# Treating the Solid Pendulum Motion by the Large Parameter Procedure

Authors: A. I. Ismail
Journal: International Journal of Aerospace Engineering
 (2020)

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## Abstract

In this paper, we consider the dynamical description of a pendulum model consists of a heavy solid connection to a nonelastic string which suspended on an elliptic path in a vertical plane. We suppose that the dimensions of the solid are large enough to the length of the suspended string, in contrast to previous works which considered that the dimensions of the body are sufficiently small to the length of the string. According to this new assumption, we define a large parameterε and apply Lagrange’s equation to construct the equations of motion for this case in terms of this large parameter. These equations give a quasi-linear system of second order with two degrees of freedom. The obtained system will be solved in terms of the generalized coordinates θ and φ using the large parameter procedure. This procedure has an advantage over the other methods because it solves the problem in a new domain when fails all other methods for solving the problem in such a domain under these conditions. It is one of the most important applications, when we study the slow spin motion of a rigid body in a Newtonian field of force under an external moment or the rotational motion of a heavy solid in a uniform gravity field or the gyroscopic motions with a sufficiently small angular velocity component about the major or the minor axis of the ellipsoid of inertia. There are many applications of this technique in aerospace science, satellites, navigations, antennas, and solar collectors. This technique is also useful in all perturbed problems in physics and mechanics, for example, the perturbed pendulum motions and the perturbed mechanical systems. The results of this paper also are useful in moving bridges and the swings. For satisfying the validation of the obtained solutions, we consider numerical considerations by one of the numerical methods and compare the obtained analytical and numerical solutions.


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## Body

## 1. Introduction

The pendulum models have attracted scientists and researchers with many descriptions of motions and their analysis as important examples in physics and theoretical and applied dynamics. The most important pendulum motions come from the moving of a heavy particle suspended a light rod which is jointed pivotally at a point on theX-axis which rotates by an angular velocity ω about the horizontal fixed axis. In [1], the authors presented the elastic pendulum problem. They derived the equations of motion and gave real-life examples for elastic pendulum motions. Equilibrium states and trivial cases are studied. In [2], a harmonically excited spring pendulum motion with 3 degrees of freedom is considered. The authors were connected to a direct transversely tuned absorber. The authors used the multiple scale method for solving the system of equations of motion. They used the phase-plane method for applying the stability of the motion. They studied the influence of the tuned absorber and the parameters of the model on the change of the motion numerically. In [3], the elastic pendulum with resonance appearing at cubic approximation in the Lagrangian function was investigated. The modulated equations of motion were obtained. The authors used the Hamiltonian reduction technique to describe the dynamical behavior of this system. In [4], the approximated damping elastic pendulum model was considered. The authors derived the governing equation with six degrees of freedom. The natural frequencies are derived from the linearized system of equations. The stability diagrams are used to show regular periodicity. Due to damping, there are unstable waves (irregular) depending on the damping level and the amount of parametric excitation for the natural frequency. The article [5] presented the rotational motion in a plane for a mathematical pendulum when its axis swinging vertically and horizontally, so the axis path is an ellipse nears a circle. The study is considered as precise spin solutions for a circular axis path case with zero gravity. The existence of the stability conditions of these solutions is considered. Supposing that the excitation amplitude is not small and the coaxial path contains a small ellipsoid. Approximated solutions for both high and low linear damping are found. A comparison of approximate and digital solutions was performed for different values of the damping factor. In [6], the governing system of two generalized coordinates of a nonlinear dynamic model with the spring pendulum damped motion is considered in presence of the flow of an inviscid fluid. The motion control system is attained by applying the equations of Lagrange. The multiple scale procedure is used to solve the equations of motion for this system. The authors obtained approximated solutions of the 2nd approximation. The resonance cases and the steady-state ones are studied. The stability procedure is given using the phase plane diagrams. In [7], the dynamics of nonlinear multiple degrees of freedom for a spring pendulum model moving in an auxiliary circle for the elliptical path are considered. Using the method of multiple scales, the authors solved the system of equations of motion. The temporal history of the solutions obtained and the phase plane projections are given to explain the dynamic behavior of the mentioned system.In [8] the authors studied the elastic pendulum motion with resonance appearing at the third order in its Lagrange’s approximation. The governing dynamical equations of motion and Hamiltonian reduction are obtained for this system. In [9], the frequencies of the elastic pendulum oscillations are in the ratio 2 : 1 which is named the pulsation. The dynamics of this problem and the modulation equations for the resonant motion with small amplitudes are obtained as 3 wave equations. The complete solutions are obtained by using the Hamiltonian reduction. The phases, amplitudes, and precession angles of the solutions are given. The validity of the obtained results is given in high accuracy by numerical experiments. The article [10] presented a dynamical behavior of an elastic pendulum model. The author assumed that the pendulum moves in a vertical plane which rotates with uniform angular velocity ω. The author used Lagrange’s equations to find the differential equations of motion. The obtained equations are reduced to a nonlinear system of 2nd order which is solved by the small parameter perturbed technique. Numerical considerations are obtained through the Matlab program. These considerations show the validity of both analytical and numerical solutions.In [11], the nonlinear multiple degrees of freedom response is investigated for a normal dynamic system given by the spring pendulum moving in an elliptical path. The Lagrange equations were used to construct the governing system of equations for the motion. The multiple scale method is used for deriving the approximated solutions of the resonance system. The steady-state motions and the stabilities of the solutions are considered. The authors in [12] studied the problem of spring pendulum dynamics in the presence of pendulum absorber using the theory of nonlinear normal patterns and asymptotic numerical procedures. They investigated the dynamics of the pendulum for both low and high vibration amplitudes. Using different methods, the stability of the vibration patterns is analyzed. The authors in [13] studied the relative periodic solutions for an elastic pendulum motion of a rigid body with a corresponding suspended point moves on the auxiliary sphere of an ellipse. The Lagrange’s function is applied to construct the equations of motion which are solved using the perturbed small parameter method. Numerical considerations are given using the Runge-Kutta method to prove the validity of the solutions.In this article, we consider a heavy solid suspended in a string which suspended to an ellipse in a uniform vertical plane. The equations of motion for this mathematical pendulum in terms of the two degrees of freedom are obtained, and a large parameter depends on the model properties is assumed. By the definition of the large parameter, the approximated periodic solutions are obtained using the large parameter perturbed procedure. The accuracy of these solutions is investigated through a numerical technique and computerized programs.


