TEMPLATE DESIGN © 2008

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TEMPLATE DESIGN © 2008

www.PosterPresentations.com

Modeling Non-Linear Behavior of Physical Pendulum

Brian Swain, Peter Chanseyha, Younes AtiiayanSanta Rosa Junior College

ABSTRACT

THEORY

SETUP

RESULTS AND DISCUSSIONS

REFERENCES

Acknowledgements

Amathematicalmodelwasdevelopedtopredict the non-linear oscillation of a physicalpendulum. The model was tested against

experimentaldataobtainedbyusingawaterfilled,long,cylindricaltubeandallowingwatertoescapefromthebottomofthetubewhileitwassetonoscillation. After making corrections for theexponentialdecayoftheoscillationamplitude,the

theorywasfoundtobeingoodagreementwiththeexperimentaldata.Asthefirstapproximation,theexponentialdecaywasassumedtobeconstantaswellasthevelocityofthewaterexitingthetube.Toachievebetteragreementbetweentheexperimental

dataandthemodel,itwasnecessarytotakeintoaccountthevariationofthewatervelocityaswellasthedampingcoefficient.

The objective of this experiment was to test amodel, which was developed for an oscillatingphysical pendulum with a changing center ofmass. Understanding the non-linear behavior ofphysical pendulum is a major step inunderstanding and controlling motion of somemechanicalsystemssuchasrobotsandcranesRef-1.In a standard physics textbook Ref-2, a physicalpendulumismodeledusingthemomentofinertia.For this model we modified the formula todevelopatimedependentangularvelocityforthesystem. We began with the differential equationdescribing the motion of damped physical

pendulumRef-3: . In this

equation, D is the distance between the pivotpoint and the center of mass, I is the moment oftheinertiaofthependulum,m isthemassofthependulum and b is the drag coefficient. The

solution to the differential equation would be:

. Since experimentally

we will be used a motion detector to analyzedisplacement of the pendulum, the previousequation can be re-written in terms ofdisplacementas:

Equation1

!!q + b !q +

mgD

Iq = 0

q = qmaxe- b t cos

mgD

It + j

æ

èç

ö

ø÷

x = xmaxe- bt cos

mgD

It +j

æ

èç

ö

ø÷

In order to model the non-linearpendulum, we must make and timedependent. To achieve this, the column of waterwas modeled as a cylinder pivoted at an offsetfrom its central diameter. The moment of inertiafor a cylinder pivoted at its center of mass wasderived using the perpendicular axis theorem to

be . The offset is accounted

for by the parallel access theorem giving us

, where is the

radiusofthecolumnofwaterand is itslength.Asthewaterdrainsfromthecylinder, becomestime dependent and can be written as:

with being the initial length of

the column of water and being the velocity ofthe water level as it drains. Also, as the water

drains, , with being the

initialdistancebetweenthelevelofthewaterandthepivotpoint.So,ourtimedependentmomentofinertiabecomes:

Equation2Putting everything together, our final

equationbecomes:

Equation3During the preliminary testing of the

modelweusedtheaveragevalueof (described

in the setup section). We believe this would be areasonable approximation of , even though

graduallydecreasesastheweightofthependulumand the speed of the pendulum decrease. Inaddition,initiallyweusedanaveragevalueforthespeed of the surface of the water’s displacement,even though graphical analysis of the water levelreveals otherwise. An attempt was made toincorporate the variation of surface velocity as afunctionoftimeforfine-tuningthemodel.

I D

Icm =1

4mR2 +

1

12mL2

I =1

4mR2 +

1

12mL2 + mD2 R

LL

L(t) = L0 - vt L0

v

D(t) = d +1

2L0 +

1

2vt d

I (t) =1

4mR2 +

1

12m(L0 - vt)2 + m d +

1

2L0 +

1

2vt

æ

èçö

ø÷

2

x(t) = xmaxe- b t cos

g d +1

2L0 +

1

2vt

æ

èçö

ø÷

1

4R2 +

1

12(L0 - vt)2 + d +

1

2L0 +

1

2vt

æ

èçö

ø÷

2× t

æ

è

çççç

ö

ø

÷÷÷÷

b

b b

Inthisexperiment,acylindricaltubewithaholeatthebottomwasusedtoinvestigatenon-linear behavior of physical pendulum. Thecylinder was filled with dyed water, which wasallowedtopouroutofthetubewhilethetubewasset to oscillation. Placing washers of variousdiametersatthebottomofthetubecontrolledtherate at which the water was flowing out of thetube. A motion detector, connected tocomputer, was placed in line with the horizontaldisplacement of the pendulum. The programLoggerPro® was used to collect and analyze thedata obtained from the motion detector. Also, acamera was placed in front of the pendulum. Avideo taken of the gradual decline of the waterlevelwasusedtomeasurehowfastthewaterwasdraining.Thisinformationhelpedustodetermine,and make some corrections. The drag coefficient,

,wasfoundbyanalyzingtheexponentialdecay

characteristics of the pendulum. This was donewhilethependulumwasfullandthenwhileitwasempty. The average value of the drag coefficientobtained from these data was used as the dragcoefficient for analyzing the oscillating motion ofthenon-linearpendulum.

b

Figure1:Experimentalset-upinmotion.

Afterconductingseveraltrialswithvariouswasherholediameters,theresultsofoneoftheexperimentaltrialsareshowninfigure2.Thedata

generatedusingEquation3issuperimposednexttotheexperimentaldata.Ingeneratingthetheoretical

data, values of b = 0.04329s-1 and a velocity

vave = 0.024m / s wereused.Theseparametersare

theaveragevaluesempiricallyobtainedasdescribed

in the setup section. In addition, measureddimensions of d = 0.033m , R = 0.02088m ,

L0 = 0.305m ,andxmax = 0.01938m wereusedfor

generatingthetheoreticalvalues.

As can be seen from this figure,experimentaldataisinitiallyinagreementwiththemodel, but as time passes, the frequency of

oscillationdeviatesnoticeablyfromthemeasureddata.Thisbehaviorismainlyduetothespeedofthewaterexitingthetubeataratethatisdecreasing.Analysis of thewater level using the graphicalanalysis program, Tracker, resulted in the time

dependentequationforthevelocityofthesurfaceofthewaterasv = 0.032 – 0.0014t .

Aftermakingthenecessarycorrectionsforthetimedependencyofthevelocity,aclosermatchbetweentheexperimentaldataandthemodeled

oscillationofthenon-linearpendulumwasfound(figure3).Webelievefurtherimprovementcanbeachievedbyreplacingtheaveragevalueof with

its time dependent equation, similar to theimprovementmadebyusingthetimedependentequationforvelocity.

b

1:Rev.Bras.Biom.,SãoPaulo,v.24,n.4,p.66-84,2010

2:Serway,RaymondA.,Jewett,JohnW.,PhysicsforScientistsand

Engineers.8thEdition.PacificGrove,Calif.;London:Brooks/Cole,2010.Print.

3:DennisG.Zill.AFirstCourseinDifferentialEquationswith

ModelingApplications.PacificGrove,CA:Brooks/ColeThomson

Learning,2001.Print.

TheauthorswishtothankDarciRosales(directorofMESA)forherinvaluablesupportthroughouttheprojectandtheChemistry/PhysicsandEngineering

departmentsandSRJCfoundationfortheirfinancialsupport.

Figure2:Modelwithouttimedependentvelocity.

Figure3:Modelwithtimedependentvelocity.