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ORIGINALARTICLEDeformationcontrolthroughfixturelayoutdesignandclampingforceoptimizationWeifangChen&LijunNi&JianbinXueReceived:2February2007/Accepted:4July2007#Springer-VerlagLondonLimited2007AbstractWorkpiecedeformationmustbecontrolledinthenumericalcontrolmachiningprocess.Fixturelayoutandclampingforcearetwomainaspectsthatinfluencethedegreeanddistributionofmachiningdeformation.Inthispaper,amulti-objectivemodelwasestablishedtoreducethedegreeofdeformationandtoincreasethedistributinguniformityofdeformation.Thefiniteelementmethodwasemployedtoanalyzethedeformation.Ageneticalgorithmwasdevelopedtosolvetheoptimizationmodel.Finally,anexampleillustratedthatasatisfactoryresultwasobtained,whichisfarsuperiortotheexperientialone.Themulti-objectivemodelcanreducethemachiningdeformationeffectivelyandimprovethedistributioncondition.KeywordsFixturelayout.Clampingforce.Geneticalgorithm.Finiteelementmethod1IntroductionFixturedesignisanimportantprocedureinmanufacturingengineering.Itiscriticaltomachiningaccuracy.Aworkpieceshouldbeconstrainedinafixtureduringmachiningwithfixtureelementssuchaslocators,clamps,andsupports.Thepositionsoflocators,clampsandsupportsshouldbestrategicallydesignedandappropriateclampingforcesshouldbeapplied.Thefixtureelementscanbeplacedanywherewithinthecandidateregionsontheworkpiecesurfaces.Clampingforcemustbelargeenoughtoholdtheworkpieceduringmachining.Typically,itreliesheavilyonthedesignersexperiencetochoosethepositionsofthefixtureelementsandtodeterminetheclampingforces.Thusthereisnoassurancethattheresultantsolutionisoptimalornearoptimalforagivenworkpiece.Consequently,thefixturelayoutandtheclampingforceoptimizationbecometwomainaspectsinfixturedesign.Thepositionsoflocatorsandclamps,andthevaluesofclampingforceshouldbeproperlyselectedandcalculatedsothattheworkpiecedeformationduetoclampingandcuttingforceisminimizedanduniformed.Theobjectiveoffixturedesignistofindanoptimallayoutorpositionsofthefixtureelementsaroundtheworkpieceandoptimalclampingforce.Inthispaper,amulti-objectiveoptimizationmethodispresentedforthefixturelayoutdesignandclampingforceoptimization.Theobjectiveistwofolded.Oneistominimizethemaximumelasticdeformationofthemachinedsurfaces,andanotheristomaximizetheuniformityofdeforma-tion.TheANSYSsoftwarepackageisusedtocalculatethedeformationoftheworkpieceundergivenclampingforceandcuttingforce.Ageneticalgorithmisdevel-oped,andthedirectsearchtoolboxofMATLABisemployedtosolvetheoptimizationproblem.Finally,acasestudyisgiventoillustratetheapplicationoftheproposedapproach.2LiteraturereviewWiththewideapplicationsofoptimizationmethodsinindustry,fixturedesignoptimizationhasgainedmoreinterestsinrecentyears.Fixturedesignoptimizationincludesfixturelayoutoptimizationandclampingforceoptimization.KingandHutterpresentedamethodforIntJAdvManufTechnolDOI10.1007/s00170-007-1153-2W.Chen:L.Ni:J.Xue(*)CollegeofMechanicalandElectronicalEngineering,NanjingUniversityofAeronauticsandAstronautics,No.29,YudaoStreet,Nanjing210016,Chinae-mail:meejbxueoptimalfixturelayoutdesignusingarigidbodymodelofthefixture-workpiecesystem1.DeMeteralsousedarigidbodymodelfortheanalysisandsynthesisofoptimalfixturelayoutsandminimumclampingforce2.Hepresentedafiniteelementmethod(FEM)basedsupportlayoutoptimizationprocedurewithcomputationallyattrac-tivequalities3.LiandMelkoteusedanonlinearprogrammingmethodandacontactelasticitymodeltosolvethelayoutoptimizationproblem4.Twoyearslater,theypresentedamethodfordeterminingtheoptimalclampingforceforamultipleclampfixturesubjectedtoquasi-staticmachiningforce5.Theyalsopresentedanoptimalsynthesisapproachoffixturelayoutandclampingforcethatconsidersworkpiecedynamicsduringmachining6.Acombinedfixturelayoutandclampingforceoptimizationprocedurewaspresented.Otherresearchers7,8usedtheFEMforfixturedesignandanalysis.Caietal.9extendedtheworkofMenassaandDeVries8toincludesynthesisoffixturelayoutforsheetmetalassembly.Qinetal.10establishedanelasticcontactmodelbetweenclampandworkpiecetooptimizetheclampingforcewithanobjectivetominimizethepositionerroroftheworkpiece.DengandMelkote11presentedamodel-basedframeworkfordeterminingtheminimumrequiredclampingforce,whichensuresthedynamicstabilityofafixturedworkpieceduringmachining.Mostoftheabovestudiesusednonlinearprogrammingmethods,whichseldomgaveglobalornear-globalopti-mumsolutions.Allofthefixturelayoutoptimizationproceduresmuststartwithaninitialfeasiblelayout.Inaddition,solutionsobtainedfromthesemodelsareverysensitivetotheinitialfeasiblefixturelayout.Theproblemoffixturedesignoptimizationisnonlinearbecausethereisnodirectanalyticalrelationshipbetweentheobjectivefunctionanddesignvariables,i.e.betweenthemachinedsurfaceerrorandthefixtureparameters(positionsoflocatorandclamp,andclampingforces).Previousresearchershadshownthatgeneticalgorithm(GA)wasausefultechniqueinsolvingsuchoptimiza-tionproblems.WuandChan12usedtheGAtodeterminethemoststaticallystablefixturelayout.IshikawaandAoyama13appliedGAtodeterminetheoptimalclampingconditionforanelasticworkpiece.Vallapuzhaetal.14usedspatialcoordinatestoencodeintheGAbasedoptimizationoffixturelayout.Theyalsopresentedthemethodologyandresultsofanextensiveinvestigationintotherelativeeffectivenessofthemaincompetingfixtureoptimizationmethods,whichshowedthatcontinuousGAyieldedthebestqualitysolutions15.KrishnakumarandMelkote16developedafixturelayoutoptimizationtechniquethatusedGAtofindthefixturelayoutthatminimizedthedeformationofthemachinedsurfaceduetoclampingandcuttingforceovertheentiretoolpath.Locatorandclamppositionswerespecifiedwithnodenumbers.Krishnakumaretal.17presentedaniterativealgorithmthatminimizedtheworkpieceelasticdeformationfortheentirecuttingprocessbyalternativelyvaryingthefixturelayoutandclampingforce.Laietal.18setupananalysismodelthattreatedlocatorandclampsasthesamefixturelayoutelementsfortheflexiblepartdeformation.Hamedi19discussedahybridlearningsystemthatusednonlinearFEAwithasupportivecombinationofartificialneuralnetwork(ANN)andGA.TheANNwasusedtocalculateworkpiecemaximumelasticdeformation,theGAwasusedtodeterminetheoptimumclampingforces.Kumar20proposedtocombinetheGAandANNtodevelopafixturedesignsystem.Kaya21usedtheGAandFEMtofindtheoptimallocatorsandclampingpositionsin2Dworkpieceandtookchipremovaleffectsintoaccount.Zhouetal.22presentedaGAbasedmethodthatoptimizedfixturelayoutandclampingforcesimulta-neously.Someofthestudiesdidnotconsidertheoptimizationofthelayoutforentiretoolpath.Someofthestudiesusednodenumbersasdesignparameters.Someofthestudiesaddressedfixturelayoutorclampingforceoptimizationmethodsbutnotbothsimultaneously.Andtherewerefewstudiestakingfrictionandchipremovalintoaccount.Theeffectsofchipremovalandfrictionalcontactcannotbeneglectedforachievingamorerealisticandaccurateworkpiece-fixturelayoutverificationanalysis23,soitisessentialtotakechipremovaleffectsandfrictioneffectintoaccounttoachieveabettermachiningaccuracy.Inthispaper,thefrictionandchipremovalaretakenintoaccounttoachievetheminimumdegreeofthemaximumdeformationofthemachinedsurfacesunderclampingandcuttingforceandtouniformthedeforma-tion.Amulti-objectiveoptimizationmodelisestablished.AnoptimizationprocessbasedonGAandFEMispresentedtofindtheoptimalfixturelayoutandclampingforce.Finally,theresultofthemulti-objectiveoptimiza-tionmodeliscomparedwiththesingleobjectiveoptimizationmethodandtheexperiencemethodforalowrigidityworkpiece.3Amulti-objectiveoptimizationmodelforfixturedesignAfeasiblefixturelayouthastosatisfythreeconstraints.First,thelocatorsandclampscannotapplytensileforcesontheworkpiece.Second,theCoulombfrictionconstraintmustbesatisfiedatallfixture-workpiececontactpoints.Thepositionsoffixtureelement-workpiececontactpointsmustbeinthecandidateregions.Foraprobleminvolvingpfixtureelement-workpiececontactsandnmachiningloadIntJAdvManufTechnolsteps,theoptimizationproblemcanbemathematicallymodeledasfollowsminmax1jj;2jj;:;jC12C12C12C12;:;njjC0C1s;j1;2;:;n1SubjecttomFnijjC21F2tiF2hiq2FniC2103posi2Vi;i1;2;:;p4wherejreferstothemaximumelasticdeformationatamachiningregioninthej-thstepofthemachiningoperation,Xnj1jC0C0C16C172C30nvuutistheaverageofjFniisthenormalforceatthei-thcontactpointisthestaticcoefficientoffrictionFti;Fhiarethetangentialforcesatthei-thcontactpointpos(i)isthei-thcontactpointV(i)isthecandidateregionofthei-thcontactpoint.TheoverallprocessisillustratedinFig.1todesignafeasiblefixturelayoutandtooptimizetheclampingforce.Themaximalcuttingforceiscalculatedincuttingmodelandtheforceissenttofiniteelementanalysis(FEA)model.OptimizationprocedurecreatessomefixturelayoutandclampingforcewhicharesenttotheFEAmodeltoo.InFEAblock,machiningdeformationunderthecuttingforceandtheclampingforceiscalculatedusingfiniteelementmethodunderacertainfixturelayout,andthedeformationisthensenttooptimizationproceduretosearchforanoptimalfixturescheme.4Fixturelayoutdesignandclampingforceoptimization4.1AgeneticalgorithmGeneticalgorithms(GA)arerobust,stochasticandheuristicoptimizationmethodsbasedonbiologicalreproductionprocesses.ThebasicideabehindGAistosimulate“survivalofthefittest”phenomena.Eachindividualcandidateinthepopulationisassignedafitnessvaluethroughafitnessfunctiontailoredtothespecificproblem.TheGAthenconductsreproduction,crossoverandmutationprocessestoeliminateunfitindividualsandthepopulationevolvestothenextgeneration.Sufficientnumberofevolutionsofthepopulationbasedontheseoperatorsleadtoanincreaseintheglobalfitnessofthepopulationandthefittestindividualrepresentsthebestsolution.TheGAproceduretooptimizefixturedesigntakesfixturelayoutandclampingforceasdesignvariablestogeneratestringswhichrepresentdifferentlayouts.Thestringsarecomparedtothechromosomesofnaturalevolution,andthestring,whichGAfindoptimal,ismappedtotheoptimalfixturedesignscheme.Inthisstudy,thegeneticalgorithmanddirectsearchtoolboxofMATLABareemployed.TheconvergenceofGAiscontrolledbythepopulationsize(Ps),theprobabilityofcrossover(Pc)andtheprobabilityofmutations(Pm).Onlywhennochangeinthebestvalueoffitnessfunctioninapopulation,Nchg,reachesapre-definedvalueNCmax,orthenumberofgenerations,N,reachesthespecifiedmaximumnumberofevolutions,Nmax.,didtheGAstop.TherearefivemainfactorsinGA,encoding,fitnessfunction,geneticoperators,controlparametersandcon-straints.Inthispaper,thesefactorsareselectedaswhatislistedinTable1.SinceGAislikelytogeneratefixturedesignstringsthatdonotcompletelyrestrainthefixturewhensubjectedtomachiningloads.ThesesolutionsareconsideredinfeasibleandthepenaltymethodisusedtodrivetheGAtoafeasiblesolution.Afixturedesignschemeisconsideredinfeasibleorunconstrainedifthereactionsatthelocatorsarenegative,inotherwords,itdoesnotsatisfytheconstraintsinequations(2)and(3).ThepenaltymethodessentiallyinvolvesMachiningProcessModelFEAOptimizationprocedurecuttingforcesfitnessOptimizationresultFixturelayoutandclampingforceFig.1Fixturelayoutandclamp-ingforceoptimizationprocessTable1SelectionofGAsparametersFactorsDescriptionEncodingRealScalingRankSelectionRemainderCrossoverIntermediateMutationUniformControlparameterSelf-adaptingIntJAdvManufTechnolassigningahighobjectivefunctionvaluetotheschemethatisinfeasible,thusdrivingittothefeasibleregioninsuccessiveiterationsofGA.Forconstraint(4),whennewindividualsaregeneratedbygeneticoperatorsortheinitialgenerationisgenerated,itisnecessarytocheckupwhethertheysatisfytheconditions.Thegenuinecandidateregionsarethoseexcludinginvalidregions.Inordertosimplifythechecking,polygonsareusedtorepresentthecandidateregionsandinvalidregions.Thevertexofthepolygonsareusedforthechecking.The“inpolygon”functioninMATLABcouldbeusedtohelpthechecking.4.2FiniteelementanalysisThesoftwarepackageofANSYSisusedforFEAcalculationsinthisstudy.Thefiniteelementmodelisasemi-elasticcontactmodelconsideringfrictioneffect,wherethematerialsareassumedlinearlyelastic.AsshowninFig.2,eachlocatororsupportisrepresentedbythreeorthogonalspringsthatproviderestrainsintheX,YandZdirectionsandeachclampissimilartolocatorbutclampingforceinnormaldirection.Thespringinnormaldirectioniscallednormalspringandtheothertwospringsarecalledtangentialsprings.ThecontactspringstiffnesscanbecalculatedaccordingtotheHerzcontacttheory8asfollowskiz16RC3iEC32i9C16C1713fiz13kizkiy6EC3i2C0vfiGfi2C0vwiGwiC16C17C01C1kiz8:5wherekiz,kix,kiyarethetangentialandnormalcontactstiffness,1RC3i1Rwi1Rfiisthenominalcontactradius,1EC3i1C0V2wiEwi1C0V2fiEfiisthenominalcontactelasticmodulus,Rwi,Rfiareradiusofthei-thworkpieceandfixtureelement,Ewi,EfiareYoungsmoduliforthei-thworkpieceandfixturematerials,wi,fiarePoissonratiosforthei-thworkpieceandfixturematerials,Gwi,Gfiareshearmoduliforthei-thworkpieceandfixturematerialsandfizisthereactionforceatthei-thcontactpointintheZdirection.Contactstiffnessvarieswiththechangeofclampingforceandfixturelayout.Areasonablelinearapproximationofthecontactstiffnesscanbeobtainedfromaleast-squaresfittotheaboveequation.Thecontinuousinterpolation,whichisusedtoapplyboundaryconditionstotheworkpieceFEAmodel,isFig.2Semi-elasticcontactmodeltakingfrictionintoaccountSpringpositionFixtureelementposition12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849Fig.3ContinuousinterpolationFig.4AhollowworkpieceTable2MachiningparametersandconditionsParameterDescriptionTypeofoperationEndmillingCutterdiameter25.4mmNumberofflutes4CutterRPM500Feed0.1016mm/toothRadialdepthofcut2.54mmAxialdepthofcut25.4mmRadialrakeangle10Helixangle30Projectionlength92.07mmIntJAdvManufTechnolillustratedinFig.3.Threefixtureelementlocationsareshownasblackcircles.Eachelementlocationissurroundedbyitsfourorsixnearestneighboringnodes.Thesesetsofnodes,whichareillustratedbyblacksquares,are37,38,31and30,9,10,11,18,17and16and26,27,34,41,40and33.Asetofspringelementsareattachedtoeachofthesenodes.Foranysetofnodes,thespringconstantiskijdijPk2hidikki6wherekijisthespringstiffnessatthej-thnodesurroundingthei-thfixtureelement,dijisthedistancebetweenthei-thfixtureelementandthej-thnodesurroundingit,kiisthespringstiffnessatthei-thfixtureelementlocation.iisthenumberofnodessurroundingthei-thfixtureelementlocation.Foreachmachiningloadstep,appropriateboundaryconditionshavetobeappliedtothefiniteelementmodeloftheworkpiece.Inthiswork,thenormalspringsareconstrainedinthethreedirections(X,Y,Z)andthetangentialspringsareconstrainedinthetangentialdirec-tions(X,Y).Clampingforcesareappliedinthenormaldirection(Z)attheclampnodes.TheentiretoolpathissimulatedforeachfixturedesignschemegeneratedbytheGAbyapplyingthepeakX,Y,Zcuttingforcessequentiallytotheelementsurfacesoverwhichthecutterpasses23.Inthiswork,chipremovalfromthetoolpathistakenintoaccount.Theremovalofthematerialduringmachiningaltersthegeometry,sodoesthestructuralstiffnessoftheworkpiece.Thus,itisnecessarytoconsiderchipremovalaffects.TheFEAmodelisanalyzedwithrespecttotoolmovementandchipremovalusingtheelementdeathtechnique.Inordertocalculatethefitnessvalueforagivenfixturedesignscheme,displacementsarestoredforeachloadstep.Thenthemaximumdisplacementisselectedasfitnessvalueforthisfixturedesignscheme.TheinteractionbetweenGAprocedureandANSYSisimplementedasfollows.Boththepositionsoflocatorsandclamps,andtheclampingforceareextractedfromrealstrings.Theseparametersarewrittentoatextfile.TheinputbatchfileofANSYScouldreadtheseparametersandcalculatethedeformationofmachinedsurfaces.ThusthefitnessvaluesinGAprocedurecanalsobewrittentoatextfileforcurrentfixturedesignscheme.ItiscostlytocomputethefitnessvaluewhentherearealargenumberofnodesinanFEMmodel.ThusitisnecessarytospeedupthecomputationforGAprocedure.Asthegenerationgoesby,chromosomesinthepopulationaregettingsimilar.Inthiswork,calculatedfitnessvaluesarestoredinaSQLServerdatabasewiththechromosomesandfitnessvalues.GAprocedurefirstchecksifcurrentchromosomesfitnessvaluehasbeencalculatedbefore,ifnot,fixturedesignschemearesenttoANSYS,otherwisefitnessvaluesaredirectlytakenfromthedatabase.ThemeshingofworkpieceFEAmodelkeepssameineverycalculatingtime.Thedifferenceamongeverycalculatingmodelistheboundaryconditions.Thus,themeshedworkpieceFEAmodelcouldbeusedrepeatedlybythe“resume”commandinANSYS.5CasestudyAnexampleofmillingfixturedesignoptimizationproblemforalowrigidityworkpiecedisplayedinpreviousresearchpapers16,18,22ispresentedinthefollowingsections.Fig.5CandidateregionsforthelocatorsandclampsTable3BoundofdesignvariablesMinimumMaximumX/mmZ/mmX/mmZ/mmL10076.238.1L276.20152.438.1L3038.176.276.2L476.238.1152.476.2C10076.276.2C276.20152.476.2F1/N06673.2F2/N06673.2IntJAdvManufTechnol5.1WorkpiecegeometryandpropertiesThegeometryandfeaturesoftheworkpieceareshowninFig.4.Thematerialofthehollowworkpieceisaluminum390withaPoissonrationof0.3andYoungsmodulusof71Gpa.Theoutlinedimensionsare152.4mm127mm76.2mm.Theonethirdtopinnerwalloftheworkpieceisundergoinganend-millingprocessanditscutterpathisalsoshowninFig.4.ThematerialoftheemployedfixtureelementsisalloysteelwithaPoissonrationof0.3andYoungsmodulusof220Gpa.5.2SimulatingandmachiningoperationAperipheralendmillingoperationiscarriedoutontheexampleworkpiece.ThemachiningparametersoftheoperationaregiveninTable2.Basedontheseparameters,themaximumvaluesofcuttingforcesthatarecalculatedandappliedaselementsurfaceloadsontheinnerwalloftheworkpieceatthecutterpositionare330.94N(tangential),398.11N(radial)and22.84N(axial).Theentiretoolpathisdiscretizedinto26loadstepsandcuttingforcedirectionsaredeterminedbythecutterposition.5.3FixturedesignplanThefixtureplanforholdingtheworkpieceinthemachiningoperationisshowninFig.5.Generally,the321locatorprincipleisusedinfixturedesign.Thebasecontrols3degrees.Onesidecontrolstwodegrees,andanotherorthogonalsidecontrolsonedegree.Here,itusesfourlocators(L1,L2,L3andL4)ontheY=0mmfacetolocatetheworkpiececontrollingtwodegrees,andtwoclamps(C1,C2)ontheoppositefacewhereY=127mm,toholdit.Ontheorthogonalside,onelocatorisneededtocontroltheremainingdegree,whichisneglectedintheoptimalmodel.Thecoordinateboundsforthelocating/clampingregionsaregiveninTable3.Sincethereisnosimplerule-of-thumbprocedurefordeterminingtheclampingforce,alargevalueoftheclampingforceof6673.2Nwasinitiallyassumedtoactate
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