency values to complete the solutions for the problem up to the third approximation.



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