I倒挡拨叉钻8.7mm孔夹具设计（全套含CAD图纸）
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上传时间：2018-01-07
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倒挡拨叉钻
mm
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夹具
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全套
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INTJADVMANUFTECHNOL2001171041132001SPRINGERVERLAGLONDONLIMITEDFIXTURECLAMPINGFORCEOPTIMISATIONANDITSIMPACTONWORKPIECELOCATIONACCURACYBLIANDSNMELKOTEGEORGEWWOODRUFFSCHOOLOFMECHANICALENGINEERING,GEORGIAINSTITUTEOFTECHNOLOGY,GEORGIA,USAWORKPIECEMOTIONARISINGFROMLOCALISEDELASTICDEFORMATIONATFIXTUREWORKPIECECONTACTSOWINGTOCLAMPINGANDMACHININGFORCESISKNOWNTOAFFECTSIGNIFICANTLYTHEWORKPIECELOCATIONACCURACYAND,HENCE,THEFINALPARTQUALITYTHISEFFECTCANBEMINIMISEDTHROUGHFIXTUREDESIGNOPTIMISATIONTHECLAMPINGFORCEISACRITICALDESIGNVARIABLETHATCANBEOPTIMISEDTOREDUCETHEWORKPIECEMOTIONTHISPAPERPRESENTSANEWMETHODFORDETERMININGTHEOPTIMUMCLAMPINGFORCESFORAMULTIPLECLAMPFIXTURESUBJECTEDTOQUASISTATICMACHININGFORCESTHEMETHODUSESELASTICCONTACTMECHANICSMODELSTOREPRESENTTHEFIXTUREWORKPIECECONTACTANDINVOLVESTHEFORMULATIONANDSOLUTIONOFAMULTIOBJECTIVECONSTRAINEDOPTIMISATIONMODELTHEIMPACTOFCLAMPINGFORCEOPTIMISATIONONWORKPIECELOCATIONACCURACYISANALYSEDTHROUGHEXAMPLESINVOLVINGA321TYPEMILLINGFIXTUREKEYWORDSELASTICCONTACTMODELLINGFIXTURECLAMPINGFORCEOPTIMISATION1INTRODUCTIONTHELOCATIONANDIMMOBILISATIONOFTHEWORKPIECEARETWOCRITICALFACTORSINMACHININGAMACHININGFIXTUREACHIEVESTHESEFUNCTIONSBYLOCATINGTHEWORKPIECEWITHRESPECTTOASUITABLEDATUM,ANDCLAMPINGTHEWORKPIECEAGAINSTITTHECLAMPINGFORCEAPPLIEDMUSTBELARGEENOUGHTORESTRAINTHEWORKPIECEMOTIONCOMPLETELYDURINGMACHININGHOWEVER,EXCESSIVECLAMPINGFORCECANINDUCEUNACCEPTABLELEVELOFWORKPIECEELASTICDISTORTION,WHICHWILLADVERSELYAFFECTITSLOCATIONAND,INTURN,THEPARTQUALITYHENCE,ITISNECESSARYTODETERMINETHEOPTIMUMCLAMPINGFORCESTHATMINIMISETHEWORKPIECELOCATIONERRORDUETOELASTICDEFORMATIONWHILESATISFYINGTHETOTALRESTRAINTREQUIREMENTPREVIOUSRESEARCHERSINTHEFIXTUREANALYSISANDSYNTHESISAREAHAVEUSEDTHEFINITEELEMENTFEMODELLINGAPPROACHORCORRESPONDENCEANDOFFPRINTREQUESTSTODRSNMELKOTE,GEORGEWWOODRUFFSCHOOLOFMECHANICALENGINEERING,GEORGIAINSTITUTEOFTECHNOLOGY,ATLANTA,GEORGIA303320405,USAEMAILSHREYESMELKOTEMEGATECHEDUTHERIGIDBODYMODELLINGAPPROACHEXTENSIVEWORKBASEDONTHEFEAPPROACHHASBEENREPORTED18WITHTHEEXCEPTIONOFDEMETER8,ACOMMONLIMITATIONOFTHISAPPROACHISTHELARGEMODELSIZEANDCOMPUTATIONCOSTALSO,MOSTOFTHEFEBASEDRESEARCHHASFOCUSEDONFIXTURELAYOUTOPTIMISATION,ANDCLAMPINGFORCEOPTIMISATIONHASNOTBEENADDRESSEDADEQUATELYSEVERALRESEARCHERSHAVEADDRESSEDFIXTURECLAMPINGFORCEOPTIMISATIONBASEDONTHERIGIDBODYMODEL911THERIGIDBODYMODELLINGAPPROACHTREATSTHEFIXTUREELEMENTANDWORKPIECEASPERFECTLYRIGIDSOLIDSDEMETER12,13USEDSCREWTHEORYTOSOLVEFORTHEMINIMUMCLAMPINGFORCETHEOVERALLPROBLEMWASFORMULATEDASALINEARPROGRAMWHOSEOBJECTIVEWASTOMINIMISETHENORMALCONTACTFORCEATEACHLOCATINGPOINTBYADJUSTINGTHECLAMPINGFORCEINTENSITYTHEEFFECTOFTHECONTACTFRICTIONFORCEWASNEGLECTEDBECAUSEOFITSRELATIVELYSMALLMAGNITUDECOMPAREDWITHTHENORMALCONTACTFORCESINCETHISAPPROACHISBASEDONTHERIGIDBODYASSUMPTION,ITCANUNIQUELYONLYHANDLE3DFIXTURINGSCHEMESTHATINVOLVENOMORETHAN6UNKNOWNSFUHANDNEE14ALSOPRESENTEDANITERATIVESEARCHBASEDMETHODTHATCOMPUTESTHEMINIMUMCLAMPINGFORCEBYASSUMINGTHATTHEFRICTIONFORCEDIRECTIONSAREKNOWNAPRIORITHEPRIMARYLIMITATIONOFTHERIGIDBODYANALYSISISTHATITISSTATICALLYINDETERMINATEWHENMORETHANSIXCONTACTFORCESAREUNKNOWNASARESULT,WORKPIECEDISPLACEMENTSCANNOTBEDETERMINEDUNIQUELYBYTHISMETHODTHISLIMITATIONMAYBEOVERCOMEBYACCOUNTINGFORTHEELASTICITYOFTHEFIXTUREWORKPIECESYSTEM15FORARELATIVELYRIGIDWORKPIECE,THELOCATIONOFTHEWORKPIECEINTHEMACHININGFIXTUREISSTRONGLYINFLUENCEDBYTHELOCALISEDELASTICDEFORMATIONATTHEFIXTURINGPOINTSHOCKENBERGERANDDEMETER16USEDEMPIRICALCONTACTFORCEDEFORMATIONRELATIONSCALLEDMETAFUNCTIONSTOSOLVEFORTHEWORKPIECERIGIDBODYDISPLACEMENTSDUETOCLAMPINGANDQUASISTATICMACHININGFORCESTHESAMEAUTHORSALSOINVESTIGATEDTHEEFFECTOFMACHININGFIXTUREDESIGNPARAMETERSONWORKPIECEDISPLACEMENT17GUIETAL18REPORTEDANELASTICCONTACTMODELFORIMPROVINGWORKPIECELOCATIONACCURACYTHROUGHOPTIMISATIONOFTHECLAMPINGFORCEHOWEVER,THEYDIDNOTADDRESSMETHODSFORCALCULATINGTHEFIXTUREWORKPIECECONTACTSTIFFNESSINADDITION,THEAPPLICATIONOFTHEIRALGORITHMFORASEQUENCEOFMACHININGLOADSREPRESENTINGAFINITETOOLPATHWASNOTDISCUSSEDLIANDMELKOTE19ANDHURTADOANDMELKOTE20USEDCONTACTMECHANICSTOFIXTURECLAMPINGFORCEOPTIMISATION105SOLVEFORTHECONTACTFORCESANDWORKPIECEDISPLACEMENTPRODUCEDBYTHEELASTICDEFORMATIONATTHEFIXTURINGPOINTSOWINGTOCLAMPINGLOADSTHEYALSODEVELOPEDMETHODSFOROPTIMISINGTHEFIXTURELAYOUT21ANDCLAMPINGFORCEUSINGTHISMETHOD22HOWEVER,CLAMPINGFORCEOPTIMISATIONFORAMULTICLAMPSYSTEMANDITSIMPACTONWORKPIECEACCURACYWERENOTCOVEREDINTHESEPAPERSTHISPAPERPRESENTSANEWALGORITHMBASEDONTHECONTACTELASTICITYMETHODFORDETERMININGTHEOPTIMUMCLAMPINGFORCESFORAMULTICLAMPFIXTUREWORKPIECESYSTEMSUBJECTEDTOQUASISTATICLOADSTHEMETHODSEEKSTOMINIMISETHEIMPACTOFWORKPIECEMOTIONDUETOCLAMPINGANDMACHININGLOADSONTHEPARTLOCATIONACCURACYBYSYSTEMATICALLYOPTIMISINGTHECLAMPINGFORCESACONTACTMECHANICSMODELISUSEDTODETERMINEASETOFCONTACTFORCESANDDISPLACEMENTS,WHICHARETHENUSEDFORTHECLAMPINGFORCEOPTIMISATIONTHECOMPLETEPROBLEMISFORMULATEDANDSOLVEDASAMULTIOBJECTIVECONSTRAINEDOPTIMISATIONPROBLEMTHEIMPACTOFCLAMPINGFORCEOPTIMISATIONONWORKPIECELOCATIONACCURACYISANALYSEDVIATWOEXAMPLESINVOLVINGA321FIXTURELAYOUTFORAMILLINGOPERATION2FIXTUREWORKPIECECONTACTMODELLING21MODELLINGASSUMPTIONSTHEMACHININGFIXTURECONSISTSOFLLOCATORSANDCCLAMPSWITHSPHERICALTIPSTHEWORKPIECEANDFIXTUREMATERIALSARELINEARLYELASTICINTHECONTACTREGION,ANDPERFECTLYRIGIDELSEWHERETHEWORKPIECEFIXTURESYSTEMISSUBJECTEDTOQUASISTATICLOADSDUETOCLAMPINGANDMACHININGTHECLAMPINGFORCEISASSUMEDTOBECONSTANTDURINGMACHININGTHISASSUMPTIONISVALIDWHENHYDRAULICORPNEUMATICCLAMPSAREUSEDINREALITY,THEELASTICITYOFTHEFIXTUREWORKPIECECONTACTREGIONISDISTRIBUTEDHOWEVER,INTHISMODELDEVELOPMENT,LUMPEDCONTACTSTIFFNESSISASSUMEDSEEFIG1THEREFORE,THECONTACTFORCEANDLOCALISEDDEFORMATIONATTHEITHFIXTURINGPOINTCANBERELATEDASFOLLOWSFIJKIJDIJ1WHEREKIJJX,Y,ZDENOTESTHECONTACTSTIFFNESSINTHETANGENTIALANDNORMALDIRECTIONSOFTHELOCALXI,YI,ZICOORDINATEFRAME,DIJFIG1ALUMPEDSPRINGFIXTUREWORKPIECECONTACTMODELXI,YI,ZI,DENOTETHELOCALCOORDINATEFRAMEATTHEITHCONTACTJX,Y,ZARETHECORRESPONDINGLOCALISEDELASTICDEFORMATIONSALONGTHEXI,YI,ANDZIAXES,RESPECTIVELY,FIJJX,J,ZREPRESENTSTHELOCALCONTACTFORCECOMPONENTSWITHFIXANDFIYBEINGTHELOCALXIANDYICOMPONENTSOFTHETANGENTIALFORCE,ANDFIZTHENORMALFORCE22WORKPIECEFIXTURECONTACTSTIFFNESSMODELTHELUMPEDCOMPLIANCEATASPHERICALTIPLOCATOR/CLAMPANDWORKPIECECONTACTISNOTLINEARBECAUSETHECONTACTRADIUSVARIESNONLINEARLYWITHTHENORMALFORCE23THECONTACTDEFORMATIONDUETOTHENORMALFORCEPIACTINGBETWEENASPHERICALTIPPEDFIXTUREELEMENTOFRADIUSRIANDAPLANARWORKPIECESURFACECANBEOBTAINEDFROMTHECLOSEDFORMHERTZIANSOLUTIONTOTHEPROBLEMOFASPHEREINDENTINGANELASTICHALFSPACEFORTHISPROBLEM,THENORMALDEFORMATIONDINISGIVENAS23,P93