, ISSN 0974-570X (Online), ISSN 0974-578 (Prin), Vol. 6; Issue No. 3; Year 05, Copyrigh 05 by CESER PUBLICATIONS Dynamic Analysis of Damped Driven Pendulum using Laplace Transform Mehod M.C. Agarana and O.O. Agboola Deparmen of Mahemaics, College of Science &Technology, Covenan Universiy, Oa, Nigeria; Email : michael.agarana@covenanuniversiy.edu.ng ABSTRACT In many recen works, many researchers have demonsraed he usefulness of dynamical sysems. In his paper, a damped driven pendulum, as a dynamical sysem, is considered. The effecs of is angular displacemen and angular driven force on he dynamics of he pendulum is analyzed. The Laplace ransform mehod is used o ransform he differenial euaion governing he moion of he pendulum ino is algebraic form and he desired resuls obained. I is observed ha angular displacemen and angular driven force affec he moion of he pendulum. Specifically i is noed ha he lower he fixed value of he angular driving force he higher he angular velociy, a various values of he angular displacemen. Keywords: Dynamic analysis, damping, damped driven pendulum, angular velociy, angular displacemen, angular driven force Mahemaics Subjec Classificaion: 3JN5. INTRODUCTION When a pendulum is aced on, boh by a velociy dependen damping force, and a periodic driving force, i can display boh ordered and chaoic behaviours, for cerain ranges of parameers [3,6,]. The pendulum is a dynamical sysem. Our invesigaion of pendulum dynamics begins wih Newon s second law of moion; which saes ha he relaionship beween an objec s mass m, is acceleraion a, and he applied force F, is F ma. If we hold damping and driving force in reserved; and he small angle approximaion of a simplified, idealized pendulum is given by he following euaion [3,8]: d I mglsin 0 If we add a dissipaive force proporional o he velociy, he governing euaion of moion will now d include he damping erm,, where is he dissipaive coefficien. If we also drive he pendulum, www.ceser.in/ijmc.hml www.ceserp.com/cp-jour www.ceserpublicaions.com
feeding in energy o resupply he energy dissipaed, he governing euaion will also include he driving force erm, making he new siuaion a very ineresing one [3,0]. Solving his governing euaion and esimaing he soluion a differen values of he parameers, using graphics give one an insigh o he dynamics of he sysem. The objecive of his paper is o analyse he dynamic effecs of he angular displacemen and angular driving force on he damped pendulum.. THE GOVERNING EQUATION AND ITS DIMENSIONLESS FORM The euaion of moion for damped, driven pendulum of mass m and lengh l can be wrien as: [,]: d d sin cos( ) () ml mgl C D A D G P where A is acceleraion, D is he damping, G is he graviaion and P is he driving force. D is he angular driving force, is he angular displacemen, is he ime, l is he lengh, m is he mass, is he dissipaion coefficien, C is he ampliude of he driving force and g is he acceleraion due o graviy. The dimensionless form of he euaion under sudy, which describes he damped driven pendulum of lengh l, and mass m, is [,, 3] d d sin acos( ) D () where he hree erms on he lef hand side represen acceleraion, damping and graviaion respecively and he erm on he righ hand side is a sinusoidal driving orue, which is made up of an ampliude g and a freuency D, is he damping parameer [3] Euaion can be wrien as sin acos( D) (3) which can also be wrien as w wsin acos( D) (4) 99
d d where w andd D ishephaseofhedrivingforceerm. Thehreedimensionsforhissysemrepresenedbyeuaion(),become w,and.inordero simplifyheresulsofhesysem,hasbeenresriceoresidewihin and,while has beenresriceoresidewihin0and.e.(4)cannowbewrienas where wcossin w w (5) (6) D (7) Depending on he values of he damping parameer and he forcing ampliude, he sysem can exhibidifferendynamics. l P D mg Fig.:Thependulum.. Laplace ransform mehod The Laplace ransform mehod was applied o solve he iniial-value problem following hree seps: (i) Taking he Laplace ransforms of boh side of he euaion; (ii) Simplifying algebraically he resul obained. (iii) Finding he inverse ransform in order oobainheunknownfuncion y(). 00
