具有加工倾角的球头铣刀切削力模型(中英)

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JournalofMaterialsProcessingTechnology

189 (2007) 85–96

Modellingofcuttingforcesinball-endmillingwithtool–surfaceinclination

PartII.In uenceofcuttingconditions,run-out,

ploughingandinclinationangle

M.Fontaine ,A.Devillez,A.Moufki,D.Dudzinski

LaboratoiredePhysiqueetM´ecaniquedesMat´eriaux,UMRCNRS7554,ISGMP,Universit´edeMetz,IleduSaulcy,57045Metz,France

Receivedinrevisedform12December2006;accepted12January2007

Abstract

Thisstudyfocusesonthein uenceoftool–workpieceinclinationoncuttingforcesinball-endmilling.Cuttingforcescalculatedfroma

thermomechanicalmodelling,presentedinpartIofthispaper[M.Fontaine,A.Moufki,A.Devillez,D.Dudzinski,Modellingofcuttingforcesinball-endmillingwithtool–surfaceinclination.PartI.Predictiveforcemodelandexperimentalvalidation,J.Mater.Process.Technol.189(2007)73–84],areherediscussedindetailandcomparedtoexperimentalresults.Theproposedmodellingofball-endmillingwasappliedtomachiningoperationswithstraighttoolpathsandvarioustool–surfaceinclinations.Bothrampingandcontouringcon gurationswerestudied.Theexperimentalresultswereobtainedfromball-endmillingtestsperformedonathree-axisCNCequippedwithaKistlerdynamometer.Theattentionisherepointedontheshapeandlevelofthecuttingforcessignals.Theevolutionofthemaximumvaluesofcuttingforcesactingonthetoolisinvestigatedinordertoidentifytheoptimuminclinationangle.In uencesofcuttingconditions,radialrun-outandploughingoncuttingforcesandcuttingstabilityarediscussed.

© 2007 Elsevier B.V. All rights reserved.

Keywords:Ball-endmilling;Cuttingforces;Cuttingconditions;Toolrun-out;Ploughing;Tool–surfaceinclination

1.Introduction

Manyauthorshaveproposedef cientmodelstopredictcut-tingforcesinmillingoperations.Someexperimentalresultsareavailableforsteels[2–15],aluminiumalloys[14–18],zincalloys[19,20]oraeronauticalalloys[6,21–23].However,thefactorswhichin uencetheseresultsarerarelydiscussed.Itiswellknownthatsomeparametersin uencesigni cantlythecut-tingforceslevelandevolution,anditisimportanttoconsidertheminmachiningmodels.Inball-endmilling,themainparam-etersarethecuttingconditions,thetoolrun-out,theploughingphenomenonandthetool–surfaceinclination.Theyaretreatedseparatelyinsomeworksbutneverconsideredtogether.

Intheliterature,theexperimentalcuttingforcesresultsforball-endmillingoperationsareoftenrelativetoasmallsetofcuttingconditions(spindlefrequency,feedvalues,depths

Correspondingauthor.Tel.:+33381666721;fax:+33381666700.E-mailaddresses:fontaine@univ-metz.fr,michael.fontaine@ens2m.fr(M.Fontaine).

0924-0136/$–seefrontmatter© 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.jmatprotec.2007.01.007

ofcut,cuttingmode).Themainreasonisthatexperimentaltestsrequirealotoftimeandareexpensive.Manyexperi-mentalworkspresentresultsforvariouscuttingspeedvaluesassociatedto xedfeedrate[3,4,6,11,13,15,18,19,21],oth-ersstudytheeffectsofbothfeedrateandspindlefrequency[7,11,12,16,17,22].Theusualdevicetomeasurecuttingforceispiezoelectricdynamometer,howeverforhighspeedmilling,itsuseislimitedbythedynamicresponseoftheusedsen-sors.Then,onlyareducednumberofstudiesdealwithhighcuttingspeeds[6,7,12,22].Someauthorsproposeresultsfordifferentaxialdepthsofcut[7,15,19,21],variousradialdepthsofcut[13,17]orbothaxialandradialdepthsofcut[3,4,6,12,16,20,22].Thedown-cuttingmodeiswidelyusedintheseworksbutitisrarelycomparedtotheup-cuttingmode[10].

Thetoolrun-outisclassicallymodelledforfacemillingoper-ations,onlysomeauthorsproposesolutionstotakeintoaccountofthiskinematicaldefaultinball-endmilling.Inthisway,therun-outisusuallyconsideredasaradialeccentricityonly(radialrun-out)[3,14]andthetiltangle(axialrun-out)israrelytakenintoaccount.Inball-endmilling,theusuallimiteddiameter

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86M.Fontaineetal./JournalofMaterialsProcessingTechnology 189 (2007) 85–96

andlengthofthetoolreduceitsin uence.Itisinterestingtointroducethisaxialrun-outtocalculateaccuratelythework-piece/cutterengagementdomaininacomplete ve-axismillingforcemodel[22].Thesepreviousworksshowedthatradialrun-outhasahugein uenceontheedgesengagementandonthecuttingforceslevel,especiallyforhighcuttingspeedsandsmallchipthicknesses.

Non-shearingphenomenaoccuraroundthecuttingedgeandcreateadditionalforcesactingonthetool.Theycanbedeter-minedandmodelledwithaslip-lineapproachinorthogonalcutting[24,25].Inball-endmilling,theyareimplicitlytakenintoaccountinmechanisticmodelsbutwithoutaspeci cidenti ca-tion.Nevertheless,someauthorsseparatetheshearingprocessfromtheedgephenomenabytheuseofshearingandedgecoef- cientsidenti edfromturningtests[21]ordirectlyfrommillingtestsbyinversemethodandlastsquareadjustmentorconvolu-tionintegralapproach[8,15,18,26,27].Ploughingseemstobethemainedgephenomenonin uencingcuttingforces,anditsidenti cationshouldberealizedfromananalyticalmodellinginordertoidentifyproperlythecuttingforcesduetolocalisedshearing.

