机械外文.docx
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机械外文.docx
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机械外文
Kinematicanddynamicsynthesisofaparallelkinematichighspeeddrillingmachine
ReuvenKatz;ZheLi
Abstract
Typically,theterm“highspeeddrilling”isrelatedtospindlecapabilityofhighcuttingspeeds.Thesuggestedhighspeeddrillingmachine(HSDM)extendsthistermtoincludeveryfastandaccuratepoint-to-pointmotions.ThenewHSDMiscomposedofaplanarparallelmechanismwithtwolinearmotorsastheinputs.Thepaperisfocusedonthekinematicanddynamicsynthesisofthisparallelkinematicmachine(PKM).Thekinematicsynthesisintroducesanewmethodologyofinputmotionplanningforidealdrillingoperationandaccuratepoint-to-pointpositioning.ThedynamicsynthesisaimsatreducingtheinputpowerofthePKMusingaspringelement.
Keywords:
Parallelkinematicmachine;Highspeeddrilling;Kinematicanddynamicsynthesis
1.Introduction
Duringtherecentyears,alargevarietyofPKMswereintroducedbyresearchinstitutesandbyindustries.Most,butnotall,ofthesemachineswerebasedonthewell-knownStewartplatform[1]configuration.Theadvantagesoftheseparallelstructuresarehighnominalloadtoweightratio,goodpositionalaccuracyandarigidstructure[2].ThemaindisadvantagesofStewarttypePKMsarethesmallworkspacerelativetotheoverallsizeofthemachineandrelativelyslowoperationspeed[3,4].Workspaceofamachinetoolisdefinedasthevolumewherethetipofthetoolcanmoveandcutmaterial.ThedesignofaplanarStewartplatformwasmentionedin[5]asanaffordablewayofretrofittingnon-CNCmachinesrequiredforplasticmouldsmachining.ThedesignofthePKM[5]allowedadjustablegeometrythatcouldhavebeenoptimallyreconfiguredforanyprescribedpath.Typically,changingthelengthofoneormorelinksinacontrolledsequencedoestheadjustmentofPKMgeometry.
TheapplicationofthePKMswith‘‘constant-lengthlinks’’forthedesignofmachinetoolsislesscommonthanthetypewith‘‘varying-lengthlinks’’.Anexcellentexampleofa‘‘constant-lengthlinks’’typeofmachineisshownin[6].Renault-AutomationComauhasbuiltthemachinenamed‘‘UraneSX’’.TheHSDMdescribedhereinutilizesaparallelmechanismwithconstant-lengthlinks.
Drillingoperationsarewellintroducedintheliterature[7].Anextensiveexperimentalstudyofhighspeeddrillingoperationsfortheautomotiveindustryisreportedin[8].Datawascollectedfromhundredscontrolleddrillingexperimentsinordertospecifytheparametersrequiredforqualitydrilling.Idealdrillingmotionsandguidelinesforperforminghighqualitydrillingwerepresentedin[9]throughtheoreticalandexperimentalstudies.InthesynthesisofthesuggestedPKM,wefollowthesuggestionsin[9].
ThedetailedmechanicalstructuresoftheproposednewPKMwereintroducedin[10,11].OnepossibleconfigurationofthemachineisshowninFig.1;ithaslargeworkspace,highspeedpoint-to-pointmotionandveryhighdrillingspeed.TheparallelmechanismprovidesY,andZaxesmotions.TheXaxismotionisprovidedbythetable.Forachievinghighspeedperformance,twolinearmotorsareusedfordriving
themechanismandahighspeedspindleisusedfordrilling.Thepurposeofthispaperistodescribenewkinematicanddynamicsynthesismethodsthataredevelopedforimprovingtheperformanceofthemachine.Throughinputmotionplanningfordrillingandpoint-to-pointpositioning,themachiningerrorwillbereducedandthequalityofthefinishedholescanbegreatlyimproved.Byaddingawell-tunedspringelementtothePKM,theinputpowercanbeminimizedsothatthesizethemachineandtheenergyconsumptioncanbereduced.Numericalsimulationsverifythecorrectnessandeffectivenessofthemethodspresentedinthispaper.
2.KinematicanddynamicequationsofmotionofthePKMmodule
TheschematicdiagramofthePKMmoduleisshowninFig.2.Inconsistentwiththemachinetoolconventions,thez-axisisalongthedirectionoftoolmovement.ThePKMmodulehastwoinputs(twolinearmotors)indicatedaspart1andpart6,andoneoutputmotionofthetool.ThepositioninganddrillingmotionofthePKMmoduleinthisapplicationischaracterizedby
(yaxismotionforpoint-to-pointpositioning)and
(zaxismotionfordrilling).MotionequationsforbothrigidbodyandelasticbodyPKMmodulearedeveloped.Therigidbodyequationsareusedforthesynthesisofinputmotionplanningofdrillingandinputpowerreduction.Theelasticbodyequationsareusedforresidualvibrationcontrolafterpoint-to-pointpositioningofthetool.
