# Application of a Large-Parameter Technique for Solving a Singular Case of a Rigid Body

Authors: A. I. Ismail
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this paper, the motion of a rigid body in a singular case of the natural frequency (ω=1/3) is considered. This case of singularity appears in the previous works due to the existence of the term ω2−1/9 in the denominator of the obtained solutions. For this reason, we solve the problem from the beginning. We assume that the body rotates about its fixed point in a Newtonian force field and construct the equations of the motion for this case when ω=1/3. We use a new procedure for solving this problem from the beginning using a large parameter ε that depends on a sufficiently small angular velocity component ro. Applying this procedure, we derive the periodic solutions of the problem and investigate the geometric interpretation of motion. The obtained analytical solutions graphically are presented using programmed data. Using the fourth-order Runge-Kutta method, we find the numerical solutions for this case aimed at determining the errors between both obtained solutions.


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## Body

## 1. Introduction

In [1], the problem of the motion of a fast coherent body around a fixed point under the influence of a Newtonian field of attraction at the value ω=1/3 of the natural frequency was studied. This anomaly appeared in [2] and is specialized in various bodies classified according to the inertia. Using the Poincare small-parameter method [3], periodic solutions—with basic zero amplitudes—of the semilinear independent system were obtained in the form of power series expansions up to the third approximation containing assumed small parameters. In [4], Poincare’s method that depends on a small parameter assumed to be inversely proportional to a high angular velocity component ro is studied. Some important other analytical and numerical solutions besides practical applications are presented in [5, 6].In our problem, we assume a slowly rotating body with low angular velocityro, so we cannot use the small-parameter technique. We must look for another technique to solve the considered problem under the new initial conditions of the motion. We assume a large parameter ε directly proportional to 1/ro [7]. Therefore, we qualify the large-parameter method [8] to solve the presented problem under the new initial conditions for the motion. The equations of motion and their reduction are derived to describe the behavior of the body at any instant in time. The first integrals of the motion are obtained. Achieving a large parameter depends on the properties of the motion, and the periodic solutions are obtained in a new domain using the large-parameter technique. The geometric interpretation of the motion is illustrated to describe Euler’s angles in a new domain depending on the time, the angular velocity, and the large parameter. Gyroscopic motions are considered for some values of moments of inertia. Many applications are given.


## 2. The Considered Problem

The semilinear system of equations of motions is obtained in suitable symbols in [7] as follows:
(1)9p¨2+p2−9ε−2Fp2,ṗ2,γ2,γ̇2,ε=0,γ¨2+γ2−ε−2ϕp2,ṗ2,γ2,γ̇2,ε=0,where
(2)p2=p2p,γ,p2γ2,⋅=ddτ,τ=tro,F=F2+ε−1F3+⋯,Fϕ,ε=cγo′′r0,0≺γo′′≺1,Fkτ=Fkp20,ṗ20,γ20,γ̇20≡Fk0,Fϕ,k=2,3.The symbols likeab denote the canceled equations; c is a constant.The following first integral is deduced as [9]
(3)γ22=1−γ″0−2+γ̇22+2ε−1νp2γ2+ν1ṗ2γ̇2+s21+ε−2ν12ṗ22−2γ̇2A1−1e1γ̇2+ṗ2s21+0.5γ̇2s11A1−y0′a−1+ν2p22+s212+2s22−s11+ε−3⋯,where
(4)Sij0=Sij0p20,ṗ20,γ20,γ̇20i,j=1,2,υ, υ1, A1, e1, yo′, and a are constants depending on the rigid body parameters.Applying the method of the large parameter [8], we obtain periodic solutions—with basic zero amplitudes—of the semilinear independent system in the form of power series expansions until the second approximation. On the use of a hypothetical large parameter, we achieve the periodic solutions in the following form:
(5)p=cγ0′′M1cos0.33τ+M2sin0.33τ+ε−1χ+ℓ1cos0.33τ+m1sin0.33τ+χ1M3cosτ+ε−2∑i=07R1icos0.33iτ+R1i′sin0.33iτ+⋯,i≠6,q=cγ0′′0.33A1−1M1sin0.33τ−M2cos0.33τ+A1−1ε−1y0′a−1+χ2M3sinτ−0.33ℓ1sin0.33τ+m1cos0.33τ+ε−2∑i=07R2icos0.33iτ+R2i′sin0.33iτ+⋯,i≠6,r=r01+0.5E2ε−2∑i=07R3icos0.33iτ+R3i′sin0.33iτ+R36cos2τ+⋯,i≠1,γ=γ0″M3cosτ+ε−1aM1cos0.33τ+M2sin0.33τ−M1cosτ+ε−2∑i=06R4icos0.33τ+R4i′sin0.33τ+R49cos3τ+⋯,i≠2,4,γ′=γ0″−M3sinτ−ε−1a0.5ν1M1sin0.33τ−M2cos0.33τ−aM1sinτ+ε−2∑i=06R5icos0.33iτ+R5i′sin0.33iτ+R59sin3τ+⋯,i≠2,4,γ″=γ0″1+ε−1M3Eacosδ+0.50.33bA1−1−acosδ+0.66τ−0.50.33bA1−1+acosδ−1.25τ+ε−2∑i=04R6icos0.33iτ+R6i′sin0.33iτ+R66cos2τ+⋯,i≠1, where M1,M2,M3 are the basic amplitudes of the initial solutions of p2,ṗ2,γ2, respectively, ℓ1,m1 are the deviations of the amplitudes of the solutions p2,ṗ2 from their basic amplitudes, E=M12+M221/2, δ=tan−1M2/M1, and c, χ, χ1, χ2, and Rji are constants that depend on the rigid body parameter and are determined with correspondence by the system (1).