Thetool–surfaceinclinationinball-endmillingwasthepur-poseofmanystudiesduringthelast10years.Itallowstocontrolthecuttingforceslevelandtheirrepartitiononthetoolbyavoidingthetooltipwhereoccursgeometricandkineticproblemsandbychangingthecontactzonebetweentoolandworkpiece.Thisinclinationcanbeusedineveryball-endmillingoperationsevenforsculpturedsurfacemachiningbutitiscommonlystudiedfromthecasesofplaneorroundsur-faces.Manyauthorsapproachthisoptimisationparameterfromexperimentaltestsandmillingmodelstoanalyseitsin uenceoncuttingforces[5,6,8–10,15,17,19,20],ontoolorpartde ectionandmachiningerror[4,10,11,28–30],onsurfacetopography[8–10,29,31,32],ontoollife[7],andevenontool/workpiececontactarea[10,11,15,19]andonchipgeometry[8,10].Somereferenceinclinationanglescanbeextractedfromthesestudiesbuttheobtainedvaluesdependontheoptimisationprocedure.Finally,tocontroltheprocess,themainpointsseemtobetheglobalcuttingforceslevel,theprocessstabilityandthesurfaceintegrity.

Inordertostudytheroleplayedbythecuttingparametersinball-endmilling,thecuttingforcespredictedfromather-momechanicalmodel[1]andobtainedfromexperimentaltestsarecomparedanddiscussed.Themodelisappliedtoball-endmillingofinclinedplanesurfacesonathree-axismachineinordertosimplifytheobservationandtocomparetheresultswiththoseavailableinliterature.Themachinedmaterialwasa42CrMo4steel,typicallyusedinplasticinjectionmouldsmanu-facturing.Itscharacteristicssuchasstrainhardening,strainratesensitivityandthermalsofteningareknown.Someresultsarepresentedforthefourreferencestrategiesinslottingandthein uencesofcuttingconditions,radialrun-out,ploughingphe-nomenonandtool–surfaceinclinationanglearediscussedfromthewholeexperimentaldataobtainedforthemodelvalidationpresentedinpartI[1].Someinclinationvaluesarepresentedtogiveinformationaboutoptimisationofglobalforceslevelandploughinglimitation.

2.Millingtestsandsimulations

Theproposedexperimentalandcalculatedresultsandthediscussionscorrespondmainlytoslottingtestswithapathinter-valsuperiortothenominaldiameterofthemill( p>12mm).Thesetestsappeartobethemostappropriatetoanalysethein u-enceofeachparameterseparatelybecausethefulltoolradialimmersionstabilizestheprocessandcreatesimportantvaluesofcuttingforces;hence,themeasuresqualityisoptimumandsomeparasiticdispersionsareavoided.2.1.Cuttingstrategies

Fig.1presentsthereferencesurfacesandthefourtestedcuttingstrategiesforslotting.Therearetwostrategiesinramp-ingwithtwodifferentdirectionsoftoolpathalongZ-axis:upwardrampingorup-ramping(Fig.1(a)),downwardrampingordown-ramping(Fig.1(b));andtwostrategiesincontouringwithtwodifferentmaincuttingmodes:down-cuttingcontouring(Fig.1(c)),andup-cuttingcontouring(Fig.1(d)).

Theuncutsurfaceisainclinedplanewithrespecttothehor-izontalreferenceplane(X,Y)aroundY-axis,andtheinclinationangleisdenotedδ(Fig.1).ThetoolZ-axisandthemachineZ-axisaremerged(three-axismillingcon guration).Therefer-encepointoftheglobalcoordinatesystem(GCS)istakenattheintersectionofthreesolidplanesatacorneroftheworkpiece,pointOinFig.1.Thelocalcoordinatesystem(LCS)associatedtothetoolis xedatthetooltipE.2.2.Experimentalsetup

Thetestswereconductedona42CrMo4steel(equivalenttoAISI4142)underdryconditionsonaverticalthree-axisCNCmillingmachine.Asix-componentsKistlerdynamome-ter(model9265B)hasbeenusedtomeasurecuttingforcescomponents.TheoutputsignalswererecordedandstockedonaPCDewetronconsolethroughaheight-channelKistlerchargeampli er.Thesignalswereselectedatthemiddleoftheworkpieceandthevoltage/timesignalsweretransformedintoforce/rotationangleones.Aspectrumanalysiswascon-ductedforeachexperimentalsignaltocheckthestability.Then,somesignalswerelightlylow-pass lteredtosuppresssomeaccelerometernoiseduetovibrations.

Atwo utedball-endmill(referenced51221)fromDiager-Industriecompanywithanominaldiameterof12mm,anominalhelixangleof17 ,andanormalrakeangleof0 onballpartwasusedintheexperiments:R0=6mm;i0=17 ;αn=0 .

Thespindlefrequencyandthefeedratewere xedatthefol-lowingvalues,respectively:Ω=5000rpm,ft=0.05mm/toothandperrevolution.TheKistlerdynamometerwasclampedinaninclinedpositiononthemachinetable(Fig.2(a)).Fivevaluesofinclinationanglewerechosen:δ=0 ,5.07 ,10.43 ,15.26 and20.17 .Thespecimenwasaparallelepipedwithaheightof20mm,alengthandawidthof50mm(Fig.2(b)).Thenormaldepthofcutwastakenconstant:dn=1.5mm.Slottestswereconductedwithapathintervalof p=15mm.

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M.Fontaineetal./JournalofMaterialsProcessingTechnology 189 (2007) 85–96

87

Fig.1.Toolpathdirectionsinthree-axisball-endmillingofaninclinedsurface.(a)Up-ramping,(b)down-ramping,(c)down-cuttingcontouring,and(d)up-cuttingcontouring.

Thetoolrun-outwasmeasuredwithadialindicator xedonthemachinetableandtouchingthecylindricalpartofthemill.2.3.Modelling

Thegeometricalandthermomechanicalmodelsusedtocal-culatethecuttingforceswerepresentedinthePartIofthiswork

[1].Themodelparametersareherereminded:

Shearanglecoef cients:A1=40 ;A2=0.5. Mainfrictioncoef cient:μf=1.04(λf=46 ).

