# Solving a Problem of Rotary Motion for a Heavy Solid Using the Large Parameter Method

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

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## Abstract

The small parameter method was applied for solving many rotational motions of heavy solids, rigid bodies, and gyroscopes for different problems which classify them according to certain initial conditions on moments of inertia and initial angular velocity components. For achieving the small parameter method, the authors have assumed that the initial angular velocity is sufficiently large. In this work, it is assumed that the initial angular velocity is sufficiently small to achieve the large parameter instead of the small one. In this manner, a lot of energy used for making the motion initially is saved. The obtained analytical periodic solutions are represented graphically using a computer program to show the geometric periodicity of the obtained solutions in some interval of time. In the end, the geometric interpretation of the stability of a motion is given.


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## Body

## 1. Introduction

Consider a heavy solid of massM rotating about a fixed point O in presence of a uniform gravity field of force [1]. The fundamental equations of motion and their three first integrals are presented and reduced to a quasilinear autonomous system having one first integral [2]. Consider that the ellipsoid of inertia of the body is arbitrary [3]. The well-known general equations of motion and their first integrals are [4](1)dpdt+A1qr=MgA−1y0γ″−z0γ′,dγdt=rγ′−qγ″,pqr,A1B1C1,ABC,γγ′γ″,xoyozo,Ap2+Bq2+Cr2−2Mgx0γ+y0γ′+z0γ″=const.,Apγ+Bqγ′+Crγ″=const.,γ2+γ′2+γ″2=1,where(2)A1=C−BA,B1=A−CB,C1=B−AC.System (1) of equations of motion represents nonlinear differential equations of the considered problem. These equations are of the first order in unknown angular velocity components p, q, and r and geometric angles γ,γ′, and γ″. The quantities A, B, and C represent the moments of inertia of the body and (x0,y0,z0) represent its gravity center. g denotes the gravity acceleration. t denotes the time of the motion. The aim is to find the solution to this system using the large parameter method [5].Let the initial value of the angular velocity componentr=ro about the moving z axis be sufficiently small. The following variables are introduced:(3)p1=pc−1γ0″,q1=qc−1γ0″,r1=rr0,γ1=γγ0″,γ1′=γ′γ0″,γ1″=γ″γ0″,τ=ro−1t,.≡ddτ,p2=p1−eλ−1−λ−1e1γ2,γ2=γ1−λ−1υp2,c=MgℓC,λ=cγo″r0,e=xo′A1bω2,e1=zo′A1/b−a−11−ω2,υ=1+B11−ω2,ω2=−A1B1,a=AC,b=BC,xo=ℓxo′,yo=ℓyo′,zo=ℓzo′,ℓ2=xo2+yo2+zo2,where r0andγ0″ are the initial values of the corresponding quantities.The nonlinear equations of motions and their first integrals (1) are reduced to a quasilinear autonomous system [6]:(4)p¨2+ω2p2=λ−2Fp2,p˙2,γ2,γ˙2,1λ,γ¨2+γ2=λ−2Φp2,p˙2,γ2,γ˙2,1λ,where(5)F=C1A1−1p2p˙22+x0′p˙2γ˙2−y0′a−1p2γ˙2−y0′A1−1A1+a−1γ2p˙2−z0′a−1p2−νe11−ω2p2−ω2p2s11+A1b−1x0′s21+O1λ+…,Φ=−1−C1A1−1p2p˙2γ˙2+x0′γ˙22−y0′γ2γ˙2−z0′b−1γ2+x0′b−1−A1−2γ2p˙22+ν1−ω2e+e1γ2−γ2s11+1+B1p2s21+O1λ+…,s11=ap202−p22+bp˙202−p˙22A12−2x0′γ20−γ2+y0′γ˙20−γ˙2,s21=ap20γ20−p2γ2−bA1−1p˙20γ˙20−p˙2γ˙2,υ2=υ−A1−1,such that p20andγ20 are the initial values of the corresponding quantities.The variablesq1,r1,γ1′, and γ1″ are obtained as follows:(6)q1=−A1−1p˙2+λ−1A1−1y0′a−1−e2γ˙2+…,r1=1+0.5λ−2s11+…,γ1′=γ˙2+λ−1υ2p˙2+…,γ1″=1+λ−1s21+λ−2s22−0.5s11+…,where(7)s22=aνp202−p22+eγ20−γ2+e1γ202−γ22+bA1−1−ν2p˙202−p˙22+a−1y0′γ˙20−γ˙2−e2γ˙202−γ˙22,e2=e1+a−1zo′.Assuming that the velocityro is sufficiently small, the parameter λ is large.