DINS9PI216RIE2D1/32WHERE1E1N2WEW1N2FEFEWANDEFAREYOUNGSMODULIFORTHEWORKPIECEANDFIXTUREMATERIALS,RESPECTIVELY,ANDNWANDNFAREPOISSONRATIOSFORTHEWORKPIECEANDFIXTUREMATERIALS,RESPECTIVELYTHETANGENTIALDEFORMATIONDITDITXORDITYINTHELOCALXIANDYITANGENTIALDIRECTIONS,RESPECTIVELYDUETOATANGENTIALFORCEQIQIXORQIYHASTHEFOLLOWINGFORM23,P217DTITQI8AIS2NFGF2NWGWD3WHEREAIS3PIRI4S1NFEF1NWEWDD1/3ANDGWANDGFARESHEARMODULIFORTHEWORKPIECEANDFIXTUREMATERIALS,RESPECTIVELYAREASONABLELINEARAPPROXIMATIONOFTHECONTACTSTIFFNESSCANBEOBTAINEDFROMALEASTSQUARESFITTOEQ2THISYIELDSTHEFOLLOWINGLINEARISEDCONTACTSTIFFNESSVALUESKIZ882S16RIE29D1/34KIXKIY4ES2NJGF2NWGWD1KIZ5INDERIVINGTHEABOVELINEARAPPROXIMATION,THENORMALFORCEPIWASASSUMEDTOVARYFROM0TO1000N,ANDTHECORRESPONDINGR2VALUEOFTHELEASTSQUARESFITWASFOUNDTOBE0943CLAMPINGFORCEOPTIMISATIONTHEGOALISTODETERMINETHESETOFOPTIMALCLAMPINGFORCESTHATWILLMINIMISETHEWORKPIECERIGIDBODYMOTIONDUETO106BLIANDSNMELKOTELOCALISEDELASTICDEFORMATIONINDUCEDBYTHECLAMPINGANDMACHININGLOADS,WHILEMAINTAININGTHEFIXTUREWORKPIECESYSTEMINQUASISTATICEQUILIBRIUMDURINGMACHININGMINIMISATIONOFTHEWORKPIECEMOTIONWILL,INTURN,REDUCETHELOCATIONERRORTHISGOALISACHIEVEDBYFORMULATINGTHEPROBLEMASAMULTIOBJECTIVECONSTRAINEDOPTIMISATIONPROBLEM,ASDESCRIBEDNEXT31OBJECTIVEFUNCTIONFORMULATIONSINCETHEWORKPIECEROTATIONDUETOFIXTURINGFORCESISOFTENQUITESMALL17THEWORKPIECELOCATIONERRORISASSUMEDTOBEDETERMINEDLARGELYBYITSRIGIDBODYTRANSLATIONDDWDXWDYWDZWT,WHEREDXW,DYW,ANDDZWARETHETHREEORTHOGONALCOMPONENTSOFDDWALONGTHEXG,YG,ANDZGAXESSEEFIG2THEWORKPIECELOCATIONERRORDUETOTHEFIXTURINGFORCESCANTHENBECALCULATEDINTERMSOFTHEL2NORMOFTHERIGIDBODYDISPLACEMENTASFOLLOWSIDDWIDXW2DYW2DZW26WHEREIIDENOTESTHEL2NORMOFAVECTORINPARTICULAR,THERESULTANTCLAMPINGFORCEACTINGONTHEWORKPIECEWILLADVERSELYAFFECTTHELOCATIONERRORWHENMULTIPLECLAMPINGFORCESAREAPPLIEDTOTHEWORKPIECE,THERESULTANTCLAMPINGFORCE,PRCPRXPRYPRZT,HASTHEFORMPRCRCPC7WHEREPCPL1PLCTISTHECLAMPINGFORCEVECTOR,RCNL1NLCTISTHECLAMPINGFORCEDIRECTIONMATRIX,NLICOSALICOSBLICOSGLITISTHECLAMPINGFORCEDIRECTIONCOSINEVECTOR,ANDALI,BLI,ANDGLIAREANGLESMADEBYTHECLAMPINGFORCEVECTORATTHEITHCLAMPINGPOINTWITHRESPECTTOTHEXG,YG,ZGCOORDINATEAXESI1,2,CINTHISPAPER,THEWORKPIECELOCATIONERRORDUETOCONTACTREGIONDEFORMATIONISASSUMEDTOBEINFLUENCEDONLYBYTHENORMALFORCEACTINGATTHELOCATORWORKPIECECONTACTSTHEFRICTIONALFORCEATTHECONTACTSISRELATIVELYSMALLANDISNEGLECTEDWHENANALYSINGTHEIMPACTOFTHECLAMPINGFORCEONTHEWORKPIECELOCATIONERRORDENOTINGTHERATIOOFTHENORMALCONTACTSTIFFNESS,KIZ,TOTHESMALLESTNORMALSTIFFNESSAMONGALLLOCATORS,KSZ,BYJII1,L,ANDASSUMINGTHATTHEWORKPIECERESTSONNX,NY,ANDNZNUMBEROFLOCATORSORIENTEDINTHEXG,FIG2WORKPIECERIGIDBODYTRANSLATIONANDROTATIONYG,ANDZGDIRECTIONS,THEEQUIVALENTCONTACTSTIFFNESSINTHEXG,YG,ANDZGDIRECTIONSCANBECALCULATEDASKSZSONXI1JID,KSZSONYI1JID,ANDKSZSONZI1JIDRESPECTIVELYSEEFIG3THEWORKPIECERIGIDBODYMOTION,DDW,DUETOCLAMPINGACTIONCANNOWBEWRITTENASDDW3PRXKSZSONXI1JIDPRYKSZSONYI1JIDPRZKSZSONZI1JID4T8THEWORKPIECEMOTION,ANDHENCETHELOCATIONERRORCANBEREDUCEDBYMINIMISINGTHEWEIGHTEDL2NORMOFTHERESULTANTCLAMPINGFORCEVECTORTHEREFORE,THEFIRSTOBJECTIVEFUNCTIONCANBEWRITTENASMINIMIZEIPRCIW11PRXONXI1JI221PRYONYI1JI221PRZONZI1JI2229NOTETHATTHEWEIGHTINGFACTORSAREPROPORTIONALTOTHEEQUIVALENTCONTACTSTIFFNESSESINTHEXG,YG,ANDZGDIRECTIONSTHECOMPONENTSOFPRCAREUNIQUELYDETERMINEDBYSOLVINGTHECONTACTELASTICITYPROBLEMUSINGTHEPRINCIPLEOFMINIMUMTOTALCOMPLEMENTARYENERGY15,23THISENSURESTHATTHECLAMPINGFORCESANDTHECORRESPONDINGLOCATORREACTIONSARE“TRUE”SOLUTIONSTOTHECONTACTPROBLEMANDYIELD“TRUE”RIGIDBODYDISPLACEMENTS,ANDTHATTHEWORKPIECEISKEPTINSTATICEQUILIBRIUMBYTHECLAMPINGFORCESATALLTIMESTHEREFORE,THEMINIMISATIONOFTHETOTALCOMPLEMENTARYENERGYFORMSTHESECONDOBJECTIVEFUNCTIONFORTHECLAMPINGFORCEOPTIMISATIONANDISGIVENBYMINIMISEUW12FOLCI1FIX2KIXOLCI1FIY2KIYOLCI1FIZ2KIZG10LTQLFIG3THEBASISFORTHEDETERMINATIONOFTHEWEIGHTINGFACTORFORTHEL2NORMCALCULATIONFIXTURECLAMPINGFORCEOPTIMISATION107WHEREUREPRESENTSTHECOMPLEMENTARYSTRAINENERGYOFTHEELASTICALLYDEFORMEDBODIES,WREPRESENTSTHECOMPLEMENTARYWORKDONEBYTHEEXTERNALFORCEANDMOMENTS,QDIAGC1XC1YC1ZCLCXCLCYCLCZISTHEDIAGONALCONTACTCOMPLIANCEMATRIX,CIJKIJ1,ANDLF1XF1YF1ZFLCXFLCYFLCZTISTHEVECTOROFALLCONTACTFORCES32FRICTIONANDSTATICEQUILIBRIUMCONSTRAINTSTHEOPTIMISATIONOBJECTIVEINEQ10ISSUBJECTTOCERTAINCONSTRAINTSANDBOUNDSFOREMOSTAMONGTHEMISTHESTATICFRICTIONCONSTRAINTATEACHCONTACTCOULOMBSFRICTIONLAWSTATESTHATFIX2FIY2MISFIZMISISTHESTATICFRICTIONCOEFFICIENTACONSERVATIVEANDLINEARISEDVERSIONOFTHISNONLINEARCONSTRAINTCANBEUSEDANDISGIVENBY19UFIXUUFIYUMISFIZ11SINCEQUASISTATICLOADSAREASSUMED,THESTATICEQUILIBRIUMOFTHEWORKPIECEISENSUREDBYINCLUDINGTHEFOLLOWINGFORCEANDMOMENTEQUILIBRIUMEQUATIONSINVECTORFORMOF012OM0WHERETHEFORCESANDMOMENTSCONSISTOFTHEMACHININGFORCES,WORKPIECEWEIGHTANDTHECONTACTFORCESINTHENORMALANDTANGENTIALDIRECTIONS33BOUNDSSINCETHEFIXTUREWORKPIECECONTACTISSTRICTLYUNILATERAL,THENORMALCONTACTFORCE,PI,CANONLYBECOMPRESSIVETHISISEXPRESSEDBYTHEFOLLOWINGBOUNDONPIPI0I1,LC13WHEREITISASSUMEDTHATNORMALFORCESDIRECTEDINTOTHEWORKPIECEAREPOSITIVEINADDITION,THENORMALCOMPRESSIVESTRESSATACONTACTCANNOTEXCEEDTHECOMPRESSIVEYIELDSTRENGTHSYOFTHEWORKPIECEMATERIALTHISUPPERBOUNDISWRITTENASPISYAII1,LC14WHEREAIISTHECONTACTAREAATTHEITHWORKPIECEFIXTURECONTACTTHECOMPLETECLAMPINGFORCEOPTIMISATIONMODELCANNOWBEWRITTENASMINIMIZEFHF1F2JHLTQLIPRCIWJ15SUBJECTTO11144ALGORITHMFORMODELSOLUTIONTHEMULTIOBJECTIVEOPTIMISATIONPROBLEMINEQ15CANBESOLVEDBYTHEECONSTRAINTMETHOD24THISMETHODIDENTIFIESONEOFTHEOBJECTIVEFUNCTIONSASPRIMARY,ANDCONVERTSTHEOTHERINTOACONSTRAINTINTHISWORK,THEMINIMISATIONOFTHECOMPLEMENTARYENERGYF1ISTREATEDASTHEPRIMARYOBJECTIVEFUNCTION,ANDTHEWEIGHTEDL2NORMOFTHERESULTANTCLAMPINGFORCEF2ISTREATEDASACONSTRAINTTHECHOICEOFF1ASTHEPRIMARYOBJECTIVEENSURESTHATAUNIQUESETOFFEASIBLECLAMPINGFORCESISSELECTEDASARESULT,THEWORKPIECEFIXTURESYSTEMISDRIVENTOASTABLESTATEIETHEMINIMUMENERGYSTATETHATALSOHASTHESMALLESTWEIGHTEDL2NORMFORTHERESULTANTCLAMPINGFORCETHECONVERSIONOFF2INTOACONSTRAINTINVOLVESSPECIFYINGTHEWEIGHTEDL2NORMTOBELESSTHANOREQUALTOE,WHEREEISANUPPERBOUNDONF2TODETERMINEASUITABLEE,ITISINITIALLYASSUMEDTHATALLCLAMPINGFORCESAREUNKNOWNTHECONTACTFORCESATTHELOCATINGANDCLAMPINGPOINTSARECOMPUTEDBYCONSIDERINGONLYTHEFIRSTOBJECTIVEFUNCTIONIEF1WHILETHISSETOFCONTACTFORCESDOESNOTNECESSARILYYIELDTHELOWESTCLAMPINGFORCES,ITISA“TRUE”FEASIBLESOLUTIONTOTHECONTACTELASTICITYPROBLEMTHATCANCOMPLETELYRESTRAINTHEWORKPIECEINTHEFIXTURETHEWEIGHTEDL2NORMOFTHESECLAMPINGFORCESISCOMPUTEDANDTAKENASTHEINITIALVALUEOFETHEREFORE,THECLAMPINGFORCEOPTIMISATIONPROBLEMINEQ15CANBEREWRITTENASMINIMIZEF1LTQL16SUBJECTTOIPRCIWE,1114ANALGORITHMSIMILARTOTHEBISECTIONMETHODFORFINDINGROOTSOFANEQUATIONISUSEDTODETERMINETHELOWESTUPPERBOUNDFORIPRCIWBYDECREASINGTHEUPPERBOUNDEASMUCHASPOSSIBLE,THEMINIMUMWEIGHTEDL2NORMOFTHERESULTANTCLAMPINGFORCEISOBTAINEDTHENUMBEROFITERATIONS,K,NEEDEDTOTERMINATETHESEARCHDEPENDSONTHEREQUIREDPREDICTIONACCURACYDANDUEU,ANDISGIVENBY25KFLOG2SUEUDDG17WHEREIDENOTESTHECEILINGFUNCTIONTHECOMPLETEALGORITHMISGIVENINFIG45DETERMINATIONOFOPTIMUMCLAMPINGFORCESDURINGMACHININGTHEALGORITHMPRESENTEDINTHEPREVIOUSSECTIONCANBEUSEDTODETERMINETHEOPTIMUMCLAMPINGFORCEFORASINGLELOADVECTORAPPLIEDTOTHEWORKPIECEHOWEVER,DURINGMILLINGTHEMAGNITUDEANDPOINTOFCUTTINGFORCEAPPLICATIONCHANGESCONTINUOUSLYALONGTHETOOLPATHTHEREFORE,ANINFINITESETOFOPTIMUMCLAMPINGFORCESCORRESPONDINGTOTHEINFINITESETOFMACHININGLOADSWILLBEOBTAINEDWITHTHEALGORITHMOFFIG4THISSUBSTANTIALLYINCREASESTHECOMPUTATIONALBURDENANDCALLSFORACRITERION/PROCEDUREFORSELECTINGASINGLESETOFCLAMPINGFORCESTHATWILLBESATISFACTORYANDOPTIMUMFORTHEENTIRETOOLPATHACONSERVATIVEAPPROACHTOADDRESSINGTHESEISSUESISDISCUSSEDNEXTCONSIDERAFINITENUMBERSAYMOFSAMPLEPOINTSALONGTHETOOLPATHYIELDINGMCORRESPONDINGSETSOFOPTIMUMCLAMPINGFORCESDENOTEDASP1OPT,P2OPT,PMOPTATEACHSAMPLING108BLIANDSNMELKOTEFIG4CLAMPINGFORCEOPTIMISATIONALGORITHMUSEDINEXAMPLE1POINT,THEFOLLOWINGFOURWORSTCASEMACHININGLOADVECTORSARECONSIDEREDFXMAXFMAXXF1YF1ZTFYMAXF2XFMAXYF2ZTFZMAXF3XF3YFMAXZT18FRMAXF4XF4YF4ZTWHEREFMAXX,FMAXY,ANDFMAXZARETHEMAXIMUMXG,YG,ANDZGCOMPONENTSOFTHEMACHININGFORCE,THESUPERSCRIPTS1,2,3OFFX,FY,ANDFZSTANDFORTHEOTHERTWOORTHOGONALMACHININGFORCECOMPONENTSCORRESPONDINGTOFMAXX,FMAXY,ANDFMAXZ,RESPECTIVELY,ANDIFRMAXIMAXFX2FY2FZ2ALTHOUGHTHEFOURWORSTCASEMACHININGLOADVECTORSWILLNOTACTONTHEWORKPIECEATTHESAMEINSTANT,THEYWILLOCCURONCEPERCUTTERREVOLUTIONATCONVENTIONALFEEDRATES,THEERRORINTRODUCEDBYAPPLYINGTHELOADVECTORSATTHESAMEPOINTWOULDBENEGLIGIBLETHEREFORE,INTHISWORK,THEFOURLOADVECTORSAREAPPLIEDATTHESAMELOCATIONBUTNOTSIMULTANEOUSLYONTHEWORKPIECECORRESPONDINGTOTHESAMPLINGINSTANTTHECLAMPINGFORCEOPTIMISATIONALGORITHMOFFIG4ISTHENUSEDTOCALCULATETHEOPTIMUMCLAMPINGFORCESCORRESPONDINGTOEACHSAMPLINGPOINTTHEOPTIMUMCLAMPINGFORCESHAVETHEFORMPIJMAXCI1JCI2JCICJTI1,MJX,Y,Z,R19WHEREPIJMAXISTHEVECTOROFOPTIMUMCLAMPINGFORCESFORTHEFOURWORSTCASEMACHININGLOADVECTORS,ANDCIKJK1,CISTHEFORCEMAGNITUDEATEACHCLAMPCORRESPONDINGTOTHEITHSAMPLEPOINTANDTHEJTHLOADSCENARIOAFTERPIJMAXISCOMPUTEDFOREACHLOADAPPLICATIONPOINT,ASINGLESETOF“OPTIMUM”CLAMPINGFORCESMUSTBESELECTEDFROMALLOFTHEOPTIMUMCLAMPINGFORCESFOUNDFOREACHCLAMPFROMALLTHESAMPLEPOINTSANDLOADINGCONDITIONSTHISISDONEBYSORTINGTHEOPTIMUMCLAMPINGFORCEMAGNITUDESATACLAMPINGPOINTFORALLLOADSCENARIOSANDSAMPLEPOINTSANDSELECTINGTHEMAXIMUMVALUE