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InternationalJournalofMachineToolsManufacture432003687–697IdentificationofcutteroffsetinendmillingwithoutapriorknowledgeofcuttingcoefficientsJ.J.J.Wang∗,C.M.ZhengDepartmentofMechanicalEngineering,NationalChengKungUniversity,Tainan,Taiwan701Received28January2002accepted21January2003AbstractThispaperpresentsamethodfortheidentificationofcutteroffsetthroughmillingforcewithoutrequiringthespecificcuttingcoefficientstobeknownaspriori.Thetotalmillingforceinthepresenceofcutteroffsetisfirstobtainedonthebasisofdualcuttingmechanisms,wherethelocalforceiscomprisedofaconstantplowingforceandalinearshearingforceproportionaltothechiploadunderthecutteroffset.Thetotalmillingforceissynthesizedthroughconvolutionandisshowntobethesumofthreeforcecomponentsthenominalchipshearingforcecomponent,theplowingforcecomponentsandtheoffsetrelatedshearingforcecomponent.Fourieranalysisoftheseforcecomponentsrevealstheeffectsofoffsetgeometryandyieldsformulasfortheidentificationofoffsetgeometry.Theidentificationprocessrequiresonlytwocuttingtestsandtheevaluationoftwoalgebraicexpressionstheshearingconstantsarefoundfromtheaverageforcesofcuttingtestsandtheoffsetgeometryisidentifiedfromtheforcecomponentatthespindlefrequency.Throughnumericalsimulationandexperimentalresults,theefficacyoftheidentificationmethodisdemonstratedtheeffectsoffeedpertoothandcuttingdepthsontheaccuracyofthemodelareinvestigatedandcriteriafortheappropriateselectionoftheseparametersaresuggested.2003ElsevierScienceLtd.Allrightsreserved.KeywordsOffsetrunoutEndmillingMillingforceShearingPloughing1.IntroductionThepresenceofcutterrunoutinmillingdegradessurfacequalityandincreasesboththeradialandthrustforcevariationonthetools,aswellasonthespindlebearingthusreducingtoolandspindlelife.Theeffectofcutterrunoutonthemillingforceanditsidentificationhasbeenthesubjectofmanystudies.Researchesin1–4havealreadyproposedmillingforcemodelswithknownrunoutgeometry.However,cutterrunoutisgenerallydifficulttopredictandmeasurebecauseitresultsfromthecomplexinprocessinteractionofthestaticrunoutandtheunbalancingdynamicsoftherotatingassembly.Althoughthemeasurementofstaticcutterrunoutcanbecarriedoutofflinewithadialindicatororotherelectronicmeans,thedynamicrunoutgeometryduringthe∗Correspondingauthor.Tel.188662757575Ext62189fax188662367231.Emailaddressjjwangmail.ncku.edu.twJ..J.J.Wang.08906955/03/seefrontmatter2003ElsevierScienceLtd.Allrightsreserved.doi10.1016/S0890695503000282millingprocesscouldbedifferentandadirectmeasurementwouldbeverydifficult.Theabilitytoidentifytheinprocessrunoutgeometrywouldbeusefulfortheusersandmanufacturersofmachinetools,toolholdersandspindlesinthemonitoringandtestingofcuttereccentricitywithinthedesiredrangeofoperatingspeed.Sincecutterrunoutresultsintheredistributionofchiploadamongthecutterflutes,thegeometryofcutterrunoutisembeddedinthemillingforce.Millingforcethusbecomesaconvenientsignalsourcefortheindirectmeasurementorestimationofcutterrunout.Armaregoetal.4usedtheirrunoutforcemodelfortheestimationofcutterrunoutthroughaniterativebestfitprocedure.Guetal.5presentedalgorithmsfortheestimationofcutterradialandaxialrunoutforthefacemillingprocessbyfittingthepeakandvalleyvaluesofthecomputedmillingforceswithassumedunknowncutterrunouttothemeasuredcuttingforces.Theresultsshowedagreatdiscrepancybetweentheestimatedrunoutandstaticallymeasuredcutterrunout.Basedontheconvolutionanalysis,Jayarametal.6presentedamethodologyfor688J..J.J.Wang,C.M.Zheng/InternationalJournalofMachineToolsManufacture432003687–697NomenclatureAn,AovectorsoftheFourierseriescoefficientsforthenominalmillingforceandoffsetrelatedforcea,N,Rhelixangle,numberofcutterflutesandcutterradiusb,r,hangular,radial,andaxialpositionsforcuttingpointsincuttercoordinatecwd,cwdcchipwidthdensityfunctionofthefirstcuttingedgeandoftheentirecutterba,bpaxialimmersionangleandflutespacingangleCWDFouriertransformofthechipwidthdensityfunction,cwdda,draxialandradialcuttingdepthft,frlocaltangentialandradialcuttingforcesperunitchipwidthfcutterangulardisplacementgx,gyXandYcomponentsofthelocalcuttingforcesperunitchipwidthgtotalcuttingforcevectorkts,krsspecificshearingforceconstantsinthetangentialandradialdirectionsktp,krpspecificplowingforceconstantsinthetangentialandradialdirectionslangularlocationofthecutteroffsetps,pp,povectorsoflocaltangentialforcesperunitwidthintheworkcoordinateduetoshearing,plowingandaxisoffseteffectsPs,Pp,PoFouriertransformsofps,pp,poqs,qp,qodirectionalmatricesfortheshearing,plowingandaxisoffsetforcesqangularpositionofcuttingpointintheworkq1,q2entry/exitanglesrmagnitudeofcutteroffsetTtransfermatrixbetweenmeasuredcuttingforcesandcuttingconstantstxfeedpertoothwnormalizedfrequencywithrespecttospindlefrequencydetectingprocessfaultsfromthespectrumofthecuttingforcesignalinfacemilling.Basedonthiswork,Akshayetal.7reportedanoptimizedapproachthatusedallthespindleharmonicsofthecuttingforcespectrumofYdirectionforradialrunoutestimationinfacemilling.Itwasshownthattheoptimizedalgorithmperformedwellevenwithtimevaryingsurface.Cuttingforcemodelsandthecutterrunoutestimationschemesin4–7havereliedonnumericalanditerativeapproachtoestimatethetotalmillingforceandcutterrunoutgeometry.ByrepresentingtheradialrunoutofeachindividualinsertasadiscreteFourierseries,SeethalerandYellowley8derivedtheFourierseriescoefficientsoftheoffsetrelatedforcefortheidentificationofradialrunoutforafacemillingcutter.However,thismodelisnotapplicablewheretherunoutmagnitudevariesalongthecutterhelicaledgesuchasthecasewithendmilloffset.Forasolidendmill,cutterrunoutveryoftenresultsfromacutteraxisoffset.Forthistypeofrunoutgeometry,WangandLiang9analyzedthechiploadkinematicsandderivedanalyticexpressionsforthechipthicknessandaveragechipthickness.Combiningthischiploadkinematicsandtheconvolutionintegrationmethod,LiangandWang10formulatedandanalyzedtheinfluenceofcutteraxisoffsetonthemillingforcesinthefrequencydomain.Basedonthisanalyticalforcemodelandtherequirementofaprioriknownexponentofthecuttingcoefficientfunction,themagnitudeandangularlocationofcutteraxisoffsetwereidentifiedthroughtheFourierseriescoefficientsofcuttingforcesatthespindlefrequency.Byextendingtheworkin10,HekmanandLiang11presentedarecursiveschemeforthemostrecentestimatesofthedynamicrunoutmagnitudeandorientationintermsofthepreviousestimates.Undersimilarconvolutionstructure,ZhengandLiang12furtherextendedthechiploadandforceanalysisfortheidentificationofcutteraxistiltinendmilling.Estimationofendmilloffsetinthesenumericaloranalyticalmodelsheretoforerequiresapriorknowledgeofthespecificcuttingcoefficients,whichisusuallynotavailableespeciallyinanindustrialenvironment.Aseriesofcuttingexperimentswouldhavetobecarriedoutinadvanceforthetoolandworkmaterialofinteresttoestablishadatabaseorexpressionsforthecuttingcoefficients.Theeffectofongoingandunknowntoolwearaddsuncertaintytotheaccuracyofthispreestablisheddata.Althoughtheinstantaneoustangentialandradialcuttingconstantscanbeestimateddirectlyfromthemeasuredmillingforceasin10–12,theexponentsofthepowerexpressionsforthecuttingcoefficientsasfunctionsofchipthicknesswouldneedtobeknowninadvance.Thisprerequisitecomesfromthefactthatthemillingforcevariationduetochiploadchangeinthepresenceofcutterrunoutisattributedtothechipshear689J..J.J.Wang,C.M.Zheng/InternationalJournalofMachineToolsManufacture432003687–697ingmechanismalone,whiletheestimatedcuttingconstantsfromtheexistingmodelarebasedonthelumpedeffectsofchipshearingandedgeplowingcuttingmechanisms.Theselumpedconstantsdependgreatlyonchipthicknessandaregenerallyexpressedaspowerfunctionsofchipthicknessoraveragechipthickness.Itwouldbemoreappropriateandhasstrongerbearingonthephysicalgroundstouseonlyshearingrelatedcuttingcoefficientsinmodelingtheforcevariationduetocutterrunout.Researchesin4,8,13–16haveusedarefinedcuttingforcemodelthatexplicitlyincludestheseparateandsimultaneouseffectsofchipshearingandedgeplowingmechanismsinthecuttingprocess.Yellowleyassumedaconstantpresenceofplowingforceandalinearshearingforceproportionaltotheinstantchipthicknessandshowedthatsuchamodelisaccuratefortheidentificationofinsertrunoutin8andforthepredictionofaveragecuttingforce13.Armarego4,14proposedthattheshearingcoefficientsbeexpressedasfunctionsofnormalrakeangle,toolinclinationangle,normalshearangle,shearstress,frictionangleandchipflowangle,andtheplowingcoefficientsexpressedasfunctionsofnormalrakeangleandcuttingvelocity.Basedonworksin4,14,Budaketal.establishedapredictivemodelforthedualcuttingcoefficientsin15andshowedthattheshearingcoefficientsandplowingcoefficientscanbetreatedasconstantswithoutlossofaccuracy.Itisthereforeproposedinthispaperthattheendmillingforcewithcutteroffsetbeformulatedusingconstantplowingandshearingcuttingcoefficients.Forcecomponentsduetothecutteroffsetcanthenbeappropriatelyaccountedforbytheshearingcuttingconstants.Theuseofconstantcoefficientswithdualcuttingmechanismswillbeshownhereintonotonlysimplifythetaskfortheidentificationoftheoffsetgeometrybutalsoyieldmoreaccurateresults.Inthenextsectiontheanalyticalforcemodelwithcutteroffsetisfirstestablished.Section3presentsthemethodsfortheestimationofcuttingconstantsandtheoffsetgeometryfollowedbythenumericalandexperimentalverificationsinSection4andconclusionsinSection5.2.MillingforceswithcutteraxisoffsetThecutterandworkcoordinateswiththeoffsetgeometryaresetupsimilarlytothosein9andareshowninFig.1.Thepositionsofcuttingedgesarerepresentedintherbhcoordinate.Thebottomofanarbitrarilychosencuttingedgeislocatedatb0andcoincideswithq0intheworkcoordinatewhenthecutterdisplacementfis0.Asthecutterrotates,thesethreeangularvariablestakeonthefollowingrelationshipFig.1.aThecutterandworkgeometryandcoordinatesinthemillingprocessandbcutteroffsetgeometry.qH11005fH11002b1TheoffsetgeometryshowninFig.1.bhasthegeometricalcenterofthecuttershiftedadistanceofrfromtherotationcenterinanangulardirectionoflfromb0.Therevisedchipthicknesswithcutteroffsetisshownin9,18tobeapproximatedbytcq,bH11005txsinqH11001tob2wheretobH11005H110022rsinpNsinbH11002lH11002pN3Consideringtheshearingandplowingcuttingmechanisms,thelocalcuttingforceperunitchipwidthisassumedtotakethefollowinglinearformwithfourcuttingconstants,H20873gtq,bgrq,bH20874H11005H20873ktskrsktsH20874tcq,bH11001H20873ktpkrpktpH208744wherektsandkrsaretheshearingconstantsrepresenting,respectively,thetangentialspecificshearingconstantandtheratioofradialtothetangentialshearingforce.ktpandkrparetheplowingconstantsrepresentingthetangentialplowingconstantandtheratioofradialtothetangentialplowingforcerespectively.Intheworkcoordinate,thelocalforcebecomesH20873gxq,bgyq,bH20874H11005H20875cosqsinqsinqH11002cosqH20876H20873gtq,bgrq,bH208745690J..J.J.Wang,C.M.Zheng/InternationalJournalofMachineToolsManufacture432003687–697CombiningEqs.2–5andrearrangingEq.5,thelocalforcescanbeexpressedasthesuperpositionofanominalcuttingforceandaforcecontributionfromthecutteroffsetH20873gxq,bgyq,bH20874H11005H20873fxqfyqH20874H11001H20873fxoq,bfyoq,bH208746inwhichH20873fxqfyqH20874H11005qspsqH11001qpppqandH20873fxoq,bfyoq,bH208747H11005tobqopoqwherepsqH11005ktstxH20873p1p2H20874H11005ktstxH20873sinqcosqsin2qH20874wq,ppqH11005ktpH20873p3p4H20874H11005ktpH20873cosqsinqH20874wq,poqH11005ktsH20873p3p4H208748H11005ktsH20873cosqsinqH20874wqandqsH11005qoH11005H208751krsH11002krs1H20876,qpH11005H208751krpH11002krp1H208769ThenominallocalforceinEq.7isexpressedasthesumoftheshearingandtheplowingcomponentsasindicatedbytheirsubscripts.Matricesqs,qpandqoreflecttheeffectsofradialcuttingconstantsonthelocalforce.Forcevectors,ps,ppandpoinEq.8arethelocaltangentialforcesperunitwidthduetotheshearing,plowingandaxisoffseteffectsrespectively.AunitheightpulsefunctionwqisincludedinEq.8torepresenttheactualradialrangeofcuttingbetweentheentryangle,q1,andtheexitangle,q2.Althoughtheentryandexitanglesarefunctionsofbasshownin9,theyareapproximatedasconstantsheretoreducethemathematicalcomplexityoftheforcemodel.Thetotalforcecanbeobtainedbyintegratingthelocalforcealongthehdirectionwithintheaxialdepthofcut.WithachangeofvariableasinEq.1,thetotalforcecanbeshowntotakeonaconvolutionintegralformgfH11005H20873gxfgyfH20874H11005H20885da0H20873gxqgyqH20874dhc10H11005H20885H11009H11002H11009H20873gxfH11002bgyfH11002bH20874cwdcbdbH11005gf∗cwdcfwhere∗denotestheconvolutionoperation.hcbinEq.10representstheaxialpositionoftheactivecuttingpointsfortheperiodicsequenceofcutterflutesandcwdcisthechipwidthdensityfunctiondefinedbydhc/db,thechipwidthperradianinb.ThesetwofunctionsareshowninFig.2andtheirexpressionsarehcbH11005H20913H11009iH11005H11002H11009hbH11002ibpwithhb11H11005H20902Rtanab,0H11349bH11349ba0,otherwisecwdcbH11005dhcdbH11005H20888H11009iH11005H11002H11009cwdbH11002ibp12withcwdbH11005dhdbH11005H20902Rtana,0H11349bH11349ba0,otherwisewherebp2p/Nistheangularspacingbetweenneighboringflutesandbadatana/Rrepresentstheaxialimmersionangleofthecutter.Indexiisthesequencenumberofthecutterflutes.SubstitutingEqs.7–9into10resultsinaconvolutionformintermsofthreedifferentforcegeneratingmechanismsFig.2.aTheaxialpositionfunctionoftheactivecuttingpointsandbthechipwidthdensityfunction.
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