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    Research

1258Chenotal/JZh~iangUnivSciA20089f9J:1258-1263

    ;JournalofZhejiangUniversitySCIENCEA

    ;ISSN1673-565X(Print);ISSN18621775(Online)

    ;www.zjueducn~zus;wwwspringedinkcom

    ;E-mail:jzus@zju.educn

    ;Researchinnon.equalizationmachiningmethodforspatialcam

    ;JunhuaCHEN.Yi-jieWU

    ;fInstituteofMechanicalandElectricalEngineering,NingboInstituteofTechnology,ZhefiangUniversity,Ningbo315010,China)

    ;(2InstituteofModernManufactureEngineering,ZhejiangUniverSi.Hangzhou31002LChina)

    ;E-mail:cjh@nit.zju.edu.cn;wyjl116@zju.edu.cn

    ;ReceivedDec.3,2007;revisionacceptedMar.24,2008

    ;Abstract:Manykindsofdeviceswithcamhavebeenwidelyusedinvariousmechanicalequipments.Howeve~

    ;nonequalizationmachiningforspatialcamtrou

    ghremainstobeadifficultproblem.Thispaperfocusesontheanalysisofrunning ;conditionsandmachiningprocessesofspatialcamwithoscillatingfollower.Wepointoutthecommonerrorsinthebiaseddistance

    ;curing.Byanalyzingthemotionofoscillatingfollowerofspatialcam,wepresentanew3Dcurveexpansionmodelofspatialcain

    ;troughoutline.Basedonthismodel,wehaveproposedamachiningmethodfortrochoidalmillingwithnonequalizationdiameter

    ;curer.Thisnewmethodhasledtoacreativeandeffectivewayfornonequalizationdiamet

    ermachiningforspatialcamwith

    ;oscillatingfollower.

    ;Keywords:Oscillatingfollower,Spatialcam,3Dcurveexpansion,Trochoidalmilling,Nonequalizationdiameter

    ;doi:10.1631/jzus.A0720097Documentcode:ACLCnumber:TH122;TG711 ;INTRODUCTION

    ;Accordingtothedifierentmotionstyrlesoftheir

    ;followers.spatia1camsaregenerallydividedintotwo

    ;categories.i.e.,camwithtranslatingfollowerandcam

    ;withoscillatingfollower.Designingandmachining

    ;forspatialcamwithtranslatingfollowerarerelatively

    ;easybasedonitsexpandedplanefigure.whereas

    ;designingandmachiningforspatialcamwithoscil

    ;latingfollowerarecomplicatedanddifficult.The

    ;latterisespeciallytruefornonequalizationdiameter

;machining(alsocalledunequivalentmachining,i.e.,

    ;whenthewidthofcamtroughisbiggerthanthatof ;machiningcutter.thetroughoutlineneedstobema

    ;chinedatsinglesidewithsmalledgecutter).During ;therapidandprecisemanufacturingofcamtrough, ;onehastoensureitsexactdimensionandsurface ;}Correspondingauthor

    ;„ProjectsupposedbytheNationaINaturalScienceFoundationof

    ;China(No.50575205).theHi.1-echResearchandDevelopmentPro.

    ;gram(863)ofChina(No.2006AA04Z233).theNaturalScience ;FoundationofZhejiangProvince(No.Y104243).andtheNatural ;ScienceFoundationofNingbo(Nos.2008A610038and200703B ;10030181.China

    ;quality.Thecommonpracticeinproductionisfirstto ;machinecoarselywithamillingcurerwhosedi

    ;ameterissmallerthantheroller‟sdiameter,thento

    ;preciselymachineitssinglesidewithsmallmilling ;cutterorgrindingwheeltoensurethatcamtroughcan ;fitinexactlywichfollowerandtoimprovethemotion ;precisionofspatialcam.Asthespatialcamfigureisa ;complicatedspatialcurvewhichcannotbeexpanded, ;thenonequalizationdiametermachiningisadi~cult ;problemandcatchestheattentionofdomesticand ;oversearesearchers.

