Research ArticleThe Dynamical Behavior of a Rigid Body RelativeEquilibrium Position

T S Amer

Department of Mathematics Faculty of Science Tanta University Tanta 3127 Egypt

Correspondence should be addressed to T S Amer tarekamersciencetantaedueg

Received 26 October 2016 Revised 12 January 2017 Accepted 24 January 2017 Published 8 March 2017

Academic Editor Manuel De Leon

Copyright copy 2017 T S Amer This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper wewill focus on the dynamical behavior of a rigid body suspended on an elastic spring as a pendulummodel with threedegrees of freedom It is assumed that the body moves in a rotating vertical plane uniformly with an arbitrary angular velocity Therelative periodic motions of this model are consideredThe governing equations of motion are obtained using Lagrangersquos equationsand represent a nonlinear system of second-order differential equations that can be solved in terms of generalized coordinatesThe numerical solutions are investigated using the fourth-order Runge-Kutta algorithms throughMatlab packagesThese solutionsare represented graphically in order to describe and discuss the behavior of the body at any instant for different values of thephysical parameters of the bodyTheobtained results have been discussed and comparedwith someprevious publishedworks Someconcluding remarks have been presented at the end of this work The importance of this work is due to its numerous applicationsin life such as the vibrations that occur in buildings and structures

1 Introduction

The pendulum models have provided the researchers with afertile source of examples in nonlinear dynamics and lately innonlinear control The most famous rigid pendulum consistsof a mass particle that is attached to one end of a masslessrigid arm and the other end of the arm is fixed to a pivotpoint that provides a rotational joint for the arm and massparticle If the arm andmass particle are constrained to movewithin a fixed plane the system is referred to as a planarone-dimension pendulum If the arm and mass particle areunconstrained the system is referred to as a spherical two-dimension pendulum The three-dimensional motion of theswinging spring is studied in [1]The resonance phenomenonthat occurs during the motion is also investigated Nonlinearnormal vibration modes of the spring pendulum and thesystem containing a pendulumabsorber are considered in [2]

The published articles on such models are very large Fewresearches view the pendulum as a rigid body Slanderedpendulum models are defined by a single rotational degreeof freedom referred to as a planar rigid rotational degree offreedom or two rotational degrees of freedom referred to asspherical rigid pendulum Control problems for planar and

spherical pendulum models have been studied by outstand-ing researchers see [3ndash6]

In [7] the motion of a variable length pendulum wasstudied to determine the characteristics of motion In [8]the process analysis method is presented as the analyticalmethod to obtain a second-order approximate solution fora simple pendulum This method does not depend on smallparameter and therefore can overcome the disadvantages andlimitations of the perturbation expansion method In [9] theauthors studied a simple pendulum with a hinge exhibitingbilinear hysteretic moment-rotation characteristics subjectedto periodic base motions They have shown that the bilinearhysteretic nature of the system becomes an effective way tolimit the growth of the response during parametric reso-nance Various perturbation techniques [10] were employedto obtain the analytical solutions formany physical problemsIn [11] the authors studied the motion of the supportedpoint of a pendulum on an ellipse and the method of smallparameter [10] was used to obtain the periodic solution ofthe equation of motion The relative periodic motion of apendulum with an elastic string was studied in [12] andgeneralized in [13] when the motion of the supported pointof a rigid body pendulum with an elastic string moves on an

HindawiAdvances in Mathematical PhysicsVolume 2017 Article ID 8070525 13 pageshttpsdoiorg10115520178070525

2 Advances in Mathematical Physics

elliptical path The equations of motion were deduced usingLagrangersquos equations and solved through the small parametermethod to obtain their solutions up to the second order ofapproximation

In [14] Amer and Bek studied the chaotic responses ofa harmonically excited spring pendulum which moves in acircular path under some conditions The obtained equa-tions of motion represent a nonautonomous system of twononlinear differential equations of two degrees of freedomThe approximate solution was obtained up to the third orderusing themultiple scalesmethod [10]The parametric controlof oscillations and rotations of a compound pendulum wasstudied in [15] An approximate asymptotic approach of thisproblem based on a combination of the averaging method[10] and the maximum principle is proposed and appliedThe limiting cases of small oscillations and rapid rotation ofa pendulum are studied in [16]

In [17] the authors studied the vibration and stability ofthe nonlinear spring pendulum to describe the motion of aship The effects of the longitudinal absorber on the systemare described through the obtained results This model ismodified in [18] by connecting the spring pendulum to thetransverse absorber So the motion has three degrees offreedomundermultiparametric excitationsThe approximatesolution is obtained using the multiple scales method up tothe second-order approximations

The nonlinear two-degree-of-freedom system has beenexamined in [19] The analytical approximate solution up tothe third order is obtained using the same previous methodAll the possible resonances of this solution are examined

The aim of this work is to investigate the motion ofa rigid body suspended on an elastic massless spring Theequations of motion are derived using Lagrangersquos equationand are considered as a nonlinear system of second-orderdifferential equations Each equation of this system dependson all the body variables with their derivatives So it isnot easy to separate these equations as explicit second-orderdifferential equations of one variable in one side In order toovercome this quandary the Mathematica program is usedConsequently the numerical solutions are achieved usingthe fourth-order Runge-Kutta procedure of ode45 solver[20] with the aid of more recent computer package forexample Matlab program Computer codes are carried out toobtain the graphical representations of the attained numericalsolutions for the different parameters of the body Thestability of the solutions is checked during the time intervalof motion Discussion of the results is presented through thecomparison between the different plots for different variablesThe importance of this problem is due to its wide applicationsin many fields such as physics and engineering applicationslike swaying buildings

2 Formulation of the Problem

This section is devoted to introduce the motion of a rigidbody suspended on an elastic massless spring as a pendulummodel So we consider119874119883119884 as a coordinates system rotatingwith angular velocity 120596 with respect to the downward axis

XO

Y

C

O1

Y1

1205931

1205932

120577

120585

120578

O2

ℎ sin 120596t

Figure 1 The rigid body pendulum

119874119884 relative to the motion of a rigid body of mass 119898 Theelastic spring is suspended to a point 1198741 where 1198741198741 =ℎ sin120596119905 at any time 119905 Let us suppose that the rigid body isattached with the spring at the point 1198742 and has the point 119862as a center ofmass1205931 denotes the deformation angle betweenthe spring and the vertical axis 11987411198841 and 1205932 refers to theangle between the straight line directed through 1198742 to 119862and the vertical Choosing an orthogonal coordinates system119862120585120578120577 of the body in which 119862120578 directed along 119874

2119862 119862120585 is

perpendicular to 119862120578 and lying in the 119874119883119884 plane while 119862120577is perpendicular to the 119874119883119884 plane (see Figure 1) Assumewithout loss of generality that the axes 119862120585 119862120578 and 119862120577 arethe principal axes of inertia of the body

The coordinates of the center of mass of the body 119909119862and119910

119862 relative to the system 119874119883119884 can be written as

119909119862= ℎ sin120596119905 + 120588 sin120593

1+ 119886 sin120593

2

119910119862= 120588 cos120593

1+ 119886 cos120593

2 119886 = 119874

2119862 (1)

where 120588 is the length of elastic string after time 119905The kinetic energy 119879 and the potential energy 119881 of the

system have the form

119879 = 12119898(ℎ120596 cos120596119905)2 + 2 + (120588

1)2 + (119886

2)2

+ 2ℎ120596 sin1205931 cos120596119905 + 2ℎ1205961205881 cos120596119905 cos1205931+ 2119886ℎ120596

2cos120596119905 cos120593

2+ 2119886

2sin (120593

1minus 1205932)

+ 211988612cos (1205931 minus 1205932)

+ 1205962 (ℎ sin120596119905 + 120588 sin1205931+ 119886 sin120593

2)2 + 119869

311989822

+ 1205962119898 (1198691sin2 120593

2+ 1198692cos2 120593

2)

119881 = 121198962 (120588 minus ℓ)

2 minus 119898119892 (120588 cos1205931 + 119886 cos1205932)

(2)

where 1198691 1198692 and 119869

3are the principal moments of inertia with

respect to the axes 119862120585 119862120578 and 119862120577 respectively 1198962 is the

Advances in Mathematical Physics 3

springrsquos constant ℓ represents the unstretched length of thestring and 119892 denotes the gravitational attraction

According to the above equations one can obtain theLagrangian of the system [21]

119871 = 119879 minus 119881 (3)An inspection of (1)ndash(3) we can observe that the Lagrangian119871 is expressed in terms of three generalized coordinates120588 1205931 1205932

and three corresponding generalized velocities 1 2

Use the following Lagrangersquos equations

119889119889119905 (

120597119871120597119894

) minus 120597119871120597119902119894

= 0119902119894 equiv (120588 1205931 1205932) 119894 equiv (

1 2)

(4)

to obtain the equations of motion in the form

+ 1198862sin (120593

1minus 1205932) minus 1198862

2cos (120593

1minus 1205932) minus 1205882

1

minus 1205962 (2ℎ sin120596119905 + 120588 sin1205931 + 119886 sin1205932) sin1205931minus 119892 cos120593

1+ 1198702 (120588 minus ℓ) = 0

1205881+ 2

1+ 1198862cos (1205931 minus 1205932) + 11988622 sin (1205931 minus 1205932)

minus 1205962 (2ℎ sin120596119905 + 120588 sin1205931+ 119886 sin120593

2) cos120593

1

+ 119892 sin1205931= 0

ℓ12+ ( minus 1205882

1) sin (120593

1minus 1205932)

+ (1205881+ 2

1) cos (120593

1minus 1205932)

minus 1205962 (2ℎ sin120596119905 + 120588 sin1205931 + 119886 sin1205932) cos (1205931 minus 1205932)+ 12059622119898 (1198692 minus 1198691) sin 21205932 + 119892 sin1205932 = 0

(5)

where

1198702 = 1198962119898

ℓ1= 1119886 (1198862 minus

1198693119898)

(6)

Here ℓ1 is the derived length of the body relative to 1198742

Equations (5) are the governing equations of motion of ourmodel that represent a nonlinear system of second-orderdifferential equations

In order to study this problem we consider that theoscillations of our system are closing to the position of therelative equilibrium So we can assume

1198691= 1198692 (7)

Hence for the relative equilibrium state the angles 12059310

and12059320are equal and then we can write

120588 = 119887 + 120585 (119905) 1205931= 1205930+ 120593 (119905)

1205932= 1205930+ 120595 (119905)

(8)

where1205930represents the value of120593

10and12059320and 119887 denotes the

pendulum stringrsquos length in the case of relative equilibriumMoreover the quantities120593

0and 119887 can be determined from the

following equations

1198702 (119887 minus ℓ) = 1205962 (119886 + 119887) sin2 1205930 + 119892 cos1205930119892 = 1205962 (119886 + 119887) cos1205930

(9)

Substituting from (8) into (5) then using (7) and (9) weget

+ 11988611120585 + 11988612120593 + 11988613120595 = 1198911

119887 + 119886 + 11988711120585 + 11988712120593 + 11988713120595 = 119891

2

ℓ1 + 119887 + 119888

11120585 + 11988812120593 + 11988813120595 = 119891

3

(10)

where

11988611 = 1198702 minus 1205962 (sin2 1205930 + 2ℎ cos1205930 sin120596119905) 11988612= 11988711988811

11988613= 11988611988811

11988711= 11988811

11988712= 1198702 (119887 minus ℓ) minus 1205962119887 cos2 1205930

11988713= minus1205962119886 cos2 120593

0

11988811= minus1205962 sin120593

0cos1205930

11988812= minus1205962119887 cos2 120593

0

11988813= 1198702 (119887 minus ℓ) + 1205962 (2ℎ sin1205930 sin120596119905 minus 119886 cos2 1205930)

(11)

1198911= (120585 + 119887) 2 + 119860120585120593 + 119886 (120595 minus 120593) + 1198862 + 119861120593120595+ 1198621

1198912= minus120585 minus 2 + (120595 minus 120593) 119886 + 119888

11 (120585 + 119887) 1205932+ (119863120585 + 2119886

13120595) 120593 + 119862

2

1198913 = (120595 minus 120593) + (1198872 + 1205852) (120593 minus 120595) minus 120585 minus 2+ 21205962ℎ sin120596119905 cos120593

0

minus [119887 + 119886 (1 minus 1205952) minus (119887 + 120585) 120593120595] 11988811minus 119892 sin120593

0+ 1205962120585 (120593 cos2 120593

0minus 120595 sin2 120593

0)

(12)

4 Advances in Mathematical Physics

119860 = minus211988811

119861 = 1198861205962 cos2 1205930

119863 = 1205962cos 2 12059301198621 = minus119887 (1198702 minus 1205962sin21205930)

+ 1205962 [2ℎ sin1205930sin120596119905 + 119886 (sin2 120596119905 + sin2 120593

0)]

+ 119892 cos1205930+ 1198702ℓ

1198622= 2ℎ1205962 sin120596119905 minus (119887 + 119886) 11988811 minus 119892 sin1205930

(13)

Our principle aim is to obtain the numerical solutionsof system (10) which consists of three nonlinear differentialequations of second-order In view of the right hand sides ofthese equations we found three functions119891

1 1198912 and1198913 givenby (12) In fact it is not easy to obtain the second derivativesof the generalized coordinates 120585 120593 and 120595 such that eachequation contains one of these derivatives only

3 Numerical Solutions

This section is devoted to discuss the numerical solutions forthe considered model in Section 2 Computer programs arecarried out to investigate the graphical representations forthese solutions to describe the motion and to illustrate thebehavior of the pendulum at any time

System (10) consists of three nonlinear differential equa-tions of second-order in terms of 120585 120593 and 120595 and is recon-sidered to obtain the numerical solutions in framework ofthe fourth-order Runge-Kutta algorithms through Matlabpackages [22] Each equation of this system includes allvariables 120585 120593 120595 and their derivatives from the first andsecond order see systems of (10) (11) (12) and (13) Sothe mentioned system is more complicated to deal with andto obtain another corresponding one consisting of second-order differential equations in terms of and explicitlyComputer codes are utilized in order to overcome thesedifficulties and to separate each of and Consequentlysystem (10) is transformed into the following system with theaid of (11) (12) and (13)

= minus 1119867 minus119886120595 minus (119887 + 120585) (11988811120585 + 11988812120593 + 11988813120595 minus 119891

3)

+ 119887 (11988711120585 + 11988712120593 + 11988713120595 + 119892 sin120593

0+ 1205962

sdot minus120585120593 cos2 1205930 minus 2ℎ sin120596119905 cos1205930+ [minus119887 + (119887 + 120585) 120593120595 + 119886 (1205952 minus 1)]sdot cos120593

0 sin1205930 + 120585120593 sin2 1205930+ 2 minus (119887 + 120585) (120593 minus 120595) 2)

+ [119886119887 minus (119887 + 120585) ℓ1] [1198621 minus 11988611120585 minus 11988612120593minus11988613120595 + (119887 + 120585) 2 + (119860120585 + 119861120595 + 1198862) 120593]

= minus 11198871198913 minus 11988811120585 minus 11988812120593 minus 11988813120595 + ℓ

1119886120595sdot [1198621 minus 11988611120585 minus 11988612120593 minus 11988613120595 + (119887 + 120585) 2

+ (119860120585 + 119861120595 + 1198862) 120593]+ 1119886119867120595 ℓ1 minus119886120595 (119887 + 120585)

sdot (1198913 minus 11988811120585 minus 11988812120593 minus 11988813120595) + 119887

sdot (11988711120585 + 11988712120593 + 11988713120595 + 119892 sin120593

0+ 1205962

sdot minus120585120593 cos2 1205930minus 2ℎ sin120596119905 cos120593

0

+ [(119887 + 120585) 120593120595 + 119886 (1205952 minus 1) minus 119887] cos1205930sin1205930

+ 120585120593 sin2 1205930 +2 minus (119887 + 120585) (120593 minus 120595) 2)+ [119886119887 minus (119887 + 120585) ℓ1] 1198621 minus 11988611120585 minus 11988612120593 minus 11988613120595+ (119887 + 120585) 2 + (119860120585 + 119861120595 + 1198862) 120593

= minus 1119867 119887 (120593 minus 120595) [119886

11120585 + 11988612120593 + 11988613120595 minus (119860120585 + 119861120595) 120593]

minus 119887 (11988711120585 + 11988712120593 + 11988713120595) + (119887 + 120585) (11988811120585 + 11988812120593 + 11988813120595)

minus 1198871198621(120593 minus 120595) + 1198871205962 cos120593

0(120585120593 cos120593

0+ 2ℎ sin120596119905)

minus (119887 + 120585) 1198913 minus 119887119892 sin1205930 + 1198871205962sdot 119886 + 119887 minus 120595 [(119887 + 120585) 120593 + 119886120595] cos1205930 sin1205930minus 119887 [2 + 120593 (1205962120585 sin2 120593

0+ 119886 (120593 minus 120595) 2)]

(14)

where

119867 = [119886119887 (1205952 minus 120593120595 minus 1) + (119887 + 120585) ℓ1] (15)

It is clear that the left hand sides of the equations ofthe previous system are given explicitly in terms of and respectively On the other hand the right hand sides arefunctions of 120585 120593 120595 and

The ode45 solver is used in order to obtain the numericalsolutions of the nonstiff ordinary differential equations of theprevious system (14) in which this solver uses a variable stepof Runge-Kutta technique [20] So we can rewrite system (14)as a system of coupled first-order differential equations asfollows

Advances in Mathematical Physics 5

A choice of the state variables for this system is

1198831 = 120585

1198832= 120593

1198833= 120595

1198834=

1198835 =

1198836=

(16)

which results in the following state-equations

1= 1198834

2 = 11988353= 1198836

(17)

Use (16) and (17) into system (14) to get

4= minus 1

119867 minus1198861198833minus (119887 + 119883

1)

sdot (119888111198831 + 119888121198832 + 119888131198833 minus 1198913)+ 119887 (119887

111198831 + 119887121198832 + 119887131198833 + 119892 sin1205930 + 1205962sdot minus11988311198832cos2 120593

0minus 2ℎ sin120596119905 cos120593

0

+ [minus119887 + (119887 + 1198831)11988321198833+ 119886 (1205952 minus 1)] cos120593

0sin1205930

+11988311198832 sin2 1205930+212 minus (119887 + 1198831) (1198832 minus 1198833) 22)+ [119886119887 minus (119887 + 120585) ℓ1] [1198621 minus 119886111198831 minus 119886121198832minus 119886131198833 + (119887 + 1198831) 22+ (119860119883

1+ 1198611198833+ 1198863

2)1198832]

5= minus 1

1198871198913 minus 119888111198831 minus 119888121198832 minus 119888131198833+ ℓ11198861198833

[1198621minus 119886111198831minus 119886121198832minus 119886131198833+ (119887 + 119883

1) 2

2

+ (1198601198831+ 1198611198833+ 1198863

2)1198832] + 1

1198861198671198833sdot ℓ1minus119886119883

3(119887 + 119883

1) (1198913minus 119888111198831minus 119888121198832minus 119888131198833)

+ 119887 (119887111198831+ 119887121198832+ 119887131198833+ 119892 sin120593

0

+ 1205962 minus11988311198832cos2 120593

0minus 2ℎ sin120596119905 cos120593

0

+ [(119887 + 1198831)11988321198833 + 119886 (11988323 minus 1) minus 119887] cos1205930 sin1205930

+11988311198832 sin2 1205930+212 minus (119887 + 1198831) (1198832 minus 1198833) 22)+ [119886119887 minus (119887 + 119883

1) ℓ1]

sdot 1198621minus 11988611120585 minus 119886121198832minus 119886131198833

+ (119887 + 1198831) 2

2 + (1198601198831+ 1198611198833+ 1198863

2)1198832

6= minus 1

119867 119887 (1198832minus 1198833)

sdot [119886111198831 + 119886121198832 + 119886131198833 minus (1198601198831 + 1198611198833)1198832]minus 119887 (119887111198831+ 119887121198832+ 119887131198833)

+ (119887 + 1198831) (119888111198831+ 119888121198832+ 119888131198833)

minus 1198871198621 (1198832 minus 1198833)+ 1198871205962 cos120593

0(11988311198832cos1205930+ 2ℎ sin120596119905)

minus (119887 + 1198831) 1198913 minus 119887119892 sin1205930+ 1198871205962 119886 + 119887 minus 119883

3[(119887 + 119883

1)1198832+ 1198861198833]

sdot cos1205930sin1205930

minus 119887 [212+ 1198832

sdot (12059621198831sin2 120593

0+ 119886 (119883

2minus 1198833) 3

2)] (18)

The following initial conditions are required to achieve thenumerical solution of (18) by using the fourth-order Runge-Kutta method of ode45 solver in framework of Matlabprogram

1198831 (0) = 00011198832 (0) = 01

1198833 (0) = 001

4 (0) = 0

5 (0) = 0

6 (0) = 0

(19)

in addition to the following physical parameters of theconsidered model

119898 = 50 kg119892 = 98m sdot sminus21198691= 8 kg sdotm2

ℓ = 07m120596 = 4 rad sdot sminus1119886 = 05m

6 Advances in Mathematical Physics

0 05 1 15

0

05

1

15

t

minus05

minus1

120585and 120585

times104

120585120585

1205930 = 0

(a)

0 05 1 15

0

5

t

1205930 = 0

minus5

minus10

minus15

minus20

minus25

120593

120593and120593

(b)

0 05 1 15

0

20

t

1205930 = 0

minus20

minus40

120595and

120595

120595120595

(c)

0

2

4

6

8

120585and 120585

times104

0 05 1 15t

120585120585

minus2

minus4

1205930 = 04

(d)

0

10

20

30

120593and120593

0 05 1 15t

120593

minus10

minus20

1205930 = 04

(e)

0

50

100

120595and

120595

0 05 1 15t

120595

120595

1205930 = 04

minus50

minus100

(f)

Figure 2 Variation of the solutions and their derivatives versus time 119905 when 119887 = 3m ℎ = 45 and 119886 = 05m (a d) show the effect of 119905 on thebehavior of 120585 and waves when 120593

0= 0 and 120593

0= 04 rad respectively (b e) show the effect of 119905 on the behavior of 120593 and waves when 120593

0= 0

and 1205930= 04 rad respectively and (c f) show the effect of 119905 on the wave that describes the behavior of 120595 and when 120593

0= 0 and 120593

0= 04 rad

respectively

119887 = (0 3)m1205930= (0 04) rad

ℎ = (25 45) 119905 = 0 997888rarr 17min

(20)

Figure 2 shows the variation of the solutions 120585 120593 120595 and theirderivatives against time 119905 when 120593

0= 0 and 120593

0=04 rad This figure is drawn at 119887 = 3m ℎ = 24 and

119886 = 05m The variations of 120585 120593 and 120595 with and respectively are illustrated in Figure 3 namely the phaseplane diagrams that are represented in Figures 3(a) 3(b)3(c) and 3(d) 3(e) 3(f) when 1205930 = 0 and 1205930 = 04 radrespectively with the same other parameters that are takeninto consideration in Figure 2

In these figures our principle aim is to investigate theeffect of increasing time on the motion of pendulum

According to the calculations depicted in Figure 2(a) wefound that when 1205930 = 0 the wave of the elongation 120585 growsup with the increasing of time till 119905 = 09min After thatboth of the elongation 120585 and its derivative fluctuate between

increasing and decreasing when time reaches 119905 = 143minThus the wave of the solution 120585 is stable see the phase planeFigure 3(a) With the passing of time one can observe that120585 and are growing quickly so the motion will be unstableafter 119905 = 143min The rage behavior of both 120585 and is dueto the weight of the rigid body and the values of the principalmoments of inertia Consequently we expect that behaviorof elongation becomes greater as observed in Figures 2(a) and2(d)Moreover the variation of the spring between stretchingand contraction is consistent with the phase plane diagramsrepresented in Figures 3(a) and 3(d)

It is worthwhile to notice fromFigure 2(b) that when time119905 increases from 119905 = 0 to 119905 = 04min the behavior of theangle 120593 increases gradually to reach the value 120593 ≃ 1 rad ≃ 57∘and then decreases slowly to reach 120593 ≃ 08 rad ≃ 46∘ duringthe time period 119905 isin ]04 09[min After 119905 = 09min thedecline of the wave becomes quickly to reach 120593 ≃ minus23 rad ≃minus132∘ at the end of time period (minus sign indicates oppositedirection) This is not possible because 120593 must belong to theinterval ] minus 1205872 1205872] So the motion of the wave is unstableas it is manifest from Figure 3(b) On the other hand increases till 119905 ≃ 02min and then fluctuates as indicated fromFigure 2(b)

Advances in Mathematical Physics 7

0 500 1000 1500 2000 2500

0

05

1

15

times104

120585

120585

1205930 = 0

minus05

minus1

(a)

0 1

0

5 1205930 = 0

minus5

minus10

minus15

minus20

minus25minus2 minus1

120593

(b)

0 2

0

20

minus20

minus40

minus2minus4

120595

120595

1205930 = 0

(c)

0 2000 4000 6000 8000 10000

0

2

4

6

8times104

120585

120585

minus2

minus4

1205930 = 04

(d)

0 02 04 06 08 1

0

10

20

30

120593

1205930 = 04

minus10

minus20

minus02

(e)

0 2 4

0

50

100

minus2minus4

120595

120595

1205930 = 04

minus50

minus100

(f)

Figure 3 The phase plane diagram when 119887 = 3m ℎ = 45 and 119886 = 05m (a d) represent the variation of the amplitude 120585 with its velocity at 120593

0= 0 and 120593

0= 04 rad respectively (b e) represent the variation of the amplitude 120593 with its velocity at 120593

0= 0 and 120593

0= 04 rad

respectively and (c f) represent the variation of the amplitude 120595 with its velocity at 1205930= 0 and 120593

0= 04 rad respectively

The graphs displayed in Figures 2(c) and 3(c) describethe variation of the (120595 and ) against time and the phaseplane diagram ( with 120595) respectively when 1205930 = 0 Itis clear that when time belongs to the period [0 043]minthe angle 120595 remains stationary and then its wave oscillatesbetween decreasing and increasing till 119905 = 143min Afterthat time the angle 120595 increases up to the end of time intervaland consequently the motion will be stable as seen fromFigure 3(c) during the period 0 lt 119905 le 143 It is obviousfrom Figure 2(c) that the behavior of remains stationary tosome extent through the time interval [0 06]min and thenoscillates between increasing and decreasing till 119905 = 17min

It should be noticed that when1205930= 04 rad the stretching

on the string 120585 increases gradually till the time 119905 becomes09min and then 120585 and oscillate between increasing anddecreasingwhen the time reaches the end of time interval seeFigure 2(d) Consequently the wave of the solution is stableas seen from the phase plane Figure 3(d)

An inspection of the graphs depicted in Figure 2(e) showsthat the wave describing the behavior of the angle 120593 increasesgradually from 120593 = 0 at 119905 = 0 to its maximum value 120593 ≃09 rad ≃ 56∘ at 119905 = 04min and then decreases slowly at119905 ≃ 1min to reach its minimum value 120593 ≃ minus019 rad ≃ minus11∘(minus sign indicates opposite direction) at 119905 ≃ 126minWith the increasing of time thewave grows again to reach thevalue 120593 ≃ 085 rad ≃ 49∘ at 119905 ≃ 139min Thus the motion is

stable as it is manifest from Figure 3(e) On the other hand increases and decreases as indicated from Figure 2(e)

Also it is remarkable from Figure 2(f) that the behaviorof the angle 120595 remains steady till 119905 = 05min then its waveoscillates between decreasing and increasing Consequentlythe motion will be stable as seen from Figure 3(f) It isworthwhile to notice also from Figure 2(f) that the behaviorof oscillates between increasing and decreasing

From the above observations we can conclude that themotion of our model is more stable when 120593

0= 04 rad than

when 1205930= 0 This highlights the importance of the effect of120593

0value on the motion It is worthwhile to notice that the

comparison between the solutions 120585 120593 and 120595 included inFigures 2(a) 2(b) and 2(c) with the corresponding Figures2(d) 2(e) and 2(f) reveals that the amplitude of the wavesdecreases when 120593

0increases from 0 to 04 rad On the other

hand the comparison between their derivatives shows thatthe amplitude of the waves increases when 120593

0increases

Figure 4 shows the variation of (120585 ) (120593 ) and (120595 ) withtime 119905 when 120593

0 changes from 0 for Figures 4(a) 4(b) and4(c) to 04 rad for Figures 4(d) 4(e) and 4(f) at the samevalues of other parameters 119887 = 3m ℎ = 45 and 119886 = 05mAccording to the calculations depicted in these figures we canconsider these figures as a rotation of the corresponding partsof Figure 3 with time to observe the bending and crossing ofthe resulting curves

8 Advances in Mathematical Physics

01000 20000

1

0051

15

t

minus1

times104

1205930 = 0

120585120585

(a)

010

0051

15

t

minus10minus20 minus1

minus2

1205930 = 0

120593

(b)

0 2020

0051

15

t

1205930 = 0

minus20minus40 minus4

minus2

120595120595

(c)

05000

1000002468

0051

15

t

times104

minus4minus2

1205930 = 04

120585120585

(d)

005

10

20

0051

15

t

minus20

1205930 = 04

120593

(e)

0 2 401000

051

15

t

minus4 minus2

1205930 = 04

minus100120595

120595

(f)

Figure 4The 3D pattern when 119887 = 3m ℎ = 45 and 119886 = 05m (a d) indicate the variation of 120585 and versus 119905 when 1205930= 0 and 120593

0= 04 rad

respectively (b e) indicate the variation of 120593 and versus 119905 when 1205930= 0 and 120593

0= 04 rad respectively and (c f) indicate the variation of 120595

and versus 119905 when 1205930= 0 and 120593

0= 04 rad respectively

01000

200001

02

minus1minus2

minus2

minus4

120585120593

120595

1205930 = 0

(a)

01

0

020

1205930 = 0

minus1

minus10minus20

minus20

minus40

times104

120585

120595

(b)

05000

100000

051

024

minus2

minus4

120595

1205930 = 04

120585

120593

(c)

0 2 4 6 8020

0

100

minus2minus4times104

120595

1205930 = 04

minus20

minus100

120585

(d)

Figure 5 The 3D diagrams when 119887 = 3m ℎ = 45 and 119886 = 05m (a c) elucidate the variation of 120585 and 120593 versus 120595 when 1205930= 0 and

