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TEMPLATE DESIGN © 2008
www.PosterPresentations.com
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.