    ;Hsieh(2007)presentedasimpleyetcompre

    ;hensivemethodforthedesignandmachiningofa ;cylindricalcamwithameshingindexingdisc.Chen ;andWu(2007)showedthafthemotionDachofta

    ;peredrolleronoscillatingbarcouldbeexpandedon ;thecylinderandpresentedamethodfordesigning ;andmachiningof3Dcurveexpansionofconicalcam ;withoscillatingfollower.LeeandLee(2007)pro

    ;posedaninterferencefreetoolpathgenerating

    ;methodforfiveaxismachiningofaspatialcam.

    ;ChenandXin(2007)appliedthe3DCurveexpansion ;systemtothecylindricalcamdesignandmanufac

    ;turing.YinetaL(2002)usedthewebbasedremote

    ;

    ;Cheneta1./dZheflangUnivSciA20089f9J:12581263

    ;designsystemtoformulatespatialcammechanisms ;basedonmathematicalmodels.Leerl998)presented ;amethodforfindingtheadmissibletoolorientation ;byconsideringbothlocalgougingandreargouging. ;Geeta1.(2006)deducedthecontoursurfaceequation

    ;ofspatialcambasedonpurerollingconditionof ;rolleronthecamcontoursurface.LiandYin(2003) ;proposedthatthecontoursurfaceofcylindricalcam ;couldbeexpandedfromlinearsurface.Grantand ;Sonirl999)workedouttheanalyticexpressionfor ;camcontoursurfaceofconicalcammechanismwith ;oscillatingtaperedrollerfollowerin3Dspaceby ;applyingthetheoryofenvelopestooneparameter ;familyofsurfaces.

    ;However,allthesestudiesarelimitedtothe ;questionashowtoestablishthecamfiguremodels. ;Theinformationfordesignandmachiningisstill ;mostlyatthestageoftheoreticalanalysis,whichhas ;seriouslylimiteditspracticalapplication.Moreover, ;precisemachiningofspatialcamwithoscillating ;followerisespeciallyhardtoachieve.

    ;Byfocusingonspatialcammechanismwithos- ;cillatingfollower,thispaperpresentsthatthemotion ;canbedividedintotwocategories.i…etheoscillating

    ;motionofoscillatingbarandtherotarymotionof ;oscillatingbarrelativetospatialcam.Withtheex- ;pansionofrotarymotionofoscillatingbar,wepro- ;poseanew3Dcurveexpansionmodelofspatialcam ;trough-outline,andbasedonit,suggestamachining ;methodfortrochoidalmillingwithnon.equalization ;diametercurer.Thisnewmethodhasledtoacreative ;andeffectivewayformachiningofspatialcamwith ;preciseoscillatingfollower.

    ;ANALYSISOFTHERIINGCOURSEOFTHE

    ;SPATIALCAMWITHOSCILLATINGFOL-

    ;LOWER

    ;Themodelofspatialcammechanismwithos-

    ;cillatingfollowerisshowninFig.1.Theoscillating ;axisofoscillatingbarintersectswiththerotaryaxisof ;spatialcamindifferentspaces.LineVisthecom- ;monperpendicularlineofthesetwoaxes,withNas ;thepointofintersectionbetweencommonperpen- ;dicularandrotaryaxisofspatialcamastheoriginof ;coordinateframes.Therotaryaxisofspatialcamisthe ;X-axisandthecommonperpendicularisthey-axis. ;1259

    ;Hereisthehypothesis:assumethatSisthedis- ;placementofrollerofoscillatingbar(mm);the ;oscillatingangleofoscillatingbar(.),,thelengthof

    ;oscillatingbar(mm),therotaryangleofspatialcam ;fo),thedistancefromtherotaryaxisofoscillating ;bartotherotaryaxisofspatialcamfi.e.,thelengthof ;commonperpendicularline)(mm).Thetroughin ;thefollowerofspatialcamisa3Dspatialfigure.It ;surroundsthespatialcaminlinewithacertainmotion ;rule.Generatingcuring,usuallyadoptedinmachin- ;ingOfconicalcamtrough,meansthatthespatialcam ;meetingthedemandofdesignisproducedinthe ;followingway:replacetaperedroller,i.e.,followerof ;spatialcammechanismwithcurerwiththesame ;figureanddimension,andthensimulatethecorrela- ;tivemotionrelationshipunderthepracticalrunning ;conditionofspatialcamandtaperedrollerfollowerin ;thewayofnumericalcontro1.