ParametersoftheJohnson–Cookconstitutivelaw:A=

˙0=0.001s 1;n=0.15;m=0.008;612MPa;B=436MPa;γ

ν=1.46;Tr=Tw=293K;Tm=1793

K.

Fig.2.Ball-endmillingtestsonaspeci cinclinedworkpiecedevice.(a)Dynamometerininclinedpositionand(b)detailoftestworkpieceandtool.

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88M.Fontaineetal./JournalofMaterialsProcessingTechnology 189 (2007) 85–96

Otherusefulmaterialparameters:ρ=7800kg/m3;c=500J/(kgK);β=0.9.

Primaryshearzonethickness:h=parisonbetweencalculatedandexperimentalresults

Figs.3–6presentthemeasuredandpredictedcuttingforcesactingonthetoolforthefourdifferentmillingstrategiesmen-tionedabove(Fig.1):upwardramping(orup-ramping)(Fig.3),downwardramping(ordown-ramping)(Fig.4),contouringbytheleft(ordown-cuttingcontouring)(Fig.5),andcontouringbytheright(orup-cuttingcontouring)(Fig.6).Thepresentedvaluesofinclinationangleδarequitehigh:15 fortheramp-ingtestsand20 forthecontouringones.Duetoatoolchange,therun-outparameters(eccentricityeandpositionangleψe),measuredwithadialindicator,aredifferentinrampingandincontouringtests.

Thecurvesshapesofthemeasuredandpredictedcuttingforcesareverysimilarforallthecon gurationswithagoodreproductionofthetoolentriesandexitsintheworkpiecemate-rial.Thediscrepancyincuttingforceslevelbetweenthetwoteethduetotoolrun-outiswellreproducedbythemodel.Thespeci cin uenceofthisoffsetisdiscussedinthenextsection.Theforceslevelslightlydiffers;smallervalueswereobtainedfromthesimulation.ThecuttingforceslevelwasalwayswellpredictedforFxandFycuttingforcecomponents(offsetunder15%),butamoreimportantamplitudeoffsetappearsonFzinparticularforthecaseofdown-ramping(upto30%)(Fig.4).Theamplitudeoffsetsaremainlyduetotherepetitiveentriesofthecuttingedgesintheworkpiecematerial,whichinducecuttinginstability.Theusedlowpass lteringdoesnotsuppressallper-turbationsbecausealltherepresentativeharmonicsofthespindlefrequencymustbeconserved.Thisinstabilitycanbeobservedonthecurvesaroundthezeroforcevalue.Itisnotwellreproducedbythesimulationbecausetheusedcuttingmodelisbasedoncontinuouscuttingprocess.ItcanbenoticedthatthemeasuredsignalsalongtheZ-axisarelightlymoreunstablethanalongX-orY-axis.Somedeformationphenomenaoccurringaroundthecuttingedgeandonitsclearanceface,inparticularploughingeffects,arealsoresponsibleoftheseexistingdiscrepancies.Thisphenomenonanditsin uencearepresented

below.

Fig.3.Measuredandpredictedcuttingforcesforup-ramping.Ω=5000rev/min,ft=0.05mm/tooth, p=12mm,e=0.01mm,ψe=80 ,andδ=15 .

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M.Fontaineetal./JournalofMaterialsProcessingTechnology 189 (2007) 85–96

89

Fig.4.Measuredandpredictedcuttingforcesfordown-ramping.Ω=5000rev/min,ft=0.05mm/tooth, p=12mm,e=0.01mm,ψe=80 ,andδ=15 .

3.In uenceofcuttingconditions,toolrun-outandploughing

3.1.In uenceofcuttingconditions

ThegloballevelofcuttingforcesdecreaseswithincreasingvaluesofcuttingvelocityV,andhighvaluesofVtendtostabilizetheprocessandthesignals.Thethermomechanicalapproachadoptedheregivesbetterresultsathighercuttingspeed[33]andthediscrepancybetweenmeasuredandcalculatedforcestendstodecreasewhenVincreases.Itcanbenotedthatthefeedratevariationaffectsdirectlythecuttingforces,indeedthesevaluesdependsproportionallyontheundeformedchipthicknesst0.Thehighesttestedvalueoffeedrate(0.2mm/rev)tendstolimittheploughingeffectbutmayincreasetoolde ectionandtoolwear(directlylinkedtothecuttingforceslevel).

Decreasingtoolpathinterval pnaturallyreducesthecut-tingforcesvaluesbuttherecordedsignalsforsmallvaluesof p(semi- nishingoperations)aremoreoftenunstable.Theseexperimentalsignalscouldbecomparedtocalculatedonesobtainedbyusinganenhancedversionofthemodeltaken

intoaccountoftoolde ectionandchattervibration.Neverthe-less,thesesemi- nishingresultsunderlinethediscrepancyofforcesrepartitionbetween“up”strategies(up-rampingandup-contouring[1])andtheothers:inthesefavourablecases,theglobalcuttingforcesleveldecreasesandtherepartitionismoreuniformonthedifferentaxes.Thisfactisveryimportanttoconsiderforthetoolde ectionlimitation.

Finally,thedown-cuttingcon gurationtendstostabilizethecuttingprocesslikeitcanbeseenonthemeasuredsignalsandtoprovidebettersurface nish,particularlyforsmallvaluesoftoolpathinterval p.Theup-cuttingcon gurationismoreef cientwhenthetooltipiswidelyusedforcutting(downwardrampingtypically)bylimitingedgephenomena,andforhighvaluesofpathinterval p(slottingorroughing).3.2.In uenceoftoolrun-out

Toolde ectionandvibrationsareavoidablebylimitingandcontrollingthecuttingconditionsandbychoosingstifftoolsandtool-holders.Butthereisanotherproblem,whichisdif culttocontrol:thetoolrun-out.Inmilling,thisgeometricaldefaultcan

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189 (2007) 85–96

Fig.5.Measuredandpredictedcuttingforcesfordown-cuttingcontouring(+Y).Ω=5000rev/min,ft=0.05mm/tooth, p=12mm,e=0.008mm,ψe=110 ,andδ=20 .

beduetotoolitself(wear,asymmetry,insertsetting,dynamicimbalanceandthermaldeformation)butitismainlyduetotheoffsetbetweenthepositionofthetoolrotationaxisandthespin-dlerotationaxis.Theconsequenceisatoolrotationaroundthespindleaxiswithaneccentricity.Thiseccentricitymodi esthetoolengagementandthelocalcuttingconditions(cuttingveloc-ityandangles).Then,therun-outhasadirecteffectonthecuttingforceslevelandvariation.Itdependsonthegeometricalqualityofthespindleandtoolholder.Itseffectisparticularlysigni cantwhentheundeformedchipthicknesst0reachessmallvalues,andinthiscase,oneorseveralcuttingedgesmaybeoutofanyworkingposition(nocutting).