1205930= 04 rad respectively and (b d) elucidate the variation of and versus when 120593

0= 0 and 120593

0= 04 rad respectively

Advances in Mathematical Physics 9

0 05 1 15

0

2

4

6

8

t

120585and 120585

times104

minus2

b = 0

120585120585

(a)

t

0 05 1 15

0

10

20

30

40

50b = 0

minus10

120593120593and120593

(b)

0 05 1 15

0

200

400

t

b = 0

minus200

minus400

minus600

120595and

120595

120595120595

(c)

Figure 6 (a) (b) and (c) explain the variation of the solutions 120585 120593 and 120595 with their derivatives and via time 119905 respectively when119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad

0 5000 10000 15000 20000

0

2

4

6

8

minus2

times104

b = 0

120585

120585

(a)

0 1 2 3

0

10

20

30

40

50b = 0

minus10

minus1

120593

(b)

0 5 10 15

0

200

400b = 0

120595

120595

minus200

minus400

minus600minus5minus10minus15

(c)

Figure 7 The phase plane diagrams between amplitudes and their velocities at 119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad (a) shows the

influence of 120585 on (b) shows the effect of 120593 on and (c) shows the variation of 120595 with

Figures 5(a) 5(c) and 5(b) 5(d) represent 3D plots thatillustrate the variation of the solutions 120585 120593 via 120595 and via respectively for different values of 120593

0when 119887 =3m ℎ = 45 and 119886 = 05m The graphs displayed in

parts of Figure 6 show the variation of (120585 ) (120593 ) and(120595 ) against time 119905 when 119887 = 0 with consideration of theparameters 1205930 = 04 rad ℎ = 45 and 119886 = 05m Thecorresponding phase plane between the amplitudes 120585 120593 120595and their derivatives is represented in parts of Figure 7Inspection of the graph depicted in Figure 6(a) shows thatwhen time 119905 increases from 0 to 045min the behavior of thesolution 120585 remains stationary and quickly growing during thetime interval 119905 isin ]045 105[min and then oscillates till theend of time interval This indicates that the motion is stableas seen from Figure 7(a) On the other side the behaviorof the derivative remains approximately stationary duringthe interval 119905 isin [0 045]min and then fluctuates with theincreasing of time see Figure 6(a)

By the same way we can observe that the wave of theangle 120593 increases through a short time to reach its maximumvalue 120593 ≃ 27 rad ≃ 155∘ at 119905 ≃ 023min taking intoconsideration that minus1205872 lt 120593 lt 1205872 and then decreasesslowly to reach its minimum value 120593 ≃ minus09 rad ≃ minus52∘at the end of time interval see Figure 6(b) This indicatesthat the motion is close to be stable as observed fromthe phase plane Figure 8(b) As seen from Figure 6(b) increases and decreases quickly during the period 119905 isin[0 01]min to reach its minimum value at the end of timeinterval

The variation of 120595 and with time is illustrated inFigure 6(c) In this figure our main goal is to examine theinfluence of time on the motion of pendulum It is clear thatthe behavior of 120595 and remains stationary (to some extent)when 119905 isin [0 05]min then their waves fluctuate till the endof time interval Consequently the motion is stable as seenfrom the phase plane diagram Figure 7(c) The comparisonbetween parts of Figure 6 with the corresponding Figures

10 Advances in Mathematical Physics

010000

2000002468

0

05

1

15

t

120585120585 minus2

b = 0

times104

(a)

t

0 1 2 30

2040

0

05

1

15

120593minus1

b = 0

(b)

t

0 1005000

05

1

15

minus10minus500 120595120595

b = 0

(c)

Figure 8 The 3D plots at 119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad (a) illustrates the variation of 120585 and via 119905 (b) illustrates the variation

of 120593 and via 119905 and (c) illustrates the variation of 120595 and via 119905

0 05 1 15

0

5000

10000

15000

t

minus5000

120585120585

120585and 120585

h = 25

(a)

0 05 1 15

0

5

10

t

minus5

minus10

minus15

120593

120593and120593

h = 25

(b)

0 05 1 15

0

20

t

120595and

120595120595120595

minus20

minus40

minus60

h = 25

(c)

Figure 9 (a) (b) and (c) demonstrate the variation of (120585 and ) (120593 and ) and (120595 and ) against time 119905 respectively at 119887 = 3m ℎ = 25119886 = 05m and 1205930= 04 rad

2(d) 2(e) and 2(f) shows that when 119887 changes from 0 to 3mthe amplitude of the waves decreases Also the motion willbe more stable when 119887 = 3m than when 119887 = 0 as seen fromthe corresponding phase plane diagrams that is Figures 3(d)3(e) 3(f) and 7(a) 7(b) 7(c) respectively

On the other hand parts of Figure 8 show 3D plots thatdescribe the variation of the solutions and their derivative viatime when 119887 = 0 ℎ = 45 120593

0= 04 rad and 119886 = 05m

The plots displayed in thementioned parts show bending andcrossing of the resulting curves

Figures 9(a) 9(b) and 9(c) show the variation of thesolutions 120585 120593 120595 and their derivatives with time 119905whenℎ = 25 for the given values of other parameters 119887 = 3m1205930= 04 rad and 119886 = 05m In view of the first part we can

conclude that when time 119905 increases each of the waves 120585 and oscillates between increasing and decreasing till 119905 = 146minand then increases gradually So the motion is stable as seenfrom Figure 10(a)

From a closer look on the second part of Figure 9(b) wecan write with the increasing of time the behavior of 120593 wave

increases to reach its maximum value 120593 ≃ 09 rad ≃ 52∘ at119905 = 043min and then decreases slowly through the period 119905 isin]043 119]min After that its behavior has a sharp declinein a few seconds (about 24 s) and then increases till the endof time period and consequently the motion is stable seeFigure 10(b)

According to the calculations depicted in Figure 9(c) wecan observe that thewaves describing120595 and decrease slowlytill 119905 = 09min and then increase and decline sharp Thephase plane Figure 10(c) shows that the behavior of 120595 is notstable

When parts of Figure 9 and their phase plane parts (ofFigure 10) are generally compared with the correspondingFigures 2(d) 2(e) and 2(f) and their phase plane Figures 3(d)3(e) and 3(f) we can observe that amplitude of the waveincreases when ℎ = 45 compared to when ℎ = 25 and themotion is more stable when ℎ = 45 An inspection of partsof Figure 11 reveals the 3D plots when ℎ = 25 with the sameother data considered in Figures 9 and 10 Figure 10 shows thevariation of the solutions 120585 120593 120595 and their derivatives

Advances in Mathematical Physics 11

0 500 1000 1500 2000

0

5000

10000

15000 h = 25

minus5000

120585

120585

(a)

0 02 04 06 08 1

0

5

10h = 25

minus5

minus10

minus15

minus02

120593

(b)

0 2

0

20h = 25

minus20

minus40

minus60

minus2minus4minus6minus8

120595

120595

(c)

Figure 10 The phase plane diagrams which portray the relation between amplitudes and their velocities at 119887 = 3m ℎ = 25 119886 = 05m and1205930= 04 rad (a) describes the influence of 120585 on (b) shows the effect of 120593 on and (c) illustrates the variation of 120595 with

01000

200005000

1000015000

0051

15

t

minus5000120585

120585

h = 25

(a)

t

005

10100

05

1

15

minus10120593

h = 25

(b)

t

0 20200

05

1

15

minus20minus40

minus60minus2minus4minus6minus8120595

120595

h = 25

(c)

Figure 11 The 3D patterns at 119887 = 3m ℎ = 25 119886 = 05m and 1205930= 04 rad (a) illustrates the variation of 120585 and versus 119905 (b) illustrates the

variation of 120593 and versus 119905 (c) illustrates the variation of 120595 and versus 119905

with time 119905 It is worthwhile to notice that the comparisonbetween Figures 4(d) 4(e) and 4(f) and Figures 11(a) 11(b)and 11(c) shows more bending and crossing of the curvesin Figures 4(d) 4(e) and 4(f) when ℎ = 45 than thecorresponding ones of Figure 11

Now we study the last case when ℎ = 0 with the sameother data 119887 = 3m 1205930 = 04 rad and 119886 = 05mThe obtainedresults are represented graphically in Figures 12(a) 12(b) and12(c) while their phase plane diagrams are given in Figures12(d) 12(e) and 12(f) At the first glance we can conclude thatthis case is not stable so it is very important to notice that thedimensionless parameter ℎmust take any value different fromzero as it is pointed in Figure 2 (ℎ = 45) and Figure 9 (ℎ =25) This elucidates the importance of ℎ parameter on themotion

4 Conclusion

A conclusion that may be made here is that the problemof the relative motion of a rigid body as a pendulum

model is investigated The governing deferential equationsare obtained using Lagrangersquos equations Mathematica pack-age was utilized in order to overcome the difficulties thatappear in the separation of the second derivatives of thegeneralized coordinates 120585 120593 and 120595 for the nonlinear system(10) Computer codes are used to obtain the numericalsolutions for system (14) These solutions are representedgraphically using Matlab program to study the influenceof the different parameters on the motion The good effectof the parameters ℎ 119887 and 120593

0on the motion is obvious

from the mentioned plots The motion of our model is morestable when the parameters ℎ 119887 and 120593

0take values run

away from zero This highlights the importance of the effectof these parameters on the motion Such results have beenconfirmed by many works such as Ismail [13] and Amer andBek [14]

Competing Interests

The author declares that they have no competing interests

12 Advances in Mathematical Physics

0 05 1 15

0

05

1

15

2

25

3

t

120585120585

120585and 120585

h = 0

times104

(a)

0 05 1 15

0

05

1

15

2

25

t

120593120593and120593

h = 0

(b)

0 05 1 15

0

5

10

15

t

120595and

120595

120595120595

h = 0

(c)

0 1000 2000 3000 4000 5000

0

05

1

15

2

25

3times104

120585

120585

h = 0

(d)

02 04 06 08 1

0

05

1

15

2

25

120593

h = 0

(e)

0 05 1 15 2

0

5

10

15

120595

120595

h = 0

(f)

Figure 12 (a b and c) explain the variation of the solutions 120585 120593 and120595with their derivatives and via time 119905 respectively when 119887 = 3mℎ = 0 119886 = 05m and 1205930= 04 rad (d e and f) illustrate the variation of the solutions against their first derivatives for the same values of the

considered parameters

References

[1] P Lynch ldquoResonant motions of the three-dimensional elasticpendulumrdquo International Journal of Non-Linear Mechanics vol37 no 2 pp 345ndash367 2002

[2] A A Klimenko Y V Mikhlin and J Awrejcewicz ldquoNonlinearnormal modes in pendulum systemsrdquoNonlinear Dynamics vol70 no 1 pp 797ndash813 2012

[3] S Mori H Nishihara and K Furuta ldquoControl of unstablemechanical system control of pendulumrdquo International Journalof Control vol 23 no 5 pp 673ndash692 1976

[4] C C Chung and J Hauser ldquoNonlinear control of a swingingpendulumrdquo Automatica A Journal of IFAC vol 31 no 6 pp851ndash862 1995

[5] A Shiriaev A Pogromsky H Ludvigsen and O Egeland ldquoOnglobal properties of passivity-based control of an inverted pen-dulumrdquo International Journal of Robust and Nonlinear Controlvol 10 no 4 pp 283ndash300 2000

[6] A S Shiriaev H Ludvigsen and O Egeland ldquoSwinging upthe spherical pendulum via stabilization of its first integralsrdquoAutomatica A Journal of IFAC the International Federation ofAutomatic Control vol 40 no 1 pp 73ndash85 2004

[7] M N Brearley ldquoThe Simple Pendulum with Uniformly Chang-ing String Lengthrdquo Proceedings of the Edinburgh MathematicalSociety vol 15 no 1 pp 61ndash66 1966

[8] S J Liao ldquoSecond-order approximate analytical solution of asimple pendulum by the process analysis methodrdquo Journal ofApplied Mechanics Transactions ASME vol 59 no 4 pp 970ndash975 1992

[9] W K Tso and K G Asmis ldquoParametric excitation of a pen-dulum with bilinear hysteresisrdquo Journal of Applied MechanicsTransactions ASME vol 37 no 4 pp 1061ndash1068 1970

[10] A H Nayfeh Perturbations Methods Wiley-VCH WeinheimGermany 2004

[11] F A El-Barki A I Ismail M O Shaker and T S AmerldquoOn the motion of the pendulum on an ellipserdquo Zeitschrift furAngewandteMathematik undMechanik vol 79 no 1 pp 65ndash721999

[12] N V Stoianov ldquoOn the relative periodic motions of a pendu-lumrdquo Journal of AppliedMathematics andMechanics vol 28 pp188ndash193 1964

[13] A I Ismail ldquoRelative periodicmotion of a rigid body pendulumon an ellipserdquo Journal of Aerospace Engineering vol 22 no 1 pp67ndash77 2009

[14] T S Amer andM A Bek ldquoChaotic responses of a harmonicallyexcited spring pendulum moving in circular pathrdquo NonlinearAnalysis Real World Applications An International Multidisci-plinary Journal vol 10 pp 3196ndash3202 2009

Advances in Mathematical Physics 13

[15] L D Akulenko ldquoParametric control of oscillations and rota-tions of a compound pendulum (a swing)rdquo Journal of AppliedMathematics and Mechanics vol 57 no 2 pp 301ndash310 1993

[16] M A Pinsky and A A Zevin ldquoOscillations of a pendulumwith a periodically varying length and a model of swingrdquoInternational Journal of Non-LinearMechanics vol 34 no 1 pp105ndash109 1999

[17] M Kamel M Eissa and A T El-Sayed ldquoVibration reductionof a nonlinear spring pendulum under multiparametric excita-tions via a longitudinal absorberrdquo Physica Scripta vol 80 no 2Article ID 025005 2009

[18] M Eissa M Kamel and A T El-Sayed ldquoVibration reduction ofmulti-parametric excited spring pendulum via a transversallytuned absorberrdquo Nonlinear Dynamics vol 61 no 1-2 pp 109ndash121 2010

[19] R Starosta G Sypniewska-Kaminska and J AwrejcewiczldquoAsymptotic analysis of kinematically excited dynamical sys-tems near resonancesrdquo Nonlinear Dynamics An InternationalJournal of Nonlinear Dynamics and Chaos in Engineering Sys-tems vol 68 no 4 pp 459ndash469 2012

[20] H MooreMatlab for Engineers Pearson 3rd edition 2012[21] M D Ardema Analytical Dynamics Theory and Applications

Springer Berlin Germany 2009[22] A Tewari Modern Control Design with Matlab and Similink

John Wiley and Sons Ltd New York NY USA 2002

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 3

springrsquos constant ℓ represents the unstretched length of thestring and 119892 denotes the gravitational attraction

According to the above equations one can obtain theLagrangian of the system [21]

119871 = 119879 minus 119881 (3)An inspection of (1)ndash(3) we can observe that the Lagrangian119871 is expressed in terms of three generalized coordinates120588 1205931 1205932

and three corresponding generalized velocities 1 2

Use the following Lagrangersquos equations

119889119889119905 (

120597119871120597119894

) minus 120597119871120597119902119894

= 0119902119894 equiv (120588 1205931 1205932) 119894 equiv (

1 2)

(4)

to obtain the equations of motion in the form

+ 1198862sin (120593

1minus 1205932) minus 1198862

2cos (120593

1minus 1205932) minus 1205882

1

minus 1205962 (2ℎ sin120596119905 + 120588 sin1205931 + 119886 sin1205932) sin1205931minus 119892 cos120593

1+ 1198702 (120588 minus ℓ) = 0

1205881+ 2

1+ 1198862cos (1205931 minus 1205932) + 11988622 sin (1205931 minus 1205932)

minus 1205962 (2ℎ sin120596119905 + 120588 sin1205931+ 119886 sin120593

2) cos120593

1

+ 119892 sin1205931= 0

ℓ12+ ( minus 1205882

1) sin (120593

1minus 1205932)

+ (1205881+ 2

1) cos (120593

1minus 1205932)

minus 1205962 (2ℎ sin120596119905 + 120588 sin1205931 + 119886 sin1205932) cos (1205931 minus 1205932)+ 12059622119898 (1198692 minus 1198691) sin 21205932 + 119892 sin1205932 = 0

(5)

where

1198702 = 1198962119898

ℓ1= 1119886 (1198862 minus

1198693119898)

(6)

Here ℓ1 is the derived length of the body relative to 1198742

Equations (5) are the governing equations of motion of ourmodel that represent a nonlinear system of second-orderdifferential equations

In order to study this problem we consider that theoscillations of our system are closing to the position of therelative equilibrium So we can assume

1198691= 1198692 (7)

Hence for the relative equilibrium state the angles 12059310

and12059320are equal and then we can write

120588 = 119887 + 120585 (119905) 1205931= 1205930+ 120593 (119905)

1205932= 1205930+ 120595 (119905)

(8)

where1205930represents the value of120593

10and12059320and 119887 denotes the

pendulum stringrsquos length in the case of relative equilibriumMoreover the quantities120593

0and 119887 can be determined from the

following equations

1198702 (119887 minus ℓ) = 1205962 (119886 + 119887) sin2 1205930 + 119892 cos1205930119892 = 1205962 (119886 + 119887) cos1205930

(9)

Substituting from (8) into (5) then using (7) and (9) weget

+ 11988611120585 + 11988612120593 + 11988613120595 = 1198911

119887 + 119886 + 11988711120585 + 11988712120593 + 11988713120595 = 119891

2

ℓ1 + 119887 + 119888

11120585 + 11988812120593 + 11988813120595 = 119891

3

(10)

where

11988611 = 1198702 minus 1205962 (sin2 1205930 + 2ℎ cos1205930 sin120596119905) 11988612= 11988711988811

11988613= 11988611988811

11988711= 11988811

11988712= 1198702 (119887 minus ℓ) minus 1205962119887 cos2 1205930

11988713= minus1205962119886 cos2 120593

0

11988811= minus1205962 sin120593

0cos1205930

11988812= minus1205962119887 cos2 120593

0

11988813= 1198702 (119887 minus ℓ) + 1205962 (2ℎ sin1205930 sin120596119905 minus 119886 cos2 1205930)

(11)

1198911= (120585 + 119887) 2 + 119860120585120593 + 119886 (120595 minus 120593) + 1198862 + 119861120593120595+ 1198621

1198912= minus120585 minus 2 + (120595 minus 120593) 119886 + 119888

11 (120585 + 119887) 1205932+ (119863120585 + 2119886

13120595) 120593 + 119862

2

1198913 = (120595 minus 120593) + (1198872 + 1205852) (120593 minus 120595) minus 120585 minus 2+ 21205962ℎ sin120596119905 cos120593

0

minus [119887 + 119886 (1 minus 1205952) minus (119887 + 120585) 120593120595] 11988811minus 119892 sin120593

0+ 1205962120585 (120593 cos2 120593

0minus 120595 sin2 120593

0)

(12)

4 Advances in Mathematical Physics

119860 = minus211988811

119861 = 1198861205962 cos2 1205930

119863 = 1205962cos 2 12059301198621 = minus119887 (1198702 minus 1205962sin21205930)

+ 1205962 [2ℎ sin1205930sin120596119905 + 119886 (sin2 120596119905 + sin2 120593

0)]

+ 119892 cos1205930+ 1198702ℓ

1198622= 2ℎ1205962 sin120596119905 minus (119887 + 119886) 11988811 minus 119892 sin1205930

(13)

Our principle aim is to obtain the numerical solutionsof system (10) which consists of three nonlinear differentialequations of second-order In view of the right hand sides ofthese equations we found three functions119891

1 1198912 and1198913 givenby (12) In fact it is not easy to obtain the second derivativesof the generalized coordinates 120585 120593 and 120595 such that eachequation contains one of these derivatives only

3 Numerical Solutions

This section is devoted to discuss the numerical solutions forthe considered model in Section 2 Computer programs arecarried out to investigate the graphical representations forthese solutions to describe the motion and to illustrate thebehavior of the pendulum at any time

System (10) consists of three nonlinear differential equa-tions of second-order in terms of 120585 120593 and 120595 and is recon-sidered to obtain the numerical solutions in framework ofthe fourth-order Runge-Kutta algorithms through Matlabpackages [22] Each equation of this system includes allvariables 120585 120593 120595 and their derivatives from the first andsecond order see systems of (10) (11) (12) and (13) Sothe mentioned system is more complicated to deal with andto obtain another corresponding one consisting of second-order differential equations in terms of and explicitlyComputer codes are utilized in order to overcome thesedifficulties and to separate each of and Consequentlysystem (10) is transformed into the following system with theaid of (11) (12) and (13)

= minus 1119867 minus119886120595 minus (119887 + 120585) (11988811120585 + 11988812120593 + 11988813120595 minus 119891

3)

+ 119887 (11988711120585 + 11988712120593 + 11988713120595 + 119892 sin120593

0+ 1205962

sdot minus120585120593 cos2 1205930 minus 2ℎ sin120596119905 cos1205930+ [minus119887 + (119887 + 120585) 120593120595 + 119886 (1205952 minus 1)]sdot cos120593

0 sin1205930 + 120585120593 sin2 1205930+ 2 minus (119887 + 120585) (120593 minus 120595) 2)

+ [119886119887 minus (119887 + 120585) ℓ1] [1198621 minus 11988611120585 minus 11988612120593minus11988613120595 + (119887 + 120585) 2 + (119860120585 + 119861120595 + 1198862) 120593]

= minus 11198871198913 minus 11988811120585 minus 11988812120593 minus 11988813120595 + ℓ

1119886120595sdot [1198621 minus 11988611120585 minus 11988612120593 minus 11988613120595 + (119887 + 120585) 2

+ (119860120585 + 119861120595 + 1198862) 120593]+ 1119886119867120595 ℓ1 minus119886120595 (119887 + 120585)

sdot (1198913 minus 11988811120585 minus 11988812120593 minus 11988813120595) + 119887

sdot (11988711120585 + 11988712120593 + 11988713120595 + 119892 sin120593

0+ 1205962

sdot minus120585120593 cos2 1205930minus 2ℎ sin120596119905 cos120593

0

+ [(119887 + 120585) 120593120595 + 119886 (1205952 minus 1) minus 119887] cos1205930sin1205930

+ 120585120593 sin2 1205930 +2 minus (119887 + 120585) (120593 minus 120595) 2)+ [119886119887 minus (119887 + 120585) ℓ1] 1198621 minus 11988611120585 minus 11988612120593 minus 11988613120595+ (119887 + 120585) 2 + (119860120585 + 119861120595 + 1198862) 120593

= minus 1119867 119887 (120593 minus 120595) [119886

11120585 + 11988612120593 + 11988613120595 minus (119860120585 + 119861120595) 120593]

minus 119887 (11988711120585 + 11988712120593 + 11988713120595) + (119887 + 120585) (11988811120585 + 11988812120593 + 11988813120595)

minus 1198871198621(120593 minus 120595) + 1198871205962 cos120593

0(120585120593 cos120593

0+ 2ℎ sin120596119905)

minus (119887 + 120585) 1198913 minus 119887119892 sin1205930 + 1198871205962sdot 119886 + 119887 minus 120595 [(119887 + 120585) 120593 + 119886120595] cos1205930 sin1205930minus 119887 [2 + 120593 (1205962120585 sin2 120593

0+ 119886 (120593 minus 120595) 2)]

(14)

where

119867 = [119886119887 (1205952 minus 120593120595 minus 1) + (119887 + 120585) ℓ1] (15)

It is clear that the left hand sides of the equations ofthe previous system are given explicitly in terms of and respectively On the other hand the right hand sides arefunctions of 120585 120593 120595 and

The ode45 solver is used in order to obtain the numericalsolutions of the nonstiff ordinary differential equations of theprevious system (14) in which this solver uses a variable stepof Runge-Kutta technique [20] So we can rewrite system (14)as a system of coupled first-order differential equations asfollows

Advances in Mathematical Physics 5

A choice of the state variables for this system is

1198831 = 120585

1198832= 120593

1198833= 120595

1198834=

1198835 =

1198836=

(16)

which results in the following state-equations

1= 1198834

2 = 11988353= 1198836

(17)

Use (16) and (17) into system (14) to get

4= minus 1

119867 minus1198861198833minus (119887 + 119883

1)

sdot (119888111198831 + 119888121198832 + 119888131198833 minus 1198913)+ 119887 (119887

111198831 + 119887121198832 + 119887131198833 + 119892 sin1205930 + 1205962sdot minus11988311198832cos2 120593

0minus 2ℎ sin120596119905 cos120593

0

+ [minus119887 + (119887 + 1198831)11988321198833+ 119886 (1205952 minus 1)] cos120593

0sin1205930

+11988311198832 sin2 1205930+212 minus (119887 + 1198831) (1198832 minus 1198833) 22)+ [119886119887 minus (119887 + 120585) ℓ1] [1198621 minus 119886111198831 minus 119886121198832minus 119886131198833 + (119887 + 1198831) 22+ (119860119883

1+ 1198611198833+ 1198863

2)1198832]

5= minus 1

1198871198913 minus 119888111198831 minus 119888121198832 minus 119888131198833+ ℓ11198861198833

[1198621minus 119886111198831minus 119886121198832minus 119886131198833+ (119887 + 119883

1) 2

2

+ (1198601198831+ 1198611198833+ 1198863

2)1198832] + 1

1198861198671198833sdot ℓ1minus119886119883

3(119887 + 119883

1) (1198913minus 119888111198831minus 119888121198832minus 119888131198833)

+ 119887 (119887111198831+ 119887121198832+ 119887131198833+ 119892 sin120593

0

+ 1205962 minus11988311198832cos2 120593

0minus 2ℎ sin120596119905 cos120593

0

+ [(119887 + 1198831)11988321198833 + 119886 (11988323 minus 1) minus 119887] cos1205930 sin1205930

+11988311198832 sin2 1205930+212 minus (119887 + 1198831) (1198832 minus 1198833) 22)+ [119886119887 minus (119887 + 119883

1) ℓ1]

sdot 1198621minus 11988611120585 minus 119886121198832minus 119886131198833

+ (119887 + 1198831) 2

2 + (1198601198831+ 1198611198833+ 1198863

2)1198832

6= minus 1

119867 119887 (1198832minus 1198833)

sdot [119886111198831 + 119886121198832 + 119886131198833 minus (1198601198831 + 1198611198833)1198832]minus 119887 (119887111198831+ 119887121198832+ 119887131198833)

+ (119887 + 1198831) (119888111198831+ 119888121198832+ 119888131198833)

minus 1198871198621 (1198832 minus 1198833)+ 1198871205962 cos120593

0(11988311198832cos1205930+ 2ℎ sin120596119905)

minus (119887 + 1198831) 1198913 minus 119887119892 sin1205930+ 1198871205962 119886 + 119887 minus 119883

3[(119887 + 119883

1)1198832+ 1198861198833]

sdot cos1205930sin1205930

minus 119887 [212+ 1198832

sdot (12059621198831sin2 120593

0+ 119886 (119883

2minus 1198833) 3

2)] (18)

The following initial conditions are required to achieve thenumerical solution of (18) by using the fourth-order Runge-Kutta method of ode45 solver in framework of Matlabprogram

1198831 (0) = 00011198832 (0) = 01

1198833 (0) = 001

4 (0) = 0

5 (0) = 0

6 (0) = 0

(19)

in addition to the following physical parameters of theconsidered model

119898 = 50 kg119892 = 98m sdot sminus21198691= 8 kg sdotm2

ℓ = 07m120596 = 4 rad sdot sminus1119886 = 05m

6 Advances in Mathematical Physics

0 05 1 15

0

05

1

15

t

minus05

minus1

120585and 120585

times104

120585120585

1205930 = 0

(a)

0 05 1 15

0

5

t

1205930 = 0

minus5

minus10

minus15

minus20

minus25

120593

120593and120593

(b)

0 05 1 15

0

20

t

1205930 = 0

minus20

minus40

120595and

120595

120595120595

(c)

0

2

4

6

8

120585and 120585

times104

0 05 1 15t

120585120585

minus2

minus4

1205930 = 04

(d)

0

10

20

30

120593and120593

0 05 1 15t

120593

minus10

minus20

1205930 = 04

(e)

0

50

100

120595and

120595

0 05 1 15t

120595

120595

1205930 = 04

minus50

minus100

(f)

Figure 2 Variation of the solutions and their derivatives versus time 119905 when 119887 = 3m ℎ = 45 and 119886 = 05m (a d) show the effect of 119905 on thebehavior of 120585 and waves when 120593

0= 0 and 120593

0= 04 rad respectively (b e) show the effect of 119905 on the behavior of 120593 and waves when 120593

0= 0

and 1205930= 04 rad respectively and (c f) show the effect of 119905 on the wave that describes the behavior of 120595 and when 120593

0= 0 and 120593

0= 04 rad

respectively

119887 = (0 3)m1205930= (0 04) rad

ℎ = (25 45) 119905 = 0 997888rarr 17min

(20)

Figure 2 shows the variation of the solutions 120585 120593 120595 and theirderivatives against time 119905 when 120593

0= 0 and 120593

0=04 rad This figure is drawn at 119887 = 3m ℎ = 24 and

119886 = 05m The variations of 120585 120593 and 120595 with and respectively are illustrated in Figure 3 namely the phaseplane diagrams that are represented in Figures 3(a) 3(b)3(c) and 3(d) 3(e) 3(f) when 1205930 = 0 and 1205930 = 04 radrespectively with the same other parameters that are takeninto consideration in Figure 2

In these figures our principle aim is to investigate theeffect of increasing time on the motion of pendulum

According to the calculations depicted in Figure 2(a) wefound that when 1205930 = 0 the wave of the elongation 120585 growsup with the increasing of time till 119905 = 09min After thatboth of the elongation 120585 and its derivative fluctuate between

increasing and decreasing when time reaches 119905 = 143minThus the wave of the solution 120585 is stable see the phase planeFigure 3(a) With the passing of time one can observe that120585 and are growing quickly so the motion will be unstableafter 119905 = 143min The rage behavior of both 120585 and is dueto the weight of the rigid body and the values of the principalmoments of inertia Consequently we expect that behaviorof elongation becomes greater as observed in Figures 2(a) and2(d)Moreover the variation of the spring between stretchingand contraction is consistent with the phase plane diagramsrepresented in Figures 3(a) and 3(d)