    ;Fig.1Mechanismofspatialcamwithoscillatingfollower ;„enthetroughofspatialcamwithoscillating

    ;f0llowerisnotwide.thereisnostrictdemandforan ;exactprecision.Amillingcurerinthesamedimen- ;sionwithoscillatingfollowerrollercouldbeused.It ;isquiteeasytogetthetroughofspatialcamwith ;oscillatingfollowerinaccordancewiththeabove ;demandsthroughnumericalcontrolmachiningby ;usingthemethodofgeneratingcuringalong ;trough-centerline.

    ;Butthismachiningmethodhasmanylimitations. ;enhigherprecisionisdemandedforthetroughof ;spatialcamwithoscillatingfollower,inotherwords, ;coarselymachiningdoesnotwork;orwhenthetrough ;ofspatialcamistoowidetofindamillingcurerinits ;dimension.itishardtoguaranteethatthewidthofthe ;obtainedtroughofspatialcamwithoscillatingfol- ;lowerisinaccordancewiththerequirementofproc- ;essingprecision.Ingeneral,thelattercaseiscalled ;

    ;Chenetal/JZhejiangUnivSciA20089(9):12581263

    ;nonequalizationdiametermachiningofspatialcam ;withoscillatingfollower.

    ;“0frSet”methodhasbeenreferredinlotsoflit—

    ;erature(LinandTsai.l996;TsayandLin.1996; ;Zhangeta1..1997;Xiaoeta1..2005).Asshownin ;Fig.2.itis”Offset”machiningtheretumpartof

    ;troughofspatialcamwithoscillatingfollower.Sup. ;posethetroughwidthisl6mm.Afterroughma

;chining.fine”Offset”machiningispracticedwitha

    ;millingcutterwhosediameteris5nllT1.

    ;30

    ;0

    ;30

    ;Fig.2Cuttermotionpathof‟‟offset‟‟machining

    widthsectionPP,betweenthe ;Onthetrough

    ;pointof”Offset”machiningandthepointoftheo—

    ;reticalfigureline(centerline)ofcamtrough,devia

    ;tionoccursintherotaryangleofcam.Itis5.as

    ;showninFig.2,namely,thespatialcamrotaryangles ;correspondingtothe‟‟Offset”machiningpointand

    ;theoreticalmachiningpointaredifferent.Neither ;millingcuttercenterlinesofeachmachiningpointare ;paralleltoeachother,norareonthesameplane. ;SectionPPcanbeapproximatelydepictedinFig.3. ;Whenthetopoftrough..widthsectionmeetsthedi.. ;mensionaldemandsthereisadeviationof5.be

    ;tweenthemillingcuttercenterlineof”Offset”ma—

    ;chiningpointandthatofthetheoreticalmachining ;pointcorrespondingtothecamrotaryangle.Asa ;result,thebottomoftroughwidthsectionissmaller

    ;thanthedimensionaldemands.Whenthishappens. ;theinterferencebetweenfollowerandtroughwould ;occur(asshowninFig.3,andleadtothesocalled

    ;deadlockphenomenon.Nishioka(2003)evaluated ;theoffseterrorofcylindricalcam.

    ;Tosolvethisproblem,thispaper,byusingthe ;3Dexpansionofspatialcamwithoscillatingfollower, ;presentsaneweffectivemachiningmethodforspatial ;camwidetroughwithoscillatingfollower,whichcan ;guaranteetheprecisionofprocessingspatialcam ;trough.

    ;Fig.3Interferencebetweenfollowerandtrough ;3DEXPANS10NOFSPATIALCAMWITH

    ;0SCILLATINGF0LL0WER

    ;SpatialcamisturningaroundwiththeaxisOO. ;Duringtherotarycourse,thedistancebetweeneach ;pointofspatialmechanism(includingspatialroller, ;followerandoscillatingbarframe)andthecommon ;perpendicularline?isinvariable.whereasitspo

    ;sitionisconstantlychangingwiththerotaryangleof ;spatialcam?.Whenweapplythekinematicalinver—

    ;sionwidelyusedinthedesignofcamcontoursurface,

    ;therollerfollowerandoscillatingbarflameshould ;reverseeveryroundaroundpointNandYZ-planeif ;thespatialcamiskeptmotionless.asshowninFig.4. ;/

    ;/

    ;,.