Withoutrun-outmodelling,theforcespredictedbythemodelareaveragevaluesandacalculationerrorisintroduced.Thiserrorcanreach15%insometestscorrespondingtoacuttingforcediscrepancyof30%betweenthetwoteeth.Introducetheradialtoolrun-outinthemodellingreducessigni cantlythediscrepancybetweenpredictedandmeasuredmaximumforcevalues.Thecuttingforceslevelisin uencedbythisgeometricaldefaultespeciallywhentheinclinationanglegrowsup.Theproposedmodellingreproducesthesametendency.Therun-outisalsoresponsibleofachatterexcitation,whichdegradesthesurfacequalityandreducestheprocessstability.More-over,thetestscorrespondingtohighvaluesofeccentricityeandofinclinationangleδpresentveryunstablesignalsandpoorsurfacequality.Thein uenceofaradialoffsetincreaseswiththeinclination,then,usingthetool–surfaceinclinationasanoptimisationparametermustbedoneforlimitedvaluesofrun-out.

Toolwearisalsogreatlyin uencedbythisdefault.Someobservationsinopticalmicroscopycon rmedanevidentdif-ferenceinwearpatternsbetweenthetwoteethforhighvaluesofeccentricity,inparticularwhentherun-outlocationangleisclosedto0 or90 .Thediscrepancybetweenthecuttingforcesactingoneachtoothgivesinformationaboutthiswearasym-metry.Thetoollifeisthenreducedononetoothandmaybeincreasedontheother.

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M.Fontaineetal./JournalofMaterialsProcessingTechnology 189 (2007) 85–96

91

Fig.6.Measuredandpredictedcuttingforcesforup-cuttingcontouring( Y).Ω=5000rev/min,ft=0.05mm/tooth, p=12mm,e=0.008mm,ψe=110 ,andδ=20 .

3.3.In uenceofploughing

Thematerial owoccurringaroundthecuttingedgeaffectsmainlytheFzforcecomponent.Thismaterial owisneglectedinthemodellingapproach,inparticular,thecuttingedgeissupposedtobeperfectlysharp.Infact,thematerial owandassociatedshearingoccurringattheclearancefaceleadtoanadditionalploughingforce.Theploughingforcelevelbecomesverysigni cantaroundthetooltipwhencuttingvelocityandundeformedchipthicknesstendtozero;inthisregiontheresul-tantploughingforcevalueishigh.Theploughingforceismainlynormaltothetoolenvelope(er-direction)andatthetooltipitsdirectionisclosedtothez-direction.Inconsequence,theFzcom-ponentismoreaffectedbythisphenomenonandthepredictedFzvaluesaresometimeslowerthanthemeasuredones.Thedownwardrampingcon gurationpresentsthemoreimportantdiscrepancybetweenthecalculatedandexperimentalFzcompo-nentvalues.Inthiscon guration,thetooltipwaswidelyusedandthentheresultantploughingforcewasincreased.Hence,theobserveddifferencebetweenmeasuredandpredictedforcesisproportionaltotheexistingploughingforce.Theexistenceofthisphenomenonisprobablythereasonwhythesimulationresultsareoftenbetterindown-rampingthaninup-ramping(Figs.3and4).Thelessin uencedforcecomponentseemstobetheoneassociatedtothefeeddirectionaxis(FxinrampingandFyincontouring).

Thecuttingedgesharpnessin uencesdirectlytheef ciencyofcuttingandthentheploughingeffect.Toolweartendstoincreasethemaincuttingforceandtheploughingforce.Itisdif- culttoenhancetoollifeandlimitedgeeffectssimultaneouslywhenselectingtoolrakeangle,clearanceangle,helixangleandedgeradius.Forthechosenmill,therakeandclearanceangleswereatsmallvaluesandthein uenceofploughingwasprobablyampli ed.

Ifploughingincreasesthecuttingforces,italsoaffectsthesurfaceintegrity(surfaceroughnessandresidualstressesgrowup)andreducesthetoollife.Thisphenomenonshouldbeavoidedasmuchaspossible.Theploughingeffectcanbe

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92M.Fontaineetal./JournalofMaterialsProcessingTechnology 189 (2007) 85–96

limitedbyincreasingthetool–surfaceinclinationinordertokeepthetooltipoutsideofcuttingposition.AccordingtothefactthatploughingoccursmainlyonFzforcecomponent,whichislessrelevantfortoolde ection,chatterandsurface nishpredictioninball-endmilling,itwasnottakenintoaccountofthisphenomenoninthismodellingapproach.4.In uenceofinclinationangle

Figs.7–10presentthethreeforcecomponentsFx,FyandFz(experimentalandcalculated)maximumvaluesversustheinclinationangleδorsurfaceslope.Thefourmillingstrate-giespreviouslypresented(Fig.1)areconsidered.Themeasuredvaluescorrespondtoaveragevalues,andinthecalculationthetoolrun-outwasnottakenintoaccount.Theobjectiveistodeterminethemostfavourableorientationofthetoolwithregardtotheworkpiecesurfaceinordertolimitde ection,chattervibrationsandtoolbreakage,andthemaximumval-uesreportedinthementioned guresareabsolutevalues.Theinclinationofthedynamometerintheclampingdevicewaslim-ited,thentheexperimentalvalueswereobtaineduntilasurfaceslopeofδ=20 ,whilethemodelresultswerecalculateduntilδ=45 .