## 3. Geometric Interpretation of Motion

In this section, we describe the motion by Euler’s anglesθ, ψ, and φ which are determined from the obtained periodic solutions (5). Since the initial system of (1) does not depend explicitly on time, then the periodic solutions are still so if t is replaced by t+t0 where t0 is an arbitrary interval of time [10]. Let t=t0 be the initial instant of time, and substituting the periodic solutions (5) into Euler’s angles, we get [11]
(6)φ0−0.5π=r0t0+⋯,tanθ0=M3,θ−θ0=ε−1Eθ1t+t0−θ1t0−ε−2cosθ0sin−1θ0θ2t+t0−θ2t0+⋯,ψ−ψ0=MgℓC−1r0−1sin−1θ00.5χ1+ν0cos2θ0sin−1θ0t+0.5ε−1r0sin−1θ0ψ1t+t0−ψ1t0+ε−2r0cosθ0sin−3θ0ψ2t+t0−ψ2t0+⋯,φ−φ0=r0−0.5χ1MgℓC−1r0−1γo′′sin−1θ0+aEr0−1ccosδγo′′+0.5MgℓC−1E2R30γo′′−ν0∗γo′′r0+acEcosδsinθ0/γo′′+MgℓC−1r0−1R60γo′′−0.17EMgℓC−1γo′′0.50.33bA1−1−aR72′cosδ+R72sinδ−0.33bA1−1+aR74′cosδ−R74sinδt+0.5ε−1γo′′φ1t+t0−φ1t0+ε−2φ2t+t0−φ2t0+⋯,where the functions θ1,θ2,θ3,ψ1,ψ2,φ1,φ2,ν0,ν0∗ and the constants Rij,Rij′ are determined from the correspondence with the system (1) and (5).


## 4. Numerical and Analytical Solutions of the Slow Spinning Motion of a Body Whenω=1/3

In this section, we will program the analytical solutions to the problem mentioned above and represent these solutions graphically. Also, the numerical solutions of the system of semilinear differential equations are considered and will be inferred by the Runge-Kutta method [12] of the fourth rank. The obtained results are represented graphically under the same conditions. In the end, we have done comparing analytical solutions with numerical ones through common graphical representations. We will denote p2,γ2 by P,J, respectively, and their derivatives to time by the symbols X1=dP/dt,Y1=dJ/dt.