    ;\

    ;/

    ;Fig.4Relativemotionpathofoscillatingbar ;AsshowninFig.5,followingthedoubledotline, ;therotationofmachineflameisexpandedtoaline ;andisperpendiculartothesurface.onwhichtheos

    ;cillatingbarisoscillating.

    ;Thedirectionoftheline(i.e.,directionof

    ;spatialcamrotaryangle)isparalleltoZ-axisinthe ;coordinatingflame.Archedmotionofoscillating ;

     ;Chenetal/JZhejiangUnivSciA20089(9):1258-1263;followerrotatingwithspatialcamcanthenbeex

    ;pandedtoa3Dcylindermotion.

    ;OscillatingmotionofoscillatingbariSdrivenby ;therotationofspatialcam.Basedonthedemandsof ;motionpathofmachining,ageneralrelationshipcan ;bededucedwithspatialcamrotaryangleasparame

    ;tersinthefollowingequation:

    ;.

    ;)

    ;Fig.5Expansionofspatialcurvedsurface ;(1)

    ;Adisplacementcurve(subsectioncurve)is ;drawnbasedonEq.(1).AsshowninFig.6(ifoscil

    ;latingangleisbetween+30.),themotionrelationship ;curvewithdisplacementofoscillatingfollowerand ;rotaryangleofspatialcamcanbeestablished. ;0

    ;Fig.6Displacementcurve

    ;The(,valueofeachpointinthedisplace

    ;mentcurveCandeterminethecorrelativepointonthe ;cylindersurface,andthevalueZ)Canbees

    ;tablishedasfollows:

    ;

    ;Thegeometricalrelationshipbetweenoscillating ;1261

    ;angleofoscillatingbarandfollowerdisplacementS

    ;iSshowninFig.7.Asseenfromthesketch.through ;thevertexofoscillatingangle,onecanmakeanarc ;ifconsideringoscillatingbarlength,asradius.which ;willintersectwithangleborderatpointsAandB.and ;thenmakeaperpendicularlineACthroughA.withC ;astheperpendicularbasepoint.ThelengthOfCiS ;equaltothedisplacementSofoscillatingfollower ;(roller)incamaxisdirectionfi.e.,X-direction),which ;correspondstotheoscillatingangle.

    ;Fig.7Geometricalrelationshipbetweenoscillating ;motionandangle

    ;Mathematicalrelationshipbetweendisplace

    ;mentofoscillatingfollowerandoscillatingangleof ;oscillatingbarcanbeexpressedas:

    ;S=lsinq~.

    ;Byproiectingthedisplacementcurveonthe ;cylinderwhoseradiusiS,.thelengthofoscillating ;bar,theobtainedcurveiSa3Dexpansionofmotion ;pathofoscillatingfollower,asshowninFig.8. ;Accordingly.thevalueforcoordinateZOfeach

    ;pointin3Dcurveisequaltothedegreeofspatialcam ;rotaryangle;andthevaluesforcoordinate-Xand ;coordinate..Yarerespectivelyequaltothecoordinate.. ;Z

    ;Fig.83Dexpansionofmotionpath

    ;t_II——II)

    ;

    ;1262Chenetal/JZhejiangUnivSciA20089f9J:1258-1263 ;valuesofarchedmotionofoscillatingtaperedroller ;followerwhenspatialcamisrotatingtodegree. ;NON—EQUALIZATIONDIAMETER”TROCHOI—

    ;DALMILLING”MACHININGMETHODOF

    ;SPATIALCAM

    ;Basedontheabove3Dcurveexpansionofmo

    ;tionpathofoscillatingfollower,cuttinggeneration ;canalsobeadoptedinthemachiningoftroughof ;spatialcamwithoscillatingfollower.