This inverse ransform, y() ishedesiredsoluionofhegivendifferenialeuaion. 3. ANALYSIS From euaion (5), i.e. wcossin w cossin, where is he angular acceleraion. (9) We consider he value of w for differen values of and afixedparameers and g.forhis paper,weconsidersomevaluesofand,henuselaplaceransformmehoofind w in each case. This can be seen in able. Also he parameers are kep consan; aking, g. 3. Iniial Condiions: Our iniial value problem is formed by subjecing he differenial euaion in (9) o he following iniial condiions: w(0) 0, w(0) 0.(0) Thesoluionofheiniialvalueproblemisobained,usingLaplaceransformmehod,as w e () when 90 and 90. Differenvaluesofheangularvelociy areobainedfordifferenvaluesofandwhere 0 and.ineachcaseheiniialcondiionsaresaisfied. 4. Effecsof,,,, g on he pendulum moion In order o compare he effecs of angulardisplacemen and angulardriving force on he pendulum moion, he following cases are considered in secions (4.), (4.), (4.3) and (4.4) : 4.. Effecs of AngularDriving force and angulardisplacemen on he undampedpendulum : FromE. () 0
d d sin cos( ) ml mgl C D where d is he dampingerm. The euaion, wihoudamping, becomes d sin cos( ) (3) ml mgl C D The dimensionlessform of he euaion, wihoudampingbecomes d sin acos( ) D (4) whichcanbewrien as sin a cos( ) D (5) hais, wacos sin 0 (6) or w acos sin (7) i.e. a cos sin cos sin (for g ) 4. Effec of AngularDisplacemen on he dampedpendulumwihou he angulardriving force WriingE. () wihou he angulardriving force, we have d d ml mgl sin 0 (8) The non-dimensionalformbecomes d d sin 0 (9) 0
sin 0 w wsin 0 w wsin sin (0) () () w sin (3) 4.3 Effec of AngularDisplacemen on he UndampedPendulumwihou he angulardriving force WriingE. () wihou he dampingerm and angulardriving force, we have d ml mgl sin 0 (4) The non-dimensionalformis d sin 0 (5) sin 0 (6) w sin (7) sin (8) dw dw d dw w w d d (9) dw w sin (30) d (3) wdw sin d w cos, (3) 03
w cos, (33) 5. NUMERICAL RESULTS AND DISCUSSION The numericalcalculaions have been carried ou for a dampeddrivenpendulum. The following values wereused : p=, =,, 0.Inableareshownhevalues,inermsofime, ofheangularvelociyforhedampeddrivenpendulumadifferenvaluesofangulardisplacemen andangulardrivenforcephase.iisobservehaheangularvelociyhashesamemagniudebu opposiedirecion,for=30,=30and=60,=60. Figures (a)-(g) are he angular velociy a various imes, for paricular values of angular displacemen and angular driving force. Figure 3, on he oher hand shows he angular velociy a various values of angular displacemen, angular driving force and ime. We can see clearly ha he curve are asympoic o paricular values of he angular velociy, which implies a some poin in ime he rae of change of angular velociy becomes zero. For various values of he angular displacemen and fixed value of angular driving force, he angular velociy for he nine special cases considered, were calculaed and are ploed in Figure 4, a a paricular ime. Also for various values of he angular driving force and fixed value of angular displacemen, he angular velociy for he eigh special cases considered, were calculaed and were ploed in Figure 5, a a paricular ime. I can be seen ha he minimum angular velociy occurred a = for differen values of. Figure 6 shows he angular velociy of undamped, wihou any driving force, a differen values of he angular displacemen. Clearly he angular velociy is highes when he angular displacemen is 0 degree, and lowes when i is -90 and 90 degrees. I can be noed from figure 5 ha he lower he fixed value of he angular driving force he higher he angular velociy, a various values of he