It is worthwhile to notice fromFigure 2(b) that when time119905 increases from 119905 = 0 to 119905 = 04min the behavior of theangle 120593 increases gradually to reach the value 120593 ≃ 1 rad ≃ 57∘and then decreases slowly to reach 120593 ≃ 08 rad ≃ 46∘ duringthe time period 119905 isin ]04 09[min After 119905 = 09min thedecline of the wave becomes quickly to reach 120593 ≃ minus23 rad ≃minus132∘ at the end of time period (minus sign indicates oppositedirection) This is not possible because 120593 must belong to theinterval ] minus 1205872 1205872] So the motion of the wave is unstableas it is manifest from Figure 3(b) On the other hand increases till 119905 ≃ 02min and then fluctuates as indicated fromFigure 2(b)

Advances in Mathematical Physics 7

0 500 1000 1500 2000 2500

0

05

1

15

times104

120585

120585

1205930 = 0

minus05

minus1

(a)

0 1

0

5 1205930 = 0

minus5

minus10

minus15

minus20

minus25minus2 minus1

120593

(b)

0 2

0

20

minus20

minus40

minus2minus4

120595

120595

1205930 = 0

(c)

0 2000 4000 6000 8000 10000

0

2

4

6

8times104

120585

120585

minus2

minus4

1205930 = 04

(d)

0 02 04 06 08 1

0

10

20

30

120593

1205930 = 04

minus10

minus20

minus02

(e)

0 2 4

0

50

100

minus2minus4

120595

120595

1205930 = 04

minus50

minus100

(f)

Figure 3 The phase plane diagram when 119887 = 3m ℎ = 45 and 119886 = 05m (a d) represent the variation of the amplitude 120585 with its velocity at 120593

0= 0 and 120593

0= 04 rad respectively (b e) represent the variation of the amplitude 120593 with its velocity at 120593

0= 0 and 120593

0= 04 rad

respectively and (c f) represent the variation of the amplitude 120595 with its velocity at 1205930= 0 and 120593

0= 04 rad respectively

The graphs displayed in Figures 2(c) and 3(c) describethe variation of the (120595 and ) against time and the phaseplane diagram ( with 120595) respectively when 1205930 = 0 Itis clear that when time belongs to the period [0 043]minthe angle 120595 remains stationary and then its wave oscillatesbetween decreasing and increasing till 119905 = 143min Afterthat time the angle 120595 increases up to the end of time intervaland consequently the motion will be stable as seen fromFigure 3(c) during the period 0 lt 119905 le 143 It is obviousfrom Figure 2(c) that the behavior of remains stationary tosome extent through the time interval [0 06]min and thenoscillates between increasing and decreasing till 119905 = 17min

It should be noticed that when1205930= 04 rad the stretching

on the string 120585 increases gradually till the time 119905 becomes09min and then 120585 and oscillate between increasing anddecreasingwhen the time reaches the end of time interval seeFigure 2(d) Consequently the wave of the solution is stableas seen from the phase plane Figure 3(d)

An inspection of the graphs depicted in Figure 2(e) showsthat the wave describing the behavior of the angle 120593 increasesgradually from 120593 = 0 at 119905 = 0 to its maximum value 120593 ≃09 rad ≃ 56∘ at 119905 = 04min and then decreases slowly at119905 ≃ 1min to reach its minimum value 120593 ≃ minus019 rad ≃ minus11∘(minus sign indicates opposite direction) at 119905 ≃ 126minWith the increasing of time thewave grows again to reach thevalue 120593 ≃ 085 rad ≃ 49∘ at 119905 ≃ 139min Thus the motion is

stable as it is manifest from Figure 3(e) On the other hand increases and decreases as indicated from Figure 2(e)

Also it is remarkable from Figure 2(f) that the behaviorof the angle 120595 remains steady till 119905 = 05min then its waveoscillates between decreasing and increasing Consequentlythe motion will be stable as seen from Figure 3(f) It isworthwhile to notice also from Figure 2(f) that the behaviorof oscillates between increasing and decreasing

From the above observations we can conclude that themotion of our model is more stable when 120593

0= 04 rad than

when 1205930= 0 This highlights the importance of the effect of120593

0value on the motion It is worthwhile to notice that the

comparison between the solutions 120585 120593 and 120595 included inFigures 2(a) 2(b) and 2(c) with the corresponding Figures2(d) 2(e) and 2(f) reveals that the amplitude of the wavesdecreases when 120593

0increases from 0 to 04 rad On the other

hand the comparison between their derivatives shows thatthe amplitude of the waves increases when 120593

0increases

Figure 4 shows the variation of (120585 ) (120593 ) and (120595 ) withtime 119905 when 120593

0 changes from 0 for Figures 4(a) 4(b) and4(c) to 04 rad for Figures 4(d) 4(e) and 4(f) at the samevalues of other parameters 119887 = 3m ℎ = 45 and 119886 = 05mAccording to the calculations depicted in these figures we canconsider these figures as a rotation of the corresponding partsof Figure 3 with time to observe the bending and crossing ofthe resulting curves

8 Advances in Mathematical Physics

01000 20000

1

0051

15

t

minus1

times104

1205930 = 0

120585120585

(a)

010

0051

15

t

minus10minus20 minus1

minus2

1205930 = 0

120593

(b)

0 2020

0051

15

t

1205930 = 0

minus20minus40 minus4

minus2

120595120595

(c)

05000

1000002468

0051

15

t

times104

minus4minus2

1205930 = 04

120585120585

(d)

005

10

20

0051

15

t

minus20

1205930 = 04

120593

(e)

0 2 401000

051

15

t

minus4 minus2

1205930 = 04

minus100120595

120595

(f)

Figure 4The 3D pattern when 119887 = 3m ℎ = 45 and 119886 = 05m (a d) indicate the variation of 120585 and versus 119905 when 1205930= 0 and 120593

0= 04 rad

respectively (b e) indicate the variation of 120593 and versus 119905 when 1205930= 0 and 120593

0= 04 rad respectively and (c f) indicate the variation of 120595

and versus 119905 when 1205930= 0 and 120593

0= 04 rad respectively

01000

200001

02

minus1minus2

minus2

minus4

120585120593

120595

1205930 = 0

(a)

01

0

020

1205930 = 0

minus1

minus10minus20

minus20

minus40

times104

120585

120595

(b)

05000

100000

051

024

minus2

minus4

120595

1205930 = 04

120585

120593

(c)

0 2 4 6 8020

0

100

minus2minus4times104

120595

1205930 = 04

minus20

minus100

120585

(d)

Figure 5 The 3D diagrams when 119887 = 3m ℎ = 45 and 119886 = 05m (a c) elucidate the variation of 120585 and 120593 versus 120595 when 1205930= 0 and

1205930= 04 rad respectively and (b d) elucidate the variation of and versus when 120593

0= 0 and 120593

0= 04 rad respectively

Advances in Mathematical Physics 9

0 05 1 15

0

2

4

6

8

t

120585and 120585

times104

minus2

b = 0

120585120585

(a)

t

0 05 1 15

0

10

20

30

40

50b = 0

minus10

120593120593and120593

(b)

0 05 1 15

0

200

400

t

b = 0

minus200

minus400

minus600

120595and

120595

120595120595

(c)

Figure 6 (a) (b) and (c) explain the variation of the solutions 120585 120593 and 120595 with their derivatives and via time 119905 respectively when119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad

0 5000 10000 15000 20000

0

2

4

6

8

minus2

times104

b = 0

120585

120585

(a)

0 1 2 3

0

10

20

30

40

50b = 0

minus10

minus1

120593

(b)

0 5 10 15

0

200

400b = 0

120595

120595

minus200

minus400

minus600minus5minus10minus15

(c)

Figure 7 The phase plane diagrams between amplitudes and their velocities at 119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad (a) shows the

influence of 120585 on (b) shows the effect of 120593 on and (c) shows the variation of 120595 with

Figures 5(a) 5(c) and 5(b) 5(d) represent 3D plots thatillustrate the variation of the solutions 120585 120593 via 120595 and via respectively for different values of 120593

0when 119887 =3m ℎ = 45 and 119886 = 05m The graphs displayed in

parts of Figure 6 show the variation of (120585 ) (120593 ) and(120595 ) against time 119905 when 119887 = 0 with consideration of theparameters 1205930 = 04 rad ℎ = 45 and 119886 = 05m Thecorresponding phase plane between the amplitudes 120585 120593 120595and their derivatives is represented in parts of Figure 7Inspection of the graph depicted in Figure 6(a) shows thatwhen time 119905 increases from 0 to 045min the behavior of thesolution 120585 remains stationary and quickly growing during thetime interval 119905 isin ]045 105[min and then oscillates till theend of time interval This indicates that the motion is stableas seen from Figure 7(a) On the other side the behaviorof the derivative remains approximately stationary duringthe interval 119905 isin [0 045]min and then fluctuates with theincreasing of time see Figure 6(a)

By the same way we can observe that the wave of theangle 120593 increases through a short time to reach its maximumvalue 120593 ≃ 27 rad ≃ 155∘ at 119905 ≃ 023min taking intoconsideration that minus1205872 lt 120593 lt 1205872 and then decreasesslowly to reach its minimum value 120593 ≃ minus09 rad ≃ minus52∘at the end of time interval see Figure 6(b) This indicatesthat the motion is close to be stable as observed fromthe phase plane Figure 8(b) As seen from Figure 6(b) increases and decreases quickly during the period 119905 isin[0 01]min to reach its minimum value at the end of timeinterval

The variation of 120595 and with time is illustrated inFigure 6(c) In this figure our main goal is to examine theinfluence of time on the motion of pendulum It is clear thatthe behavior of 120595 and remains stationary (to some extent)when 119905 isin [0 05]min then their waves fluctuate till the endof time interval Consequently the motion is stable as seenfrom the phase plane diagram Figure 7(c) The comparisonbetween parts of Figure 6 with the corresponding Figures

10 Advances in Mathematical Physics

010000

2000002468

0

05

1

15

t

120585120585 minus2

b = 0

times104

(a)

t

0 1 2 30

2040

0

05

1

15

120593minus1

b = 0

(b)

t

0 1005000

05

1

15

minus10minus500 120595120595

b = 0

(c)

Figure 8 The 3D plots at 119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad (a) illustrates the variation of 120585 and via 119905 (b) illustrates the variation

of 120593 and via 119905 and (c) illustrates the variation of 120595 and via 119905

0 05 1 15

0

5000

10000

15000

t

minus5000

120585120585

120585and 120585

h = 25

(a)

0 05 1 15

0

5

10

t

minus5

minus10

minus15

120593

120593and120593

h = 25

(b)

0 05 1 15

0

20

t

120595and

120595120595120595

minus20

minus40

minus60

h = 25

(c)

Figure 9 (a) (b) and (c) demonstrate the variation of (120585 and ) (120593 and ) and (120595 and ) against time 119905 respectively at 119887 = 3m ℎ = 25119886 = 05m and 1205930= 04 rad

2(d) 2(e) and 2(f) shows that when 119887 changes from 0 to 3mthe amplitude of the waves decreases Also the motion willbe more stable when 119887 = 3m than when 119887 = 0 as seen fromthe corresponding phase plane diagrams that is Figures 3(d)3(e) 3(f) and 7(a) 7(b) 7(c) respectively

On the other hand parts of Figure 8 show 3D plots thatdescribe the variation of the solutions and their derivative viatime when 119887 = 0 ℎ = 45 120593

0= 04 rad and 119886 = 05m

The plots displayed in thementioned parts show bending andcrossing of the resulting curves

Figures 9(a) 9(b) and 9(c) show the variation of thesolutions 120585 120593 120595 and their derivatives with time 119905whenℎ = 25 for the given values of other parameters 119887 = 3m1205930= 04 rad and 119886 = 05m In view of the first part we can

conclude that when time 119905 increases each of the waves 120585 and oscillates between increasing and decreasing till 119905 = 146minand then increases gradually So the motion is stable as seenfrom Figure 10(a)

From a closer look on the second part of Figure 9(b) wecan write with the increasing of time the behavior of 120593 wave

increases to reach its maximum value 120593 ≃ 09 rad ≃ 52∘ at119905 = 043min and then decreases slowly through the period 119905 isin]043 119]min After that its behavior has a sharp declinein a few seconds (about 24 s) and then increases till the endof time period and consequently the motion is stable seeFigure 10(b)

According to the calculations depicted in Figure 9(c) wecan observe that thewaves describing120595 and decrease slowlytill 119905 = 09min and then increase and decline sharp Thephase plane Figure 10(c) shows that the behavior of 120595 is notstable

When parts of Figure 9 and their phase plane parts (ofFigure 10) are generally compared with the correspondingFigures 2(d) 2(e) and 2(f) and their phase plane Figures 3(d)3(e) and 3(f) we can observe that amplitude of the waveincreases when ℎ = 45 compared to when ℎ = 25 and themotion is more stable when ℎ = 45 An inspection of partsof Figure 11 reveals the 3D plots when ℎ = 25 with the sameother data considered in Figures 9 and 10 Figure 10 shows thevariation of the solutions 120585 120593 120595 and their derivatives

Advances in Mathematical Physics 11

0 500 1000 1500 2000

0

5000

10000

15000 h = 25

minus5000

120585

120585

(a)

0 02 04 06 08 1

0

5

10h = 25

minus5

minus10

minus15

minus02

120593

(b)

0 2

0

20h = 25

minus20

minus40

minus60

minus2minus4minus6minus8

120595

120595

(c)

Figure 10 The phase plane diagrams which portray the relation between amplitudes and their velocities at 119887 = 3m ℎ = 25 119886 = 05m and1205930= 04 rad (a) describes the influence of 120585 on (b) shows the effect of 120593 on and (c) illustrates the variation of 120595 with

01000

200005000

1000015000

0051

15

t

minus5000120585

120585

h = 25

(a)

t

005

10100

05

1

15

minus10120593

h = 25

(b)

t

0 20200

05

1

15

minus20minus40

minus60minus2minus4minus6minus8120595

120595

h = 25

(c)

Figure 11 The 3D patterns at 119887 = 3m ℎ = 25 119886 = 05m and 1205930= 04 rad (a) illustrates the variation of 120585 and versus 119905 (b) illustrates the

variation of 120593 and versus 119905 (c) illustrates the variation of 120595 and versus 119905

with time 119905 It is worthwhile to notice that the comparisonbetween Figures 4(d) 4(e) and 4(f) and Figures 11(a) 11(b)and 11(c) shows more bending and crossing of the curvesin Figures 4(d) 4(e) and 4(f) when ℎ = 45 than thecorresponding ones of Figure 11

Now we study the last case when ℎ = 0 with the sameother data 119887 = 3m 1205930 = 04 rad and 119886 = 05mThe obtainedresults are represented graphically in Figures 12(a) 12(b) and12(c) while their phase plane diagrams are given in Figures12(d) 12(e) and 12(f) At the first glance we can conclude thatthis case is not stable so it is very important to notice that thedimensionless parameter ℎmust take any value different fromzero as it is pointed in Figure 2 (ℎ = 45) and Figure 9 (ℎ =25) This elucidates the importance of ℎ parameter on themotion

4 Conclusion

A conclusion that may be made here is that the problemof the relative motion of a rigid body as a pendulum

model is investigated The governing deferential equationsare obtained using Lagrangersquos equations Mathematica pack-age was utilized in order to overcome the difficulties thatappear in the separation of the second derivatives of thegeneralized coordinates 120585 120593 and 120595 for the nonlinear system(10) Computer codes are used to obtain the numericalsolutions for system (14) These solutions are representedgraphically using Matlab program to study the influenceof the different parameters on the motion The good effectof the parameters ℎ 119887 and 120593

0on the motion is obvious

from the mentioned plots The motion of our model is morestable when the parameters ℎ 119887 and 120593

0take values run

away from zero This highlights the importance of the effectof these parameters on the motion Such results have beenconfirmed by many works such as Ismail [13] and Amer andBek [14]

Competing Interests

The author declares that they have no competing interests

12 Advances in Mathematical Physics

0 05 1 15

0

05

1

15

2

25

3

t

120585120585

120585and 120585

h = 0

times104

(a)

0 05 1 15

0

05

1

15

2

25

t

120593120593and120593

h = 0

(b)

0 05 1 15

0

5

10

15

t

120595and

120595

120595120595

h = 0

(c)

0 1000 2000 3000 4000 5000

0

05

1

15

2

25

3times104

120585

120585

h = 0

(d)

02 04 06 08 1

0

05

1

15

2

25

120593

h = 0

(e)

0 05 1 15 2

0

5

10

15

120595

120595

h = 0

(f)

Figure 12 (a b and c) explain the variation of the solutions 120585 120593 and120595with their derivatives and via time 119905 respectively when 119887 = 3mℎ = 0 119886 = 05m and 1205930= 04 rad (d e and f) illustrate the variation of the solutions against their first derivatives for the same values of the

considered parameters

References

[1] P Lynch ldquoResonant motions of the three-dimensional elasticpendulumrdquo International Journal of Non-Linear Mechanics vol37 no 2 pp 345ndash367 2002

[2] A A Klimenko Y V Mikhlin and J Awrejcewicz ldquoNonlinearnormal modes in pendulum systemsrdquoNonlinear Dynamics vol70 no 1 pp 797ndash813 2012

[3] S Mori H Nishihara and K Furuta ldquoControl of unstablemechanical system control of pendulumrdquo International Journalof Control vol 23 no 5 pp 673ndash692 1976

[4] C C Chung and J Hauser ldquoNonlinear control of a swingingpendulumrdquo Automatica A Journal of IFAC vol 31 no 6 pp851ndash862 1995

[5] A Shiriaev A Pogromsky H Ludvigsen and O Egeland ldquoOnglobal properties of passivity-based control of an inverted pen-dulumrdquo International Journal of Robust and Nonlinear Controlvol 10 no 4 pp 283ndash300 2000

[6] A S Shiriaev H Ludvigsen and O Egeland ldquoSwinging upthe spherical pendulum via stabilization of its first integralsrdquoAutomatica A Journal of IFAC the International Federation ofAutomatic Control vol 40 no 1 pp 73ndash85 2004

[7] M N Brearley ldquoThe Simple Pendulum with Uniformly Chang-ing String Lengthrdquo Proceedings of the Edinburgh MathematicalSociety vol 15 no 1 pp 61ndash66 1966

[8] S J Liao ldquoSecond-order approximate analytical solution of asimple pendulum by the process analysis methodrdquo Journal ofApplied Mechanics Transactions ASME vol 59 no 4 pp 970ndash975 1992

[9] W K Tso and K G Asmis ldquoParametric excitation of a pen-dulum with bilinear hysteresisrdquo Journal of Applied MechanicsTransactions ASME vol 37 no 4 pp 1061ndash1068 1970

[10] A H Nayfeh Perturbations Methods Wiley-VCH WeinheimGermany 2004

[11] F A El-Barki A I Ismail M O Shaker and T S AmerldquoOn the motion of the pendulum on an ellipserdquo Zeitschrift furAngewandteMathematik undMechanik vol 79 no 1 pp 65ndash721999

[12] N V Stoianov ldquoOn the relative periodic motions of a pendu-lumrdquo Journal of AppliedMathematics andMechanics vol 28 pp188ndash193 1964

[13] A I Ismail ldquoRelative periodicmotion of a rigid body pendulumon an ellipserdquo Journal of Aerospace Engineering vol 22 no 1 pp67ndash77 2009

[14] T S Amer andM A Bek ldquoChaotic responses of a harmonicallyexcited spring pendulum moving in circular pathrdquo NonlinearAnalysis Real World Applications An International Multidisci-plinary Journal vol 10 pp 3196ndash3202 2009

Advances in Mathematical Physics 13

[15] L D Akulenko ldquoParametric control of oscillations and rota-tions of a compound pendulum (a swing)rdquo Journal of AppliedMathematics and Mechanics vol 57 no 2 pp 301ndash310 1993

[16] M A Pinsky and A A Zevin ldquoOscillations of a pendulumwith a periodically varying length and a model of swingrdquoInternational Journal of Non-LinearMechanics vol 34 no 1 pp105ndash109 1999

[17] M Kamel M Eissa and A T El-Sayed ldquoVibration reductionof a nonlinear spring pendulum under multiparametric excita-tions via a longitudinal absorberrdquo Physica Scripta vol 80 no 2Article ID 025005 2009

[18] M Eissa M Kamel and A T El-Sayed ldquoVibration reduction ofmulti-parametric excited spring pendulum via a transversallytuned absorberrdquo Nonlinear Dynamics vol 61 no 1-2 pp 109ndash121 2010

[19] R Starosta G Sypniewska-Kaminska and J AwrejcewiczldquoAsymptotic analysis of kinematically excited dynamical sys-tems near resonancesrdquo Nonlinear Dynamics An InternationalJournal of Nonlinear Dynamics and Chaos in Engineering Sys-tems vol 68 no 4 pp 459ndash469 2012

[20] H MooreMatlab for Engineers Pearson 3rd edition 2012[21] M D Ardema Analytical Dynamics Theory and Applications

Springer Berlin Germany 2009[22] A Tewari Modern Control Design with Matlab and Similink

John Wiley and Sons Ltd New York NY USA 2002

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Advances in Mathematical Physics

119860 = minus211988811

119861 = 1198861205962 cos2 1205930

119863 = 1205962cos 2 12059301198621 = minus119887 (1198702 minus 1205962sin21205930)

+ 1205962 [2ℎ sin1205930sin120596119905 + 119886 (sin2 120596119905 + sin2 120593

0)]

+ 119892 cos1205930+ 1198702ℓ

1198622= 2ℎ1205962 sin120596119905 minus (119887 + 119886) 11988811 minus 119892 sin1205930

(13)

Our principle aim is to obtain the numerical solutionsof system (10) which consists of three nonlinear differentialequations of second-order In view of the right hand sides ofthese equations we found three functions119891

1 1198912 and1198913 givenby (12) In fact it is not easy to obtain the second derivativesof the generalized coordinates 120585 120593 and 120595 such that eachequation contains one of these derivatives only

3 Numerical Solutions

This section is devoted to discuss the numerical solutions forthe considered model in Section 2 Computer programs arecarried out to investigate the graphical representations forthese solutions to describe the motion and to illustrate thebehavior of the pendulum at any time

System (10) consists of three nonlinear differential equa-tions of second-order in terms of 120585 120593 and 120595 and is recon-sidered to obtain the numerical solutions in framework ofthe fourth-order Runge-Kutta algorithms through Matlabpackages [22] Each equation of this system includes allvariables 120585 120593 120595 and their derivatives from the first andsecond order see systems of (10) (11) (12) and (13) Sothe mentioned system is more complicated to deal with andto obtain another corresponding one consisting of second-order differential equations in terms of and explicitlyComputer codes are utilized in order to overcome thesedifficulties and to separate each of and Consequentlysystem (10) is transformed into the following system with theaid of (11) (12) and (13)

= minus 1119867 minus119886120595 minus (119887 + 120585) (11988811120585 + 11988812120593 + 11988813120595 minus 119891

3)

+ 119887 (11988711120585 + 11988712120593 + 11988713120595 + 119892 sin120593

0+ 1205962

sdot minus120585120593 cos2 1205930 minus 2ℎ sin120596119905 cos1205930+ [minus119887 + (119887 + 120585) 120593120595 + 119886 (1205952 minus 1)]sdot cos120593

0 sin1205930 + 120585120593 sin2 1205930+ 2 minus (119887 + 120585) (120593 minus 120595) 2)

+ [119886119887 minus (119887 + 120585) ℓ1] [1198621 minus 11988611120585 minus 11988612120593minus11988613120595 + (119887 + 120585) 2 + (119860120585 + 119861120595 + 1198862) 120593]

= minus 11198871198913 minus 11988811120585 minus 11988812120593 minus 11988813120595 + ℓ

1119886120595sdot [1198621 minus 11988611120585 minus 11988612120593 minus 11988613120595 + (119887 + 120585) 2

+ (119860120585 + 119861120595 + 1198862) 120593]+ 1119886119867120595 ℓ1 minus119886120595 (119887 + 120585)

sdot (1198913 minus 11988811120585 minus 11988812120593 minus 11988813120595) + 119887

sdot (11988711120585 + 11988712120593 + 11988713120595 + 119892 sin120593

0+ 1205962

sdot minus120585120593 cos2 1205930minus 2ℎ sin120596119905 cos120593

0

+ [(119887 + 120585) 120593120595 + 119886 (1205952 minus 1) minus 119887] cos1205930sin1205930

+ 120585120593 sin2 1205930 +2 minus (119887 + 120585) (120593 minus 120595) 2)+ [119886119887 minus (119887 + 120585) ℓ1] 1198621 minus 11988611120585 minus 11988612120593 minus 11988613120595+ (119887 + 120585) 2 + (119860120585 + 119861120595 + 1198862) 120593

= minus 1119867 119887 (120593 minus 120595) [119886

11120585 + 11988612120593 + 11988613120595 minus (119860120585 + 119861120595) 120593]

minus 119887 (11988711120585 + 11988712120593 + 11988713120595) + (119887 + 120585) (11988811120585 + 11988812120593 + 11988813120595)

minus 1198871198621(120593 minus 120595) + 1198871205962 cos120593

0(120585120593 cos120593

0+ 2ℎ sin120596119905)

minus (119887 + 120585) 1198913 minus 119887119892 sin1205930 + 1198871205962sdot 119886 + 119887 minus 120595 [(119887 + 120585) 120593 + 119886120595] cos1205930 sin1205930minus 119887 [2 + 120593 (1205962120585 sin2 120593

0+ 119886 (120593 minus 120595) 2)]

(14)

where

119867 = [119886119887 (1205952 minus 120593120595 minus 1) + (119887 + 120585) ℓ1] (15)

It is clear that the left hand sides of the equations ofthe previous system are given explicitly in terms of and respectively On the other hand the right hand sides arefunctions of 120585 120593 120595 and

The ode45 solver is used in order to obtain the numericalsolutions of the nonstiff ordinary differential equations of theprevious system (14) in which this solver uses a variable stepof Runge-Kutta technique [20] So we can rewrite system (14)as a system of coupled first-order differential equations asfollows

Advances in Mathematical Physics 5

A choice of the state variables for this system is

1198831 = 120585

1198832= 120593

1198833= 120595

1198834=

1198835 =

1198836=

(16)

which results in the following state-equations

1= 1198834

2 = 11988353= 1198836

(17)

Use (16) and (17) into system (14) to get

4= minus 1

119867 minus1198861198833minus (119887 + 119883

1)

sdot (119888111198831 + 119888121198832 + 119888131198833 minus 1198913)+ 119887 (119887

111198831 + 119887121198832 + 119887131198833 + 119892 sin1205930 + 1205962sdot minus11988311198832cos2 120593

0minus 2ℎ sin120596119905 cos120593

0

+ [minus119887 + (119887 + 1198831)11988321198833+ 119886 (1205952 minus 1)] cos120593

0sin1205930

+11988311198832 sin2 1205930+212 minus (119887 + 1198831) (1198832 minus 1198833) 22)+ [119886119887 minus (119887 + 120585) ℓ1] [1198621 minus 119886111198831 minus 119886121198832minus 119886131198833 + (119887 + 1198831) 22+ (119860119883

1+ 1198611198833+ 1198863

2)1198832]

5= minus 1

1198871198913 minus 119888111198831 minus 119888121198832 minus 119888131198833+ ℓ11198861198833

[1198621minus 119886111198831minus 119886121198832minus 119886131198833+ (119887 + 119883

1) 2

2

+ (1198601198831+ 1198611198833+ 1198863

2)1198832] + 1

1198861198671198833sdot ℓ1minus119886119883

3(119887 + 119883

1) (1198913minus 119888111198831minus 119888121198832minus 119888131198833)

+ 119887 (119887111198831+ 119887121198832+ 119887131198833+ 119892 sin120593

0

+ 1205962 minus11988311198832cos2 120593

0minus 2ℎ sin120596119905 cos120593

0

+ [(119887 + 1198831)11988321198833 + 119886 (11988323 minus 1) minus 119887] cos1205930 sin1205930

+11988311198832 sin2 1205930+212 minus (119887 + 1198831) (1198832 minus 1198833) 22)+ [119886119887 minus (119887 + 119883

1) ℓ1]

sdot 1198621minus 11988611120585 minus 119886121198832minus 119886131198833

+ (119887 + 1198831) 2

2 + (1198601198831+ 1198611198833+ 1198863

2)1198832

6= minus 1

119867 119887 (1198832minus 1198833)

sdot [119886111198831 + 119886121198832 + 119886131198833 minus (1198601198831 + 1198611198833)1198832]minus 119887 (119887111198831+ 119887121198832+ 119887131198833)

+ (119887 + 1198831) (119888111198831+ 119888121198832+ 119888131198833)

minus 1198871198621 (1198832 minus 1198833)+ 1198871205962 cos120593

0(11988311198832cos1205930+ 2ℎ sin120596119905)

minus (119887 + 1198831) 1198913 minus 119887119892 sin1205930+ 1198871205962 119886 + 119887 minus 119883

3[(119887 + 119883

1)1198832+ 1198861198833]

sdot cos1205930sin1205930

minus 119887 [212+ 1198832

sdot (12059621198831sin2 120593

0+ 119886 (119883

2minus 1198833) 3

2)] (18)

The following initial conditions are required to achieve thenumerical solution of (18) by using the fourth-order Runge-Kutta method of ode45 solver in framework of Matlabprogram

1198831 (0) = 00011198832 (0) = 01

1198833 (0) = 001

4 (0) = 0

5 (0) = 0

6 (0) = 0

(19)

in addition to the following physical parameters of theconsidered model

119898 = 50 kg119892 = 98m sdot sminus21198691= 8 kg sdotm2

ℓ = 07m120596 = 4 rad sdot sminus1119886 = 05m

6 Advances in Mathematical Physics

0 05 1 15

0

05

1

15

t

minus05

minus1

120585and 120585

times104

120585120585

1205930 = 0

(a)

0 05 1 15

0

5

t

1205930 = 0

minus5

minus10

minus15

minus20

minus25

120593

120593and120593

(b)

0 05 1 15

0

20

t

1205930 = 0

minus20

minus40

120595and

120595

120595120595

(c)

0

2

4

6

8

120585and 120585

times104

0 05 1 15t

120585120585

minus2

minus4

1205930 = 04

(d)

0

10

20

30

120593and120593

0 05 1 15t

120593

minus10

minus20

1205930 = 04

(e)

