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1、Beadmovingalongathin,rigid,wire.RodolfoR.Rosales,DepartmentofMathematics,MassachusettsInst.ofTechnology,Cambridge,Massachusetts,MA02139October17,2004AbstractAnequationdescribingthemotionofabeadalongarigidwireisderived.Firstthecasewithnofrictionisconsidered,andaLagrangianformulationisusedtoderive
2、theequation.Nextasimplecorrectionfortheeffectoffrictionisaddedtotheequation.Finally,weconsiderthecasewherefrictiondominatesoverinertia,andusethistoreducetheorderofthesystem.Contents1Equationswithnofriction.2Principleofleastaction..................................3Equationforthemotionofthebead....
3、.......................4Beadonaverticalrotatinghoop.............................4Problem1.1:Beadonanhorizontalrotatinghoop..................4Wireonverticalplane,rotatingaroundaverticalplane................5Problem1.2:Wiresrotatingwithvariablerates....................5Parametricinstabilities.....
4、.............................5TheEulerLagrangeequation:derivation.........................6Problem1.3:Hoopmovingupanddowninaverticalplane.............6Parametricstabilization..................................6Wirerestrictedtoafixedverticalplane.........................72Addfrictiontotheequations.
5、7Equationofmotionwithfrictionincluded........................8Addfrictiontotheverticalrotatinghoop........................8Wirerestrictedtoafixedverticalplane.........................8Wiremovingrigidlyupanddown.............................9Slidingwiresandfriction.................................9
6、1RosalesBeadmovingalongathin,rigid,wire.23Frictiondominatesinertia.103.1Nondimensionalequations.................................103.2Limitoflargefriction...................................104ProblemAnswers.111Equationswithnofriction.ConsiderthemotionofabeadofmassMmovingalongathinrigidwire,undert
7、heinfluenceofgravity.Let(x,y,z)beasystemofCartesiancoordinates,withztheverticaldirectionandzincreasingwithheight.Letgbetheaccelerationofgravity.Describethewireinparametricform,asfollows:x=X(s,t),y=Y(s,t),andz=Z(s,t),(1.1)wher