,CMAXK,ASGIVENINEQ20CMAXKCIKJK1,C20ONCETHISISCOMPLETE,ASETOFOPTIMISEDCLAMPINGFORCESPOPTCMAX1CMAX2CMAXCTISOBTAINEDTHESEFORCESMUSTBEVERIFIEDFORTHEIRABILITYTOENSURESTATICEQUILIBRIUMOFTHEWORKPIECEFIXTURESYSTEMOTHERWISE,MORESAMPLINGPOINTSARESELECTEDANDTHEAFOREMENTIONEDPROCEDUREREPEATEDINTHISFASHION,THE“OPTIMUM”CLAMPINGFORCE,POPT,CANBEDETERMINEDFORTHEENTIRETOOLPATHFIGURE5SUMMARISESTHEALGORITHMJUSTDESCRIBEDNOTETHATALTHOUGHTHISAPPROACHISCONSERVATIVE,ITPROVIDESASYSTEMATICWAYOFDETERMININGASETOFCLAMPINGFORCESTHATMINIMISETHEWORKPIECELOCATIONERROR6IMPACTONWORKPIECELOCATIONACCURACYITISOFINTERESTTOEVALUATETHEIMPACTOFTHECLAMPINGFORCEALGORITHMPRESENTEDEARLIERONTHEWORKPIECELOCATIONACCURACYTHEWORKPIECEISFIRSTPLACEDONTHEFIXTUREBASEPLATEINCONTACTWITHTHELOCATORSCLAMPINGFORCESARETHENAPPLIEDTOPUSHTHEWORKPIECEAGAINSTTHELOCATORSCONSEQUENTLY,LOCALISEDDEFORMATIONSOCCURATEACHWORKPIECEFIXTURECONTACT,CAUSINGTHEWORKPIECETOTRANSLATEANDROTATEINTHEFIXTURESUBSEQUENTLY,THEQUASISTATICMACHININGLOADISAPPLIEDCAUSINGADDITIONALMOTIONOFTHEWORKPIECEINTHEFIXTURETHEWORKPIECERIGIDBODYMOTIONISDEFINEDBYITSTRANSLATIONDDWDXWDYWDZWTANDROTATIONDUWDUWXDUWYDUWZTABOUTTHEXG,YG,ANDZGAXESSEEFIG2ASNOTEDEARLIER,THEWORKPIECERIGIDBODYMOTIONARISESFROMTHELOCALISEDDEFORMATION,DIDIXDIYDIZT,ATEACHFIXTURINGPOINTASSUMINGTHATRIXIYIZITDESCRIBESTHEPOSITIONVECTOROFTHEITHLOCATINGPOINTRELATIVETOTHEWORKPIECECENTREOFMASS,THECOORDINATETRANSFORMATIONTHEOREMCANBEUSEDTOEXPRESSDIINTERMSOFTHEWORKPIECETRANSLATION,DDWDXWDYWDZW,ANDWORKPIECEROTATION,DUWDUWXDUWYDUWZT,ASFOLLOWSDIR1ITRDUWRIDDWRI21WHERER1IDENOTESTHEROTATIONMATRIXDESCRIBINGTHEORIENTATIONOFTHELOCALXI,YI,ZICOORDINATEFRAMEATTHEITHCONTACTRELATIVETOTHEGLOBALCOORDINATEFRAMEANDRDUWCISAROTATIONMATRIXFIXTURECLAMPINGFORCEOPTIMISATION109FIG5THEALGORITHMUSEDINEXAMPLE2FIG6THEWORKPIECELOCATIONERRORDRMDUETOCLAMPINGANDMACHININGDEFININGTHEORIENTATIONOFTHEWORKPIECEFIXEDCOORDINATEFRAMERELATIVETOTHEGLOBALCOORDINATEFRAMEASSUMINGTHATWORKPIECEROTATIONWITHINTHEFIXTUREDUETOCLAMPING,DUW,ISSMALL,RDUWCANBEAPPROXIMATEDASRDUWF1DUWZDUWYDUWZ1DUWXDUWYDUWX1G22EQUATION21CANNOWBEREWRITTENASDIR1ITBIQ23WHEREBI31000ZIYI010ZI0XI001YIXI04ISTHETRANSFORMATIONMATRIXOBTAINEDAFTERREARRANGINGEQ21,ANDQDXWDYWDZWDUWXDUWYDUWZTISTHEWORKPIECERIGIDBODYMOTIONVECTORDUETOCLAMPINGA
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