0

50

100

120595and

120595

0 05 1 15t

120595

120595

1205930 = 04

minus50

minus100

(f)

Figure 2 Variation of the solutions and their derivatives versus time 119905 when 119887 = 3m ℎ = 45 and 119886 = 05m (a d) show the effect of 119905 on thebehavior of 120585 and waves when 120593

0= 0 and 120593

0= 04 rad respectively (b e) show the effect of 119905 on the behavior of 120593 and waves when 120593

0= 0

and 1205930= 04 rad respectively and (c f) show the effect of 119905 on the wave that describes the behavior of 120595 and when 120593

0= 0 and 120593

0= 04 rad

respectively

119887 = (0 3)m1205930= (0 04) rad

ℎ = (25 45) 119905 = 0 997888rarr 17min

(20)

Figure 2 shows the variation of the solutions 120585 120593 120595 and theirderivatives against time 119905 when 120593

0= 0 and 120593

0=04 rad This figure is drawn at 119887 = 3m ℎ = 24 and

119886 = 05m The variations of 120585 120593 and 120595 with and respectively are illustrated in Figure 3 namely the phaseplane diagrams that are represented in Figures 3(a) 3(b)3(c) and 3(d) 3(e) 3(f) when 1205930 = 0 and 1205930 = 04 radrespectively with the same other parameters that are takeninto consideration in Figure 2

In these figures our principle aim is to investigate theeffect of increasing time on the motion of pendulum

According to the calculations depicted in Figure 2(a) wefound that when 1205930 = 0 the wave of the elongation 120585 growsup with the increasing of time till 119905 = 09min After thatboth of the elongation 120585 and its derivative fluctuate between

increasing and decreasing when time reaches 119905 = 143minThus the wave of the solution 120585 is stable see the phase planeFigure 3(a) With the passing of time one can observe that120585 and are growing quickly so the motion will be unstableafter 119905 = 143min The rage behavior of both 120585 and is dueto the weight of the rigid body and the values of the principalmoments of inertia Consequently we expect that behaviorof elongation becomes greater as observed in Figures 2(a) and2(d)Moreover the variation of the spring between stretchingand contraction is consistent with the phase plane diagramsrepresented in Figures 3(a) and 3(d)

It is worthwhile to notice fromFigure 2(b) that when time119905 increases from 119905 = 0 to 119905 = 04min the behavior of theangle 120593 increases gradually to reach the value 120593 ≃ 1 rad ≃ 57∘and then decreases slowly to reach 120593 ≃ 08 rad ≃ 46∘ duringthe time period 119905 isin ]04 09[min After 119905 = 09min thedecline of the wave becomes quickly to reach 120593 ≃ minus23 rad ≃minus132∘ at the end of time period (minus sign indicates oppositedirection) This is not possible because 120593 must belong to theinterval ] minus 1205872 1205872] So the motion of the wave is unstableas it is manifest from Figure 3(b) On the other hand increases till 119905 ≃ 02min and then fluctuates as indicated fromFigure 2(b)

Advances in Mathematical Physics 7

0 500 1000 1500 2000 2500

0

05

1

15

times104

120585

120585

1205930 = 0

minus05

minus1

(a)

0 1

0

5 1205930 = 0

minus5

minus10

minus15

minus20

minus25minus2 minus1

120593

(b)

0 2

0

20

minus20

minus40

minus2minus4

120595

120595

1205930 = 0

(c)

0 2000 4000 6000 8000 10000

0

2

4

6

8times104

120585

120585

minus2

minus4

1205930 = 04

(d)

0 02 04 06 08 1

0

10

20

30

120593

1205930 = 04

minus10

minus20

minus02

(e)

0 2 4

0

50

100

minus2minus4

120595

120595

1205930 = 04

minus50

minus100

(f)

Figure 3 The phase plane diagram when 119887 = 3m ℎ = 45 and 119886 = 05m (a d) represent the variation of the amplitude 120585 with its velocity at 120593

0= 0 and 120593

0= 04 rad respectively (b e) represent the variation of the amplitude 120593 with its velocity at 120593

0= 0 and 120593

0= 04 rad

respectively and (c f) represent the variation of the amplitude 120595 with its velocity at 1205930= 0 and 120593

0= 04 rad respectively

The graphs displayed in Figures 2(c) and 3(c) describethe variation of the (120595 and ) against time and the phaseplane diagram ( with 120595) respectively when 1205930 = 0 Itis clear that when time belongs to the period [0 043]minthe angle 120595 remains stationary and then its wave oscillatesbetween decreasing and increasing till 119905 = 143min Afterthat time the angle 120595 increases up to the end of time intervaland consequently the motion will be stable as seen fromFigure 3(c) during the period 0 lt 119905 le 143 It is obviousfrom Figure 2(c) that the behavior of remains stationary tosome extent through the time interval [0 06]min and thenoscillates between increasing and decreasing till 119905 = 17min

It should be noticed that when1205930= 04 rad the stretching

on the string 120585 increases gradually till the time 119905 becomes09min and then 120585 and oscillate between increasing anddecreasingwhen the time reaches the end of time interval seeFigure 2(d) Consequently the wave of the solution is stableas seen from the phase plane Figure 3(d)

An inspection of the graphs depicted in Figure 2(e) showsthat the wave describing the behavior of the angle 120593 increasesgradually from 120593 = 0 at 119905 = 0 to its maximum value 120593 ≃09 rad ≃ 56∘ at 119905 = 04min and then decreases slowly at119905 ≃ 1min to reach its minimum value 120593 ≃ minus019 rad ≃ minus11∘(minus sign indicates opposite direction) at 119905 ≃ 126minWith the increasing of time thewave grows again to reach thevalue 120593 ≃ 085 rad ≃ 49∘ at 119905 ≃ 139min Thus the motion is

stable as it is manifest from Figure 3(e) On the other hand increases and decreases as indicated from Figure 2(e)

Also it is remarkable from Figure 2(f) that the behaviorof the angle 120595 remains steady till 119905 = 05min then its waveoscillates between decreasing and increasing Consequentlythe motion will be stable as seen from Figure 3(f) It isworthwhile to notice also from Figure 2(f) that the behaviorof oscillates between increasing and decreasing

From the above observations we can conclude that themotion of our model is more stable when 120593

0= 04 rad than

when 1205930= 0 This highlights the importance of the effect of120593

0value on the motion It is worthwhile to notice that the

comparison between the solutions 120585 120593 and 120595 included inFigures 2(a) 2(b) and 2(c) with the corresponding Figures2(d) 2(e) and 2(f) reveals that the amplitude of the wavesdecreases when 120593

0increases from 0 to 04 rad On the other

hand the comparison between their derivatives shows thatthe amplitude of the waves increases when 120593

0increases

Figure 4 shows the variation of (120585 ) (120593 ) and (120595 ) withtime 119905 when 120593

0 changes from 0 for Figures 4(a) 4(b) and4(c) to 04 rad for Figures 4(d) 4(e) and 4(f) at the samevalues of other parameters 119887 = 3m ℎ = 45 and 119886 = 05mAccording to the calculations depicted in these figures we canconsider these figures as a rotation of the corresponding partsof Figure 3 with time to observe the bending and crossing ofthe resulting curves

8 Advances in Mathematical Physics

01000 20000

1

0051

15

t

minus1

times104

1205930 = 0

120585120585

(a)

010

0051

15

t

minus10minus20 minus1

minus2

1205930 = 0

120593

(b)

0 2020

0051

15

t

1205930 = 0

minus20minus40 minus4

minus2

120595120595

(c)

05000

1000002468

0051

15

t

times104

minus4minus2

1205930 = 04

120585120585

(d)

005

10

20

0051

15

t

minus20

1205930 = 04

120593

(e)

0 2 401000

051

15

t

minus4 minus2

1205930 = 04

minus100120595

120595

(f)

Figure 4The 3D pattern when 119887 = 3m ℎ = 45 and 119886 = 05m (a d) indicate the variation of 120585 and versus 119905 when 1205930= 0 and 120593

0= 04 rad

respectively (b e) indicate the variation of 120593 and versus 119905 when 1205930= 0 and 120593

0= 04 rad respectively and (c f) indicate the variation of 120595

and versus 119905 when 1205930= 0 and 120593

0= 04 rad respectively

01000

200001

02

minus1minus2

minus2

minus4

120585120593

120595

1205930 = 0

(a)

01

0

020

1205930 = 0

minus1

minus10minus20

minus20

minus40

times104

120585

120595

(b)

05000

100000

051

024

minus2

minus4

120595

1205930 = 04

120585

120593

(c)

0 2 4 6 8020

0

100

minus2minus4times104

120595

1205930 = 04

minus20

minus100

120585

(d)

Figure 5 The 3D diagrams when 119887 = 3m ℎ = 45 and 119886 = 05m (a c) elucidate the variation of 120585 and 120593 versus 120595 when 1205930= 0 and

1205930= 04 rad respectively and (b d) elucidate the variation of and versus when 120593

0= 0 and 120593

0= 04 rad respectively

Advances in Mathematical Physics 9

0 05 1 15

0

2

4

6

8

t

120585and 120585

times104

minus2

b = 0

120585120585

(a)

t

0 05 1 15

0

10

20

30

40

50b = 0

minus10

120593120593and120593

(b)

0 05 1 15

0

200

400

t

b = 0

minus200

minus400

minus600

120595and

120595

120595120595

(c)

Figure 6 (a) (b) and (c) explain the variation of the solutions 120585 120593 and 120595 with their derivatives and via time 119905 respectively when119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad

0 5000 10000 15000 20000

0

2

4

6

8

minus2

times104

b = 0

120585

120585

(a)

0 1 2 3

0

10

20

30

40

50b = 0

minus10

minus1

120593

(b)

0 5 10 15

0

200

400b = 0

120595

120595

minus200

minus400

minus600minus5minus10minus15

(c)

Figure 7 The phase plane diagrams between amplitudes and their velocities at 119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad (a) shows the

influence of 120585 on (b) shows the effect of 120593 on and (c) shows the variation of 120595 with

Figures 5(a) 5(c) and 5(b) 5(d) represent 3D plots thatillustrate the variation of the solutions 120585 120593 via 120595 and via respectively for different values of 120593

0when 119887 =3m ℎ = 45 and 119886 = 05m The graphs displayed in

parts of Figure 6 show the variation of (120585 ) (120593 ) and(120595 ) against time 119905 when 119887 = 0 with consideration of theparameters 1205930 = 04 rad ℎ = 45 and 119886 = 05m Thecorresponding phase plane between the amplitudes 120585 120593 120595and their derivatives is represented in parts of Figure 7Inspection of the graph depicted in Figure 6(a) shows thatwhen time 119905 increases from 0 to 045min the behavior of thesolution 120585 remains stationary and quickly growing during thetime interval 119905 isin ]045 105[min and then oscillates till theend of time interval This indicates that the motion is stableas seen from Figure 7(a) On the other side the behaviorof the derivative remains approximately stationary duringthe interval 119905 isin [0 045]min and then fluctuates with theincreasing of time see Figure 6(a)

By the same way we can observe that the wave of theangle 120593 increases through a short time to reach its maximumvalue 120593 ≃ 27 rad ≃ 155∘ at 119905 ≃ 023min taking intoconsideration that minus1205872 lt 120593 lt 1205872 and then decreasesslowly to reach its minimum value 120593 ≃ minus09 rad ≃ minus52∘at the end of time interval see Figure 6(b) This indicatesthat the motion is close to be stable as observed fromthe phase plane Figure 8(b) As seen from Figure 6(b) increases and decreases quickly during the period 119905 isin[0 01]min to reach its minimum value at the end of timeinterval

The variation of 120595 and with time is illustrated inFigure 6(c) In this figure our main goal is to examine theinfluence of time on the motion of pendulum It is clear thatthe behavior of 120595 and remains stationary (to some extent)when 119905 isin [0 05]min then their waves fluctuate till the endof time interval Consequently the motion is stable as seenfrom the phase plane diagram Figure 7(c) The comparisonbetween parts of Figure 6 with the corresponding Figures

10 Advances in Mathematical Physics

010000

2000002468

0

05

1

15

t

120585120585 minus2

b = 0

times104

(a)

t

0 1 2 30

2040

0

05

1

15

120593minus1

b = 0

(b)

t

0 1005000

05

1

15

minus10minus500 120595120595

b = 0

(c)

Figure 8 The 3D plots at 119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad (a) illustrates the variation of 120585 and via 119905 (b) illustrates the variation

of 120593 and via 119905 and (c) illustrates the variation of 120595 and via 119905

0 05 1 15

0

5000

10000

15000

t

minus5000

120585120585

120585and 120585

h = 25

(a)

0 05 1 15

0

5

10

t

minus5

minus10

minus15

120593

120593and120593

h = 25

(b)

0 05 1 15

0

20

t

120595and

120595120595120595

minus20

minus40

minus60

h = 25

(c)

Figure 9 (a) (b) and (c) demonstrate the variation of (120585 and ) (120593 and ) and (120595 and ) against time 119905 respectively at 119887 = 3m ℎ = 25119886 = 05m and 1205930= 04 rad

2(d) 2(e) and 2(f) shows that when 119887 changes from 0 to 3mthe amplitude of the waves decreases Also the motion willbe more stable when 119887 = 3m than when 119887 = 0 as seen fromthe corresponding phase plane diagrams that is Figures 3(d)3(e) 3(f) and 7(a) 7(b) 7(c) respectively

On the other hand parts of Figure 8 show 3D plots thatdescribe the variation of the solutions and their derivative viatime when 119887 = 0 ℎ = 45 120593

0= 04 rad and 119886 = 05m

The plots displayed in thementioned parts show bending andcrossing of the resulting curves

Figures 9(a) 9(b) and 9(c) show the variation of thesolutions 120585 120593 120595 and their derivatives with time 119905whenℎ = 25 for the given values of other parameters 119887 = 3m1205930= 04 rad and 119886 = 05m In view of the first part we can

conclude that when time 119905 increases each of the waves 120585 and oscillates between increasing and decreasing till 119905 = 146minand then increases gradually So the motion is stable as seenfrom Figure 10(a)

From a closer look on the second part of Figure 9(b) wecan write with the increasing of time the behavior of 120593 wave

increases to reach its maximum value 120593 ≃ 09 rad ≃ 52∘ at119905 = 043min and then decreases slowly through the period 119905 isin]043 119]min After that its behavior has a sharp declinein a few seconds (about 24 s) and then increases till the endof time period and consequently the motion is stable seeFigure 10(b)

According to the calculations depicted in Figure 9(c) wecan observe that thewaves describing120595 and decrease slowlytill 119905 = 09min and then increase and decline sharp Thephase plane Figure 10(c) shows that the behavior of 120595 is notstable

When parts of Figure 9 and their phase plane parts (ofFigure 10) are generally compared with the correspondingFigures 2(d) 2(e) and 2(f) and their phase plane Figures 3(d)3(e) and 3(f) we can observe that amplitude of the waveincreases when ℎ = 45 compared to when ℎ = 25 and themotion is more stable when ℎ = 45 An inspection of partsof Figure 11 reveals the 3D plots when ℎ = 25 with the sameother data considered in Figures 9 and 10 Figure 10 shows thevariation of the solutions 120585 120593 120595 and their derivatives

Advances in Mathematical Physics 11

0 500 1000 1500 2000

0

5000

10000

15000 h = 25

minus5000

120585

120585

(a)

0 02 04 06 08 1

0

5

10h = 25

minus5

minus10

minus15

minus02

120593

(b)

0 2

0

20h = 25

minus20

minus40

minus60

minus2minus4minus6minus8

120595

120595

(c)

Figure 10 The phase plane diagrams which portray the relation between amplitudes and their velocities at 119887 = 3m ℎ = 25 119886 = 05m and1205930= 04 rad (a) describes the influence of 120585 on (b) shows the effect of 120593 on and (c) illustrates the variation of 120595 with

01000

200005000

1000015000

0051

15

t

minus5000120585

120585

h = 25

(a)

t

005

10100

05

1

15

minus10120593

h = 25

(b)

t

0 20200

05

1

15

minus20minus40

minus60minus2minus4minus6minus8120595

120595

h = 25

(c)

Figure 11 The 3D patterns at 119887 = 3m ℎ = 25 119886 = 05m and 1205930= 04 rad (a) illustrates the variation of 120585 and versus 119905 (b) illustrates the

variation of 120593 and versus 119905 (c) illustrates the variation of 120595 and versus 119905

with time 119905 It is worthwhile to notice that the comparisonbetween Figures 4(d) 4(e) and 4(f) and Figures 11(a) 11(b)and 11(c) shows more bending and crossing of the curvesin Figures 4(d) 4(e) and 4(f) when ℎ = 45 than thecorresponding ones of Figure 11

Now we study the last case when ℎ = 0 with the sameother data 119887 = 3m 1205930 = 04 rad and 119886 = 05mThe obtainedresults are represented graphically in Figures 12(a) 12(b) and12(c) while their phase plane diagrams are given in Figures12(d) 12(e) and 12(f) At the first glance we can conclude thatthis case is not stable so it is very important to notice that thedimensionless parameter ℎmust take any value different fromzero as it is pointed in Figure 2 (ℎ = 45) and Figure 9 (ℎ =25) This elucidates the importance of ℎ parameter on themotion

4 Conclusion

A conclusion that may be made here is that the problemof the relative motion of a rigid body as a pendulum

model is investigated The governing deferential equationsare obtained using Lagrangersquos equations Mathematica pack-age was utilized in order to overcome the difficulties thatappear in the separation of the second derivatives of thegeneralized coordinates 120585 120593 and 120595 for the nonlinear system(10) Computer codes are used to obtain the numericalsolutions for system (14) These solutions are representedgraphically using Matlab program to study the influenceof the different parameters on the motion The good effectof the parameters ℎ 119887 and 120593

0on the motion is obvious

from the mentioned plots The motion of our model is morestable when the parameters ℎ 119887 and 120593

0take values run

away from zero This highlights the importance of the effectof these parameters on the motion Such results have beenconfirmed by many works such as Ismail [13] and Amer andBek [14]

Competing Interests

The author declares that they have no competing interests

12 Advances in Mathematical Physics

0 05 1 15

0

05

1

15

2

25

3

t

120585120585

120585and 120585

h = 0

times104

(a)

0 05 1 15

0

05

1

15

2

25

t

120593120593and120593

h = 0

(b)

0 05 1 15

0

5

10

15

t

120595and

120595

120595120595

h = 0

(c)

0 1000 2000 3000 4000 5000

0

05

1

15

2

25

3times104

120585

120585

h = 0

(d)

02 04 06 08 1

0

05

1

15

2

25

120593

h = 0

(e)

0 05 1 15 2

0

5

10

15

120595

120595

h = 0

(f)

Figure 12 (a b and c) explain the variation of the solutions 120585 120593 and120595with their derivatives and via time 119905 respectively when 119887 = 3mℎ = 0 119886 = 05m and 1205930= 04 rad (d e and f) illustrate the variation of the solutions against their first derivatives for the same values of the

considered parameters

References

[1] P Lynch ldquoResonant motions of the three-dimensional elasticpendulumrdquo International Journal of Non-Linear Mechanics vol37 no 2 pp 345ndash367 2002

[2] A A Klimenko Y V Mikhlin and J Awrejcewicz ldquoNonlinearnormal modes in pendulum systemsrdquoNonlinear Dynamics vol70 no 1 pp 797ndash813 2012

[3] S Mori H Nishihara and K Furuta ldquoControl of unstablemechanical system control of pendulumrdquo International Journalof Control vol 23 no 5 pp 673ndash692 1976

[4] C C Chung and J Hauser ldquoNonlinear control of a swingingpendulumrdquo Automatica A Journal of IFAC vol 31 no 6 pp851ndash862 1995

[5] A Shiriaev A Pogromsky H Ludvigsen and O Egeland ldquoOnglobal properties of passivity-based control of an inverted pen-dulumrdquo International Journal of Robust and Nonlinear Controlvol 10 no 4 pp 283ndash300 2000

[6] A S Shiriaev H Ludvigsen and O Egeland ldquoSwinging upthe spherical pendulum via stabilization of its first integralsrdquoAutomatica A Journal of IFAC the International Federation ofAutomatic Control vol 40 no 1 pp 73ndash85 2004

[7] M N Brearley ldquoThe Simple Pendulum with Uniformly Chang-ing String Lengthrdquo Proceedings of the Edinburgh MathematicalSociety vol 15 no 1 pp 61ndash66 1966

[8] S J Liao ldquoSecond-order approximate analytical solution of asimple pendulum by the process analysis methodrdquo Journal ofApplied Mechanics Transactions ASME vol 59 no 4 pp 970ndash975 1992

[9] W K Tso and K G Asmis ldquoParametric excitation of a pen-dulum with bilinear hysteresisrdquo Journal of Applied MechanicsTransactions ASME vol 37 no 4 pp 1061ndash1068 1970

[10] A H Nayfeh Perturbations Methods Wiley-VCH WeinheimGermany 2004

[11] F A El-Barki A I Ismail M O Shaker and T S AmerldquoOn the motion of the pendulum on an ellipserdquo Zeitschrift furAngewandteMathematik undMechanik vol 79 no 1 pp 65ndash721999

[12] N V Stoianov ldquoOn the relative periodic motions of a pendu-lumrdquo Journal of AppliedMathematics andMechanics vol 28 pp188ndash193 1964

[13] A I Ismail ldquoRelative periodicmotion of a rigid body pendulumon an ellipserdquo Journal of Aerospace Engineering vol 22 no 1 pp67ndash77 2009

[14] T S Amer andM A Bek ldquoChaotic responses of a harmonicallyexcited spring pendulum moving in circular pathrdquo NonlinearAnalysis Real World Applications An International Multidisci-plinary Journal vol 10 pp 3196ndash3202 2009

Advances in Mathematical Physics 13

[15] L D Akulenko ldquoParametric control of oscillations and rota-tions of a compound pendulum (a swing)rdquo Journal of AppliedMathematics and Mechanics vol 57 no 2 pp 301ndash310 1993

[16] M A Pinsky and A A Zevin ldquoOscillations of a pendulumwith a periodically varying length and a model of swingrdquoInternational Journal of Non-LinearMechanics vol 34 no 1 pp105ndash109 1999

[17] M Kamel M Eissa and A T El-Sayed ldquoVibration reductionof a nonlinear spring pendulum under multiparametric excita-tions via a longitudinal absorberrdquo Physica Scripta vol 80 no 2Article ID 025005 2009

[18] M Eissa M Kamel and A T El-Sayed ldquoVibration reduction ofmulti-parametric excited spring pendulum via a transversallytuned absorberrdquo Nonlinear Dynamics vol 61 no 1-2 pp 109ndash121 2010

[19] R Starosta G Sypniewska-Kaminska and J AwrejcewiczldquoAsymptotic analysis of kinematically excited dynamical sys-tems near resonancesrdquo Nonlinear Dynamics An InternationalJournal of Nonlinear Dynamics and Chaos in Engineering Sys-tems vol 68 no 4 pp 459ndash469 2012

[20] H MooreMatlab for Engineers Pearson 3rd edition 2012[21] M D Ardema Analytical Dynamics Theory and Applications

Springer Berlin Germany 2009[22] A Tewari Modern Control Design with Matlab and Similink

John Wiley and Sons Ltd New York NY USA 2002

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 5

A choice of the state variables for this system is

1198831 = 120585

1198832= 120593

1198833= 120595

1198834=

1198835 =

1198836=

(16)

which results in the following state-equations

1= 1198834

2 = 11988353= 1198836

(17)

Use (16) and (17) into system (14) to get

4= minus 1

119867 minus1198861198833minus (119887 + 119883

1)

sdot (119888111198831 + 119888121198832 + 119888131198833 minus 1198913)+ 119887 (119887

111198831 + 119887121198832 + 119887131198833 + 119892 sin1205930 + 1205962sdot minus11988311198832cos2 120593

0minus 2ℎ sin120596119905 cos120593

0

+ [minus119887 + (119887 + 1198831)11988321198833+ 119886 (1205952 minus 1)] cos120593

0sin1205930

+11988311198832 sin2 1205930+212 minus (119887 + 1198831) (1198832 minus 1198833) 22)+ [119886119887 minus (119887 + 120585) ℓ1] [1198621 minus 119886111198831 minus 119886121198832minus 119886131198833 + (119887 + 1198831) 22+ (119860119883

1+ 1198611198833+ 1198863

2)1198832]

5= minus 1

1198871198913 minus 119888111198831 minus 119888121198832 minus 119888131198833+ ℓ11198861198833

[1198621minus 119886111198831minus 119886121198832minus 119886131198833+ (119887 + 119883

1) 2

2

+ (1198601198831+ 1198611198833+ 1198863

2)1198832] + 1

1198861198671198833sdot ℓ1minus119886119883

3(119887 + 119883

1) (1198913minus 119888111198831minus 119888121198832minus 119888131198833)

+ 119887 (119887111198831+ 119887121198832+ 119887131198833+ 119892 sin120593

0

+ 1205962 minus11988311198832cos2 120593

0minus 2ℎ sin120596119905 cos120593

0

+ [(119887 + 1198831)11988321198833 + 119886 (11988323 minus 1) minus 119887] cos1205930 sin1205930

+11988311198832 sin2 1205930+212 minus (119887 + 1198831) (1198832 minus 1198833) 22)+ [119886119887 minus (119887 + 119883

1) ℓ1]

sdot 1198621minus 11988611120585 minus 119886121198832minus 119886131198833

+ (119887 + 1198831) 2

2 + (1198601198831+ 1198611198833+ 1198863

2)1198832

6= minus 1

119867 119887 (1198832minus 1198833)

sdot [119886111198831 + 119886121198832 + 119886131198833 minus (1198601198831 + 1198611198833)1198832]minus 119887 (119887111198831+ 119887121198832+ 119887131198833)

+ (119887 + 1198831) (119888111198831+ 119888121198832+ 119888131198833)

minus 1198871198621 (1198832 minus 1198833)+ 1198871205962 cos120593

0(11988311198832cos1205930+ 2ℎ sin120596119905)

minus (119887 + 1198831) 1198913 minus 119887119892 sin1205930+ 1198871205962 119886 + 119887 minus 119883

3[(119887 + 119883

1)1198832+ 1198861198833]

sdot cos1205930sin1205930

minus 119887 [212+ 1198832

sdot (12059621198831sin2 120593

0+ 119886 (119883

2minus 1198833) 3

2)] (18)

The following initial conditions are required to achieve thenumerical solution of (18) by using the fourth-order Runge-Kutta method of ode45 solver in framework of Matlabprogram

1198831 (0) = 00011198832 (0) = 01

1198833 (0) = 001

4 (0) = 0

5 (0) = 0

6 (0) = 0

(19)

in addition to the following physical parameters of theconsidered model

119898 = 50 kg119892 = 98m sdot sminus21198691= 8 kg sdotm2

ℓ = 07m120596 = 4 rad sdot sminus1119886 = 05m

6 Advances in Mathematical Physics

0 05 1 15

0

05

1

15

t

minus05

minus1

120585and 120585

times104

120585120585

1205930 = 0

(a)

0 05 1 15

0

5

t

1205930 = 0

minus5

minus10

minus15

minus20

minus25

120593

120593and120593

(b)

0 05 1 15

0

20

t

1205930 = 0

minus20

minus40

120595and

120595

120595120595

(c)

0

2

4

6

8

120585and 120585

times104

0 05 1 15t

120585120585

minus2

minus4

1205930 = 04

(d)

0

10

20

30

120593and120593

0 05 1 15t

120593

minus10

minus20

1205930 = 04

(e)

0

50

100

120595and

120595

0 05 1 15t

120595

120595

1205930 = 04

minus50

minus100

(f)

Figure 2 Variation of the solutions and their derivatives versus time 119905 when 119887 = 3m ℎ = 45 and 119886 = 05m (a d) show the effect of 119905 on thebehavior of 120585 and waves when 120593

0= 0 and 120593

0= 04 rad respectively (b e) show the effect of 119905 on the behavior of 120593 and waves when 120593

0= 0

and 1205930= 04 rad respectively and (c f) show the effect of 119905 on the wave that describes the behavior of 120595 and when 120593

0= 0 and 120593

0= 04 rad

respectively

119887 = (0 3)m1205930= (0 04) rad

ℎ = (25 45) 119905 = 0 997888rarr 17min

(20)

Figure 2 shows the variation of the solutions 120585 120593 120595 and theirderivatives against time 119905 when 120593

0= 0 and 120593

0=04 rad This figure is drawn at 119887 = 3m ℎ = 24 and

119886 = 05m The variations of 120585 120593 and 120595 with and respectively are illustrated in Figure 3 namely the phaseplane diagrams that are represented in Figures 3(a) 3(b)3(c) and 3(d) 3(e) 3(f) when 1205930 = 0 and 1205930 = 04 radrespectively with the same other parameters that are takeninto consideration in Figure 2

In these figures our principle aim is to investigate theeffect of increasing time on the motion of pendulum

According to the calculations depicted in Figure 2(a) wefound that when 1205930 = 0 the wave of the elongation 120585 growsup with the increasing of time till 119905 = 09min After thatboth of the elongation 120585 and its derivative fluctuate between

increasing and decreasing when time reaches 119905 = 143minThus the wave of the solution 120585 is stable see the phase planeFigure 3(a) With the passing of time one can observe that120585 and are growing quickly so the motion will be unstableafter 119905 = 143min The rage behavior of both 120585 and is dueto the weight of the rigid body and the values of the principalmoments of inertia Consequently we expect that behaviorof elongation becomes greater as observed in Figures 2(a) and2(d)Moreover the variation of the spring between stretchingand contraction is consistent with the phase plane diagramsrepresented in Figures 3(a) and 3(d)

It is worthwhile to notice fromFigure 2(b) that when time119905 increases from 119905 = 0 to 119905 = 04min the behavior of theangle 120593 increases gradually to reach the value 120593 ≃ 1 rad ≃ 57∘and then decreases slowly to reach 120593 ≃ 08 rad ≃ 46∘ duringthe time period 119905 isin ]04 09[min After 119905 = 09min thedecline of the wave becomes quickly to reach 120593 ≃ minus23 rad ≃minus132∘ at the end of time period (minus sign indicates oppositedirection) This is not possible because 120593 must belong to theinterval ] minus 1205872 1205872] So the motion of the wave is unstableas it is manifest from Figure 3(b) On the other hand increases till 119905 ≃ 02min and then fluctuates as indicated fromFigure 2(b)