## 2. Formulation of the Problem

Let us consider the coordinate systemOXY on which X-axis is horizontal and Y-axis is downward vertical. A rigid body with mass m is suspended by a string of length ℓ which attached to a point O1 moving on the ellipse. The point Q on the circle, which has the same center of the ellipse and its radius equals b such that the line AQ is normal to the largest axis of the ellipse, is called the corresponding point of the point O1 on that circle. When the point O1 moves on the ellipse, the point Q moves on the circle with a constant angular velocity ω (i.e., the angle between the line OQ and the Y-axis must depend on time t only). Let us assume that the angle between the straight line passing through O2 and the center of gravity C of the body and the vertical will be denoted by φ, the angle of deflection of the string from the vertical straight line passing through O1 will be denoted by θ, and the distance between the center of gravity of the body and the point O2 will be denoted by h (i.e., h=O2C) (see Figure 1). Choosing the body system of coordinates Cξηζ such that Cη passes through O2, Cξ is orthogonal to Cη and lying in the plane OXY, while Cζ is orthogonal to OXY.Figure 1
The pendulum model.We assume that the axes of the systemCξηζ represent the principal axes of inertia of the body. The principal moment of inertia concerning Cξ will be denoted by J. The motion of the body starts at the instant t=0 in the anticlockwise direction; after time t, the point O1 is expected to make an angle θ=ωt with the vertical axis.The coordinate of the pointC is (xc,yc) which takes the form
(1)xc=asinωt+ℓsinθ+hsinφ,yc=bcosωt+ℓcosθ+hcosφ.The kinetic energy of the system is(2)T=12mẋc2+ẏc2+12Jφ̇2,which takes the form
(3)L=12mℓ2θ̇2+h2φ̇2+ω2a2cos2ωt+b2sin2ωt+2ℓhθ̇φ̇cosθ−φ+2ℓωθ̇acosωtcosθ+bsinωtsinθ+2hωφ̇acosφcosωt+bsinφsinωt+12Jφ̇2.The potential energy of the system can be expressed as follows:(4)π=−mgℓcosθ+hcosφ.The Lagrangian of the system is [14]
(5)L=T−π.It can be written as follows:(6)L=12mℓ2θ̇2+h2φ̇2+ω2a2cos2ωt+b2sin2ωt+2ℓhθ̇φ̇cosθ−φ+2ℓωθ̇acosωtcosθ+bsinωtsinθ+2hωφ̇acosφcosωt+bsinφsinωt+12Jφ̇2+mgℓcosθ+hcosφLet us define the following parameters.(7)ε=hℓ≻≻1,εδ=aℓ,εσ=bℓ,where δ=a/h, σ=b/h, and ε is a large parameter.We introduce the variables(8)Φ=ε−1φ=ℓhφ,Θ=ε−1θ=ℓhθ,τ=ωt,so that d/dt=ωd/dτ and d/dτ=′.The new Lagrangian function of the system can be written as follows:(9)L∗=12mω2ℓ2ε−2θ′2+ω2h2ε−2φ′2+ω2a2cos2τ+b2sin2τ+2ω2ℓhε−2θ′φ′cosε−1θcosε−1φ+sinε−1θsinε−1φ+2ℓε−1ω2θ′acosτcosε−1θ+bsinτsinε−1θ+2hε−1ω2φ′acosτcosε−1φ+bsinτsinε−1φ+12Jε−2ω2φ′2+mgℓcosθ+hcosε−1φ.For a small value ofε−1θ,ε−1φ, we can expand the sines and cosines in the form of power series expansions to obtain
(10)L∗=12mω2ℓ2ε−2θ′2+h2ε−2φ′2+a2cos2τ+b2sin2τ+2ℓhε−2θ′φ′1−ε−2θ221−ε−2φ22+ε−1θ−ε−3θ36ε−1φ−ε−3φ36+2ℓε−1θ′acosτ1−ε−2θ22+bsinτε−1θ−ε−3θ36+2hε−1φ′acosτ1−ε−2φ22+bsinτε−1φ−ε−3φ36+12Jε−2ω2φ′2+mgℓ1−ε−2θ22+h1−ε−2φ22.


## 3. Equations of Motion

According to Lagrange’s equations [15],
(11)ddτ∂L∗∂qi′−∂L∗∂qi=0,i=1,2,where qi are the generalized coordinates and qi′ are the generalized velocities, and we take the coordinates q1=θ and q2=φ.Introducing the parameters(12)ωn2=gℓω2,γ=ℓ2h2+J/m,Ω2=ghh2+J/m,where g is the acceleration of gravity; the system of differential equations of motion can be written as follows:
(13)θ″+ωn2θ=δsinτ−ε−1θ″+θσcosτ−12ε−2θ2δsinτ+ε−312φ″θ2+φ″2φ2−θφφ″−φ′2θ+φ′2φ+θ36σcosτ,(14)φ″+Ω2φ=ε−1γ−θ″+δsinτ+ε−3γ12θ″θ2+12θ″φ2−θ″θφ+θ′2θ−θ′2φ−δ2φ2sinτ−σφcosτ.The expressionsΘ and Φ are expected to be functions of a large parameter ε and dependent on the values of γ, ωn, and Ω.