### 4.1. The Analytical Solutions

Let us introduce the variables [13]:
(7)p=cγ0′′p1,γ=γ0′′γ1,then, using (7) and (5), we get
(8)p1=M1cos0.33τ+M2sin0.33τ+ε−1χ+ℓ1cos0.33τ+m1sin0.33τ+χ1M3cosτ+ε−2∑i=07R1icos0.33iτ+R1i′sin0.33iτ+⋯,γ1=M3cosτ+ε−1aM1cos0.33τ+M2sin0.33τ−M1cosτ+ε−2∑i=06R4icos0.33iτ+R4i′sin0.33iτ+R49cos3τ+⋯,thus, the analytical solutions p2,γ2 are achieved using a computer program (see Table 1). These solutions appear in the form of numerical values dependent on time and the parameters of the coherent body. Assume the following data of the parameters of the slow spinning gyro obtained for this problem:
(9)A=2.4,B=2.4,C=3.2,x0=0.0,y0=0.0,z0=3.0,γ0″=0.3,λ=0.5,r0=0.000005,T=18787751.42E−6.Table 1
The analytical solutions whenω=1/3.
tPJX1Y101.1547013.1797946.67E−012.13E−11101.5441122.5763525.72E−01-1.86373201.8664799.95E−014.53E−01-3.02009302.107803−9.64E−013.15E−01-3.03018402.257608-2.557021.62E−01-1.89017502.309387-3.179632.64E−03−3.28E−02602.260894-2.59541−1.57E−011.837096702.114234-1.02612−3.10E−013.009679801.8757749.33E−01−4.49E−013.039946901.5558682.537413−5.69E−011.9164091001.1684073.179119−6.64E−016.55E−021107.30E−012.614198−7.30E−01-1.810271202.60E−011.057064−7.65E−01-2.99895130−2.21E−01−9.01E−01−7.66E−01-3.04939140-6.92E-01-2.51754-7.34E-01-1.94245150-1.13404-3.17828−6.71E−01−9.82E−02160-1.52634-2.63271−5.78E−011.783242170-1.85237-1.0879−4.60E−012.987903180-2.097978.70E−01−3.22E−013.058513190-2.252482.497397−1.70E−011.968273200-2.309183.177095−1.06E−021.31E−01210-2.265622.6509351.49E−01-1.75603220-2.123691.1186173.02E−01-2.97654230-1.88955−8.38E−014.43E−01-3.06731240-1.57337-2.476995.64E−01-1.99389250-1.18886-3.175586.60E−01−1.64E−01260−7.53E−01-2.668887.28E−011.72863270−2.84E−01-1.149227.64E−012.9648592801.97E−018.07E−017.67E−013.0757832906.70E−012.456327.37E−012.0193023001.1132563.1737236.74E−011.96E−01We obtain the following graphical representations of the analytical solutions (see Figures1 and 2).Figure 1
The analytical solutionP against the time t.Figure 2
The analytical solutionJ against the time t.


### 4.2. The Numerical Solutions

Using the semilinear system (1), using the same data (9) and the initial values for the analytical solutions, and then applying the fourth-order Runge-Kutta method through another program, we get the numerical solutions as in Table 2, while the representations of their graphs are shown (see Figures 3 and 4).Table 2
The numerical solutions whenω=1/3.
tPJX1Y101.1547013.1797946.67E−012.13E−11101.5443192.5764095.73E−01-1.863584201.8668849.95E−014.54E−01-3.019916302.108389−9.63E−013.15E−01-3.030202402.258349-2.5565211.62E−01-1.890568502.310252-3.1794122.81E−03−3.35E−02602.261844-2.595745−1.57E−011.8362702.115229-1.027045−3.10E−013.00906801.8767719.31E−01−4.49E−013.039994901.5568242.536283−5.69E−011.9172921001.1692813.178677−6.64E−016.70E−021107.31E−012.61479−7.31E−01-1.8086151202.61E−011.058642−7.65E−01-2.997872130−2.20E−01−8.99E−01−7.67E−01-3.049446140−6.92E−01-2.515767−7.35E−01-1.943798150-1.13402-3.177588−6.71E−01−1.01E−01160-1.52653-2.633542−5.78E−011.780833170-1.85276-1.090117−4.60E−012.986351180-2.098558.67E−01−3.22E−013.058559190-2.253212.494973−1.70E−011.970085200-2.310043.176146−1.07E−021.34E−01210-2.266572.6519981.49E−01-1.752857220-2.124691.1214683.02E−01-2.9745230-1.89055−8.35E−014.43E−01-3.06733240-1.57433-2.4739055.64E−01-1.996149250-1.18975-3.1743516.60E−01−1.67E−01260−7.54E−01-2.6701577.28E−011.724689270−2.85E−01-1.1526897.64E−012.962322801.97E−018.02E−017.67E−013.0757592906.70E−012.4525647.37E−012.0219883001.1132223.1722036.75E−012.01E−01Figure 3
The numerical solutionP against the time t.Figure 4
The numerical solutionJ against the time t.To verify the accuracy of the analytical solutions and numerical ones, we compare them graphically (see Figures5 and 6).Figure 5
The analytical and numerical solutionsP against the time t.Figure 6
The analytical and numerical solutionsJ against the time t.This comparison showed that the deviations between analytical and numerical solutions are very small and can be neglected, and this confirms the accuracy of analytical large-parameter technique and numerical one.From the figures, we deduce that the periodicity of the solutionJ is faster than that for the solution P. Moreover, the amplitude for solution J is larger than that for the solution P.