4.1.Rampingcon gurations

Inupwardramping(orup-ramping)(Fig.7),thevaluesdeducedfromsimulationshowverygoodcorrelationwiththeexperimentaltendencies.Thediscrepancybetweenthetwoseriesofvaluesisacceptableanditdecreaseswheninclinationangleincreases.Forthevaluesδ=15 and20 ,theresultsareverysimilar.Theevolutionsobtainedbysimulationdataover20 areinthesametrendandtheymaybeconsideredasacceptableprediction.ThediscrepancybetweenmeasuredandcalculatedvaluesisverysmallonFxcomponent(<10%)(Fig.7(a)),reason-ableonFycomponent(<15%)(Fig.7(b)),andmoreaccentuatedonFzcomponent(<25%)(Fig.7(c)),particularlyforsmallvaluesofinclinationangleδ.Asexplainedpreviously,thedis-crepancyonFzisduetoploughingphenomenon,whichisquiteimportantforsmallvaluesofinclinationangleδ,andvanishesprogressivelywithincreasingvaluesofδ.Thetool–surfaceincli-nationavoidstheuseofthetooltipforcuttingandlimitstheresultantploughingforceactingmainlyonZ-axis.Thevaria-tionsofFxandFyforcecomponentsaresmallwithapparentlyminimalvaluesforaninclinationofabout20 ,theFzcomponentdecreasescontinuouslywiththeincreasinginclination.Then,highvaluesoftheinclinationanglearefavourablefortoollifebuttoavoidtoolde ectionandchatteravalueofδ=20 shouldbepreferred.Itcanbenotedthatforthisvalue,theploughingin uenceonFzisreduced.

Indownwardramping(ordown-ramping)(Fig.8),thesametendenciesareretrievedonFx(Fig.8(a))andFy(Fig.8(b)).Thediscrepanciesbetweencalculatedandmeasuredvaluesaremoreimportantbutremainmoderate(<15%onX-axisand<20%onY-axis).Theagreementseemstobeverygoodforaninclinationangleδ=20 .However,thediscrepancyontheZ-axiscompo-nent(Fig.8(c))isstillimportantandincreaseswithincreasinginclination(33–55%).Indown-ramping,thetooltipisalwaysincuttingpositionthentheploughingeffectisalwaysactingonthetool.Theploughingforcedirectionvarieswiththecontactzonelocationbetweentoolenvelopeandworkpiece.Noexper-imentalresultsareavailableforaninclinationanglesuperiorto20 .Nevertheless,forslopevaluessuperiorto30 thedis-crepancybetweenexperimentsandmodelshouldincrease

for

Fig.7.In uenceofsurfaceinclinationangleδonmeasuredandpredictedcuttingforcesforup-ramping:(a)maximumvalueofFxfunctionofδ,(b)maximumvalueofFyfunctionofδ,and(c)maximumvalueofFzfunctionofδ.

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Fig.8.In uenceofsurfaceinclinationangleδonmeasuredandpredictedcuttingforcesfordown-ramping:(a)maximumvalueofFxfunctionofδ,(b)maximumvalueofFyfunctionofδ,and(c)maximumvalueofFzfunctionofδ.

FxandFyanddecreaseforFz.Theoptimuminclinationangleseemstobeδ=25 fortoolde ection(smallforcevaluesonX-andZ-axis)butnotprobablyfortoollifeandsurfaceintegritybecauseofhighvaluesofFzcomponentandofploughingforce.Poorsurfacequalitywasinfactobservedforδ=15 and20 .Smallvaluesofinclinationseemtobepreferableforcontrollingcuttingforcesactingontoolaxisz.Inthiscon guration,thechoiceoftheinclinationangledependsmoreonoptimisationobjectives.

Theseresultsindicatethattheupwardstrategyinrampingprovidesabetterrepartitionofcuttingforcesonthetoolandlimitstheploughingeffectaroundthetooltip.Itallowsmainlyanimprovementofcuttingstabilityandsurfacequality.Theresultsalsocon rmthattheFxcomponent,which

corresponds

Fig.9.In uenceofsurfaceinclinationangleδonmeasuredandpredictedcuttingforcesfordown-cuttingcontouring(+Y):(a)maximumvalueofFxfunctionofδ,(b)maximumvalueofFyfunctionofδ,and(c)maximumvalueofFzfunctionofδ.

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189 (2007) 85–96

Fig.10.In uenceofsurfaceinclinationangleδonmeasuredandpredictedcuttingforcesforup-cuttingcontouring( Y):(a)maximumvalueofFxfunctionofδ,(b)maximumvalueofFyfunctionofδ,and(c)maximumvalueofFzfunctionofδ.

tothemainfeeddirectionofthetool,islessin uencedbytheploughingphenomenon.4.2.Contouringcon gurations

Incontouringbytheleft(ordown-cuttingcontouring)(Fig.9),thetool–surfaceinclinationalsofavoursthemodelrele-vance.Thediscrepancybetweenmeasuredandpredictedvaluesisofthesameorderaspreviously(<20%)onX-axis(Fig.9(a))andY-axis(Fig.9(b)),eveniftheconvergenceseemstooccurlaterfortheFycomponent.ThediscrepancyontheFzcom-ponentismoreaccentuated(<30%)anddecreasesslowly.Forthenormaldepthofcutof1.5mm,theinclinationanglehastotakevaluesupto30 toavoidtheuseofthetooltipforcutting.Then,inregardoframpingresults,thein uenceofploughingisextendedbutitsdistributiononthethreeaxesisverysimilar.Theoptimuminclinationangleseemstooccurbetween0 and10 tolimittoolde ectionandbetween20 and30 toreduceedgephenomena.