Advances in Mathematical Physics 7

0 500 1000 1500 2000 2500

0

05

1

15

times104

120585

120585

1205930 = 0

minus05

minus1

(a)

0 1

0

5 1205930 = 0

minus5

minus10

minus15

minus20

minus25minus2 minus1

120593

(b)

0 2

0

20

minus20

minus40

minus2minus4

120595

120595

1205930 = 0

(c)

0 2000 4000 6000 8000 10000

0

2

4

6

8times104

120585

120585

minus2

minus4

1205930 = 04

(d)

0 02 04 06 08 1

0

10

20

30

120593

1205930 = 04

minus10

minus20

minus02

(e)

0 2 4

0

50

100

minus2minus4

120595

120595

1205930 = 04

minus50

minus100

(f)

Figure 3 The phase plane diagram when 119887 = 3m ℎ = 45 and 119886 = 05m (a d) represent the variation of the amplitude 120585 with its velocity at 120593

0= 0 and 120593

0= 04 rad respectively (b e) represent the variation of the amplitude 120593 with its velocity at 120593

0= 0 and 120593

0= 04 rad

respectively and (c f) represent the variation of the amplitude 120595 with its velocity at 1205930= 0 and 120593

0= 04 rad respectively

The graphs displayed in Figures 2(c) and 3(c) describethe variation of the (120595 and ) against time and the phaseplane diagram ( with 120595) respectively when 1205930 = 0 Itis clear that when time belongs to the period [0 043]minthe angle 120595 remains stationary and then its wave oscillatesbetween decreasing and increasing till 119905 = 143min Afterthat time the angle 120595 increases up to the end of time intervaland consequently the motion will be stable as seen fromFigure 3(c) during the period 0 lt 119905 le 143 It is obviousfrom Figure 2(c) that the behavior of remains stationary tosome extent through the time interval [0 06]min and thenoscillates between increasing and decreasing till 119905 = 17min

It should be noticed that when1205930= 04 rad the stretching

on the string 120585 increases gradually till the time 119905 becomes09min and then 120585 and oscillate between increasing anddecreasingwhen the time reaches the end of time interval seeFigure 2(d) Consequently the wave of the solution is stableas seen from the phase plane Figure 3(d)

An inspection of the graphs depicted in Figure 2(e) showsthat the wave describing the behavior of the angle 120593 increasesgradually from 120593 = 0 at 119905 = 0 to its maximum value 120593 ≃09 rad ≃ 56∘ at 119905 = 04min and then decreases slowly at119905 ≃ 1min to reach its minimum value 120593 ≃ minus019 rad ≃ minus11∘(minus sign indicates opposite direction) at 119905 ≃ 126minWith the increasing of time thewave grows again to reach thevalue 120593 ≃ 085 rad ≃ 49∘ at 119905 ≃ 139min Thus the motion is

stable as it is manifest from Figure 3(e) On the other hand increases and decreases as indicated from Figure 2(e)

Also it is remarkable from Figure 2(f) that the behaviorof the angle 120595 remains steady till 119905 = 05min then its waveoscillates between decreasing and increasing Consequentlythe motion will be stable as seen from Figure 3(f) It isworthwhile to notice also from Figure 2(f) that the behaviorof oscillates between increasing and decreasing

From the above observations we can conclude that themotion of our model is more stable when 120593

0= 04 rad than

when 1205930= 0 This highlights the importance of the effect of120593

0value on the motion It is worthwhile to notice that the

comparison between the solutions 120585 120593 and 120595 included inFigures 2(a) 2(b) and 2(c) with the corresponding Figures2(d) 2(e) and 2(f) reveals that the amplitude of the wavesdecreases when 120593

0increases from 0 to 04 rad On the other

hand the comparison between their derivatives shows thatthe amplitude of the waves increases when 120593

0increases

Figure 4 shows the variation of (120585 ) (120593 ) and (120595 ) withtime 119905 when 120593

0 changes from 0 for Figures 4(a) 4(b) and4(c) to 04 rad for Figures 4(d) 4(e) and 4(f) at the samevalues of other parameters 119887 = 3m ℎ = 45 and 119886 = 05mAccording to the calculations depicted in these figures we canconsider these figures as a rotation of the corresponding partsof Figure 3 with time to observe the bending and crossing ofthe resulting curves

8 Advances in Mathematical Physics

01000 20000

1

0051

15

t

minus1

times104

1205930 = 0

120585120585

(a)

010

0051

15

t

minus10minus20 minus1

minus2

1205930 = 0

120593

(b)

0 2020

0051

15

t

1205930 = 0

minus20minus40 minus4

minus2

120595120595

(c)

05000

1000002468

0051

15

t

times104

minus4minus2

1205930 = 04

120585120585

(d)

005

10

20

0051

15

t

minus20

1205930 = 04

120593

(e)

0 2 401000

051

15

t

minus4 minus2

1205930 = 04

minus100120595

120595

(f)

Figure 4The 3D pattern when 119887 = 3m ℎ = 45 and 119886 = 05m (a d) indicate the variation of 120585 and versus 119905 when 1205930= 0 and 120593

0= 04 rad

respectively (b e) indicate the variation of 120593 and versus 119905 when 1205930= 0 and 120593

0= 04 rad respectively and (c f) indicate the variation of 120595

and versus 119905 when 1205930= 0 and 120593

0= 04 rad respectively

01000

200001

02

minus1minus2

minus2

minus4

120585120593

120595

1205930 = 0

(a)

01

0

020

1205930 = 0

minus1

minus10minus20

minus20

minus40

times104

120585

120595

(b)

05000

100000

051

024

minus2

minus4

120595

1205930 = 04

120585

120593

(c)

0 2 4 6 8020

0

100

minus2minus4times104

120595

1205930 = 04

minus20

minus100

120585

(d)

Figure 5 The 3D diagrams when 119887 = 3m ℎ = 45 and 119886 = 05m (a c) elucidate the variation of 120585 and 120593 versus 120595 when 1205930= 0 and

1205930= 04 rad respectively and (b d) elucidate the variation of and versus when 120593

0= 0 and 120593

0= 04 rad respectively

Advances in Mathematical Physics 9

0 05 1 15

0

2

4

6

8

t

120585and 120585

times104

minus2

b = 0

120585120585

(a)

t

0 05 1 15

0

10

20

30

40

50b = 0

minus10

120593120593and120593

(b)

0 05 1 15

0

200

400

t

b = 0

minus200

minus400

minus600

120595and

120595

120595120595

(c)

Figure 6 (a) (b) and (c) explain the variation of the solutions 120585 120593 and 120595 with their derivatives and via time 119905 respectively when119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad

0 5000 10000 15000 20000

0

2

4

6

8

minus2

times104

b = 0

120585

120585

(a)

0 1 2 3

0

10

20

30

40

50b = 0

minus10

minus1

120593

(b)

0 5 10 15

0

200

400b = 0

120595

120595

minus200

minus400

minus600minus5minus10minus15

(c)

Figure 7 The phase plane diagrams between amplitudes and their velocities at 119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad (a) shows the

influence of 120585 on (b) shows the effect of 120593 on and (c) shows the variation of 120595 with

Figures 5(a) 5(c) and 5(b) 5(d) represent 3D plots thatillustrate the variation of the solutions 120585 120593 via 120595 and via respectively for different values of 120593

0when 119887 =3m ℎ = 45 and 119886 = 05m The graphs displayed in

parts of Figure 6 show the variation of (120585 ) (120593 ) and(120595 ) against time 119905 when 119887 = 0 with consideration of theparameters 1205930 = 04 rad ℎ = 45 and 119886 = 05m Thecorresponding phase plane between the amplitudes 120585 120593 120595and their derivatives is represented in parts of Figure 7Inspection of the graph depicted in Figure 6(a) shows thatwhen time 119905 increases from 0 to 045min the behavior of thesolution 120585 remains stationary and quickly growing during thetime interval 119905 isin ]045 105[min and then oscillates till theend of time interval This indicates that the motion is stableas seen from Figure 7(a) On the other side the behaviorof the derivative remains approximately stationary duringthe interval 119905 isin [0 045]min and then fluctuates with theincreasing of time see Figure 6(a)

By the same way we can observe that the wave of theangle 120593 increases through a short time to reach its maximumvalue 120593 ≃ 27 rad ≃ 155∘ at 119905 ≃ 023min taking intoconsideration that minus1205872 lt 120593 lt 1205872 and then decreasesslowly to reach its minimum value 120593 ≃ minus09 rad ≃ minus52∘at the end of time interval see Figure 6(b) This indicatesthat the motion is close to be stable as observed fromthe phase plane Figure 8(b) As seen from Figure 6(b) increases and decreases quickly during the period 119905 isin[0 01]min to reach its minimum value at the end of timeinterval

The variation of 120595 and with time is illustrated inFigure 6(c) In this figure our main goal is to examine theinfluence of time on the motion of pendulum It is clear thatthe behavior of 120595 and remains stationary (to some extent)when 119905 isin [0 05]min then their waves fluctuate till the endof time interval Consequently the motion is stable as seenfrom the phase plane diagram Figure 7(c) The comparisonbetween parts of Figure 6 with the corresponding Figures

10 Advances in Mathematical Physics

010000

2000002468

0

05

1

15

t

120585120585 minus2

b = 0

times104

(a)

t

0 1 2 30

2040

0

05

1

15

120593minus1

b = 0

(b)

t

0 1005000

05

1

15

minus10minus500 120595120595

b = 0

(c)

Figure 8 The 3D plots at 119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad (a) illustrates the variation of 120585 and via 119905 (b) illustrates the variation

of 120593 and via 119905 and (c) illustrates the variation of 120595 and via 119905

0 05 1 15

0

5000

10000

15000

t

minus5000

120585120585

120585and 120585

h = 25

(a)

0 05 1 15

0

5

10

t

minus5

minus10

minus15

120593

120593and120593

h = 25

(b)

0 05 1 15

0

20

t

120595and

120595120595120595

minus20

minus40

minus60

h = 25

(c)

Figure 9 (a) (b) and (c) demonstrate the variation of (120585 and ) (120593 and ) and (120595 and ) against time 119905 respectively at 119887 = 3m ℎ = 25119886 = 05m and 1205930= 04 rad

2(d) 2(e) and 2(f) shows that when 119887 changes from 0 to 3mthe amplitude of the waves decreases Also the motion willbe more stable when 119887 = 3m than when 119887 = 0 as seen fromthe corresponding phase plane diagrams that is Figures 3(d)3(e) 3(f) and 7(a) 7(b) 7(c) respectively

On the other hand parts of Figure 8 show 3D plots thatdescribe the variation of the solutions and their derivative viatime when 119887 = 0 ℎ = 45 120593

0= 04 rad and 119886 = 05m

The plots displayed in thementioned parts show bending andcrossing of the resulting curves

Figures 9(a) 9(b) and 9(c) show the variation of thesolutions 120585 120593 120595 and their derivatives with time 119905whenℎ = 25 for the given values of other parameters 119887 = 3m1205930= 04 rad and 119886 = 05m In view of the first part we can

conclude that when time 119905 increases each of the waves 120585 and oscillates between increasing and decreasing till 119905 = 146minand then increases gradually So the motion is stable as seenfrom Figure 10(a)

From a closer look on the second part of Figure 9(b) wecan write with the increasing of time the behavior of 120593 wave

increases to reach its maximum value 120593 ≃ 09 rad ≃ 52∘ at119905 = 043min and then decreases slowly through the period 119905 isin]043 119]min After that its behavior has a sharp declinein a few seconds (about 24 s) and then increases till the endof time period and consequently the motion is stable seeFigure 10(b)

According to the calculations depicted in Figure 9(c) wecan observe that thewaves describing120595 and decrease slowlytill 119905 = 09min and then increase and decline sharp Thephase plane Figure 10(c) shows that the behavior of 120595 is notstable

When parts of Figure 9 and their phase plane parts (ofFigure 10) are generally compared with the correspondingFigures 2(d) 2(e) and 2(f) and their phase plane Figures 3(d)3(e) and 3(f) we can observe that amplitude of the waveincreases when ℎ = 45 compared to when ℎ = 25 and themotion is more stable when ℎ = 45 An inspection of partsof Figure 11 reveals the 3D plots when ℎ = 25 with the sameother data considered in Figures 9 and 10 Figure 10 shows thevariation of the solutions 120585 120593 120595 and their derivatives

Advances in Mathematical Physics 11

0 500 1000 1500 2000

0

5000

10000

15000 h = 25

minus5000

120585

120585

(a)

0 02 04 06 08 1

0

5

10h = 25

minus5

minus10

minus15

minus02

120593

(b)

0 2

0

20h = 25

minus20

minus40

minus60

minus2minus4minus6minus8

120595

120595

(c)

Figure 10 The phase plane diagrams which portray the relation between amplitudes and their velocities at 119887 = 3m ℎ = 25 119886 = 05m and1205930= 04 rad (a) describes the influence of 120585 on (b) shows the effect of 120593 on and (c) illustrates the variation of 120595 with

01000

200005000

1000015000

0051

15

t

minus5000120585

120585

h = 25

(a)

t

005

10100

05

1

15

minus10120593

h = 25

(b)

t

0 20200

05

1

15

minus20minus40

minus60minus2minus4minus6minus8120595

120595

h = 25

(c)

Figure 11 The 3D patterns at 119887 = 3m ℎ = 25 119886 = 05m and 1205930= 04 rad (a) illustrates the variation of 120585 and versus 119905 (b) illustrates the

variation of 120593 and versus 119905 (c) illustrates the variation of 120595 and versus 119905

with time 119905 It is worthwhile to notice that the comparisonbetween Figures 4(d) 4(e) and 4(f) and Figures 11(a) 11(b)and 11(c) shows more bending and crossing of the curvesin Figures 4(d) 4(e) and 4(f) when ℎ = 45 than thecorresponding ones of Figure 11

Now we study the last case when ℎ = 0 with the sameother data 119887 = 3m 1205930 = 04 rad and 119886 = 05mThe obtainedresults are represented graphically in Figures 12(a) 12(b) and12(c) while their phase plane diagrams are given in Figures12(d) 12(e) and 12(f) At the first glance we can conclude thatthis case is not stable so it is very important to notice that thedimensionless parameter ℎmust take any value different fromzero as it is pointed in Figure 2 (ℎ = 45) and Figure 9 (ℎ =25) This elucidates the importance of ℎ parameter on themotion

4 Conclusion

A conclusion that may be made here is that the problemof the relative motion of a rigid body as a pendulum

model is investigated The governing deferential equationsare obtained using Lagrangersquos equations Mathematica pack-age was utilized in order to overcome the difficulties thatappear in the separation of the second derivatives of thegeneralized coordinates 120585 120593 and 120595 for the nonlinear system(10) Computer codes are used to obtain the numericalsolutions for system (14) These solutions are representedgraphically using Matlab program to study the influenceof the different parameters on the motion The good effectof the parameters ℎ 119887 and 120593

0on the motion is obvious

from the mentioned plots The motion of our model is morestable when the parameters ℎ 119887 and 120593

0take values run

away from zero This highlights the importance of the effectof these parameters on the motion Such results have beenconfirmed by many works such as Ismail [13] and Amer andBek [14]

Competing Interests

The author declares that they have no competing interests

12 Advances in Mathematical Physics

0 05 1 15

0

05

1

15

2

25

3

t

120585120585

120585and 120585

h = 0

times104

(a)

0 05 1 15

0

05

1

15

2

25

t

120593120593and120593

h = 0

(b)

0 05 1 15

0

5

10

15

t

120595and

120595

120595120595

h = 0

(c)

0 1000 2000 3000 4000 5000

0

05

1

15

2

25

3times104

120585

120585

h = 0

(d)

02 04 06 08 1

0

05

1

15

2

25

120593

h = 0

(e)

0 05 1 15 2

0

5

10

15

120595

120595

h = 0

(f)

Figure 12 (a b and c) explain the variation of the solutions 120585 120593 and120595with their derivatives and via time 119905 respectively when 119887 = 3mℎ = 0 119886 = 05m and 1205930= 04 rad (d e and f) illustrate the variation of the solutions against their first derivatives for the same values of the

considered parameters

References

[1] P Lynch ldquoResonant motions of the three-dimensional elasticpendulumrdquo International Journal of Non-Linear Mechanics vol37 no 2 pp 345ndash367 2002

[2] A A Klimenko Y V Mikhlin and J Awrejcewicz ldquoNonlinearnormal modes in pendulum systemsrdquoNonlinear Dynamics vol70 no 1 pp 797ndash813 2012

[3] S Mori H Nishihara and K Furuta ldquoControl of unstablemechanical system control of pendulumrdquo International Journalof Control vol 23 no 5 pp 673ndash692 1976

[4] C C Chung and J Hauser ldquoNonlinear control of a swingingpendulumrdquo Automatica A Journal of IFAC vol 31 no 6 pp851ndash862 1995

[5] A Shiriaev A Pogromsky H Ludvigsen and O Egeland ldquoOnglobal properties of passivity-based control of an inverted pen-dulumrdquo International Journal of Robust and Nonlinear Controlvol 10 no 4 pp 283ndash300 2000

[6] A S Shiriaev H Ludvigsen and O Egeland ldquoSwinging upthe spherical pendulum via stabilization of its first integralsrdquoAutomatica A Journal of IFAC the International Federation ofAutomatic Control vol 40 no 1 pp 73ndash85 2004

[7] M N Brearley ldquoThe Simple Pendulum with Uniformly Chang-ing String Lengthrdquo Proceedings of the Edinburgh MathematicalSociety vol 15 no 1 pp 61ndash66 1966

[8] S J Liao ldquoSecond-order approximate analytical solution of asimple pendulum by the process analysis methodrdquo Journal ofApplied Mechanics Transactions ASME vol 59 no 4 pp 970ndash975 1992

[9] W K Tso and K G Asmis ldquoParametric excitation of a pen-dulum with bilinear hysteresisrdquo Journal of Applied MechanicsTransactions ASME vol 37 no 4 pp 1061ndash1068 1970

[10] A H Nayfeh Perturbations Methods Wiley-VCH WeinheimGermany 2004

[11] F A El-Barki A I Ismail M O Shaker and T S AmerldquoOn the motion of the pendulum on an ellipserdquo Zeitschrift furAngewandteMathematik undMechanik vol 79 no 1 pp 65ndash721999

[12] N V Stoianov ldquoOn the relative periodic motions of a pendu-lumrdquo Journal of AppliedMathematics andMechanics vol 28 pp188ndash193 1964

[13] A I Ismail ldquoRelative periodicmotion of a rigid body pendulumon an ellipserdquo Journal of Aerospace Engineering vol 22 no 1 pp67ndash77 2009

[14] T S Amer andM A Bek ldquoChaotic responses of a harmonicallyexcited spring pendulum moving in circular pathrdquo NonlinearAnalysis Real World Applications An International Multidisci-plinary Journal vol 10 pp 3196ndash3202 2009

Advances in Mathematical Physics 13

[15] L D Akulenko ldquoParametric control of oscillations and rota-tions of a compound pendulum (a swing)rdquo Journal of AppliedMathematics and Mechanics vol 57 no 2 pp 301ndash310 1993

[16] M A Pinsky and A A Zevin ldquoOscillations of a pendulumwith a periodically varying length and a model of swingrdquoInternational Journal of Non-LinearMechanics vol 34 no 1 pp105ndash109 1999

[17] M Kamel M Eissa and A T El-Sayed ldquoVibration reductionof a nonlinear spring pendulum under multiparametric excita-tions via a longitudinal absorberrdquo Physica Scripta vol 80 no 2Article ID 025005 2009

[18] M Eissa M Kamel and A T El-Sayed ldquoVibration reduction ofmulti-parametric excited spring pendulum via a transversallytuned absorberrdquo Nonlinear Dynamics vol 61 no 1-2 pp 109ndash121 2010

[19] R Starosta G Sypniewska-Kaminska and J AwrejcewiczldquoAsymptotic analysis of kinematically excited dynamical sys-tems near resonancesrdquo Nonlinear Dynamics An InternationalJournal of Nonlinear Dynamics and Chaos in Engineering Sys-tems vol 68 no 4 pp 459ndash469 2012

[20] H MooreMatlab for Engineers Pearson 3rd edition 2012[21] M D Ardema Analytical Dynamics Theory and Applications

Springer Berlin Germany 2009[22] A Tewari Modern Control Design with Matlab and Similink

John Wiley and Sons Ltd New York NY USA 2002

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Advances in Mathematical Physics

0 05 1 15

0

05

1

15

t

minus05

minus1

120585and 120585

times104

120585120585

1205930 = 0

(a)

0 05 1 15

0

5

t

1205930 = 0

minus5

minus10

minus15

minus20

minus25

120593

120593and120593

(b)

0 05 1 15

0

20

t

1205930 = 0

minus20

minus40

120595and

120595

120595120595

(c)

0

2

4

6

8

120585and 120585

times104

0 05 1 15t

120585120585

minus2

minus4

1205930 = 04

(d)

0

10

20

30

120593and120593

0 05 1 15t

120593

minus10

minus20

1205930 = 04

(e)

0

50

100

120595and

120595

0 05 1 15t

120595

120595

1205930 = 04

minus50

minus100

(f)

Figure 2 Variation of the solutions and their derivatives versus time 119905 when 119887 = 3m ℎ = 45 and 119886 = 05m (a d) show the effect of 119905 on thebehavior of 120585 and waves when 120593

0= 0 and 120593

0= 04 rad respectively (b e) show the effect of 119905 on the behavior of 120593 and waves when 120593

0= 0

and 1205930= 04 rad respectively and (c f) show the effect of 119905 on the wave that describes the behavior of 120595 and when 120593

0= 0 and 120593

0= 04 rad

respectively

119887 = (0 3)m1205930= (0 04) rad

ℎ = (25 45) 119905 = 0 997888rarr 17min

(20)

Figure 2 shows the variation of the solutions 120585 120593 120595 and theirderivatives against time 119905 when 120593

0= 0 and 120593

0=04 rad This figure is drawn at 119887 = 3m ℎ = 24 and

119886 = 05m The variations of 120585 120593 and 120595 with and respectively are illustrated in Figure 3 namely the phaseplane diagrams that are represented in Figures 3(a) 3(b)3(c) and 3(d) 3(e) 3(f) when 1205930 = 0 and 1205930 = 04 radrespectively with the same other parameters that are takeninto consideration in Figure 2

In these figures our principle aim is to investigate theeffect of increasing time on the motion of pendulum

According to the calculations depicted in Figure 2(a) wefound that when 1205930 = 0 the wave of the elongation 120585 growsup with the increasing of time till 119905 = 09min After thatboth of the elongation 120585 and its derivative fluctuate between

increasing and decreasing when time reaches 119905 = 143minThus the wave of the solution 120585 is stable see the phase planeFigure 3(a) With the passing of time one can observe that120585 and are growing quickly so the motion will be unstableafter 119905 = 143min The rage behavior of both 120585 and is dueto the weight of the rigid body and the values of the principalmoments of inertia Consequently we expect that behaviorof elongation becomes greater as observed in Figures 2(a) and2(d)Moreover the variation of the spring between stretchingand contraction is consistent with the phase plane diagramsrepresented in Figures 3(a) and 3(d)

It is worthwhile to notice fromFigure 2(b) that when time119905 increases from 119905 = 0 to 119905 = 04min the behavior of theangle 120593 increases gradually to reach the value 120593 ≃ 1 rad ≃ 57∘and then decreases slowly to reach 120593 ≃ 08 rad ≃ 46∘ duringthe time period 119905 isin ]04 09[min After 119905 = 09min thedecline of the wave becomes quickly to reach 120593 ≃ minus23 rad ≃minus132∘ at the end of time period (minus sign indicates oppositedirection) This is not possible because 120593 must belong to theinterval ] minus 1205872 1205872] So the motion of the wave is unstableas it is manifest from Figure 3(b) On the other hand increases till 119905 ≃ 02min and then fluctuates as indicated fromFigure 2(b)

Advances in Mathematical Physics 7

0 500 1000 1500 2000 2500

0

05

1

15

times104

120585

120585

1205930 = 0

minus05

minus1

(a)

0 1

0

5 1205930 = 0

minus5

minus10

minus15

minus20

minus25minus2 minus1

120593

(b)

0 2

0

20

minus20

minus40

minus2minus4

120595

120595

1205930 = 0

(c)

0 2000 4000 6000 8000 10000

0

2

4

6

8times104

120585

120585

minus2

minus4

1205930 = 04

(d)

0 02 04 06 08 1

0

10

20

30

120593

1205930 = 04

minus10

minus20

minus02

(e)

0 2 4

0

50

100

minus2minus4

120595

120595

1205930 = 04

minus50

minus100

(f)

Figure 3 The phase plane diagram when 119887 = 3m ℎ = 45 and 119886 = 05m (a d) represent the variation of the amplitude 120585 with its velocity at 120593

0= 0 and 120593

0= 04 rad respectively (b e) represent the variation of the amplitude 120593 with its velocity at 120593

0= 0 and 120593

0= 04 rad

respectively and (c f) represent the variation of the amplitude 120595 with its velocity at 1205930= 0 and 120593

0= 04 rad respectively

The graphs displayed in Figures 2(c) and 3(c) describethe variation of the (120595 and ) against time and the phaseplane diagram ( with 120595) respectively when 1205930 = 0 Itis clear that when time belongs to the period [0 043]minthe angle 120595 remains stationary and then its wave oscillatesbetween decreasing and increasing till 119905 = 143min Afterthat time the angle 120595 increases up to the end of time intervaland consequently the motion will be stable as seen fromFigure 3(c) during the period 0 lt 119905 le 143 It is obviousfrom Figure 2(c) that the behavior of remains stationary tosome extent through the time interval [0 06]min and thenoscillates between increasing and decreasing till 119905 = 17min

It should be noticed that when1205930= 04 rad the stretching

on the string 120585 increases gradually till the time 119905 becomes09min and then 120585 and oscillate between increasing anddecreasingwhen the time reaches the end of time interval seeFigure 2(d) Consequently the wave of the solution is stableas seen from the phase plane Figure 3(d)

An inspection of the graphs depicted in Figure 2(e) showsthat the wave describing the behavior of the angle 120593 increasesgradually from 120593 = 0 at 119905 = 0 to its maximum value 120593 ≃09 rad ≃ 56∘ at 119905 = 04min and then decreases slowly at119905 ≃ 1min to reach its minimum value 120593 ≃ minus019 rad ≃ minus11∘(minus sign indicates opposite direction) at 119905 ≃ 126minWith the increasing of time thewave grows again to reach thevalue 120593 ≃ 085 rad ≃ 49∘ at 119905 ≃ 139min Thus the motion is

stable as it is manifest from Figure 3(e) On the other hand increases and decreases as indicated from Figure 2(e)

Also it is remarkable from Figure 2(f) that the behaviorof the angle 120595 remains steady till 119905 = 05min then its waveoscillates between decreasing and increasing Consequentlythe motion will be stable as seen from Figure 3(f) It isworthwhile to notice also from Figure 2(f) that the behaviorof oscillates between increasing and decreasing

From the above observations we can conclude that themotion of our model is more stable when 120593

0= 04 rad than

when 1205930= 0 This highlights the importance of the effect of120593

0value on the motion It is worthwhile to notice that the

comparison between the solutions 120585 120593 and 120595 included inFigures 2(a) 2(b) and 2(c) with the corresponding Figures2(d) 2(e) and 2(f) reveals that the amplitude of the wavesdecreases when 120593

0increases from 0 to 04 rad On the other

hand the comparison between their derivatives shows thatthe amplitude of the waves increases when 120593

0increases

Figure 4 shows the variation of (120585 ) (120593 ) and (120595 ) withtime 119905 when 120593

0 changes from 0 for Figures 4(a) 4(b) and4(c) to 04 rad for Figures 4(d) 4(e) and 4(f) at the samevalues of other parameters 119887 = 3m ℎ = 45 and 119886 = 05mAccording to the calculations depicted in these figures we canconsider these figures as a rotation of the corresponding partsof Figure 3 with time to observe the bending and crossing ofthe resulting curves

8 Advances in Mathematical Physics

01000 20000

1

0051

15

t

minus1

times104

1205930 = 0

120585120585

(a)

010

0051

15

t

minus10minus20 minus1

minus2

1205930 = 0

120593

(b)

0 2020

0051

15

t

1205930 = 0

minus20minus40 minus4

minus2

120595120595

(c)

05000

1000002468

0051

15

t

times104

minus4minus2

1205930 = 04

120585120585

(d)

005

10

20

0051

15

t

minus20

1205930 = 04

120593

(e)

0 2 401000

051

15

t

minus4 minus2

1205930 = 04

minus100120595

120595

(f)

Figure 4The 3D pattern when 119887 = 3m ℎ = 45 and 119886 = 05m (a d) indicate the variation of 120585 and versus 119905 when 1205930= 0 and 120593

0= 04 rad

respectively (b e) indicate the variation of 120593 and versus 119905 when 1205930= 0 and 120593

0= 04 rad respectively and (c f) indicate the variation of 120595

and versus 119905 when 1205930= 0 and 120593

0= 04 rad respectively

01000

200001

02

minus1minus2

minus2

minus4

120585120593

120595

1205930 = 0

(a)

01

0

020

1205930 = 0

minus1

minus10minus20

minus20

minus40

times104

120585

120595

(b)

05000

100000

051

024

minus2

minus4

120595

1205930 = 04

120585

120593

(c)

0 2 4 6 8020

0

100

minus2minus4times104

120595

1205930 = 04

minus20

minus100

120585

(d)

Figure 5 The 3D diagrams when 119887 = 3m ℎ = 45 and 119886 = 05m (a c) elucidate the variation of 120585 and 120593 versus 120595 when 1205930= 0 and

1205930= 04 rad respectively and (b d) elucidate the variation of and versus when 120593

0= 0 and 120593

0= 04 rad respectively

Advances in Mathematical Physics 9

0 05 1 15

0

2

4

6

8

t

120585and 120585

times104

minus2

b = 0

120585120585

(a)

t

0 05 1 15

0

10

20

30

40

50b = 0

minus10

120593120593and120593

(b)

0 05 1 15

0

200

400

t

b = 0

minus200

minus400

minus600

120595and

120595

120595120595

(c)