## 2. Construction of Periodic Solutions, with Zero Basic Amplitudes

In this section, the periodic solutions, with zero basic amplitudes [7], of the autonomous system (4) are achieved and the large parameter method is applied. Without loss of generality of solutions, it is considered that(8)p20,0=p˙20,0=γ˙20,1λ=0.Consider the generating system1/λ=0, that is, (λ⟶∞), of (4) in the form:(9)p¨20+ω2p20=0,γ¨20+γ20=0,with a period T0=2πn. There are three possibilities of the values of frequency ω which are 1−ω=1; 2−ω=m/n where m and n are primes; 3−ω equals an irrational number.Consider the case whenω=m/n, then the solution of the generating system (9) becomes(10)p20=a0∗cosωτ,γ20=b0∗cosτ,where a0∗ and b0∗ are the constants to be determined. The autonomous system (4) has periodic solutions with a period T0+α, where α is a function of 1/λ such that α0=0. These solutions are reduced to the generating ones (10) when 1/λ=0λ⟶∞ and written in the form:(11)p2=a∗cosψ+∑n=1N1λnpn∗a∗,ψ+O1λN+1,γ2=b∗cosϕ+∑n=1N1λnγn∗a∗,ϕ+O1λN+1.With initial conditions:(12)p20,1λ=a∗=a0∗+a∗1λ,γ20,1λ=b∗=b0∗+b∗1λ,γ˙20,1λ=0,where a∗1/λ=0 and b∗1/λ=0 when 1/λ=0.From the first integral (4) and initial conditions (12), one has the following:(13)0<b0∗=1−γ0″21/2γ0″−1<∞,b∗1λ=−1λνa0∗+a∗1λ+….Leta∗, ψ, and ϕ change with time according to(14)a˙∗=∑n=1N1λnAn∗a∗+O1λN+1,(15)ψ˙=ω+∑n=1N1λnψna∗+O1λN+1,(16)ϕ˙=1+∑n=1N1λnϕna∗+O1λN+1.The following derivatives are obtained:(17)p˙2=−a∗ωsinψ+O1λ,γ˙2=−b∗sinϕ+O1λ,p¨2=−a∗ω2cosψ+1λω2∂2p1∗∂ψ2−2a∗ωψ1cosψ−2ωA1∗sinψ+1λ22ωA1∗∂2p1∗∂a∗∂ψ−2ωA2∗+A1∗ψ1sinψ+A1∗dA1∗da∗cosψ+ω2∂2p2∗∂ψ2+2ωψ1∂2p1∗∂ψ2−a∗ψ12+2ωψ2cosψ−a∗A1∗sinψdψ1da∗+O1λ3,γ¨2=−b∗cosϕ+1λ∂2γ1∗∂ϕ2−2b∗ϕ1cosϕ+1λ2∂2γ2∗∂ϕ2+2ϕ1∂2γ1∗∂ϕ2−b∗ϕ12+2ϕ2cosϕ+2A1∗∂2γ1∗∂a∗∂ϕ−b∗A1∗dϕ1da∗sinϕ+O1λ3.From (5), (7), (11), and (17), it is obtained that(18)s110=aa0∗2cos2ψ0−cos2ψ−bA1−2a0∗2ω2sin2ψ−2b0∗x0′cosϕ0−cosϕ+y0′sinϕ,s210=a0∗b0∗acosψ0cosϕ0−cosψcosϕ+bA1−1ωsinψsinϕ,s220=aνa0∗2cos2ψ0−cos2ψ+eb0∗cosϕ0−cosϕ+e1b0∗2cos2ϕ0−cos2ϕ+bA1−1ν2a0∗2ω2sin2ψ+a−1y0′b0∗sinϕ+e2b0∗2sin2ϕ,where ψ0andϕ0 are the initial values of the corresponding quantities.Using (5), (11), and (17), the following is obtained:(19)F0=C1A1−1a0∗3ω2cosψsin2ψ+ωa0∗b0∗x0′sinψsinϕ+a−1a0∗b0∗y0′cosψsinϕ+ωA1−1A1+a−1a0∗b0∗y0′sinψcosϕ−z0′a−1a0∗cosψ−νe11−ω2a0∗cosψ−ω2a0∗cosψaa0∗2cos2ψ0−cos2ψ−bA1−2a0∗2ω2sin2ψ−2b0∗x0′cosϕ0−cosϕ+y0′sinϕ+A1b−1x0′a0∗b0∗acosψ0cosϕ0−cosψcosϕ+bA1−1ωsinψsinϕ,Φ0=12C1−1A1−1ωa0∗2b0∗sin2ψsinϕ+12x0′b0∗21−cos2ϕ+12y0′b0∗2sin2ϕ−z0′b−1b0∗cosϕ+x0′b−1−12A1−2ω2a0∗2b0∗1−cos2ψcosϕ+νe1−ω2+νe11−ω2b0∗cosϕ−aa0∗2b0∗cos2ψ0cosϕ+12aa0∗2b0∗1+cos2ψcosϕ+12bA1−2ω2a0∗2b0∗1−cos2ψcosϕ+2x0′b0∗2cosϕ0cosϕ−x0′b0∗21+cos2ϕ+y0′b0∗2sin2ϕ+a0∗2b0∗1+B1bA1−1ωsinψsinϕ+acosψ0cosϕ0−cosψcosϕcosψ.Substituting (11), (17), and (19) into system (4) and equating coefficients of similar power terms of 1/λ, the following is