angular displacemen. Also he smaller he fixed value of he angular displacemen he smaller he angular velociy, a various values of angular driving force, a a paricular ime. This implies here is a negaive correlaion beween he angular velociy and he angular driving force. Similarly, here is a posiive correlaion beween he angular velociy and he angular displacemen. Furhermore, he angular velociy is highes when he angular displacemen is 0 degree. Inuiively, we can imagine ha he angular velociy occurs when he oscillaing sysem (pendulum) has reached he euilibrium posiion (angular displacemen euals zero) and is abou o overshoo. 04
Table. The values of w for differen values of and f w e 0 0 3 e 30 30 45 0 e 45 45 0 30 60 0 60 30 0 60 60 3 e 0 90 0 90 0 0 90 90 e 80 360 e -80 80 e 0 80 e -30 30 3 e 60 70 3 e (a) (b) 05
(c) (d) (e) (f) (g) Figure (a) (g): Graphs of angularvelociy w agains ime. 06
Figure 3. The angularvelociy of he pendulum a differen values of, f and ime. Figure 4. The angular velociy of he damped driven pendulum for fixed angular displacemen and differen values of angular driving force a a paricular ime 07
Figure 5.The angular velociy of he damped driven pendulum for fixed angular driving force and differen values of angular displacemen a a paricular ime Figure 6. The angularvelociy of undampedpendulum a differen values of angulardisplacemen 08
6. CONCLUSION This paper has oulinedamehod o analyse he dynamics of dampeddrivenpendulum. Also o compare he dynamics of he cases of undamped, driven and damped bu no drivenpendulums. I isshownhaangularvelociy and angular force phase are negaivelycorrelaed, whilehereis a posiive correlaionbeween he angularvelociy and he angulardisplacemen. In he case of whenangulardisplacemeniszero, he angularvelociyassumedishighes value. I isalsoobserveha he angularacceleraion of he dynamic sysem becomeszero a some poin in ime. Concerning he angularvelociy ; iisnoiceha he value of he angularvelociyis minimum when he driving force is radian. The sudy has conribued o scienific knowledge by showing he effecs of damping and driving force on he dynamics of pendulum as a dynamical sysem. Also he effec of angle of displacemen on he angular velociy, in addiion o relaionships of he parameers of a dynamical sysem have been oulined. This sudy has herefore shown ha hisdynamical sysem phenomenonis of greapracical imporance. REFERENCES [] Baker, G.L., Gollins, J.P., 996, Chaoic Dynamics : An inroducion, Cambridge UniversiyPress, NY. [] Davies, B., 999, Exploring Chaos: Theory and Experimen, Persues Books, MA. [3] Davidson, G.D., 0, The dampeddrivenpendulum: Bifurcaion analysis of experimenal daa, The Division of Mahemaics and Naural Sciences, Reed College. [4] Tsneg, Z.S., 008, The Laplace Transform. [5] Russell, D.A., 04, Acousics and Vibraion Animaions, Graduae Program in Acousics, The Pennysylvania Sae Universiy. [6] Richard Fizparic, 006, The simple pendulum, farside.ph.uexas.edu/eaching/ 30/Lecures/node40.hml [7] David Darling, Classical Mechanics, The Worlds of David Darling, Encyclopedia of AlernaeEnergy www.daviddarling.info>encyclopedia of Science [8] Waler Lewin, 999, Pendulum Euaion, MiopenCourseware, Massachuses Insiue of Technology [9] Peer Dourmashkin, J. David Liser, David Prichad and Bernard Surrow, 004, Pendulums and Collisions, Miopencourseware, Massachuses Insiue of Technology [0] Gray D. Davidso, 0, The DampedDrivenPendulum : Bifurcaion Analysis of Experimenal Daa. A Thesis for he Division of Mahemaics and Naural Sciences, Reed College. [] John H. Hubbard, The forceddampedpendulum: Chaos, Complicaion and Conrol. hp://www.mah.cornell.edu/-hubbard/pendulum.pdf. [] Dehui Qui, Min Rao, QirunHuo, Jie Yang, 03, DynamicalModeling and Sliding Mode Conrol of Invered Pendulum Sysems, Inernaional Journal of AppliedMahemaics and Saisics, 45(5), 35-333. 09