Incontouringbytheright(orup-cuttingcontouring)(Fig.10),thepredictedvaluesarecorrectandimprovewithincreasingvaluesofδ,andthediscrepancyissimilar(<20%)thanindown-cuttingcontouringontheFxandFycomponents(Fig.10(a)and(b),respectively).Nevertheless,itcanbenoticedasmalldiscrep-ancyandaquickerconvergenceonY-axis.TheFzcomponentpresentsmoreoffset(<35%)andasimilarevolutionthaninthepreviouscon guration.Theincreaseofdiscrepancybetween15 and20 isdif culttoanalysebecauseofabsenceofexperi-mentaldataaftertheinclinationof20 .Incomparisonwiththedown-cuttingcon guration,thecurvesslopesaresimilarbutthemaximumlevelofFyforceishigherandtheFxforcetendstodecreasecontinuously.Here,aninclinationangleδof15

seemstobefavourabletocontrolbothcuttingforceslevelandploughingeffect.

Theseresultsforthecontouringcon gurationsindicatethatthecuttingforcesdistributionsonthetoolarelessuniformthaninrampingandthatthein uenceofploughingisaveraged.Theyalsounderlinethespeci cityofeachcuttingmodeevenifthesemi- nishingtestsprovidemoreinformationinthis eldbecauseofthelimitedpathinterval( p=1.5mm)andthenthelimitedchipslength.5.Globaldiscussion

Thein uenceofcuttingconditionswasstudiedandsomeusualtendencieswereretrieved:

theglobalforcesleveltendstodecreasewithincreasingcut-tingspeed;

thecuttingforcesareproportionallyaffectedbythefeedvelocity;

thedown-cuttingmodeismorefavourableforsmallpathintervalsanddepthsofcut;

theupcuttingismorefavourableforsomeslottingtestspar-ticularlywhenthetoolendiswidelyengagedinworkpiecematerial.Ithasbeenalsoobservedamorefavourabledistributionofcuttingforcesinup-rampingandup-contouringstrategiesevenforsmallvaluesofpathinterval.

Thein uenceofradialtoolrun-outoncuttingforcesleveliswellreproducedandthatvalidatesthemodellingofrun-outwitharadialeccentricityonly.Theforcediscrepancybetweenthetwoteethcanreach30%forslottingtestsand50%forsemi-

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Table1

Referencevaluesofoptimuminclinationangleδ

Optimisationcriteria

Limitingmaximumcuttingforces

Upwardramping0 or20 Downwardramping25

Contouringbytheleft0 or10 Contouringbytheright

0 or15

nishingtests.Degradationofsurfacequalityandoftoollifewasalsoobservedwithincreasingvaluesofrun-outeccentric-ity.Thetoolrun-outincreasesalsochattervibrationsobservedfromaspectrumanalysisconductedoneachmeasuredsignal.Thevariationofinclinationangleunderlinedthefactthatthein uenceofradialtoolrun-outincreaseswiththeinclinationangle.Forallthesereasons,itisessentialtotakeintoaccountoftoolrun-outinanymillingforcemodelsandthisdefaulthastobecontrolledandavoidedasfaraspossibleinordertouseproperlythetool–surfaceinclinationasanoptimisationparameter.

Theexistingdiscrepancybetweenmeasuredandcalculatedvaluesismainlyduetopersistentchatterassociatedtothedis-continuouscuttingprocessinmillingandtoploughingeffectwhichparticularlyoccurswhenthecuttingedgesharpnessdecreasesandwhencuttingvelocityandundeformedchipthick-nesstendtozero.TheseconditionsarecombinedinthetoolendregionandithasbeenshownthattheresultantploughingforceactingmainlyonthetoolaxisforcecomponentFz.Ploughingaffectscuttingforcesbutalsocuttingstability,toolwearon ankfaceandsurfaceintegrity.Itisthenimportanttoavoidthisphe-nomenonbyselectinganappropriatetool–surfaceinclination,andinthisview,themodelcanbeusedtoidentifytheeffectofploughingfrommeasureddata.

Theresultspresentedhereunderlinethatthemoreef cientstrategyisup-ramping,thisisduetoabetterdistributionofcuttingforcesonthecuttingedges.Inaddition,thetoolendisoutoftheworkingzone,andploughingeffectsareavoided.Theworststrategyisthedown-rampingcon gurationbecauseofploughingandinstability.Thecontouringstrategiesaremoredependentontheoptimisationcriteria.SomereferencevaluesofoptimuminclinationangleδareproposedinTable1foreachslottingcon gurations.Thesevaluesarecomparablewiththoseavailableinliteratureforsimilarcases[19,20,34].Thestrategiescorrespondingtointermediatemillingtoolpathswerenottestedherebutrecentworksindicatethatthesecasesgiveintermediatecuttingforceslevelsandshapes[10,15].6.Conclusion

Thecuttingforcespredictedfromathermomechanicalmodelappliedtothree-axisball-endmillingofaninclinedsurfaceareherepresentedandcomparedwithexperimentaldataobtainedfromslottingtestsandmeasuredwithadynamometer.Thether-momechanicalmillingmodelisdetailedinapreviouspaper,partIofthiswork[1].Someresultsexamplesarepresentedinordertoshowtheshapeandlevelofcuttingforcesforboth

LimitingploughingeffectLimitingbotheffects15–20 20 0 –20–30 10 15–20

15

measuredandcalculatedinthefourreferenceslottingtestscon gurations:upwardramping,downwardramping,andcon-touringbytheleftandbytheright.Thein uencesofthemainparametersresponsibleofcuttingforcesevolutionanddiscrep-ancybetweenexperimentalandsimulationdataarediscussed.Theseparametersarecuttingconditions,toolrun-out,plough-ingandtool–surfaceinclinationangle.Inparticular,thespeci cin uenceofthetool–surfaceinclinationwasstudiedfromthevariationofmaximumcuttingforcesleveloneachaxisinregardoftheworkpieceinclinationangledenotedδ.Thisworkcon- rmedthattheeffectofploughingisdirectlylinkedtotheuseofthetooltipregionincuttingandthatitispossibletolimitthisnegativeeffectwiththetool–surfaceinclination.Acknowledgements

