ORIGINALARTICLEColdextrusionofalongtrapeziumsplineanditsforminganalysisYuanAnfuReceived:21August2007/Accepted:10March2008/Publishedonline:15April2008#Springer-VerlagLondonLimited2008AbstractThemachiningofalongtrapeziumsplineisdifficultduetoitsstiffness.Inthispaper,aspecialcoldextrusiontechniquehasbeenadoptedonthebasisofanalyzingitsthree-dimensionalvelocityfieldandsimula-tionusingthesoftwareprogrammeDeform-3D5.0tomakeaqualifiedtrapeziumsplineof12840with16teeth.KeywordsColdextrusion.Trapeziumspline.Forminglimit.Deform1IntroductionTherectanglesplineismainlyusedtoholdheavyloadsduetoitsthickroot.Thereforeitiswidelyusedtotransmitpowerintheautoindustry.Thispapersmainlyfocusesontheformingofalongtrapeziumspline,whichisadifficultproblemtosolveinthemachiningindustry.Traditionalmachiningmethods,suchasmilling,hobbling,etc.,arenotcapableofformingsuchsplineinbatchesduetotheirlowefficiencyandquality；thereforespecialmanufacturingmethodssuchastwisting,extruding,etc.havebecomemoreandmorewidelyused.Still,forlongtrapeziumspline,asshowninFig.1,bothofthemhavesomedifficultiestoformaqualifiedpart17.2DetailsofparttobeformedThedrawingofatrapeziumsplinetobeformedreferstoFig.1.Itsdetailsareasfollows:Teeth:16ThickofTooth:1:3Material:20CrGBElasticModule:205GpaPossionRatio:0:29Density:7850KgC14m3YieldStrength:685MpaTopdia:oftooth:Dt12:8mmRootdia:oftooth:10:86mm3StepstosolveTherearetwoproblemswhichneedtobesolvedinordertoformaqualifiedproduct:3.1StructureofthediecavityThestructureofthediecavitydirectlyinfluencesdeforma-tionoftheworkpieceandstressandstraindistributionduringextrusion.ThereforeitisnecessarytodesignareasonablediestructureaccordingtotheactualformingIntJAdvManufTechnol(2009)41:461467DOI10.1007/s00170-008-1480-yY.Anfu(*)InformationandControlCollege,NanjingUniversityofInformationScience&Technology,No.114PanchengNewStreetPukouDistrict,Nanjing,JiangsuProvince210044,Chinae-mail:conditionstoobtainaminimumformingforce.Inthispaper,basedontheanalysistothethree-dimensionalvelocityfieldinmetalextrusionaccordingtoassumptionofisoextrusionratioflow,areasonablediestructurecanbedesigned.AndthenformingofextrusionunderthesameconditionswithactualextrusionissimulatedwithDeform.Finallytheextrusiondieismachinedaccordingtotheabovetworesults.3.2StiffnessofparttobeformedAstothethinlongtheworkpiecetobeformedinthispaper,howtokeepenoughstiffnessoftheworkpieceduringcoldextrusionbecomesanoverwhelmingproblem.Soinactualextrusion,traditionalpushforceischangedtoapullforceand,atthesametime,supportedbyspecialmechanism.Thesetechniquesaboveareprovedtobeeffective.4AnalysisofextrusionInthispaper,allanalysisismadeonthebasisofopendieextrusion.4.1ShapefunctionofsplineshaftDuetothesymmetryofthetrapeziumspline,onlyhalfthestructureofitsshapeisdrawninFig.2.Accordingtotheworkingprincipleoftrapeziumspline,itsshapefunctionofsurfacecanbeobtainedasfollows:fr；ZgC0RbC138Z=LRb1whereLValidlengthofworkingzoneofdiecavity；RbRadiusofbasecircle,i.e.,radiusofrawbar；g()isdefinedasfollows:gRt0C2081SR81C2082Rr81C20828:2whereRtRadiusoftopcircleofsplineRrRadiusofrootcircleofsplineSR()Equationoftrapeziumsplinewhichisdescribedasfollows:SR:Rrrsin81C0RtrsinC082RtRrsin81C0823whereakPressureangleofthesplinetPolaranglewhentopcircleofsplineispolarradiusFig.2ShapefunctionoftrapeziumsplineanditscoordinationsystemFig.3DeformingzoneoftrapeziumsplineFig.1Partdrawingtobeformed462IntJAdvManufTechnol(2009)41:4614674.2ActualanglesAccordingtoactualdimensionsoftrapeziumspline,Eqs.(1,2and3)canbetransferredasfollows:815:830；826:8808212:30ZoneI:fr；ZRtC0RbZ=LRb4ZoneII:fr；Z:rRtRrsin81C082Rrsin81C0RtsinC082ZZ:8:174.4PowercalculationbyupperboundanalysisDuringextrusion,therearethreepowersofdeformationpowerinsidetheworkpiece,frictionpowerandshearpower,thatis:PTPdPtPf18wherePd,Pt,Pfarepowersofdeformationpowerinsidetheworkpiece,shearpowerandfrictionpoweronthecontactingsurfaceofthesplinerespectively.T