Computer-Simulation-of-Stick-Slip-Friction-in-Mech

willcanceltheswing.Stoppinganywherebetweenthesepointswillresultinresidualoscillation.Thevariationinoscillationamplitudewithtransportvelocityfortheswing-freetrajectoryisbelievedtobeprimarilyduetosmallrandomoscillationoftheshellbeforetransportmotionwasbegun.Thisviolatesthezeroinitialconditionsconstraintoftheanalysis.Theserandomoscillationswerecausedbyaircurrentsfromthebuildingventilatingsystem.ConclusionsAmethodhasbeenshownfortransportingsuspendedobjectswithapath-controlledrobotmanipulatorsuchthattheobjectsarestationaryattheendofthemotion.Thismethodrequiresamanipulatorcapableofconstant-velocitystraight-linemotionandanaccelerationtimethatissmallcomparedtothenaturalperiodofthesuspendedobject.ThemethodhasbeensuccessfullyusedwithaPUMA600manipulatortotransporthalf-sizewoodenmodelsof8-inchartilleryshellspriortoinsertingthemintoaperforatedpallet.Reference1Shames,I.H.,EngineeringMechanics,Vol.II,Dynamics,SecondEdition,Prentice-Hall,Inc.,EnglewoodCliffs,NewJersey,1966.ComputerSimulationofStick-SlipFrictioninMechanicalDynamicSystemsDeanKarnopp1Stick-slipfrictionispresenttosomedegreeinalmostallactuatorsandmechanismsandisoftenresponsibleforper­formancelimitations.Simulationofstick-slipfrictionisdifficultbecauseofstronglynonlinearbehaviorinthevicinityofzerovelocity.Astraightforwardmethodforrepresentingandsimulatingfrictioneffectsispresented.Truezerovelocitystickingisrepresentedwithoutequationreformulationortheintroductionofnumericalstiffnessproblems.1IntroductionIfafrictionforceFfisplottedversustherelativevelocityVbetweentwocontactingsurfaces,avarietyofrelationshipsappearinpractice.Bystick-slipfriction,wemeanhereadryfrictionlawsuchasthatshowninFig.1(a)inwhichFfatV=0canachievealargervaluethanatmoderatevaluesofV.Stick-slipfrictionoftenresultsinundesirableunstabledynamicbehavior.Thechatteringofmechanismsandhuntingofspeedgovernorsareoftencausedbystick-slipfrictionforexample.AsFig.1(a)makesclear,itisdifficulttolinearizeadryfrictionlawnearV=0.Thismeansthatlineardynamicmodelscannotbeexpectedtopredicteffectssuchaschat­teringduringslewing,stickingbehavioratfinalpositioning,orlimitcyclingaroundafinalpositionwhichcanbecausedbystick-slipfriction.ItislogicaltousecomputersimulationstostudytheeffectsDepartmentofMechanicalEngineering,UniversityofCalifornia,Davis,Calif.95616.ContributedbytheDynamicSystemsandControlDivisionofTHEAMERICANSOCIETYOFMECHANICALENGINEERS.ManuscriptreceivedatASMEHead­quarters,December20,1984.Fig.1Stick-slipfrictionlawsofstick-slipfriction,buttherepresentationofafrictionlawlikethatshowninFig.1(a)isnotentirelystraightforward.Neitherofthecausalinput-outputformsFj=Fy(V)orV=V{Fj)isunique.Figure1(b)showsaversionofthefrictionlawinwhichFfisauniquefunctionofV,butthislawhasphysicalandnumericalproblems.Anotherapproachinvolvesswitchinginput-outputcausalities.Whenthesystemslips,V^0andFy=Ff(V).AtV=0,thesystemsticksandiytakesonvaluesdeterminedbyotherelementsofthesystemmodel.IfFsexceedsthebreakawayforcelevel,thenthesystemswitchesbacktotheslipmode.Hereweshowhowtopresenttruestick-slipbehaviorwithconstantcausalityandhencewithasinglesetofstateequations.Usefulresultsareobtainedwithoutexcessivelyshorttimesteps.100/Vol.107,MARCH1985TransactionsoftheASMECopyright©1985byASMEDownloaded01May2012to138.25.78.25.RedistributionsubjecttoASMElicenseorcopyright;see~P,VFy»///rT7-CO.)T:*\—^0—^'IT7-*a)Fig.2MassacteduponbyanetforceandfrictionIITheBasicModelofStick-SlipFrictionThefrictionlawtobeusedissketchedinFig.1(c).Inasense,thefrictionforceFfisalwaysafunctionofV.Aregionofsmallvelocityisdefinedas-DVVDV.Outsidethisregion,FfisanarbitraryfunctionofV.InsidethesmallregionsurroundingV=0,weconsiderVtobezero.(Thefiniteregionisnecessaryfordigitalcomputationsinceanexactvalueofzerowillnotbecomputed.)InsidetheV=0region,FjisdeterminedbyotherforcesinthesysteminsuchawaythatVremainsintheregionuntilthebreakawayvalueofforceisreached.TheconceptisbestillustratedusingthesimpleexampleofFig.2.Considerthemass,m,withvelocityV,momentumP,andfrictionforceFf.TheforceFisthenetforceonmfromtherestofthesystem.Thebondgraphofpart(b)showsthestructureofthesystem(seereference[1]).AlthoughtheconstantcausalityFf=Ff(V)isassumed,theactivebond(orsignalflow)showsthattheforceFalsohasaninfluenceonthecomputationofFf.Theextra2-portO-junctionmerelyservestodefineFastheactivebondvariable.TheblockdiagramofFig.2(c)correspondstothebondgraphofpart(b).NotethatFfiscomposedoftwoparts.If+DVV-DV,thenFf=Fslip.Theothercomponent,alimitedversionofS'Fstickiszerointhiscasesincetheswitchvariable,S,vanishesintheslipregion.Ontheotherhand,inthestickregion,-DVV+DV,FsUp=0andS=1.SoFf-Fstick,butlimitedtothebreakawaylevels±FH.K4K-ecuF^rft,)Cc)Fig.3Frictioninvolvingrelativemotionv0m^Pi.Vi^^o-^VVV•m-iFz(t)7/Fry//-r/j//t/7-~Zyfr?//?\FT-(a)CJIsiv.1flfy6-O^fi&V,o-^1It«.!'—T(t)o*llov^o(a)5z=iCO-S2=014-7*0Sy=ovS2=^Cd)Si-0(c)Fig.4ComplexexamplewithinteractingfrictionforcesJournalofDynamicSystems,Measurement,andControlMARCH1985,Vol.107/101Downloaded01May2012to138.25.78.25.Redistribu