Figure 6 (a) (b) and (c) explain the variation of the solutions 120585 120593 and 120595 with their derivatives and via time 119905 respectively when119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad

0 5000 10000 15000 20000

0

2

4

6

8

minus2

times104

b = 0

120585

120585

(a)

0 1 2 3

0

10

20

30

40

50b = 0

minus10

minus1

120593

(b)

0 5 10 15

0

200

400b = 0

120595

120595

minus200

minus400

minus600minus5minus10minus15

(c)

Figure 7 The phase plane diagrams between amplitudes and their velocities at 119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad (a) shows the

influence of 120585 on (b) shows the effect of 120593 on and (c) shows the variation of 120595 with

Figures 5(a) 5(c) and 5(b) 5(d) represent 3D plots thatillustrate the variation of the solutions 120585 120593 via 120595 and via respectively for different values of 120593

0when 119887 =3m ℎ = 45 and 119886 = 05m The graphs displayed in

parts of Figure 6 show the variation of (120585 ) (120593 ) and(120595 ) against time 119905 when 119887 = 0 with consideration of theparameters 1205930 = 04 rad ℎ = 45 and 119886 = 05m Thecorresponding phase plane between the amplitudes 120585 120593 120595and their derivatives is represented in parts of Figure 7Inspection of the graph depicted in Figure 6(a) shows thatwhen time 119905 increases from 0 to 045min the behavior of thesolution 120585 remains stationary and quickly growing during thetime interval 119905 isin ]045 105[min and then oscillates till theend of time interval This indicates that the motion is stableas seen from Figure 7(a) On the other side the behaviorof the derivative remains approximately stationary duringthe interval 119905 isin [0 045]min and then fluctuates with theincreasing of time see Figure 6(a)

By the same way we can observe that the wave of theangle 120593 increases through a short time to reach its maximumvalue 120593 ≃ 27 rad ≃ 155∘ at 119905 ≃ 023min taking intoconsideration that minus1205872 lt 120593 lt 1205872 and then decreasesslowly to reach its minimum value 120593 ≃ minus09 rad ≃ minus52∘at the end of time interval see Figure 6(b) This indicatesthat the motion is close to be stable as observed fromthe phase plane Figure 8(b) As seen from Figure 6(b) increases and decreases quickly during the period 119905 isin[0 01]min to reach its minimum value at the end of timeinterval

The variation of 120595 and with time is illustrated inFigure 6(c) In this figure our main goal is to examine theinfluence of time on the motion of pendulum It is clear thatthe behavior of 120595 and remains stationary (to some extent)when 119905 isin [0 05]min then their waves fluctuate till the endof time interval Consequently the motion is stable as seenfrom the phase plane diagram Figure 7(c) The comparisonbetween parts of Figure 6 with the corresponding Figures

10 Advances in Mathematical Physics

010000

2000002468

0

05

1

15

t

120585120585 minus2

b = 0

times104

(a)

t

0 1 2 30

2040

0

05

1

15

120593minus1

b = 0

(b)

t

0 1005000

05

1

15

minus10minus500 120595120595

b = 0

(c)

Figure 8 The 3D plots at 119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad (a) illustrates the variation of 120585 and via 119905 (b) illustrates the variation

of 120593 and via 119905 and (c) illustrates the variation of 120595 and via 119905

0 05 1 15

0

5000

10000

15000

t

minus5000

120585120585

120585and 120585

h = 25

(a)

0 05 1 15

0

5

10

t

minus5

minus10

minus15

120593

120593and120593

h = 25

(b)

0 05 1 15

0

20

t

120595and

120595120595120595

minus20

minus40

minus60

h = 25

(c)

Figure 9 (a) (b) and (c) demonstrate the variation of (120585 and ) (120593 and ) and (120595 and ) against time 119905 respectively at 119887 = 3m ℎ = 25119886 = 05m and 1205930= 04 rad

2(d) 2(e) and 2(f) shows that when 119887 changes from 0 to 3mthe amplitude of the waves decreases Also the motion willbe more stable when 119887 = 3m than when 119887 = 0 as seen fromthe corresponding phase plane diagrams that is Figures 3(d)3(e) 3(f) and 7(a) 7(b) 7(c) respectively

On the other hand parts of Figure 8 show 3D plots thatdescribe the variation of the solutions and their derivative viatime when 119887 = 0 ℎ = 45 120593

0= 04 rad and 119886 = 05m

The plots displayed in thementioned parts show bending andcrossing of the resulting curves

Figures 9(a) 9(b) and 9(c) show the variation of thesolutions 120585 120593 120595 and their derivatives with time 119905whenℎ = 25 for the given values of other parameters 119887 = 3m1205930= 04 rad and 119886 = 05m In view of the first part we can

conclude that when time 119905 increases each of the waves 120585 and oscillates between increasing and decreasing till 119905 = 146minand then increases gradually So the motion is stable as seenfrom Figure 10(a)

From a closer look on the second part of Figure 9(b) wecan write with the increasing of time the behavior of 120593 wave

increases to reach its maximum value 120593 ≃ 09 rad ≃ 52∘ at119905 = 043min and then decreases slowly through the period 119905 isin]043 119]min After that its behavior has a sharp declinein a few seconds (about 24 s) and then increases till the endof time period and consequently the motion is stable seeFigure 10(b)

According to the calculations depicted in Figure 9(c) wecan observe that thewaves describing120595 and decrease slowlytill 119905 = 09min and then increase and decline sharp Thephase plane Figure 10(c) shows that the behavior of 120595 is notstable

When parts of Figure 9 and their phase plane parts (ofFigure 10) are generally compared with the correspondingFigures 2(d) 2(e) and 2(f) and their phase plane Figures 3(d)3(e) and 3(f) we can observe that amplitude of the waveincreases when ℎ = 45 compared to when ℎ = 25 and themotion is more stable when ℎ = 45 An inspection of partsof Figure 11 reveals the 3D plots when ℎ = 25 with the sameother data considered in Figures 9 and 10 Figure 10 shows thevariation of the solutions 120585 120593 120595 and their derivatives

Advances in Mathematical Physics 11

0 500 1000 1500 2000

0

5000

10000

15000 h = 25

minus5000

120585

120585

(a)

0 02 04 06 08 1

0

5

10h = 25

minus5

minus10

minus15

minus02

120593

(b)

0 2

0

20h = 25

minus20

minus40

minus60

minus2minus4minus6minus8

120595

120595

(c)

Figure 10 The phase plane diagrams which portray the relation between amplitudes and their velocities at 119887 = 3m ℎ = 25 119886 = 05m and1205930= 04 rad (a) describes the influence of 120585 on (b) shows the effect of 120593 on and (c) illustrates the variation of 120595 with

01000

200005000

1000015000

0051

15

t

minus5000120585

120585

h = 25

(a)

t

005

10100

05

1

15

minus10120593

h = 25

(b)

t

0 20200

05

1

15

minus20minus40

minus60minus2minus4minus6minus8120595

120595

h = 25

(c)

Figure 11 The 3D patterns at 119887 = 3m ℎ = 25 119886 = 05m and 1205930= 04 rad (a) illustrates the variation of 120585 and versus 119905 (b) illustrates the

variation of 120593 and versus 119905 (c) illustrates the variation of 120595 and versus 119905

with time 119905 It is worthwhile to notice that the comparisonbetween Figures 4(d) 4(e) and 4(f) and Figures 11(a) 11(b)and 11(c) shows more bending and crossing of the curvesin Figures 4(d) 4(e) and 4(f) when ℎ = 45 than thecorresponding ones of Figure 11

Now we study the last case when ℎ = 0 with the sameother data 119887 = 3m 1205930 = 04 rad and 119886 = 05mThe obtainedresults are represented graphically in Figures 12(a) 12(b) and12(c) while their phase plane diagrams are given in Figures12(d) 12(e) and 12(f) At the first glance we can conclude thatthis case is not stable so it is very important to notice that thedimensionless parameter ℎmust take any value different fromzero as it is pointed in Figure 2 (ℎ = 45) and Figure 9 (ℎ =25) This elucidates the importance of ℎ parameter on themotion

4 Conclusion

A conclusion that may be made here is that the problemof the relative motion of a rigid body as a pendulum

model is investigated The governing deferential equationsare obtained using Lagrangersquos equations Mathematica pack-age was utilized in order to overcome the difficulties thatappear in the separation of the second derivatives of thegeneralized coordinates 120585 120593 and 120595 for the nonlinear system(10) Computer codes are used to obtain the numericalsolutions for system (14) These solutions are representedgraphically using Matlab program to study the influenceof the different parameters on the motion The good effectof the parameters ℎ 119887 and 120593

0on the motion is obvious

from the mentioned plots The motion of our model is morestable when the parameters ℎ 119887 and 120593

0take values run

away from zero This highlights the importance of the effectof these parameters on the motion Such results have beenconfirmed by many works such as Ismail [13] and Amer andBek [14]

Competing Interests

The author declares that they have no competing interests

12 Advances in Mathematical Physics

0 05 1 15

0

05

1

15

2

25

3

t

120585120585

120585and 120585

h = 0

times104

(a)

0 05 1 15

0

05

1

15

2

25

t

120593120593and120593

h = 0

(b)

0 05 1 15

0

5

10

15

t

120595and

120595

120595120595

h = 0

(c)

0 1000 2000 3000 4000 5000

0

05

1

15

2

25

3times104

120585

120585

h = 0

(d)

02 04 06 08 1

0

05

1

15

2

25

120593

h = 0

(e)

0 05 1 15 2

0

5

10

15

120595

120595

h = 0

(f)

Figure 12 (a b and c) explain the variation of the solutions 120585 120593 and120595with their derivatives and via time 119905 respectively when 119887 = 3mℎ = 0 119886 = 05m and 1205930= 04 rad (d e and f) illustrate the variation of the solutions against their first derivatives for the same values of the

considered parameters

References

[1] P Lynch ldquoResonant motions of the three-dimensional elasticpendulumrdquo International Journal of Non-Linear Mechanics vol37 no 2 pp 345ndash367 2002

[2] A A Klimenko Y V Mikhlin and J Awrejcewicz ldquoNonlinearnormal modes in pendulum systemsrdquoNonlinear Dynamics vol70 no 1 pp 797ndash813 2012

[3] S Mori H Nishihara and K Furuta ldquoControl of unstablemechanical system control of pendulumrdquo International Journalof Control vol 23 no 5 pp 673ndash692 1976

[4] C C Chung and J Hauser ldquoNonlinear control of a swingingpendulumrdquo Automatica A Journal of IFAC vol 31 no 6 pp851ndash862 1995

[5] A Shiriaev A Pogromsky H Ludvigsen and O Egeland ldquoOnglobal properties of passivity-based control of an inverted pen-dulumrdquo International Journal of Robust and Nonlinear Controlvol 10 no 4 pp 283ndash300 2000

[6] A S Shiriaev H Ludvigsen and O Egeland ldquoSwinging upthe spherical pendulum via stabilization of its first integralsrdquoAutomatica A Journal of IFAC the International Federation ofAutomatic Control vol 40 no 1 pp 73ndash85 2004

[7] M N Brearley ldquoThe Simple Pendulum with Uniformly Chang-ing String Lengthrdquo Proceedings of the Edinburgh MathematicalSociety vol 15 no 1 pp 61ndash66 1966

[8] S J Liao ldquoSecond-order approximate analytical solution of asimple pendulum by the process analysis methodrdquo Journal ofApplied Mechanics Transactions ASME vol 59 no 4 pp 970ndash975 1992

[9] W K Tso and K G Asmis ldquoParametric excitation of a pen-dulum with bilinear hysteresisrdquo Journal of Applied MechanicsTransactions ASME vol 37 no 4 pp 1061ndash1068 1970

[10] A H Nayfeh Perturbations Methods Wiley-VCH WeinheimGermany 2004

[11] F A El-Barki A I Ismail M O Shaker and T S AmerldquoOn the motion of the pendulum on an ellipserdquo Zeitschrift furAngewandteMathematik undMechanik vol 79 no 1 pp 65ndash721999

[12] N V Stoianov ldquoOn the relative periodic motions of a pendu-lumrdquo Journal of AppliedMathematics andMechanics vol 28 pp188ndash193 1964

[13] A I Ismail ldquoRelative periodicmotion of a rigid body pendulumon an ellipserdquo Journal of Aerospace Engineering vol 22 no 1 pp67ndash77 2009

[14] T S Amer andM A Bek ldquoChaotic responses of a harmonicallyexcited spring pendulum moving in circular pathrdquo NonlinearAnalysis Real World Applications An International Multidisci-plinary Journal vol 10 pp 3196ndash3202 2009

Advances in Mathematical Physics 13

[15] L D Akulenko ldquoParametric control of oscillations and rota-tions of a compound pendulum (a swing)rdquo Journal of AppliedMathematics and Mechanics vol 57 no 2 pp 301ndash310 1993

[16] M A Pinsky and A A Zevin ldquoOscillations of a pendulumwith a periodically varying length and a model of swingrdquoInternational Journal of Non-LinearMechanics vol 34 no 1 pp105ndash109 1999

[17] M Kamel M Eissa and A T El-Sayed ldquoVibration reductionof a nonlinear spring pendulum under multiparametric excita-tions via a longitudinal absorberrdquo Physica Scripta vol 80 no 2Article ID 025005 2009

[18] M Eissa M Kamel and A T El-Sayed ldquoVibration reduction ofmulti-parametric excited spring pendulum via a transversallytuned absorberrdquo Nonlinear Dynamics vol 61 no 1-2 pp 109ndash121 2010

[19] R Starosta G Sypniewska-Kaminska and J AwrejcewiczldquoAsymptotic analysis of kinematically excited dynamical sys-tems near resonancesrdquo Nonlinear Dynamics An InternationalJournal of Nonlinear Dynamics and Chaos in Engineering Sys-tems vol 68 no 4 pp 459ndash469 2012

[20] H MooreMatlab for Engineers Pearson 3rd edition 2012[21] M D Ardema Analytical Dynamics Theory and Applications

Springer Berlin Germany 2009[22] A Tewari Modern Control Design with Matlab and Similink

John Wiley and Sons Ltd New York NY USA 2002

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 7

0 500 1000 1500 2000 2500

0

05

1

15

times104

120585

120585

1205930 = 0

minus05

minus1

(a)

0 1

0

5 1205930 = 0

minus5

minus10

minus15

minus20

minus25minus2 minus1

120593

(b)

0 2

0

20

minus20

minus40

minus2minus4

120595

120595

1205930 = 0

(c)

0 2000 4000 6000 8000 10000

0

2

4

6

8times104

120585

120585

minus2

minus4

1205930 = 04

(d)

0 02 04 06 08 1

0

10

20

30

120593

1205930 = 04

minus10

minus20

minus02

(e)

0 2 4

0

50

100

minus2minus4

120595

120595

1205930 = 04

minus50

minus100

(f)

Figure 3 The phase plane diagram when 119887 = 3m ℎ = 45 and 119886 = 05m (a d) represent the variation of the amplitude 120585 with its velocity at 120593

0= 0 and 120593

0= 04 rad respectively (b e) represent the variation of the amplitude 120593 with its velocity at 120593

0= 0 and 120593

0= 04 rad

respectively and (c f) represent the variation of the amplitude 120595 with its velocity at 1205930= 0 and 120593

0= 04 rad respectively

The graphs displayed in Figures 2(c) and 3(c) describethe variation of the (120595 and ) against time and the phaseplane diagram ( with 120595) respectively when 1205930 = 0 Itis clear that when time belongs to the period [0 043]minthe angle 120595 remains stationary and then its wave oscillatesbetween decreasing and increasing till 119905 = 143min Afterthat time the angle 120595 increases up to the end of time intervaland consequently the motion will be stable as seen fromFigure 3(c) during the period 0 lt 119905 le 143 It is obviousfrom Figure 2(c) that the behavior of remains stationary tosome extent through the time interval [0 06]min and thenoscillates between increasing and decreasing till 119905 = 17min

It should be noticed that when1205930= 04 rad the stretching

on the string 120585 increases gradually till the time 119905 becomes09min and then 120585 and oscillate between increasing anddecreasingwhen the time reaches the end of time interval seeFigure 2(d) Consequently the wave of the solution is stableas seen from the phase plane Figure 3(d)

An inspection of the graphs depicted in Figure 2(e) showsthat the wave describing the behavior of the angle 120593 increasesgradually from 120593 = 0 at 119905 = 0 to its maximum value 120593 ≃09 rad ≃ 56∘ at 119905 = 04min and then decreases slowly at119905 ≃ 1min to reach its minimum value 120593 ≃ minus019 rad ≃ minus11∘(minus sign indicates opposite direction) at 119905 ≃ 126minWith the increasing of time thewave grows again to reach thevalue 120593 ≃ 085 rad ≃ 49∘ at 119905 ≃ 139min Thus the motion is

stable as it is manifest from Figure 3(e) On the other hand increases and decreases as indicated from Figure 2(e)

Also it is remarkable from Figure 2(f) that the behaviorof the angle 120595 remains steady till 119905 = 05min then its waveoscillates between decreasing and increasing Consequentlythe motion will be stable as seen from Figure 3(f) It isworthwhile to notice also from Figure 2(f) that the behaviorof oscillates between increasing and decreasing

From the above observations we can conclude that themotion of our model is more stable when 120593

0= 04 rad than

when 1205930= 0 This highlights the importance of the effect of120593

0value on the motion It is worthwhile to notice that the

comparison between the solutions 120585 120593 and 120595 included inFigures 2(a) 2(b) and 2(c) with the corresponding Figures2(d) 2(e) and 2(f) reveals that the amplitude of the wavesdecreases when 120593

0increases from 0 to 04 rad On the other

hand the comparison between their derivatives shows thatthe amplitude of the waves increases when 120593

0increases

Figure 4 shows the variation of (120585 ) (120593 ) and (120595 ) withtime 119905 when 120593

0 changes from 0 for Figures 4(a) 4(b) and4(c) to 04 rad for Figures 4(d) 4(e) and 4(f) at the samevalues of other parameters 119887 = 3m ℎ = 45 and 119886 = 05mAccording to the calculations depicted in these figures we canconsider these figures as a rotation of the corresponding partsof Figure 3 with time to observe the bending and crossing ofthe resulting curves

8 Advances in Mathematical Physics

01000 20000

1

0051

15

t

minus1

times104

1205930 = 0

120585120585

(a)

010

0051

15

t

minus10minus20 minus1

minus2

1205930 = 0

120593

(b)

0 2020

0051

15

t

1205930 = 0

minus20minus40 minus4

minus2

120595120595

(c)

05000

1000002468

0051

15

t

times104

minus4minus2

1205930 = 04

120585120585

(d)

005

10

20

0051

15

t

minus20

1205930 = 04

120593

(e)

0 2 401000

051

15

t

minus4 minus2

1205930 = 04

minus100120595

120595

(f)

Figure 4The 3D pattern when 119887 = 3m ℎ = 45 and 119886 = 05m (a d) indicate the variation of 120585 and versus 119905 when 1205930= 0 and 120593

0= 04 rad

respectively (b e) indicate the variation of 120593 and versus 119905 when 1205930= 0 and 120593

0= 04 rad respectively and (c f) indicate the variation of 120595

and versus 119905 when 1205930= 0 and 120593

0= 04 rad respectively

01000

200001

02

minus1minus2

minus2

minus4

120585120593

120595

1205930 = 0

(a)

01

0

020

1205930 = 0

minus1

minus10minus20

minus20

minus40

times104

120585

120595

(b)

05000

100000

051

024

minus2

minus4

120595

1205930 = 04

120585

120593

(c)

0 2 4 6 8020

0

100

minus2minus4times104

120595

1205930 = 04

minus20

minus100

120585

(d)

Figure 5 The 3D diagrams when 119887 = 3m ℎ = 45 and 119886 = 05m (a c) elucidate the variation of 120585 and 120593 versus 120595 when 1205930= 0 and

1205930= 04 rad respectively and (b d) elucidate the variation of and versus when 120593

0= 0 and 120593

0= 04 rad respectively

Advances in Mathematical Physics 9

0 05 1 15

0

2

4

6

8

t

120585and 120585

times104

minus2

b = 0

120585120585

(a)

t

0 05 1 15

0

10

20

30

40

50b = 0

minus10

120593120593and120593

(b)

0 05 1 15

0

200

400

t

b = 0

minus200

minus400

minus600

120595and

120595

120595120595

(c)

Figure 6 (a) (b) and (c) explain the variation of the solutions 120585 120593 and 120595 with their derivatives and via time 119905 respectively when119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad

0 5000 10000 15000 20000

0

2

4

6

8

minus2

times104

b = 0

120585

120585

(a)

0 1 2 3

0

10

20

30

40

50b = 0

minus10

minus1

120593

(b)

0 5 10 15

0

200

400b = 0

120595

120595

minus200

minus400

minus600minus5minus10minus15

(c)

Figure 7 The phase plane diagrams between amplitudes and their velocities at 119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad (a) shows the

influence of 120585 on (b) shows the effect of 120593 on and (c) shows the variation of 120595 with

Figures 5(a) 5(c) and 5(b) 5(d) represent 3D plots thatillustrate the variation of the solutions 120585 120593 via 120595 and via respectively for different values of 120593

0when 119887 =3m ℎ = 45 and 119886 = 05m The graphs displayed in

parts of Figure 6 show the variation of (120585 ) (120593 ) and(120595 ) against time 119905 when 119887 = 0 with consideration of theparameters 1205930 = 04 rad ℎ = 45 and 119886 = 05m Thecorresponding phase plane between the amplitudes 120585 120593 120595and their derivatives is represented in parts of Figure 7Inspection of the graph depicted in Figure 6(a) shows thatwhen time 119905 increases from 0 to 045min the behavior of thesolution 120585 remains stationary and quickly growing during thetime interval 119905 isin ]045 105[min and then oscillates till theend of time interval This indicates that the motion is stableas seen from Figure 7(a) On the other side the behaviorof the derivative remains approximately stationary duringthe interval 119905 isin [0 045]min and then fluctuates with theincreasing of time see Figure 6(a)

By the same way we can observe that the wave of theangle 120593 increases through a short time to reach its maximumvalue 120593 ≃ 27 rad ≃ 155∘ at 119905 ≃ 023min taking intoconsideration that minus1205872 lt 120593 lt 1205872 and then decreasesslowly to reach its minimum value 120593 ≃ minus09 rad ≃ minus52∘at the end of time interval see Figure 6(b) This indicatesthat the motion is close to be stable as observed fromthe phase plane Figure 8(b) As seen from Figure 6(b) increases and decreases quickly during the period 119905 isin[0 01]min to reach its minimum value at the end of timeinterval

The variation of 120595 and with time is illustrated inFigure 6(c) In this figure our main goal is to examine theinfluence of time on the motion of pendulum It is clear thatthe behavior of 120595 and remains stationary (to some extent)when 119905 isin [0 05]min then their waves fluctuate till the endof time interval Consequently the motion is stable as seenfrom the phase plane diagram Figure 7(c) The comparisonbetween parts of Figure 6 with the corresponding Figures

10 Advances in Mathematical Physics

010000

2000002468

0

05

1

15

t

120585120585 minus2

b = 0

times104

(a)

t

0 1 2 30

2040

0

05

1

15

120593minus1

b = 0

(b)

t

0 1005000

05

1

15

minus10minus500 120595120595

b = 0

(c)

Figure 8 The 3D plots at 119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad (a) illustrates the variation of 120585 and via 119905 (b) illustrates the variation

of 120593 and via 119905 and (c) illustrates the variation of 120595 and via 119905

0 05 1 15

0

5000

10000

15000

t

minus5000

120585120585

120585and 120585

h = 25

(a)

0 05 1 15

0

5

10

t

minus5

minus10

minus15

120593

120593and120593

h = 25

(b)

0 05 1 15

0

20

t

120595and

120595120595120595

minus20

minus40

minus60

h = 25

(c)

Figure 9 (a) (b) and (c) demonstrate the variation of (120585 and ) (120593 and ) and (120595 and ) against time 119905 respectively at 119887 = 3m ℎ = 25119886 = 05m and 1205930= 04 rad

2(d) 2(e) and 2(f) shows that when 119887 changes from 0 to 3mthe amplitude of the waves decreases Also the motion willbe more stable when 119887 = 3m than when 119887 = 0 as seen fromthe corresponding phase plane diagrams that is Figures 3(d)3(e) 3(f) and 7(a) 7(b) 7(c) respectively

On the other hand parts of Figure 8 show 3D plots thatdescribe the variation of the solutions and their derivative viatime when 119887 = 0 ℎ = 45 120593

0= 04 rad and 119886 = 05m

The plots displayed in thementioned parts show bending andcrossing of the resulting curves

Figures 9(a) 9(b) and 9(c) show the variation of thesolutions 120585 120593 120595 and their derivatives with time 119905whenℎ = 25 for the given values of other parameters 119887 = 3m1205930= 04 rad and 119886 = 05m In view of the first part we can

conclude that when time 119905 increases each of the waves 120585 and oscillates between increasing and decreasing till 119905 = 146minand then increases gradually So the motion is stable as seenfrom Figure 10(a)

From a closer look on the second part of Figure 9(b) wecan write with the increasing of time the behavior of 120593 wave

increases to reach its maximum value 120593 ≃ 09 rad ≃ 52∘ at119905 = 043min and then decreases slowly through the period 119905 isin]043 119]min After that its behavior has a sharp declinein a few seconds (about 24 s) and then increases till the endof time period and consequently the motion is stable seeFigure 10(b)

According to the calculations depicted in Figure 9(c) wecan observe that thewaves describing120595 and decrease slowlytill 119905 = 09min and then increase and decline sharp Thephase plane Figure 10(c) shows that the behavior of 120595 is notstable

When parts of Figure 9 and their phase plane parts (ofFigure 10) are generally compared with the correspondingFigures 2(d) 2(e) and 2(f) and their phase plane Figures 3(d)3(e) and 3(f) we can observe that amplitude of the waveincreases when ℎ = 45 compared to when ℎ = 25 and themotion is more stable when ℎ = 45 An inspection of partsof Figure 11 reveals the 3D plots when ℎ = 25 with the sameother data considered in Figures 9 and 10 Figure 10 shows thevariation of the solutions 120585 120593 120595 and their derivatives

Advances in Mathematical Physics 11

0 500 1000 1500 2000

0

5000

10000

15000 h = 25

minus5000

120585

120585

(a)

0 02 04 06 08 1

0

5

10h = 25

minus5

minus10

minus15

minus02

120593

(b)

0 2

0

20h = 25

minus20

minus40

minus60

minus2minus4minus6minus8

120595

120595

(c)

Figure 10 The phase plane diagrams which portray the relation between amplitudes and their velocities at 119887 = 3m ℎ = 25 119886 = 05m and1205930= 04 rad (a) describes the influence of 120585 on (b) shows the effect of 120593 on and (c) illustrates the variation of 120595 with

01000

200005000

1000015000

0051

15

t

minus5000120585

120585

h = 25

(a)

t

005

10100

05

1

15

minus10120593

h = 25

(b)

t

0 20200

05

1

15

minus20minus40

minus60minus2minus4minus6minus8120595

120595

h = 25

(c)

Figure 11 The 3D patterns at 119887 = 3m ℎ = 25 119886 = 05m and 1205930= 04 rad (a) illustrates the variation of 120585 and versus 119905 (b) illustrates the

variation of 120593 and versus 119905 (c) illustrates the variation of 120595 and versus 119905

with time 119905 It is worthwhile to notice that the comparisonbetween Figures 4(d) 4(e) and 4(f) and Figures 11(a) 11(b)and 11(c) shows more bending and crossing of the curvesin Figures 4(d) 4(e) and 4(f) when ℎ = 45 than thecorresponding ones of Figure 11

Now we study the last case when ℎ = 0 with the sameother data 119887 = 3m 1205930 = 04 rad and 119886 = 05mThe obtainedresults are represented graphically in Figures 12(a) 12(b) and12(c) while their phase plane diagrams are given in Figures12(d) 12(e) and 12(f) At the first glance we can conclude thatthis case is not stable so it is very important to notice that thedimensionless parameter ℎmust take any value different fromzero as it is pointed in Figure 2 (ℎ = 45) and Figure 9 (ℎ =25) This elucidates the importance of ℎ parameter on themotion

4 Conclusion

A conclusion that may be made here is that the problemof the relative motion of a rigid body as a pendulum

model is investigated The governing deferential equationsare obtained using Lagrangersquos equations Mathematica pack-age was utilized in order to overcome the difficulties thatappear in the separation of the second derivatives of thegeneralized coordinates 120585 120593 and 120595 for the nonlinear system(10) Computer codes are used to obtain the numericalsolutions for system (14) These solutions are representedgraphically using Matlab program to study the influenceof the different parameters on the motion The good effectof the parameters ℎ 119887 and 120593

0on the motion is obvious

from the mentioned plots The motion of our model is morestable when the parameters ℎ 119887 and 120593

0take values run

away from zero This highlights the importance of the effectof these parameters on the motion Such results have beenconfirmed by many works such as Ismail [13] and Amer andBek [14]

Competing Interests

The author declares that they have no competing interests

12 Advances in Mathematical Physics

0 05 1 15

0

05

1

15

2

25

3

t

120585120585

120585and 120585

h = 0

times104

(a)

0 05 1 15

0

05

1

15

2

25

t

120593120593and120593

h = 0

(b)

0 05 1 15

0

5

10

15

t

120595and

120595

120595120595

h = 0

(c)

0 1000 2000 3000 4000 5000

0

05

1

15

2

25

3times104

120585

120585

h = 0

(d)

02 04 06 08 1

0

05

1

15

2

25

120593

h = 0

(e)

0 05 1 15 2

0

5

10

15

120595

120595

h = 0

(f)

Figure 12 (a b and c) explain the variation of the solutions 120585 120593 and120595with their derivatives and via time 119905 respectively when 119887 = 3mℎ = 0 119886 = 05m and 1205930= 04 rad (d e and f) illustrate the variation of the solutions against their first derivatives for the same values of the

considered parameters

References

[1] P Lynch ldquoResonant motions of the three-dimensional elasticpendulumrdquo International Journal of Non-Linear Mechanics vol37 no 2 pp 345ndash367 2002

[2] A A Klimenko Y V Mikhlin and J Awrejcewicz ldquoNonlinearnormal modes in pendulum systemsrdquoNonlinear Dynamics vol70 no 1 pp 797ndash813 2012