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。。。

具有加工倾角的球头铣刀切削力模型 Part II.切削条件，跳动，犁切和倾斜角的影响

M. Fontaine, A. Moufki, A. Devillez, D. Dudzinski 修订日期: 2006 - 12 – 12 ；收稿日期: 2007 - 01 – 12

摘要：

本文着重对球头铣刀刀具轴线和工件加工表面之间的加工倾角造成对切削力的影响进行了理论研究。从热力切割模型计算得到的切削力，已经在本论文的第一部分进行了阐述[M. Fontaine, A. Moufki, A. Devillez, D. Dudzinski，球头铣刀加工倾角的切削力模型Part I. 预测力模型和实验验证，J. Mater. Process. Technol. 189(2007) 73–84]，在这里我们将展开详细的讨论并且与实验结果进行比较。前文提出的球头铣刀模型可应用于直线刀具轨迹和变化的加工倾角的机械加工中。本文则同时对斜面和轮廓配置进行了研究。研究中的实验结果是在一台装配Kistler测力计的3轴数控铣床上进行测试的，注意这里指出的形状和切削力信号水平。为了确定最佳倾角，我们对刀具作用最大的切削力的变化做了研究。全文对切削条件、径向跳动、犁切现象和切削稳定性造成的影响分别进行了讨论。

© 2007年科学时报社版权所有。

关键词：球头铣刀；切削力；切削条件；刀具跳动；犁切；加工倾角

1. 引言

许多学者提出了有效的模型来预测铣削加工中产生的切削力。某些实验结果可用于钢材[2-15]，铝合金[14-18]，锌合金[19,20]或航空合金[6,21-23]。但是，很少有人去讨论影响实验结果的因素。众所周知，一些参数显著地影响着切削力的大小和变化，并且重要的是要考虑他们的加工模式。对于球头铣刀，主要影响的参数是切

。。。

削条件、刀具的跳动、犁切现象和加工倾角。在加工时，我们总是把它们分开对待而不是综合考虑。

在以往的文献中，球头铣削加工所得到的切削力实验结果往往只是考虑了一部分切削条件（主轴频率，进给量，切削深度，切削模式）带来的影响。究其原因主要是实验测试需要花费大量的时间和经费。许多基于实验的文献表明对于变化的切削速度值与额定进给率相关[3,4,6,11,13,15,18,19,21]，而有其他文献研究表明是受进给率和主轴运动频率的双重影响[7，11，12,16,17,22]。通常的装置测量切削力是压电式测力机，但是对于高速铣削，它的使用是受传感器动态响应的限制的。然而，针对高速铣削研究的人数却在减少。有些作者提出是由于轴向切削深度[7,15,19,21]或是径向切削深度[13,17]，也有提出是两者共同影响的结果[3,4,6,12,16,20,22]。在过去研究的过程中，主要还是应用一些旧的切削模式，却鲜有人将它与最新的切削模式进行对比。

刀具跳动是传统地以面铣削加工为蓝本的，仅有少部分学者提出一些解决方案将球头铣削中刀具的运动考虑进来。这样，跳动就通常被认为仅是径向偏心（径向跳动）[3,14]，而摆角（轴向跳动）却很少考虑。在球头铣刀中，有限的刀具的直径和长度减少了自身对跳动的影响。在一个完整的五轴铣削力模型中，利用轴向跳动精确计算工件加工表面与刀具的接触面是非常有趣的[22]。这些过去的文献表明，径向跳动对边缘的接触面和切削力大小有着巨大的影响，尤其是对于高切削速度和低切削厚度的情况。

非剪切现象出现在尖端，并产生额外的力作用在刀具上。它们可以由正交切削下产生的滑移线所决定[24，25]。在球头铣刀中，它们曾被隐晦地考虑到机械模型，但并没有进行具体的说明。然而，一些学者通过利用剪切和从转向实验中获得的边缘系数[21]将剪切过程从边缘现象中分离出去，或者直接通过铣削实验反推然后进行二乘法调整或卷积积分法[8,15,18,26,27]来实现。我们可以分析模型认识到，犁切似乎就是主要影响切削力的边缘现象，同时也确定了由于局部剪切会造成额外的切削力的事实。

过去的10年里，有很多关于球头铣刀的加工倾角研究的文章。有研究表明可以通过注意发生刀尖的几何和动力学问题、改变工具和工件之间的接触面来控制切削力大小并使得它在刀具上得以再分配。倾角可以用在各种球头铣削的加工中，

。。。

即使是对于复杂曲面，但通常我们只关心由平面或圆构成的曲面。许多学者通过实验测试和与原有的铣刀模型获得优化参数,从而可以分析它们对切削力

[5,6,8–10,15,17,19,20]

、刀具或零件的变形和加工误差[4,10,11,28–30]、表面形貌[8–10,29,31,32]、刀

具寿命[7]、甚至是刀具与工件接触区域的大小[10,11,15,19]和切削形状[8,10]的影响。一些参考倾角可以从这些研究中提取，但是想获得有价值的数据则需要优化工序。为了控制这个过程，重点似乎是取决于整个过程切削力的大小以及工艺稳定性和表面完整性。

为了研究球头铣刀的切削参数所扮演的角色，我们对由形变模型预测[1]出的切削力和从实验测试获得的切削力进行了比较和讨论。为了简化观察和方便与文献中已获得的可用的数据进行比较，在这里我们只对三轴铣床切削斜面上的球头铣刀进行研究。加工材料是42CrMo4钢,通常用于注塑模具的制造，对它的特点，如：加工硬化、应变率敏感性和热软化，我们都非常了解。某些结果给出了四个开槽的参考方法，并且从全部实验数据中得到的关于切削条件、径向跳动、犁切现象和加工倾角对切削力的影响也验证了Part I中的模型的合理性[1]。本文对球头铣刀刀具轴线和工件加工表面之间的倾角进行了理论研究，为优化全过程切削力大小以及抑制犁切现象提供了依据。

2. 铣削实验及模拟

拟议的实验和计算结果以及相关研究讨论主要对应于某一区间公称直径的铣刀(Dp>12mm)的开槽试验。这些测试似乎是最适合单独分析每个参数的影响，因为充分的刀具径向切深下进给量稳定整个切削的过程，并且可以得到过程中切削力的变化规律；因此，这种方法的测量质量是最好的，离散现象也同时得以避免。 2.1.