    ;AsshowninFig.9,bymovingandrotatingmo

    ;tionpathofoscillatingfollower,CAMsoftwarecan ;becarriedoutaccordingto(LinandTsai.1996).This ;paperwillfurtherdiscussnonequalizationdiameter

    ;generatingcuttingofspatialcam.

    ;Fig.9Sketchmapof3Dmotionpathoffollower ;Cycloidalmilling

;Cycloidalmillingisanewschemeofcuttermo

    ;tionpath.Thesocalledcycloiddescribesapath

    ;formedwhenafixedpointinthecircleisrolling ;alongthecurveofthecircle.Asthecutterisalways ;runningalongafixedcurvaturecurveduringthe

    ;cuttingprocess,themotionofcuttercanbekeptatan ;accordantfeedingspeed(Wueta1.,2006;Jtmgand

    ;Psang,2007;Stanislav,2007).

    ;Thecuttermotionpathofthetrochoidalmilling. ;exploitedonthebasisofcycloidalmilling.isex

    ;tremclysimilartothatofcycloidalmilling.Thedif- ;ferenceonlyisthatthecuttermotionpathoftro

    ;choidalmillingiscomposedofbeelinesfi_e.troch ;step)andcircles(asshowninFig.101.whenthe ;valueoftrochstepistiny.trochoidalmillinghasan ;obviousadvantageinprogramming,sinceit1seasier ;touseashorterprogram,andthereforecanimprove ;theprocessingspeedofthenumericalcontrolsystem. ;Fig.10Trochoidalmilling

    ;Nonequalizationdiameter”imitatedtrochoidal

    ;milling”ofspatialcamtrough

    ;Ifthebottomoftroughwidthsectionissmaller ;thanthedimensionaldemandswhen”Offset”method

    ;isapplied.aninterferenceofoscillatingfollowerand ;troughwouldformleadingtoadeadlockphenome

    ;noninfollowermotion.

    ;Toavoidthisproblem,weproposeanon

    ;equalizationdiameter”trochoidalmilling”.Alongthe

    ;troughcenterlinefi_e..thetheoreticallineofspatial ;camfigure),thecutterrunsonetrochstepbyone ;circle,andthenanothertrochstepbyanothercircle. ;Thelengthofeachtrochstepcouldbedifferent.but ;thecirclesareequirota1.Thesmallerthevochstepis. ;thehighertheprecisionis.AsshowninFig.11.when ;thecutterfinishesonetrochstep.thecambilletwill ;pauseitsrotatingmotion.Whenthemillingcutteris ;makingcirclingmotionintheoscillatingplaneof ;oscillatingbar,thecenterofthecircleisinthetrough ;centerline.Thecambilletisimmovableduringthe ;processofcirclingmotion,astheaxesofmilling ;cutterofcorrespondingmachiningpointsisparallelto ;eachother,asshowninPPofFig.11.Thenthe

    ;deadlockphenomenoncouldbeavoided.

    ;Inaccordancewiththe3Dexpansionofspatial

;Fig.11Imitatedtrochoidalmilling

    ;/,,

    ;/)0l\

    ;

    ;Cheneta1./JZhejiangUnivSciA20089f9J:1258-1263 ;camtrough,thecuttermotionpathofnonequaliza

    ;tiondiametercanbeestablishedasshowninFig.12. ;Inordertoimprovetheprecisionoftrough,eachtroch ;stepshouldbesmallenough.Therequiredprecision ;ofmachiningspatialcamtroughcanbeensuredby ;changingthesizeofthecircle.

    ;Fig.12PracticalCAMprogramming

    ;C0NCLUS10N

    ;Byanalyzingtheerrorsinthe”offset”machining,

    ;wehaveproposedamachiningmethodfortrochoidal ;millingwithnonequalizationdiametercutter,andthe ;programmingandmachiningwith”trochoidalmill-

    ;ing”isnotonlyadaptabletospatialcamwithoscil—

    ;latingfollower,butalsoadaptabletospatialcamwith ;ordinaryfollower.Inthissense,itisauniversal

    equalizationdiameterprogrammingmethodof ;non

    ;cam.Bygivingd

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