[3] S Mori H Nishihara and K Furuta ldquoControl of unstablemechanical system control of pendulumrdquo International Journalof Control vol 23 no 5 pp 673ndash692 1976

[4] C C Chung and J Hauser ldquoNonlinear control of a swingingpendulumrdquo Automatica A Journal of IFAC vol 31 no 6 pp851ndash862 1995

[5] A Shiriaev A Pogromsky H Ludvigsen and O Egeland ldquoOnglobal properties of passivity-based control of an inverted pen-dulumrdquo International Journal of Robust and Nonlinear Controlvol 10 no 4 pp 283ndash300 2000

[6] A S Shiriaev H Ludvigsen and O Egeland ldquoSwinging upthe spherical pendulum via stabilization of its first integralsrdquoAutomatica A Journal of IFAC the International Federation ofAutomatic Control vol 40 no 1 pp 73ndash85 2004

[7] M N Brearley ldquoThe Simple Pendulum with Uniformly Chang-ing String Lengthrdquo Proceedings of the Edinburgh MathematicalSociety vol 15 no 1 pp 61ndash66 1966

[8] S J Liao ldquoSecond-order approximate analytical solution of asimple pendulum by the process analysis methodrdquo Journal ofApplied Mechanics Transactions ASME vol 59 no 4 pp 970ndash975 1992

[9] W K Tso and K G Asmis ldquoParametric excitation of a pen-dulum with bilinear hysteresisrdquo Journal of Applied MechanicsTransactions ASME vol 37 no 4 pp 1061ndash1068 1970

[10] A H Nayfeh Perturbations Methods Wiley-VCH WeinheimGermany 2004

[11] F A El-Barki A I Ismail M O Shaker and T S AmerldquoOn the motion of the pendulum on an ellipserdquo Zeitschrift furAngewandteMathematik undMechanik vol 79 no 1 pp 65ndash721999

[12] N V Stoianov ldquoOn the relative periodic motions of a pendu-lumrdquo Journal of AppliedMathematics andMechanics vol 28 pp188ndash193 1964

[13] A I Ismail ldquoRelative periodicmotion of a rigid body pendulumon an ellipserdquo Journal of Aerospace Engineering vol 22 no 1 pp67ndash77 2009

[14] T S Amer andM A Bek ldquoChaotic responses of a harmonicallyexcited spring pendulum moving in circular pathrdquo NonlinearAnalysis Real World Applications An International Multidisci-plinary Journal vol 10 pp 3196ndash3202 2009

Advances in Mathematical Physics 13

[15] L D Akulenko ldquoParametric control of oscillations and rota-tions of a compound pendulum (a swing)rdquo Journal of AppliedMathematics and Mechanics vol 57 no 2 pp 301ndash310 1993

[16] M A Pinsky and A A Zevin ldquoOscillations of a pendulumwith a periodically varying length and a model of swingrdquoInternational Journal of Non-LinearMechanics vol 34 no 1 pp105ndash109 1999

[17] M Kamel M Eissa and A T El-Sayed ldquoVibration reductionof a nonlinear spring pendulum under multiparametric excita-tions via a longitudinal absorberrdquo Physica Scripta vol 80 no 2Article ID 025005 2009

[18] M Eissa M Kamel and A T El-Sayed ldquoVibration reduction ofmulti-parametric excited spring pendulum via a transversallytuned absorberrdquo Nonlinear Dynamics vol 61 no 1-2 pp 109ndash121 2010

[19] R Starosta G Sypniewska-Kaminska and J AwrejcewiczldquoAsymptotic analysis of kinematically excited dynamical sys-tems near resonancesrdquo Nonlinear Dynamics An InternationalJournal of Nonlinear Dynamics and Chaos in Engineering Sys-tems vol 68 no 4 pp 459ndash469 2012

[20] H MooreMatlab for Engineers Pearson 3rd edition 2012[21] M D Ardema Analytical Dynamics Theory and Applications

Springer Berlin Germany 2009[22] A Tewari Modern Control Design with Matlab and Similink

John Wiley and Sons Ltd New York NY USA 2002

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Advances in Mathematical Physics

01000 20000

1

0051

15

t

minus1

times104

1205930 = 0

120585120585

(a)

010

0051

15

t

minus10minus20 minus1

minus2

1205930 = 0

120593

(b)

0 2020

0051

15

t

1205930 = 0

minus20minus40 minus4

minus2

120595120595

(c)

05000

1000002468

0051

15

t

times104

minus4minus2

1205930 = 04

120585120585

(d)

005

10

20

0051

15

t

minus20

1205930 = 04

120593

(e)

0 2 401000

051

15

t

minus4 minus2

1205930 = 04

minus100120595

120595

(f)

Figure 4The 3D pattern when 119887 = 3m ℎ = 45 and 119886 = 05m (a d) indicate the variation of 120585 and versus 119905 when 1205930= 0 and 120593

0= 04 rad

respectively (b e) indicate the variation of 120593 and versus 119905 when 1205930= 0 and 120593

0= 04 rad respectively and (c f) indicate the variation of 120595

and versus 119905 when 1205930= 0 and 120593

0= 04 rad respectively

01000

200001

02

minus1minus2

minus2

minus4

120585120593

120595

1205930 = 0

(a)

01

0

020

1205930 = 0

minus1

minus10minus20

minus20

minus40

times104

120585

120595

(b)

05000

100000

051

024

minus2

minus4

120595

1205930 = 04

120585

120593

(c)

0 2 4 6 8020

0

100

minus2minus4times104

120595

1205930 = 04

minus20

minus100

120585

(d)

Figure 5 The 3D diagrams when 119887 = 3m ℎ = 45 and 119886 = 05m (a c) elucidate the variation of 120585 and 120593 versus 120595 when 1205930= 0 and

1205930= 04 rad respectively and (b d) elucidate the variation of and versus when 120593

0= 0 and 120593

0= 04 rad respectively

Advances in Mathematical Physics 9

0 05 1 15

0

2

4

6

8

t

120585and 120585

times104

minus2

b = 0

120585120585

(a)

t

0 05 1 15

0

10

20

30

40

50b = 0

minus10

120593120593and120593

(b)

0 05 1 15

0

200

400

t

b = 0

minus200

minus400

minus600

120595and

120595

120595120595

(c)

Figure 6 (a) (b) and (c) explain the variation of the solutions 120585 120593 and 120595 with their derivatives and via time 119905 respectively when119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad

0 5000 10000 15000 20000

0

2

4

6

8

minus2

times104

b = 0

120585

120585

(a)

0 1 2 3

0

10

20

30

40

50b = 0

minus10

minus1

120593

(b)

0 5 10 15

0

200

400b = 0

120595

120595

minus200

minus400

minus600minus5minus10minus15

(c)

Figure 7 The phase plane diagrams between amplitudes and their velocities at 119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad (a) shows the

influence of 120585 on (b) shows the effect of 120593 on and (c) shows the variation of 120595 with

Figures 5(a) 5(c) and 5(b) 5(d) represent 3D plots thatillustrate the variation of the solutions 120585 120593 via 120595 and via respectively for different values of 120593

0when 119887 =3m ℎ = 45 and 119886 = 05m The graphs displayed in

parts of Figure 6 show the variation of (120585 ) (120593 ) and(120595 ) against time 119905 when 119887 = 0 with consideration of theparameters 1205930 = 04 rad ℎ = 45 and 119886 = 05m Thecorresponding phase plane between the amplitudes 120585 120593 120595and their derivatives is represented in parts of Figure 7Inspection of the graph depicted in Figure 6(a) shows thatwhen time 119905 increases from 0 to 045min the behavior of thesolution 120585 remains stationary and quickly growing during thetime interval 119905 isin ]045 105[min and then oscillates till theend of time interval This indicates that the motion is stableas seen from Figure 7(a) On the other side the behaviorof the derivative remains approximately stationary duringthe interval 119905 isin [0 045]min and then fluctuates with theincreasing of time see Figure 6(a)

By the same way we can observe that the wave of theangle 120593 increases through a short time to reach its maximumvalue 120593 ≃ 27 rad ≃ 155∘ at 119905 ≃ 023min taking intoconsideration that minus1205872 lt 120593 lt 1205872 and then decreasesslowly to reach its minimum value 120593 ≃ minus09 rad ≃ minus52∘at the end of time interval see Figure 6(b) This indicatesthat the motion is close to be stable as observed fromthe phase plane Figure 8(b) As seen from Figure 6(b) increases and decreases quickly during the period 119905 isin[0 01]min to reach its minimum value at the end of timeinterval

The variation of 120595 and with time is illustrated inFigure 6(c) In this figure our main goal is to examine theinfluence of time on the motion of pendulum It is clear thatthe behavior of 120595 and remains stationary (to some extent)when 119905 isin [0 05]min then their waves fluctuate till the endof time interval Consequently the motion is stable as seenfrom the phase plane diagram Figure 7(c) The comparisonbetween parts of Figure 6 with the corresponding Figures

10 Advances in Mathematical Physics

010000

2000002468

0

05

1

15

t

120585120585 minus2

b = 0

times104

(a)

t

0 1 2 30

2040

0

05

1

15

120593minus1

b = 0

(b)

t

0 1005000

05

1

15

minus10minus500 120595120595

b = 0

(c)

Figure 8 The 3D plots at 119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad (a) illustrates the variation of 120585 and via 119905 (b) illustrates the variation

of 120593 and via 119905 and (c) illustrates the variation of 120595 and via 119905

0 05 1 15

0

5000

10000

15000

t

minus5000

120585120585

120585and 120585

h = 25

(a)

0 05 1 15

0

5

10

t

minus5

minus10

minus15

120593

120593and120593

h = 25

(b)

0 05 1 15

0

20

t

120595and

120595120595120595

minus20

minus40

minus60

h = 25

(c)

Figure 9 (a) (b) and (c) demonstrate the variation of (120585 and ) (120593 and ) and (120595 and ) against time 119905 respectively at 119887 = 3m ℎ = 25119886 = 05m and 1205930= 04 rad

2(d) 2(e) and 2(f) shows that when 119887 changes from 0 to 3mthe amplitude of the waves decreases Also the motion willbe more stable when 119887 = 3m than when 119887 = 0 as seen fromthe corresponding phase plane diagrams that is Figures 3(d)3(e) 3(f) and 7(a) 7(b) 7(c) respectively

On the other hand parts of Figure 8 show 3D plots thatdescribe the variation of the solutions and their derivative viatime when 119887 = 0 ℎ = 45 120593

0= 04 rad and 119886 = 05m

The plots displayed in thementioned parts show bending andcrossing of the resulting curves

Figures 9(a) 9(b) and 9(c) show the variation of thesolutions 120585 120593 120595 and their derivatives with time 119905whenℎ = 25 for the given values of other parameters 119887 = 3m1205930= 04 rad and 119886 = 05m In view of the first part we can

conclude that when time 119905 increases each of the waves 120585 and oscillates between increasing and decreasing till 119905 = 146minand then increases gradually So the motion is stable as seenfrom Figure 10(a)

From a closer look on the second part of Figure 9(b) wecan write with the increasing of time the behavior of 120593 wave

increases to reach its maximum value 120593 ≃ 09 rad ≃ 52∘ at119905 = 043min and then decreases slowly through the period 119905 isin]043 119]min After that its behavior has a sharp declinein a few seconds (about 24 s) and then increases till the endof time period and consequently the motion is stable seeFigure 10(b)

According to the calculations depicted in Figure 9(c) wecan observe that thewaves describing120595 and decrease slowlytill 119905 = 09min and then increase and decline sharp Thephase plane Figure 10(c) shows that the behavior of 120595 is notstable

When parts of Figure 9 and their phase plane parts (ofFigure 10) are generally compared with the correspondingFigures 2(d) 2(e) and 2(f) and their phase plane Figures 3(d)3(e) and 3(f) we can observe that amplitude of the waveincreases when ℎ = 45 compared to when ℎ = 25 and themotion is more stable when ℎ = 45 An inspection of partsof Figure 11 reveals the 3D plots when ℎ = 25 with the sameother data considered in Figures 9 and 10 Figure 10 shows thevariation of the solutions 120585 120593 120595 and their derivatives

Advances in Mathematical Physics 11

0 500 1000 1500 2000

0

5000

10000

15000 h = 25

minus5000

120585

120585

(a)

0 02 04 06 08 1

0

5

10h = 25

minus5

minus10

minus15

minus02

120593

(b)

0 2

0

20h = 25

minus20

minus40

minus60

minus2minus4minus6minus8

120595

120595

(c)

Figure 10 The phase plane diagrams which portray the relation between amplitudes and their velocities at 119887 = 3m ℎ = 25 119886 = 05m and1205930= 04 rad (a) describes the influence of 120585 on (b) shows the effect of 120593 on and (c) illustrates the variation of 120595 with

01000

200005000

1000015000

0051

15

t

minus5000120585

120585

h = 25

(a)

t

005

10100

05

1

15

minus10120593

h = 25

(b)

t

0 20200

05

1

15

minus20minus40

minus60minus2minus4minus6minus8120595

120595

h = 25

(c)

Figure 11 The 3D patterns at 119887 = 3m ℎ = 25 119886 = 05m and 1205930= 04 rad (a) illustrates the variation of 120585 and versus 119905 (b) illustrates the

variation of 120593 and versus 119905 (c) illustrates the variation of 120595 and versus 119905

with time 119905 It is worthwhile to notice that the comparisonbetween Figures 4(d) 4(e) and 4(f) and Figures 11(a) 11(b)and 11(c) shows more bending and crossing of the curvesin Figures 4(d) 4(e) and 4(f) when ℎ = 45 than thecorresponding ones of Figure 11

Now we study the last case when ℎ = 0 with the sameother data 119887 = 3m 1205930 = 04 rad and 119886 = 05mThe obtainedresults are represented graphically in Figures 12(a) 12(b) and12(c) while their phase plane diagrams are given in Figures12(d) 12(e) and 12(f) At the first glance we can conclude thatthis case is not stable so it is very important to notice that thedimensionless parameter ℎmust take any value different fromzero as it is pointed in Figure 2 (ℎ = 45) and Figure 9 (ℎ =25) This elucidates the importance of ℎ parameter on themotion

4 Conclusion

A conclusion that may be made here is that the problemof the relative motion of a rigid body as a pendulum

model is investigated The governing deferential equationsare obtained using Lagrangersquos equations Mathematica pack-age was utilized in order to overcome the difficulties thatappear in the separation of the second derivatives of thegeneralized coordinates 120585 120593 and 120595 for the nonlinear system(10) Computer codes are used to obtain the numericalsolutions for system (14) These solutions are representedgraphically using Matlab program to study the influenceof the different parameters on the motion The good effectof the parameters ℎ 119887 and 120593

0on the motion is obvious

from the mentioned plots The motion of our model is morestable when the parameters ℎ 119887 and 120593

0take values run

away from zero This highlights the importance of the effectof these parameters on the motion Such results have beenconfirmed by many works such as Ismail [13] and Amer andBek [14]

Competing Interests

The author declares that they have no competing interests

12 Advances in Mathematical Physics

0 05 1 15

0

05

1

15

2

25

3

t

120585120585

120585and 120585

h = 0

times104

(a)

0 05 1 15

0

05

1

15

2

25

t

120593120593and120593

h = 0

(b)

0 05 1 15

0

5

10

15

t

120595and

120595

120595120595

h = 0

(c)

0 1000 2000 3000 4000 5000

0

05

1

15

2

25

3times104

120585

120585

h = 0

(d)

02 04 06 08 1

0

05

1

15

2

25

120593

h = 0

(e)

0 05 1 15 2

0

5

10

15

120595

120595

h = 0

(f)

Figure 12 (a b and c) explain the variation of the solutions 120585 120593 and120595with their derivatives and via time 119905 respectively when 119887 = 3mℎ = 0 119886 = 05m and 1205930= 04 rad (d e and f) illustrate the variation of the solutions against their first derivatives for the same values of the

considered parameters

References

[1] P Lynch ldquoResonant motions of the three-dimensional elasticpendulumrdquo International Journal of Non-Linear Mechanics vol37 no 2 pp 345ndash367 2002

[2] A A Klimenko Y V Mikhlin and J Awrejcewicz ldquoNonlinearnormal modes in pendulum systemsrdquoNonlinear Dynamics vol70 no 1 pp 797ndash813 2012

[3] S Mori H Nishihara and K Furuta ldquoControl of unstablemechanical system control of pendulumrdquo International Journalof Control vol 23 no 5 pp 673ndash692 1976

[4] C C Chung and J Hauser ldquoNonlinear control of a swingingpendulumrdquo Automatica A Journal of IFAC vol 31 no 6 pp851ndash862 1995

[5] A Shiriaev A Pogromsky H Ludvigsen and O Egeland ldquoOnglobal properties of passivity-based control of an inverted pen-dulumrdquo International Journal of Robust and Nonlinear Controlvol 10 no 4 pp 283ndash300 2000

[6] A S Shiriaev H Ludvigsen and O Egeland ldquoSwinging upthe spherical pendulum via stabilization of its first integralsrdquoAutomatica A Journal of IFAC the International Federation ofAutomatic Control vol 40 no 1 pp 73ndash85 2004

[7] M N Brearley ldquoThe Simple Pendulum with Uniformly Chang-ing String Lengthrdquo Proceedings of the Edinburgh MathematicalSociety vol 15 no 1 pp 61ndash66 1966

[8] S J Liao ldquoSecond-order approximate analytical solution of asimple pendulum by the process analysis methodrdquo Journal ofApplied Mechanics Transactions ASME vol 59 no 4 pp 970ndash975 1992

[9] W K Tso and K G Asmis ldquoParametric excitation of a pen-dulum with bilinear hysteresisrdquo Journal of Applied MechanicsTransactions ASME vol 37 no 4 pp 1061ndash1068 1970

[10] A H Nayfeh Perturbations Methods Wiley-VCH WeinheimGermany 2004

[11] F A El-Barki A I Ismail M O Shaker and T S AmerldquoOn the motion of the pendulum on an ellipserdquo Zeitschrift furAngewandteMathematik undMechanik vol 79 no 1 pp 65ndash721999

[12] N V Stoianov ldquoOn the relative periodic motions of a pendu-lumrdquo Journal of AppliedMathematics andMechanics vol 28 pp188ndash193 1964

[13] A I Ismail ldquoRelative periodicmotion of a rigid body pendulumon an ellipserdquo Journal of Aerospace Engineering vol 22 no 1 pp67ndash77 2009

[14] T S Amer andM A Bek ldquoChaotic responses of a harmonicallyexcited spring pendulum moving in circular pathrdquo NonlinearAnalysis Real World Applications An International Multidisci-plinary Journal vol 10 pp 3196ndash3202 2009

Advances in Mathematical Physics 13

[15] L D Akulenko ldquoParametric control of oscillations and rota-tions of a compound pendulum (a swing)rdquo Journal of AppliedMathematics and Mechanics vol 57 no 2 pp 301ndash310 1993

[16] M A Pinsky and A A Zevin ldquoOscillations of a pendulumwith a periodically varying length and a model of swingrdquoInternational Journal of Non-LinearMechanics vol 34 no 1 pp105ndash109 1999

[17] M Kamel M Eissa and A T El-Sayed ldquoVibration reductionof a nonlinear spring pendulum under multiparametric excita-tions via a longitudinal absorberrdquo Physica Scripta vol 80 no 2Article ID 025005 2009

[18] M Eissa M Kamel and A T El-Sayed ldquoVibration reduction ofmulti-parametric excited spring pendulum via a transversallytuned absorberrdquo Nonlinear Dynamics vol 61 no 1-2 pp 109ndash121 2010

[19] R Starosta G Sypniewska-Kaminska and J AwrejcewiczldquoAsymptotic analysis of kinematically excited dynamical sys-tems near resonancesrdquo Nonlinear Dynamics An InternationalJournal of Nonlinear Dynamics and Chaos in Engineering Sys-tems vol 68 no 4 pp 459ndash469 2012

[20] H MooreMatlab for Engineers Pearson 3rd edition 2012[21] M D Ardema Analytical Dynamics Theory and Applications

Springer Berlin Germany 2009[22] A Tewari Modern Control Design with Matlab and Similink

John Wiley and Sons Ltd New York NY USA 2002

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 9

0 05 1 15

0

2

4

6

8

t

120585and 120585

times104

minus2

b = 0

120585120585

(a)

t

0 05 1 15

0

10

20

30

40

50b = 0

minus10

120593120593and120593

(b)

0 05 1 15

0

200

400

t

b = 0

minus200

minus400

minus600

120595and

120595

120595120595

(c)

Figure 6 (a) (b) and (c) explain the variation of the solutions 120585 120593 and 120595 with their derivatives and via time 119905 respectively when119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad

0 5000 10000 15000 20000

0

2

4

6

8

minus2

times104

b = 0

120585

120585

(a)

0 1 2 3

0

10

20

30

40

50b = 0

minus10

minus1

120593

(b)

0 5 10 15

0

200

400b = 0

120595

120595

minus200

minus400

minus600minus5minus10minus15

(c)

Figure 7 The phase plane diagrams between amplitudes and their velocities at 119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad (a) shows the

influence of 120585 on (b) shows the effect of 120593 on and (c) shows the variation of 120595 with

Figures 5(a) 5(c) and 5(b) 5(d) represent 3D plots thatillustrate the variation of the solutions 120585 120593 via 120595 and via respectively for different values of 120593

0when 119887 =3m ℎ = 45 and 119886 = 05m The graphs displayed in

parts of Figure 6 show the variation of (120585 ) (120593 ) and(120595 ) against time 119905 when 119887 = 0 with consideration of theparameters 1205930 = 04 rad ℎ = 45 and 119886 = 05m Thecorresponding phase plane between the amplitudes 120585 120593 120595and their derivatives is represented in parts of Figure 7Inspection of the graph depicted in Figure 6(a) shows thatwhen time 119905 increases from 0 to 045min the behavior of thesolution 120585 remains stationary and quickly growing during thetime interval 119905 isin ]045 105[min and then oscillates till theend of time interval This indicates that the motion is stableas seen from Figure 7(a) On the other side the behaviorof the derivative remains approximately stationary duringthe interval 119905 isin [0 045]min and then fluctuates with theincreasing of time see Figure 6(a)

By the same way we can observe that the wave of theangle 120593 increases through a short time to reach its maximumvalue 120593 ≃ 27 rad ≃ 155∘ at 119905 ≃ 023min taking intoconsideration that minus1205872 lt 120593 lt 1205872 and then decreasesslowly to reach its minimum value 120593 ≃ minus09 rad ≃ minus52∘at the end of time interval see Figure 6(b) This indicatesthat the motion is close to be stable as observed fromthe phase plane Figure 8(b) As seen from Figure 6(b) increases and decreases quickly during the period 119905 isin[0 01]min to reach its minimum value at the end of timeinterval

The variation of 120595 and with time is illustrated inFigure 6(c) In this figure our main goal is to examine theinfluence of time on the motion of pendulum It is clear thatthe behavior of 120595 and remains stationary (to some extent)when 119905 isin [0 05]min then their waves fluctuate till the endof time interval Consequently the motion is stable as seenfrom the phase plane diagram Figure 7(c) The comparisonbetween parts of Figure 6 with the corresponding Figures

10 Advances in Mathematical Physics

010000

2000002468

0

05

1

15

t

120585120585 minus2

b = 0

times104

(a)

t

0 1 2 30

2040

0

05

1

15

120593minus1

b = 0

(b)

t

0 1005000

05

1

15

minus10minus500 120595120595

b = 0

(c)

Figure 8 The 3D plots at 119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad (a) illustrates the variation of 120585 and via 119905 (b) illustrates the variation

of 120593 and via 119905 and (c) illustrates the variation of 120595 and via 119905

0 05 1 15

0

5000

10000

15000

t

minus5000

120585120585

120585and 120585

h = 25

(a)

0 05 1 15

0

5

10

t

minus5

minus10

minus15

120593

120593and120593

h = 25

(b)

0 05 1 15

0

20

t

120595and

120595120595120595

minus20

minus40

minus60

h = 25

(c)

Figure 9 (a) (b) and (c) demonstrate the variation of (120585 and ) (120593 and ) and (120595 and ) against time 119905 respectively at 119887 = 3m ℎ = 25119886 = 05m and 1205930= 04 rad

2(d) 2(e) and 2(f) shows that when 119887 changes from 0 to 3mthe amplitude of the waves decreases Also the motion willbe more stable when 119887 = 3m than when 119887 = 0 as seen fromthe corresponding phase plane diagrams that is Figures 3(d)3(e) 3(f) and 7(a) 7(b) 7(c) respectively

On the other hand parts of Figure 8 show 3D plots thatdescribe the variation of the solutions and their derivative viatime when 119887 = 0 ℎ = 45 120593

0= 04 rad and 119886 = 05m

The plots displayed in thementioned parts show bending andcrossing of the resulting curves

Figures 9(a) 9(b) and 9(c) show the variation of thesolutions 120585 120593 120595 and their derivatives with time 119905whenℎ = 25 for the given values of other parameters 119887 = 3m1205930= 04 rad and 119886 = 05m In view of the first part we can

conclude that when time 119905 increases each of the waves 120585 and oscillates between increasing and decreasing till 119905 = 146minand then increases gradually So the motion is stable as seenfrom Figure 10(a)

From a closer look on the second part of Figure 9(b) wecan write with the increasing of time the behavior of 120593 wave

increases to reach its maximum value 120593 ≃ 09 rad ≃ 52∘ at119905 = 043min and then decreases slowly through the period 119905 isin]043 119]min After that its behavior has a sharp declinein a few seconds (about 24 s) and then increases till the endof time period and consequently the motion is stable seeFigure 10(b)

According to the calculations depicted in Figure 9(c) wecan observe that thewaves describing120595 and decrease slowlytill 119905 = 09min and then increase and decline sharp Thephase plane Figure 10(c) shows that the behavior of 120595 is notstable

When parts of Figure 9 and their phase plane parts (ofFigure 10) are generally compared with the correspondingFigures 2(d) 2(e) and 2(f) and their phase plane Figures 3(d)3(e) and 3(f) we can observe that amplitude of the waveincreases when ℎ = 45 compared to when ℎ = 25 and themotion is more stable when ℎ = 45 An inspection of partsof Figure 11 reveals the 3D plots when ℎ = 25 with the sameother data considered in Figures 9 and 10 Figure 10 shows thevariation of the solutions 120585 120593 120595 and their derivatives

Advances in Mathematical Physics 11

0 500 1000 1500 2000

0

5000

10000

15000 h = 25

minus5000

120585

120585

(a)

0 02 04 06 08 1

0

5

10h = 25

minus5

minus10

minus15

minus02

120593

(b)

0 2

0

20h = 25

minus20

minus40

minus60

minus2minus4minus6minus8

120595

120595

(c)

Figure 10 The phase plane diagrams which portray the relation between amplitudes and their velocities at 119887 = 3m ℎ = 25 119886 = 05m and1205930= 04 rad (a) describes the influence of 120585 on (b) shows the effect of 120593 on and (c) illustrates the variation of 120595 with

01000

200005000

1000015000

0051

15

t

minus5000120585

120585

h = 25

(a)

t

005

10100

05

1

15

minus10120593

h = 25

(b)

t

0 20200

05

1

15

minus20minus40

minus60minus2minus4minus6minus8120595

120595

h = 25

(c)

Figure 11 The 3D patterns at 119887 = 3m ℎ = 25 119886 = 05m and 1205930= 04 rad (a) illustrates the variation of 120585 and versus 119905 (b) illustrates the

variation of 120593 and versus 119905 (c) illustrates the variation of 120595 and versus 119905

with time 119905 It is worthwhile to notice that the comparisonbetween Figures 4(d) 4(e) and 4(f) and Figures 11(a) 11(b)and 11(c) shows more bending and crossing of the curvesin Figures 4(d) 4(e) and 4(f) when ℎ = 45 than thecorresponding ones of Figure 11

Now we study the last case when ℎ = 0 with the sameother data 119887 = 3m 1205930 = 04 rad and 119886 = 05mThe obtainedresults are represented graphically in Figures 12(a) 12(b) and12(c) while their phase plane diagrams are given in Figures12(d) 12(e) and 12(f) At the first glance we can conclude thatthis case is not stable so it is very important to notice that thedimensionless parameter ℎmust take any value different fromzero as it is pointed in Figure 2 (ℎ = 45) and Figure 9 (ℎ =25) This elucidates the importance of ℎ parameter on themotion

4 Conclusion

A conclusion that may be made here is that the problemof the relative motion of a rigid body as a pendulum

model is investigated The governing deferential equationsare obtained using Lagrangersquos equations Mathematica pack-age was utilized in order to overcome the difficulties thatappear in the separation of the second derivatives of thegeneralized coordinates 120585 120593 and 120595 for the nonlinear system(10) Computer codes are used to obtain the numericalsolutions for system (14) These solutions are representedgraphically using Matlab program to study the influenceof the different parameters on the motion The good effectof the parameters ℎ 119887 and 120593

0on the motion is obvious

from the mentioned plots The motion of our model is morestable when the parameters ℎ 119887 and 120593

0take values run

away from zero This highlights the importance of the effectof these parameters on the motion Such results have beenconfirmed by many works such as Ismail [13] and Amer andBek [14]

Competing Interests

The author declares that they have no competing interests

12 Advances in Mathematical Physics

0 05 1 15

0

05

1

15

2

25

3

t

120585120585

120585and 120585

h = 0

times104

(a)

0 05 1 15

0

05

1

15

2

25

t

120593120593and120593

h = 0

(b)

0 05 1 15

0

5

10

15

t

120595and

120595

120595120595

h = 0

(c)

0 1000 2000 3000 4000 5000

0

05

1

15

2

25

3times104

120585

120585

h = 0

(d)

02 04 06 08 1

0

05

1

15

2

25

120593

h = 0

(e)

0 05 1 15 2

0

5

10

15

120595

120595

h = 0

(f)

Figure 12 (a b and c) explain the variation of the solutions 120585 120593 and120595with their derivatives and via time 119905 respectively when 119887 = 3mℎ = 0 119886 = 05m and 1205930= 04 rad (d e and f) illustrate the variation of the solutions against their first derivatives for the same values of the

considered parameters

References

[1] P Lynch ldquoResonant motions of the three-dimensional elasticpendulumrdquo International Journal of Non-Linear Mechanics vol37 no 2 pp 345ndash367 2002