obtained:(20)∂2p1∗∂ψ2+p1∗=2a0∗ωψ1cosψ+2A1∗ωsinψ,∂2γ1∗∂ϕ2+γ1∗=2b0∗ϕ1cosϕ,∂2p2∗∂ψ2+p2∗=2A2∗ωsinψ+a0∗ω22ωψ2+14ω2C1A1−1a0∗2+34ω2aa0∗2−aa0∗2−z0′a−1−νe11−ω2+14bA1−2a0∗2ω4+2ω2x0′b0∗cosϕ0cosψ+14a0∗3a−C1A1−1−ω2bA1−2cos3ψ+aa0∗ω2x0′A1b−1b0∗cosψ0cosϕ0+x0′a0∗b0∗1ω−aA12bω2−1cosϕ−ψ−x0′a0∗b0∗1+1ω+aA12bω2cosϕ+ψ+y0′a0∗b0∗1+12aω2−A1+a−12A1ωsinϕ−ψ+y0′a0∗b0∗1+12aω2+A1+a−12A1ωsinϕ+ψ,∂2γ2∗∂ϕ2+γ2∗=2ϕ2−z0′b−1+12A1−2ω2a0∗2b−1+νe11−ω2−aa0∗2cos2ψ0−12aB1a0∗2+2x0′b0∗cosϕ0b0∗cosϕ−12x0′b0∗2+x0′b−1+νe1−ω2+1+B1aa0∗2b0∗cosψ0cosϕ0cosψ−32x0′b0∗2cos2ϕ+32y0′b0∗2sin2ϕ+a0∗212A1−2ω21−b−12aB1+A1−1ωb0∗b−1cos2ψ−ϕ+12A1−2ω21−b−12aB1−A1−1ωb0∗b−1cos2ψ+ϕ.Canceling the singular terms [8] from (20), one gets(21)ψ1=A1∗=ϕ1=A2∗=0,ψ2=12ω−14ω2C1A1−1a0∗2−34ω2aa0∗2+aa0∗2+z0′a−1+νe11−ω2−14bA1−2ω4a0∗2−2ω2x0′b0∗cosϕ0,ϕ2=12z0′b−1−12A1−2ω2a0∗2b−1−νe11−ω2+aa0∗212B1+cos2ψ0−2x0′b0∗cosϕ0.Substituting (21) into (14), (15), and (16) and integrating, it is obtained that(22)a∗=a0∗arbitrary const.,ψ=ωτ+121λ2−14ωC1A1−1a0∗2−34ωaa0∗2+aa0∗2ω−1+z01a−1ω−1+νe1ω−1−ω−14bA1−2ω3a0∗2−2ωx0′b0∗cosϕ0τ,ϕ=τ+121λ2z0′b−1−12A1−2ω2a0∗2b−1−νe11−ω2+aa0∗21+12B1−2x0′b0∗τ.From the previous results, the following is obtained:(23)ψ0=ψ0=0,ϕ0=ϕ0=0.Making use of (21), (22), (11), and (13), the periodic solutions p2andγ2 of the autonomous system are deduced. Using (6), (18), (22), and (23), the following periodic solutions, with zero basic amplitudes, are obtained:(24)p1=−x0′λbB1+1λe1b0∗cosτ+…,q1=y0′λaA1+1λe1A1−1b0∗sinτ+…,r1=1−1λ2b0∗x0′1−cosτ+y0′sinτ+…,γ1=b0∗cosτ+…,γ1′=−b0∗sinτ+…,γ1″=1+1λ2b0∗1−a−1x0′+12b0∗2z0′a−ba+b−1+b0∗1−b−1y0′sinτ−b0∗1−a−1x0′cosτ−12b0∗2z0′a−ba+b−1cos2τ+…,where the correction of the period α becomes(25)α1λ=2λπnb0∗x0′−z0′+….


## 3. Conclusion

It is concluded that the method of the small parameter failed to solve this problem under the studied conditionr0 which is sufficiently small because achieving the solutions by this method depends on assuming sufficiently large angular velocity r0 to define the small parameter (ε) proportional to (1/ro). With the sufficiently small assumption, the choosing of the small parameter (ε) is impossible, and so the author had to look for another technique.The large parameter technique is the only one that solves this problem under the studied condition. The advantage of this method is that you save an enormous amount of energy given to the body at the start of the motion. The presented method proves the ability to solve this problem when the component of the angular velocity about the movingz-axis is sufficiently small. Under this technique, gyroscopic motions are obtained under low energy initially instead of high energy in using the small parameter technique. It is clear about the periodicity of the solutions p2 and γ2 from Figures 1 and 2 in a defined interval of time. The simple smooth closed curves with different amplitudes of the solution p2 against γ2 show the stability [9] of the motions, see Figure 3.Figure 1
The periodic motionp2 against the time t.Figure 2
The periodic motionγ2 against the time t.Figure 3
The stability of the solutionp2 against γ2.



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