## 4. Approximate Periodic Solutions

Now, to find the perturbed solution of the nonresonance case up to the second approximation, we apply the method of the large parameter [16] in the following form.
(15)Θτ,ε=θ0τ+ε−1θ1τ+ε−2θ2τ+⋯,Φτ,ε=φ0τ+ε−1φ1τ+ε−2φ2τ+⋯.Substituting from (15) into (13) and (14) and equating to zero each coefficient of 1/ε, we find thatCoefficient of1/ε0(16)θ0″+ωn2θ0=δsinτ,φ0″+Ω2φ0=0.Coefficient of1/ε(17)θ1″+ωn2θ1=−φ0″−θ0σcosτ,φ1″+Ω2φ1=γδsinτ−θ0″.Coefficient of1/ε2(18)θ2″+ωn2θ2=−φ1″−θ1σcosτ−12δθ02sinτ,φ2″+Ω2φ2=−γθ1″.Coefficient of1/ε3(19)θ3″+ωn2θ3=−φ2″−θ2σcosτ+δθ02θ1sinτ−12φ0″θ02−12φ0″φ02+θ0φ0φ0″+φ′02θ0−φ′02φ0−θ036σcosτ,(20)φ3″+Ω2φ3=γ−θ2″+12θ0″θ02+12θ0″φ02−θ0″θ0φ0+θ′02θ0−θ′02φ0−δ2φ02sinτ−σφ0cosτ.Making use of the equations (15)-(20), the required periodic solutions take the form
(21)Θτ,ε=δωn2−1sinτ+ε−1Ω2Aωn2−Ω2cosΩτ+Ω2Bωn2−Ω2sinΩτ−σδ2ωn2−1ωn2−4sin2τ+ε−2γδωn2ωn2−12Ω2−1sinτ−Ω2σA2ωn2−Ω2ωn2−1+Ω2cos1+Ωτ−Ω2σAcosΩ−1τ2ωn2−Ω2ωn2−Ω−12−Ω2BσsinΩ+1τ2ωn2−Ω2ωn2−Ω+12−Ω2BσsinΩ−1τ2ωn2−Ω2ωn2−Ω−12+σ2δcosτ4ωn2−12ωn2−4−σ2δcos3τ4ωn2−1ωn2−4ωn2−9−3δ3sinτ8ωn2−13+δ3sin3τ8ωn2−1ωn2−9,(22)Φτ,ε=AcosΩτ+BsinΩτ+ε−1γδωn2Ω2−1ωn2−1sinτ−ε−22γσδsin2τωn2−1ωn2−4Ω2−4.Neglecting the secular terms [17] in the formulas (21) and (22), the constants A and B will vanish; then,
(23)Θτ,ε=δωn2−1sinτ−ε−1σδsin2τ2ωn2−1ωn2−4+ε−2γδωn2ωn2−12Ω2−1sinτ+σ2δcosτ4ωn2−12ωn2−4−σ2δcos3τ4ωn2−12ωn2−4ωn2−9−3δ38ωn2−13sinτ+δ38ωn2−12ωn2−9sin3τ+⋯,Φτ,ε=ε−1γδωn2sinτωn2−1Ω2−1−ε−22γσδsin2τωn2−1ωn2−4Ω2−4+⋯.


## 5. Discussion of the Results

In this subsection, we give a parametric analysis of the obtained results for the behavior of the obtained analytical solutionsΘ and Φ as functions dependent on the time τ, the valuesγ, ωn, Ω, δ, and σ, and the large parameter ε; that is,
(24)Θ=Θτ,γ,ωn,Ω,δ,σ,ε,Φ=Φτ,γ,ωn,Ω,δ,σ,ε.We note that the domain of the obtained solutions under the assumed conditions is as follows:(25)Θ=Θτ,γ⟶0,ωn⟶∞,Ω⟶0,δ⟶0,σ⟶0,ε⟶∞,Φ=Φτ,γ⟶0,ωn⟶∞,Ω⟶0,δ⟶0,σ⟶0,ε⟶∞,while the domain of the previous corresponding cases is as follows:
(26)Θ=Θτ,γ⟶∞,ωn⟶0,Ω⟶∞,δ⟶∞,σ⟶∞,ε⟶0,Φ=Φτ,γ⟶∞,ωn⟶0,Ω⟶∞,δ⟶∞,σ⟶∞,ε⟶0.This means that the obtained solutions are treated in a new domain which is considered as a complement space for the previous work domains.In what follow, we study the validity of both analytical and numerical solutions. Using one of the numerical methods, we obtain numerical solutions and give more analysis of the results. The graphical representations for both solutions are obtained through computer programming.