## 4.1. The Analytical Solutions

Let us introduce the variables [13]:
(7)p=cγ0′′p1,γ=γ0′′γ1,then, using (7) and (5), we get
(8)p1=M1cos0.33τ+M2sin0.33τ+ε−1χ+ℓ1cos0.33τ+m1sin0.33τ+χ1M3cosτ+ε−2∑i=07R1icos0.33iτ+R1i′sin0.33iτ+⋯,γ1=M3cosτ+ε−1aM1cos0.33τ+M2sin0.33τ−M1cosτ+ε−2∑i=06R4icos0.33iτ+R4i′sin0.33iτ+R49cos3τ+⋯,thus, the analytical solutions p2,γ2 are achieved using a computer program (see Table 1). These solutions appear in the form of numerical values dependent on time and the parameters of the coherent body. Assume the following data of the parameters of the slow spinning gyro obtained for this problem:
(9)A=2.4,B=2.4,C=3.2,x0=0.0,y0=0.0,z0=3.0,γ0″=0.3,λ=0.5,r0=0.000005,T=18787751.42E−6.Table 1
The analytical solutions whenω=1/3.
tPJX1Y101.1547013.1797946.67E−012.13E−11101.5441122.5763525.72E−01-1.86373201.8664799.95E−014.53E−01-3.02009302.107803−9.64E−013.15E−01-3.03018402.257608-2.557021.62E−01-1.89017502.309387-3.179632.64E−03−3.28E−02602.260894-2.59541−1.57E−011.837096702.114234-1.02612−3.10E−013.009679801.8757749.33E−01−4.49E−013.039946901.5558682.537413−5.69E−011.9164091001.1684073.179119−6.64E−016.55E−021107.30E−012.614198−7.30E−01-1.810271202.60E−011.057064−7.65E−01-2.99895130−2.21E−01−9.01E−01−7.66E−01-3.04939140-6.92E-01-2.51754-7.34E-01-1.94245150-1.13404-3.17828−6.71E−01−9.82E−02160-1.52634-2.63271−5.78E−011.783242170-1.85237-1.0879−4.60E−012.987903180-2.097978.70E−01−3.22E−013.058513190-2.252482.497397−1.70E−011.968273200-2.309183.177095−1.06E−021.31E−01210-2.265622.6509351.49E−01-1.75603220-2.123691.1186173.02E−01-2.97654230-1.88955−8.38E−014.43E−01-3.06731240-1.57337-2.476995.64E−01-1.99389250-1.18886-3.175586.60E−01−1.64E−01260−7.53E−01-2.668887.28E−011.72863270−2.84E−01-1.149227.64E−012.9648592801.97E−018.07E−017.67E−013.0757832906.70E−012.456327.37E−012.0193023001.1132563.1737236.74E−011.96E−01We obtain the following graphical representations of the analytical solutions (see Figures1 and 2).Figure 1
The analytical solutionP against the time t.Figure 2
The analytical solutionJ against the time t.