2.1.EquationsofmotionofthePKMmodulewithrigidlinks
Usingcomplex-numberrepresentationofmechanisms[12],thekinematicequationsofthetoolunit(indicatedaspart3whichincludestheplatform,thespindle
andthetool)aredevelopedasfollows.Thedisplacementofthetoolis
and
wherebisthedistancebetweenpointBandpointC,risthelengthoflinkAB(thelengthsoflinkAB,CDandCEareequal).Thevelocityofthetoolis
where
Theaccelerationofthetoolis
where
ThedynamicequationsofthePKMmodulearedevelopedusingLagrange’sequationofthesecondkind[13]asshowninEq.(7).
whereTisthetotalkineticenergyofthesystem;
and
arethegeneralizedcoordinatesandvelocities;
isthegeneralizedforcecorrespondingto
.kisthenumberoftheindependentgeneralizedcoordinatesofthesystem.Here,k=2,q1=y1andq2=y6.Afterderivation,Eq.(7)canbeexpressedas
wherenisthenumberofthemovinglinks;
aremassandmassmomentofinertiaoflinki;
arethecoordinatesofthecenterofmassoflinki;hiistherotationangleoflinkiinthePKMmodule.Thegeneralizedforce
canbedeterminedby
whereVisthepotentialenergyandF’iarethenonpotentialforces.ForthedrillingoperationofthePKMmodule,wehave
whereFcutisthecuttingforce,F1andF6aretheinputforcesexertedonthePKMbythelinearmotors.Eqs.
(1)to(10)formthekinematicanddynamicequationsofthePKMmodulewithrigidlinks.
2.2.EquationsofmotionofthePKMmodulewithelasticlinks
Thedynamicdifferentialequationsofacompliantmechanismcanbederivedusingthefiniteelementmethodandtaketheformof
where[M],[C]and[K]aresystemmass,dampingandstiffnessmatrix,respectively;{D}isthesetofgeneralizedcoordinatesrepresentingthetranslationandrotationdeformationsateachelementnodeinglobalcoordinatesystem;{R}isthesetofgeneralizedexternalforcescorrespondingto{D};nisthenumberofthegeneralizedcoordinates(elasticdegreesoffreedomofthemechanism).InourFEAmodel,weuseframeelementshowninFig.3inwhichEIeisthebendingstiffness(Eisthemodulusofelasticityofthematerial,Ieisthemomentofinertia),qisthematerialdensity,leis
theoriginallengthoftheelement.
arenodaldisplacementsexpressedinlocalcoordinatesystem(x,y).Themassmatrixandstiffnessmatrixfortheframeelementwillbe66symmetricmatriceswhichcanbederivedfromthekineticenergyandstrainenergyexpressionsasEqs.(12)and(13)
whereTisthekineticenergyandUisthestrainenergyoftheelement;
arethelinear123456andangulardeformationsofthenodeattheelementlocalcoordinatesystem.Detailedderivationscanbefoundin[14].Typically,acompliantmechanismisdiscretizedintomanyelementsasinfiniteelementanalysis.Eachelementisassociatedwithamassandastiffnessmatrix.Eachelementhasitsownlocalcoordinatesystem.Wecombinetheelementmassandstiffnessmatricesofallelementsandperformcoordinatetransformationsnecessarytotransformtheelementlocalcoordinatesystemtoglobalcoordinatesystem.Thisgivesthesystemmass[M]andstiffness[K]matrices.Capturingthedampingcharacteristicsinacompliantsystemisnotsostraightforward.Eventhough,inmanyapplications,dampingmaybesmallbutitseffectonthesystemstabilityanddynamicresponse,especiallyintheresonanceregion,canbesignificant.Thedampingmatrix[C]canbewrittenasalinearcombinationofthemassandstiffnessmatrices[15]toformtheproportionaldamping[C]whichisexpressedas
whereaandbaretwopositivecoefficientswhichareusuallydeterminedbyexperiment.Analternatemethod[16]ofrepresentingthedampingmatrixisexpressing[C]as
Theelementof[C’]isdefinedas
wheresignKij=(Kij/|Kij|),KijandMijaretheelementsof[K]and[M],ζisthedampingratioofthematerial.
Thegeneralizedforceinaframeelementisdefinedas
whereFjandMjarethejthexternalforceandmomentincludingtheinertiaforceandmomentontheelementactingat(xj,yj),andmisthenumberoftheexternalforcesactingontheelement.Theelementgeneralizedforces
arethencombinedtoformthesystemgeneralizedforce{R}.Thesecondorderordinarydifferentialequationsofmotionofthesystem,Eq.(11),canbedirectlyintegratedwithanumericalmethodsuchasRunge-Kuttamethod.ForthePKMwestudied,eachlinkwasdiscretedas15frameelements.BothMatlabandADAMSsoftwareareusedforprogrammingandsolvingtheseequations.
3.Inputmotionplanningfordrilling
Supposeweknowtheidealmotionfunctionofthedrillingtool.Howtodeterminetheinputmotormotionsothattheidealtoolmotioncanberealizediscriticalforhighqualitydrillings.Thecreatedexplicitinputmotionfunctionalsoprovidesthenecessaryinformationformachinecontrols.Accordingtothestudydonein[9],thedrillingprocesscanbedividedintothreephases:
entrancephase,middlephase,andexitphase.Inordertoincreasetheproductivityandqualityofthedrilling,manyoperationconstraintssuchasminimumtoollifec
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