切削方法

图1中提出了参考平面和4个切割开槽的方法。Z-轴刀轨的两个不同方向的斜面的切割方法:斜向上铣削[图.1(a)]；斜向下铣削[图.1(b)]；两种不同主切削模式下的轮廓切割方法:顺铣轮廓[图.1(c)]；逆铣轮廓[图.1(d)]。未切削的表面是一个在水平(X-Y)参考平面Y方向的斜面，倾角用d表示[图(1)]。刀具Z轴与机床Z轴统一(三轴铣削配置)。全局坐标系(GCS)的参考点取为工件加工表面角落三轴交点处，如图1中的点O。与刀具固结的局部坐标系(LCS)取为刀具上的点E。

。。。

图1. 倾斜表面上三轴球头铣刀铣削方向

a.斜向上铣削 b.斜向下铣削 c. 顺铣轮廓 d. 逆铣轮廓

2.2. 实验装置及参数设定

这个实验是在干燥条件下的立式三轴数控铣床进行的，工件材料为42CrMo4

钢（即AISI4142)。切削力测量仪器则是一台六组件的Kistler测力计（9265B型）。输出信号被记录存储在由德维创电脑控制的高通道Kistler电荷放大器中。在工件中部截取信号，并将电压/时间信号转化为力量/旋转角度。频谱分析是检查实验信号的稳定性，低通滤波信号可以抑制由于振动产生的加速仪的噪声。

实验中采用了Diager-Industrie公司的有两个凹槽的球头铣刀(参考51221)，其公称直径为12mm，公称螺旋角为17°，法向前角为0°，即R0=6mm；

i0=17°;an=0°。

主轴的频率和进给量被限定为:W=5000rpm，ft=0.05mm/牙和每转。这个Kistler测力计装夹在在一个床身台面的倾斜位置[图2(a)]。五个倾角值分别取：

。。。

d=0°、5.07°、10.43°、15.26°和20.17°。试样规格：高度为20mm,宽度和长度为

50mm[图2(b)]。垂直切削深度:dn=1.5mm。槽长：p=15mm。

刀具的跳动量是由固定于机身台面并且与铣刀的圆柱部分接触的刻度盘测量

的。

图2. 倾斜的制件设备的球头铣刀铣削测试 a. 倾斜位置的测力计 b. 工件、刀具细节图

2.3. 模型制造

用来算切削力的几何和热力学模型已经在本文的Part I中提出[1]。模型参数如

下：

l 剪切角系数：A1=40°；A2=0.5。 l 主摩擦系数：mf=1.04(lf=46°)。

&=0.001s-1；l Johnson-Cook参数的本构关系：A=612MPa；B=436MPa；r

n=0.15；m=0.008；v=1.46；Tr=Tw=293K；Tm=1793K。

l 其他有用的材料参数：r=7800kg/m3；c=500J(kgK)；b=0.9。 l 主剪切带厚度：h=0.025mm。 2.4.

计算结果与实验结果的比较

图3-6呈现了上述四种不同切削方法下作用在刀具上的测量与预测切削力的关系[图1] ：斜向上铣削[图3]；斜向下铣削[图4]；顺铣轮廓（向左轮廓切割）[图5]； 逆铣轮廓（向右轮廓切割）[图6]。 采用的倾角d值很高：斜切d=15°；

。。。

轮廓切割d=20°。 由于刀具的变动，由千分表所测量的跳动量(离心率e和位置角y

e)，在两种情况下是有区别的。

图

3. 测量及预测斜向上铣削的切削力.

。。。

图

4. 测量及预测斜向下铣削的切削力.

测量和预测的切削力关系曲线中所有的数据与刀具作用在工件材料上的输入、输出值非常相似。由于刀具跳动与模型的吻合度造成了在两齿间的切削力大小的差异。它的具体影响在下一节中讨论。切削力略有不同；从模拟仿真中获得的值更小。切削力大小总能准确地预测出两切削力分量Fx、Fy（偏差<15%），而一个更重要的振幅偏差在特定条件（斜向下铣削）下则会出现在Fz上(偏差>30%)[图4]。振幅偏差主要是由于工件材料的切削边缘存在重叠区,这同时也导致了切削的不稳定。所用的低通滤波并不能抑制所有的干扰,因为所有主轴变频的具有代表性的谐波的必须保留。这个不稳定性在曲线的零值附近得以体现。它没能与模型三分贴合,因为模型中使用的切割模式是基于连续切割过程的。可以看

。。。

到，测得的沿Z轴的信号不稳定性没有沿Y轴的严重。发生在切削边缘以及后角端面的畸变现象，特别是犁切的影响都是造成这些差异的因素。这种现象及其影响

具体介绍如下。

图5. 测量及预测顺铣轮廓时的切削力（+Y）.

W

=5000rev/min,ft=0.05mm/tooth,Dp=12mm,e=0.008mm,ye=110°,d=20°.

3. 切削条件、刀具跳动和犁切现象的影响

3.1.

切削条件的影响

切削力合力会随着切削速度v值的增加而减少，并且v值越高过程和信号越倾向于稳定。在高速切削[33]以及当v值的增加测量、计算得到的力随着减小之间存在矛盾的情况下，我们这里所采取的热机械加工方法能给出更好的结果。 可以注意

。。。

到，进给率的变化直接地影响着切削力，当然这些值与切削厚度t0是成比例的。 实验中采用的最高的进给率（0.2mm/rev）有限制犁切现象的影响的趋势，但是也许会增加刀具挠曲或刀具磨损(直接与切削力大小相关)。

图6. 测量及预测逆铣轮廓时的切削力（-Y）.

W

=5000rev/min,ft=0.05mm/tooth,Dp=12mm,e=0.008mm,ye=110°,d=20°.

减少刀轨行距Dp自然地降低了切削力值,但Dp值较小时（半精加工），记录信号往往不稳定。这些实验信号，我们可能会拿来与从考虑刀具挠曲和颤振的改进版的模型中获取的计算值进行比较。然而,这些半精加工的结果使得“向上”的方法(斜向上铣削和逆铣轮廓[1])和其他方法的差异更加突出：在这些有利的情况下,切削力合力减小,并且在各轴上力的分配也更加均匀。这实际上主要得益于考虑了

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