## 4.2. The Numerical Solutions

Using the semilinear system (1), using the same data (9) and the initial values for the analytical solutions, and then applying the fourth-order Runge-Kutta method through another program, we get the numerical solutions as in Table 2, while the representations of their graphs are shown (see Figures 3 and 4).Table 2
The numerical solutions whenω=1/3.
tPJX1Y101.1547013.1797946.67E−012.13E−11101.5443192.5764095.73E−01-1.863584201.8668849.95E−014.54E−01-3.019916302.108389−9.63E−013.15E−01-3.030202402.258349-2.5565211.62E−01-1.890568502.310252-3.1794122.81E−03−3.35E−02602.261844-2.595745−1.57E−011.8362702.115229-1.027045−3.10E−013.00906801.8767719.31E−01−4.49E−013.039994901.5568242.536283−5.69E−011.9172921001.1692813.178677−6.64E−016.70E−021107.31E−012.61479−7.31E−01-1.8086151202.61E−011.058642−7.65E−01-2.997872130−2.20E−01−8.99E−01−7.67E−01-3.049446140−6.92E−01-2.515767−7.35E−01-1.943798150-1.13402-3.177588−6.71E−01−1.01E−01160-1.52653-2.633542−5.78E−011.780833170-1.85276-1.090117−4.60E−012.986351180-2.098558.67E−01−3.22E−013.058559190-2.253212.494973−1.70E−011.970085200-2.310043.176146−1.07E−021.34E−01210-2.266572.6519981.49E−01-1.752857220-2.124691.1214683.02E−01-2.9745230-1.89055−8.35E−014.43E−01-3.06733240-1.57433-2.4739055.64E−01-1.996149250-1.18975-3.1743516.60E−01−1.67E−01260−7.54E−01-2.6701577.28E−011.724689270−2.85E−01-1.1526897.64E−012.962322801.97E−018.02E−017.67E−013.0757592906.70E−012.4525647.37E−012.0219883001.1132223.1722036.75E−012.01E−01Figure 3
The numerical solutionP against the time t.Figure 4
The numerical solutionJ against the time t.To verify the accuracy of the analytical solutions and numerical ones, we compare them graphically (see Figures5 and 6).Figure 5
The analytical and numerical solutionsP against the time t.Figure 6
The analytical and numerical solutionsJ against the time t.This comparison showed that the deviations between analytical and numerical solutions are very small and can be neglected, and this confirms the accuracy of analytical large-parameter technique and numerical one.From the figures, we deduce that the periodicity of the solutionJ is faster than that for the solution P. Moreover, the amplitude for solution J is larger than that for the solution P.


## 5. Result and Discussion

The analytical and numerical solutions, their derivatives, and the comparisons between them are given in Tables1 and 2 and Figures 1–6 which lead us to the following analysis and discussion. Solution P rotates slowly, but solution J rotates fastly. The amplitude of P is between −3,3 while the amplitude of J is between −4,4; that is, the amplitude of J is larger than that for P.


## 6. Conclusions

We conclude that the equations of motion for a singular case excluded from the previous works [14] are obtained and reduced to a semilinear system of the second order of two variables. New initial conditions of the motion are assumed like the sufficiently small angular velocity of the body given initially about the major of the minor axis of the ellipsoid of inertia. Due to this assumption, we obtain a new parameter named the large parameter. A new procedure named the large-parameter technique is given for solving the obtained system. The geometric interpretations of motions are illustrated in this case. Using the fourth-order Runge-Kutta method, we obtained the numerical solutions for the problem. A comparison between the analytical and the numerical solutions is carried out to prove the accuracy of both techniques and solutions obtained. The errors between both techniques are very small and can be neglected. It is possible to enter the third component of the gyroscopic moment vector on the movement of the body and deduce the effect of this rotation on the body as well as its engineering interpretation. All anomalies that appeared in previous researches [15, 16] have been treated. The Lagrange gyroscope as a very special case could also be deduced from the solutions obtained [17]. The considered techniques give the solutions in a new domain of motion which is the reflection of one of the previous works. The numerical and analytical procedures which are presented in this article can be applied for solving other problems by reflecting the parameters of the initial condition of the motion [18–20]. The critical points for this work are summarized as follows:
(1)
The natural frequency valueω=1/3 treats a permanent singular case in the previous works which cannot be removed(2)
The angular velocity componentro about the z-axis is assumed to be sufficiently small instead of sufficiently large value in the previous works(3)
The solutionsp,q,r; γ,γ′,γ″, and Euler’s angles θ,ψ,φ are determined in a new domain τ⟶∞,ro⟶0,ε⟶∞ instead of τ⟶0,ro→∞,ε⟶0 in the previous works(4)
The required initial energy for the motion is low compared with that in the previous works, and the spin is weak which gives a weak oscillation case(5)
The parameterε is large instead of a small one in the previous works(6)
The body mass center is slightly displaced from the origin



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