[2] A A Klimenko Y V Mikhlin and J Awrejcewicz ldquoNonlinearnormal modes in pendulum systemsrdquoNonlinear Dynamics vol70 no 1 pp 797ndash813 2012

[3] S Mori H Nishihara and K Furuta ldquoControl of unstablemechanical system control of pendulumrdquo International Journalof Control vol 23 no 5 pp 673ndash692 1976

[4] C C Chung and J Hauser ldquoNonlinear control of a swingingpendulumrdquo Automatica A Journal of IFAC vol 31 no 6 pp851ndash862 1995

[5] A Shiriaev A Pogromsky H Ludvigsen and O Egeland ldquoOnglobal properties of passivity-based control of an inverted pen-dulumrdquo International Journal of Robust and Nonlinear Controlvol 10 no 4 pp 283ndash300 2000

[6] A S Shiriaev H Ludvigsen and O Egeland ldquoSwinging upthe spherical pendulum via stabilization of its first integralsrdquoAutomatica A Journal of IFAC the International Federation ofAutomatic Control vol 40 no 1 pp 73ndash85 2004

[7] M N Brearley ldquoThe Simple Pendulum with Uniformly Chang-ing String Lengthrdquo Proceedings of the Edinburgh MathematicalSociety vol 15 no 1 pp 61ndash66 1966

[8] S J Liao ldquoSecond-order approximate analytical solution of asimple pendulum by the process analysis methodrdquo Journal ofApplied Mechanics Transactions ASME vol 59 no 4 pp 970ndash975 1992

[9] W K Tso and K G Asmis ldquoParametric excitation of a pen-dulum with bilinear hysteresisrdquo Journal of Applied MechanicsTransactions ASME vol 37 no 4 pp 1061ndash1068 1970

[10] A H Nayfeh Perturbations Methods Wiley-VCH WeinheimGermany 2004

[11] F A El-Barki A I Ismail M O Shaker and T S AmerldquoOn the motion of the pendulum on an ellipserdquo Zeitschrift furAngewandteMathematik undMechanik vol 79 no 1 pp 65ndash721999

[12] N V Stoianov ldquoOn the relative periodic motions of a pendu-lumrdquo Journal of AppliedMathematics andMechanics vol 28 pp188ndash193 1964

[13] A I Ismail ldquoRelative periodicmotion of a rigid body pendulumon an ellipserdquo Journal of Aerospace Engineering vol 22 no 1 pp67ndash77 2009

[14] T S Amer andM A Bek ldquoChaotic responses of a harmonicallyexcited spring pendulum moving in circular pathrdquo NonlinearAnalysis Real World Applications An International Multidisci-plinary Journal vol 10 pp 3196ndash3202 2009

Advances in Mathematical Physics 13

[15] L D Akulenko ldquoParametric control of oscillations and rota-tions of a compound pendulum (a swing)rdquo Journal of AppliedMathematics and Mechanics vol 57 no 2 pp 301ndash310 1993

[16] M A Pinsky and A A Zevin ldquoOscillations of a pendulumwith a periodically varying length and a model of swingrdquoInternational Journal of Non-LinearMechanics vol 34 no 1 pp105ndash109 1999

[17] M Kamel M Eissa and A T El-Sayed ldquoVibration reductionof a nonlinear spring pendulum under multiparametric excita-tions via a longitudinal absorberrdquo Physica Scripta vol 80 no 2Article ID 025005 2009

[18] M Eissa M Kamel and A T El-Sayed ldquoVibration reduction ofmulti-parametric excited spring pendulum via a transversallytuned absorberrdquo Nonlinear Dynamics vol 61 no 1-2 pp 109ndash121 2010

[19] R Starosta G Sypniewska-Kaminska and J AwrejcewiczldquoAsymptotic analysis of kinematically excited dynamical sys-tems near resonancesrdquo Nonlinear Dynamics An InternationalJournal of Nonlinear Dynamics and Chaos in Engineering Sys-tems vol 68 no 4 pp 459ndash469 2012

[20] H MooreMatlab for Engineers Pearson 3rd edition 2012[21] M D Ardema Analytical Dynamics Theory and Applications

Springer Berlin Germany 2009[22] A Tewari Modern Control Design with Matlab and Similink

John Wiley and Sons Ltd New York NY USA 2002

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Advances in Mathematical Physics

010000

2000002468

0

05

1

15

t

120585120585 minus2

b = 0

times104

(a)

t

0 1 2 30

2040

0

05

1

15

120593minus1

b = 0

(b)

t

0 1005000

05

1

15

minus10minus500 120595120595

b = 0

(c)

Figure 8 The 3D plots at 119887 = 0 ℎ = 45 119886 = 05m and 1205930= 04 rad (a) illustrates the variation of 120585 and via 119905 (b) illustrates the variation

of 120593 and via 119905 and (c) illustrates the variation of 120595 and via 119905

0 05 1 15

0

5000

10000

15000

t

minus5000

120585120585

120585and 120585

h = 25

(a)

0 05 1 15

0

5

10

t

minus5

minus10

minus15

120593

120593and120593

h = 25

(b)

0 05 1 15

0

20

t

120595and

120595120595120595

minus20

minus40

minus60

h = 25

(c)

Figure 9 (a) (b) and (c) demonstrate the variation of (120585 and ) (120593 and ) and (120595 and ) against time 119905 respectively at 119887 = 3m ℎ = 25119886 = 05m and 1205930= 04 rad

2(d) 2(e) and 2(f) shows that when 119887 changes from 0 to 3mthe amplitude of the waves decreases Also the motion willbe more stable when 119887 = 3m than when 119887 = 0 as seen fromthe corresponding phase plane diagrams that is Figures 3(d)3(e) 3(f) and 7(a) 7(b) 7(c) respectively

On the other hand parts of Figure 8 show 3D plots thatdescribe the variation of the solutions and their derivative viatime when 119887 = 0 ℎ = 45 120593

0= 04 rad and 119886 = 05m

The plots displayed in thementioned parts show bending andcrossing of the resulting curves

Figures 9(a) 9(b) and 9(c) show the variation of thesolutions 120585 120593 120595 and their derivatives with time 119905whenℎ = 25 for the given values of other parameters 119887 = 3m1205930= 04 rad and 119886 = 05m In view of the first part we can

conclude that when time 119905 increases each of the waves 120585 and oscillates between increasing and decreasing till 119905 = 146minand then increases gradually So the motion is stable as seenfrom Figure 10(a)

From a closer look on the second part of Figure 9(b) wecan write with the increasing of time the behavior of 120593 wave

increases to reach its maximum value 120593 ≃ 09 rad ≃ 52∘ at119905 = 043min and then decreases slowly through the period 119905 isin]043 119]min After that its behavior has a sharp declinein a few seconds (about 24 s) and then increases till the endof time period and consequently the motion is stable seeFigure 10(b)

According to the calculations depicted in Figure 9(c) wecan observe that thewaves describing120595 and decrease slowlytill 119905 = 09min and then increase and decline sharp Thephase plane Figure 10(c) shows that the behavior of 120595 is notstable

When parts of Figure 9 and their phase plane parts (ofFigure 10) are generally compared with the correspondingFigures 2(d) 2(e) and 2(f) and their phase plane Figures 3(d)3(e) and 3(f) we can observe that amplitude of the waveincreases when ℎ = 45 compared to when ℎ = 25 and themotion is more stable when ℎ = 45 An inspection of partsof Figure 11 reveals the 3D plots when ℎ = 25 with the sameother data considered in Figures 9 and 10 Figure 10 shows thevariation of the solutions 120585 120593 120595 and their derivatives

Advances in Mathematical Physics 11

0 500 1000 1500 2000

0

5000

10000

15000 h = 25

minus5000

120585

120585

(a)

0 02 04 06 08 1

0

5

10h = 25

minus5

minus10

minus15

minus02

120593

(b)

0 2

0

20h = 25

minus20

minus40

minus60

minus2minus4minus6minus8

120595

120595

(c)

Figure 10 The phase plane diagrams which portray the relation between amplitudes and their velocities at 119887 = 3m ℎ = 25 119886 = 05m and1205930= 04 rad (a) describes the influence of 120585 on (b) shows the effect of 120593 on and (c) illustrates the variation of 120595 with

01000

200005000

1000015000

0051

15

t

minus5000120585

120585

h = 25

(a)

t

005

10100

05

1

15

minus10120593

h = 25

(b)

t

0 20200

05

1

15

minus20minus40

minus60minus2minus4minus6minus8120595

120595

h = 25

(c)

Figure 11 The 3D patterns at 119887 = 3m ℎ = 25 119886 = 05m and 1205930= 04 rad (a) illustrates the variation of 120585 and versus 119905 (b) illustrates the

variation of 120593 and versus 119905 (c) illustrates the variation of 120595 and versus 119905

with time 119905 It is worthwhile to notice that the comparisonbetween Figures 4(d) 4(e) and 4(f) and Figures 11(a) 11(b)and 11(c) shows more bending and crossing of the curvesin Figures 4(d) 4(e) and 4(f) when ℎ = 45 than thecorresponding ones of Figure 11

Now we study the last case when ℎ = 0 with the sameother data 119887 = 3m 1205930 = 04 rad and 119886 = 05mThe obtainedresults are represented graphically in Figures 12(a) 12(b) and12(c) while their phase plane diagrams are given in Figures12(d) 12(e) and 12(f) At the first glance we can conclude thatthis case is not stable so it is very important to notice that thedimensionless parameter ℎmust take any value different fromzero as it is pointed in Figure 2 (ℎ = 45) and Figure 9 (ℎ =25) This elucidates the importance of ℎ parameter on themotion

4 Conclusion

A conclusion that may be made here is that the problemof the relative motion of a rigid body as a pendulum

model is investigated The governing deferential equationsare obtained using Lagrangersquos equations Mathematica pack-age was utilized in order to overcome the difficulties thatappear in the separation of the second derivatives of thegeneralized coordinates 120585 120593 and 120595 for the nonlinear system(10) Computer codes are used to obtain the numericalsolutions for system (14) These solutions are representedgraphically using Matlab program to study the influenceof the different parameters on the motion The good effectof the parameters ℎ 119887 and 120593

0on the motion is obvious

from the mentioned plots The motion of our model is morestable when the parameters ℎ 119887 and 120593

0take values run

away from zero This highlights the importance of the effectof these parameters on the motion Such results have beenconfirmed by many works such as Ismail [13] and Amer andBek [14]

Competing Interests

The author declares that they have no competing interests

12 Advances in Mathematical Physics

0 05 1 15

0

05

1

15

2

25

3

t

120585120585

120585and 120585

h = 0

times104

(a)

0 05 1 15

0

05

1

15

2

25

t

120593120593and120593

h = 0

(b)

0 05 1 15

0

5

10

15

t

120595and

120595

120595120595

h = 0

(c)

0 1000 2000 3000 4000 5000

0

05

1

15

2

25

3times104

120585

120585

h = 0

(d)

02 04 06 08 1

0

05

1

15

2

25

120593

h = 0

(e)

0 05 1 15 2

0

5

10

15

120595

120595

h = 0

(f)

Figure 12 (a b and c) explain the variation of the solutions 120585 120593 and120595with their derivatives and via time 119905 respectively when 119887 = 3mℎ = 0 119886 = 05m and 1205930= 04 rad (d e and f) illustrate the variation of the solutions against their first derivatives for the same values of the

considered parameters

References

[1] P Lynch ldquoResonant motions of the three-dimensional elasticpendulumrdquo International Journal of Non-Linear Mechanics vol37 no 2 pp 345ndash367 2002

[2] A A Klimenko Y V Mikhlin and J Awrejcewicz ldquoNonlinearnormal modes in pendulum systemsrdquoNonlinear Dynamics vol70 no 1 pp 797ndash813 2012

[3] S Mori H Nishihara and K Furuta ldquoControl of unstablemechanical system control of pendulumrdquo International Journalof Control vol 23 no 5 pp 673ndash692 1976

[4] C C Chung and J Hauser ldquoNonlinear control of a swingingpendulumrdquo Automatica A Journal of IFAC vol 31 no 6 pp851ndash862 1995

[5] A Shiriaev A Pogromsky H Ludvigsen and O Egeland ldquoOnglobal properties of passivity-based control of an inverted pen-dulumrdquo International Journal of Robust and Nonlinear Controlvol 10 no 4 pp 283ndash300 2000

[6] A S Shiriaev H Ludvigsen and O Egeland ldquoSwinging upthe spherical pendulum via stabilization of its first integralsrdquoAutomatica A Journal of IFAC the International Federation ofAutomatic Control vol 40 no 1 pp 73ndash85 2004

[7] M N Brearley ldquoThe Simple Pendulum with Uniformly Chang-ing String Lengthrdquo Proceedings of the Edinburgh MathematicalSociety vol 15 no 1 pp 61ndash66 1966

[8] S J Liao ldquoSecond-order approximate analytical solution of asimple pendulum by the process analysis methodrdquo Journal ofApplied Mechanics Transactions ASME vol 59 no 4 pp 970ndash975 1992

[9] W K Tso and K G Asmis ldquoParametric excitation of a pen-dulum with bilinear hysteresisrdquo Journal of Applied MechanicsTransactions ASME vol 37 no 4 pp 1061ndash1068 1970

[10] A H Nayfeh Perturbations Methods Wiley-VCH WeinheimGermany 2004

[11] F A El-Barki A I Ismail M O Shaker and T S AmerldquoOn the motion of the pendulum on an ellipserdquo Zeitschrift furAngewandteMathematik undMechanik vol 79 no 1 pp 65ndash721999

[12] N V Stoianov ldquoOn the relative periodic motions of a pendu-lumrdquo Journal of AppliedMathematics andMechanics vol 28 pp188ndash193 1964

[13] A I Ismail ldquoRelative periodicmotion of a rigid body pendulumon an ellipserdquo Journal of Aerospace Engineering vol 22 no 1 pp67ndash77 2009

[14] T S Amer andM A Bek ldquoChaotic responses of a harmonicallyexcited spring pendulum moving in circular pathrdquo NonlinearAnalysis Real World Applications An International Multidisci-plinary Journal vol 10 pp 3196ndash3202 2009

Advances in Mathematical Physics 13

[15] L D Akulenko ldquoParametric control of oscillations and rota-tions of a compound pendulum (a swing)rdquo Journal of AppliedMathematics and Mechanics vol 57 no 2 pp 301ndash310 1993

[16] M A Pinsky and A A Zevin ldquoOscillations of a pendulumwith a periodically varying length and a model of swingrdquoInternational Journal of Non-LinearMechanics vol 34 no 1 pp105ndash109 1999

[17] M Kamel M Eissa and A T El-Sayed ldquoVibration reductionof a nonlinear spring pendulum under multiparametric excita-tions via a longitudinal absorberrdquo Physica Scripta vol 80 no 2Article ID 025005 2009

[18] M Eissa M Kamel and A T El-Sayed ldquoVibration reduction ofmulti-parametric excited spring pendulum via a transversallytuned absorberrdquo Nonlinear Dynamics vol 61 no 1-2 pp 109ndash121 2010

[19] R Starosta G Sypniewska-Kaminska and J AwrejcewiczldquoAsymptotic analysis of kinematically excited dynamical sys-tems near resonancesrdquo Nonlinear Dynamics An InternationalJournal of Nonlinear Dynamics and Chaos in Engineering Sys-tems vol 68 no 4 pp 459ndash469 2012

[20] H MooreMatlab for Engineers Pearson 3rd edition 2012[21] M D Ardema Analytical Dynamics Theory and Applications

Springer Berlin Germany 2009[22] A Tewari Modern Control Design with Matlab and Similink

John Wiley and Sons Ltd New York NY USA 2002

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 11

0 500 1000 1500 2000

0

5000

10000

15000 h = 25

minus5000

120585

120585

(a)

0 02 04 06 08 1

0

5

10h = 25

minus5

minus10

minus15

minus02

120593

(b)

0 2

0

20h = 25

minus20

minus40

minus60

minus2minus4minus6minus8

120595

120595

(c)

Figure 10 The phase plane diagrams which portray the relation between amplitudes and their velocities at 119887 = 3m ℎ = 25 119886 = 05m and1205930= 04 rad (a) describes the influence of 120585 on (b) shows the effect of 120593 on and (c) illustrates the variation of 120595 with

01000

200005000

1000015000

0051

15

t

minus5000120585

120585

h = 25

(a)

t

005

10100

05

1

15

minus10120593

h = 25

(b)

t

0 20200

05

1

15

minus20minus40

minus60minus2minus4minus6minus8120595

120595

h = 25

(c)

Figure 11 The 3D patterns at 119887 = 3m ℎ = 25 119886 = 05m and 1205930= 04 rad (a) illustrates the variation of 120585 and versus 119905 (b) illustrates the

variation of 120593 and versus 119905 (c) illustrates the variation of 120595 and versus 119905

with time 119905 It is worthwhile to notice that the comparisonbetween Figures 4(d) 4(e) and 4(f) and Figures 11(a) 11(b)and 11(c) shows more bending and crossing of the curvesin Figures 4(d) 4(e) and 4(f) when ℎ = 45 than thecorresponding ones of Figure 11

Now we study the last case when ℎ = 0 with the sameother data 119887 = 3m 1205930 = 04 rad and 119886 = 05mThe obtainedresults are represented graphically in Figures 12(a) 12(b) and12(c) while their phase plane diagrams are given in Figures12(d) 12(e) and 12(f) At the first glance we can conclude thatthis case is not stable so it is very important to notice that thedimensionless parameter ℎmust take any value different fromzero as it is pointed in Figure 2 (ℎ = 45) and Figure 9 (ℎ =25) This elucidates the importance of ℎ parameter on themotion

4 Conclusion

A conclusion that may be made here is that the problemof the relative motion of a rigid body as a pendulum

model is investigated The governing deferential equationsare obtained using Lagrangersquos equations Mathematica pack-age was utilized in order to overcome the difficulties thatappear in the separation of the second derivatives of thegeneralized coordinates 120585 120593 and 120595 for the nonlinear system(10) Computer codes are used to obtain the numericalsolutions for system (14) These solutions are representedgraphically using Matlab program to study the influenceof the different parameters on the motion The good effectof the parameters ℎ 119887 and 120593

0on the motion is obvious

from the mentioned plots The motion of our model is morestable when the parameters ℎ 119887 and 120593

0take values run

away from zero This highlights the importance of the effectof these parameters on the motion Such results have beenconfirmed by many works such as Ismail [13] and Amer andBek [14]

Competing Interests

The author declares that they have no competing interests

12 Advances in Mathematical Physics

0 05 1 15

0

05

1

15

2

25

3

t

120585120585

120585and 120585

h = 0

times104

(a)

0 05 1 15

0

05

1

15

2

25

t

120593120593and120593

h = 0

(b)

0 05 1 15

0

5

10

15

t

120595and

120595

120595120595

h = 0

(c)

0 1000 2000 3000 4000 5000

0

05

1

15

2

25

3times104

120585

120585

h = 0

(d)

02 04 06 08 1

0

05

1

15

2

25

120593

h = 0

(e)

0 05 1 15 2

0

5

10

15

120595

120595

h = 0

(f)

Figure 12 (a b and c) explain the variation of the solutions 120585 120593 and120595with their derivatives and via time 119905 respectively when 119887 = 3mℎ = 0 119886 = 05m and 1205930= 04 rad (d e and f) illustrate the variation of the solutions against their first derivatives for the same values of the

considered parameters

References

[1] P Lynch ldquoResonant motions of the three-dimensional elasticpendulumrdquo International Journal of Non-Linear Mechanics vol37 no 2 pp 345ndash367 2002

[2] A A Klimenko Y V Mikhlin and J Awrejcewicz ldquoNonlinearnormal modes in pendulum systemsrdquoNonlinear Dynamics vol70 no 1 pp 797ndash813 2012

[3] S Mori H Nishihara and K Furuta ldquoControl of unstablemechanical system control of pendulumrdquo International Journalof Control vol 23 no 5 pp 673ndash692 1976

[4] C C Chung and J Hauser ldquoNonlinear control of a swingingpendulumrdquo Automatica A Journal of IFAC vol 31 no 6 pp851ndash862 1995

[5] A Shiriaev A Pogromsky H Ludvigsen and O Egeland ldquoOnglobal properties of passivity-based control of an inverted pen-dulumrdquo International Journal of Robust and Nonlinear Controlvol 10 no 4 pp 283ndash300 2000

[6] A S Shiriaev H Ludvigsen and O Egeland ldquoSwinging upthe spherical pendulum via stabilization of its first integralsrdquoAutomatica A Journal of IFAC the International Federation ofAutomatic Control vol 40 no 1 pp 73ndash85 2004

[7] M N Brearley ldquoThe Simple Pendulum with Uniformly Chang-ing String Lengthrdquo Proceedings of the Edinburgh MathematicalSociety vol 15 no 1 pp 61ndash66 1966

[8] S J Liao ldquoSecond-order approximate analytical solution of asimple pendulum by the process analysis methodrdquo Journal ofApplied Mechanics Transactions ASME vol 59 no 4 pp 970ndash975 1992

[9] W K Tso and K G Asmis ldquoParametric excitation of a pen-dulum with bilinear hysteresisrdquo Journal of Applied MechanicsTransactions ASME vol 37 no 4 pp 1061ndash1068 1970

[10] A H Nayfeh Perturbations Methods Wiley-VCH WeinheimGermany 2004

[11] F A El-Barki A I Ismail M O Shaker and T S AmerldquoOn the motion of the pendulum on an ellipserdquo Zeitschrift furAngewandteMathematik undMechanik vol 79 no 1 pp 65ndash721999

[12] N V Stoianov ldquoOn the relative periodic motions of a pendu-lumrdquo Journal of AppliedMathematics andMechanics vol 28 pp188ndash193 1964

[13] A I Ismail ldquoRelative periodicmotion of a rigid body pendulumon an ellipserdquo Journal of Aerospace Engineering vol 22 no 1 pp67ndash77 2009

[14] T S Amer andM A Bek ldquoChaotic responses of a harmonicallyexcited spring pendulum moving in circular pathrdquo NonlinearAnalysis Real World Applications An International Multidisci-plinary Journal vol 10 pp 3196ndash3202 2009

Advances in Mathematical Physics 13

[15] L D Akulenko ldquoParametric control of oscillations and rota-tions of a compound pendulum (a swing)rdquo Journal of AppliedMathematics and Mechanics vol 57 no 2 pp 301ndash310 1993

[16] M A Pinsky and A A Zevin ldquoOscillations of a pendulumwith a periodically varying length and a model of swingrdquoInternational Journal of Non-LinearMechanics vol 34 no 1 pp105ndash109 1999

[17] M Kamel M Eissa and A T El-Sayed ldquoVibration reductionof a nonlinear spring pendulum under multiparametric excita-tions via a longitudinal absorberrdquo Physica Scripta vol 80 no 2Article ID 025005 2009

[18] M Eissa M Kamel and A T El-Sayed ldquoVibration reduction ofmulti-parametric excited spring pendulum via a transversallytuned absorberrdquo Nonlinear Dynamics vol 61 no 1-2 pp 109ndash121 2010

[19] R Starosta G Sypniewska-Kaminska and J AwrejcewiczldquoAsymptotic analysis of kinematically excited dynamical sys-tems near resonancesrdquo Nonlinear Dynamics An InternationalJournal of Nonlinear Dynamics and Chaos in Engineering Sys-tems vol 68 no 4 pp 459ndash469 2012

[20] H MooreMatlab for Engineers Pearson 3rd edition 2012[21] M D Ardema Analytical Dynamics Theory and Applications

Springer Berlin Germany 2009[22] A Tewari Modern Control Design with Matlab and Similink

John Wiley and Sons Ltd New York NY USA 2002

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Advances in Mathematical Physics

0 05 1 15

0

05

1

15

2

25

3

t

120585120585

120585and 120585

h = 0

times104

(a)

0 05 1 15

0

05

1

15

2

25

t

120593120593and120593

h = 0

(b)

0 05 1 15

0

5

10

15

t

120595and

120595

120595120595

h = 0

(c)

0 1000 2000 3000 4000 5000

0

05

1

15

2

25

3times104

120585

120585

h = 0

(d)

02 04 06 08 1

0

05

1

15

2

25

120593

h = 0

(e)

0 05 1 15 2

0

5

10

15

120595

120595

h = 0

(f)

Figure 12 (a b and c) explain the variation of the solutions 120585 120593 and120595with their derivatives and via time 119905 respectively when 119887 = 3mℎ = 0 119886 = 05m and 1205930= 04 rad (d e and f) illustrate the variation of the solutions against their first derivatives for the same values of the

considered parameters

References

[1] P Lynch ldquoResonant motions of the three-dimensional elasticpendulumrdquo International Journal of Non-Linear Mechanics vol37 no 2 pp 345ndash367 2002

[2] A A Klimenko Y V Mikhlin and J Awrejcewicz ldquoNonlinearnormal modes in pendulum systemsrdquoNonlinear Dynamics vol70 no 1 pp 797ndash813 2012

[3] S Mori H Nishihara and K Furuta ldquoControl of unstablemechanical system control of pendulumrdquo International Journalof Control vol 23 no 5 pp 673ndash692 1976

[4] C C Chung and J Hauser ldquoNonlinear control of a swingingpendulumrdquo Automatica A Journal of IFAC vol 31 no 6 pp851ndash862 1995

[5] A Shiriaev A Pogromsky H Ludvigsen and O Egeland ldquoOnglobal properties of passivity-based control of an inverted pen-dulumrdquo International Journal of Robust and Nonlinear Controlvol 10 no 4 pp 283ndash300 2000

[6] A S Shiriaev H Ludvigsen and O Egeland ldquoSwinging upthe spherical pendulum via stabilization of its first integralsrdquoAutomatica A Journal of IFAC the International Federation ofAutomatic Control vol 40 no 1 pp 73ndash85 2004

[7] M N Brearley ldquoThe Simple Pendulum with Uniformly Chang-ing String Lengthrdquo Proceedings of the Edinburgh MathematicalSociety vol 15 no 1 pp 61ndash66 1966

[8] S J Liao ldquoSecond-order approximate analytical solution of asimple pendulum by the process analysis methodrdquo Journal ofApplied Mechanics Transactions ASME vol 59 no 4 pp 970ndash975 1992

[9] W K Tso and K G Asmis ldquoParametric excitation of a pen-dulum with bilinear hysteresisrdquo Journal of Applied MechanicsTransactions ASME vol 37 no 4 pp 1061ndash1068 1970

[10] A H Nayfeh Perturbations Methods Wiley-VCH WeinheimGermany 2004

[11] F A El-Barki A I Ismail M O Shaker and T S AmerldquoOn the motion of the pendulum on an ellipserdquo Zeitschrift furAngewandteMathematik undMechanik vol 79 no 1 pp 65ndash721999

[12] N V Stoianov ldquoOn the relative periodic motions of a pendu-lumrdquo Journal of AppliedMathematics andMechanics vol 28 pp188ndash193 1964

[13] A I Ismail ldquoRelative periodicmotion of a rigid body pendulumon an ellipserdquo Journal of Aerospace Engineering vol 22 no 1 pp67ndash77 2009

[14] T S Amer andM A Bek ldquoChaotic responses of a harmonicallyexcited spring pendulum moving in circular pathrdquo NonlinearAnalysis Real World Applications An International Multidisci-plinary Journal vol 10 pp 3196ndash3202 2009

Advances in Mathematical Physics 13

[15] L D Akulenko ldquoParametric control of oscillations and rota-tions of a compound pendulum (a swing)rdquo Journal of AppliedMathematics and Mechanics vol 57 no 2 pp 301ndash310 1993

[16] M A Pinsky and A A Zevin ldquoOscillations of a pendulumwith a periodically varying length and a model of swingrdquoInternational Journal of Non-LinearMechanics vol 34 no 1 pp105ndash109 1999

[17] M Kamel M Eissa and A T El-Sayed ldquoVibration reductionof a nonlinear spring pendulum under multiparametric excita-tions via a longitudinal absorberrdquo Physica Scripta vol 80 no 2Article ID 025005 2009

[18] M Eissa M Kamel and A T El-Sayed ldquoVibration reduction ofmulti-parametric excited spring pendulum via a transversallytuned absorberrdquo Nonlinear Dynamics vol 61 no 1-2 pp 109ndash121 2010

[19] R Starosta G Sypniewska-Kaminska and J AwrejcewiczldquoAsymptotic analysis of kinematically excited dynamical sys-tems near resonancesrdquo Nonlinear Dynamics An InternationalJournal of Nonlinear Dynamics and Chaos in Engineering Sys-tems vol 68 no 4 pp 459ndash469 2012

[20] H MooreMatlab for Engineers Pearson 3rd edition 2012[21] M D Ardema Analytical Dynamics Theory and Applications

Springer Berlin Germany 2009[22] A Tewari Modern Control Design with Matlab and Similink

John Wiley and Sons Ltd New York NY USA 2002

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 13

[15] L D Akulenko ldquoParametric control of oscillations and rota-tions of a compound pendulum (a swing)rdquo Journal of AppliedMathematics and Mechanics vol 57 no 2 pp 301ndash310 1993

[16] M A Pinsky and A A Zevin ldquoOscillations of a pendulumwith a periodically varying length and a model of swingrdquoInternational Journal of Non-LinearMechanics vol 34 no 1 pp105ndash109 1999

[17] M Kamel M Eissa and A T El-Sayed ldquoVibration reductionof a nonlinear spring pendulum under multiparametric excita-tions via a longitudinal absorberrdquo Physica Scripta vol 80 no 2Article ID 025005 2009

[18] M Eissa M Kamel and A T El-Sayed ldquoVibration reduction ofmulti-parametric excited spring pendulum via a transversallytuned absorberrdquo Nonlinear Dynamics vol 61 no 1-2 pp 109ndash121 2010

[19] R Starosta G Sypniewska-Kaminska and J AwrejcewiczldquoAsymptotic analysis of kinematically excited dynamical sys-tems near resonancesrdquo Nonlinear Dynamics An InternationalJournal of Nonlinear Dynamics and Chaos in Engineering Sys-tems vol 68 no 4 pp 459ndash469 2012

[20] H MooreMatlab for Engineers Pearson 3rd edition 2012[21] M D Ardema Analytical Dynamics Theory and Applications

Springer Berlin Germany 2009[22] A Tewari Modern Control Design with Matlab and Similink

John Wiley and Sons Ltd New York NY USA 2002

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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