## 6. Computerized Data

This section is devoted to ascertaining the accuracy of the solutions being considered in Sections 3 and 4. Computer programs are developed for the representation of the obtained analytical solutionsθ and φ and their derivatives in a definite period. On the other hand, the fourth-order Runge-Kutta method [18] is applied to solve the autonomous system (13) and (14) for obtaining the numerical solutions and their derivatives and investigating their graphical representations. The characteristic curves for both the analytical and numerical solutions and their derivatives appear in Figures 2–5. From Figures 2 and 3, we deduce that when h increases, the number of the waves and their amplitudes of the solutions φ and φ′ decrease and vice versa, while from Figures 4 and 5, we deduce that when h increases, the number of the waves is the same as the solutions θ and θ′ but their amplitudes decrease and vice versa. The phasing diagram procedures [19] for the stability of the solutions are given in Figures 6–11. From these figures, we deduce that the solutions and their derivatives are stable but their amplitudes decrease with increasing h. In all stability figures, the curves take the cardioid form except Figure 9. We deduce from Figures 2–5 that the analytical and numerical solutions are in full agreement, and Figures 6–11 show that the stability diagrams are largely identical. That is, the numerical results are largely correspondence with the analytical ones which prove the validation of both solutions and the used techniques.Figure 2
The analytical and numerical solutionsφ against τ at h=400, 200, and 50.Figure 3
The analytical and numerical angular velocitiesφ′ against τ at h=400, 200, and 50.Figure 4
The analytical and numerical solutionsθ against τ at h=400, 200, and 50.Figure 5
The analytical and numerical angular velocitiesθ′ against τ at h=400, 200, and 50.Figure 6
The analytical and numerical stabilitiesθ′ against θ at h=50.Figure 7
The analytical and numerical stabilitiesθ′ against θ at h=200.Figure 8
The analytical and numerical stabilitiesθ′ against θ at h=400.Figure 9
The analytical and numerical stabilitiesφ′ against φ at h=50.Figure 10
The analytical and numerical stabilitiesφ′ against φ at h=200.Figure 11
The analytical and numerical stabilitiesφ′ against φ at h=400.


## 7. Conclusion

This motion is very important as a mathematical pendulum model for many problems in fluid materials and gas translation in big and small tanks and is considered as a generalization for the problem studied in [11]. The description of the motion of this pendulum model is given. The equations of motion are obtained in terms of two degrees of freedom. New conditions for the motion are considered when the length of the string and radii of the ellipse are sufficiently small comparing to the dimensions of the solid body. A large parameter ε is constructed such that ε=h/ℓ. Using the large parameter technique, we solve this problem in terms of two degrees of freedom θ and φ. The large parameter procedure gives us the chance to study the motion in new conditions and a new domain of the solutions. The obtained analytical solutions are represented graphically through a computer program. On the other hand, the fourth-order Runge-Kutta numerical technique is used to show the validity of the solutions. The obtained phasing diagrams prove full agreement of the obtained analytical and numerical solutions. We deduce also that the change of the parameters of the solid body affects the behavior of the obtained solutions. For example, the increase in h for the solutions of φ and φ′ gives a decrease in the number and amplitude of the waves and vice versa, while the increase of h for the solutions θ and θ′ gives a decrease in the amplitudes of these solutions with the same number of the waves. When h→∞, we have a straightened wave of the motion and the number of the waves is infinite.



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