General 2-D Tolerance Analysis of Mechanical Assem

General2-DToleranceAnalysisofMechanicalAssemblieswithSmallKinematicAdjustmentsKennethW.ChaseJinsongGaoSpencerP.MaglebyDepartmentofMechanicalEngineeringBrighamYoungUniversityAbstractAssemblytoleranceanalysisisakeyelementinindustryforimprovingproductqualityandreducingoverallcost.Itprovidesaquantitativedesigntoolforpredictingtheeffectsofmanufacturingvariationonperformanceandcost.Itpromotesconcurrentengineeringbybringingengineeringrequirementsandmanufacturingrequirementstogetherinacommonmodel.Anewmethod,calledtheDirectLinearizationMethod(DLM),ispresentedfortoleranceanalysisof2-Dmechanicalassemblieswhichgeneralizesvectorloop-basedmodelstoincludesmallkinematicadjustments.Ithasasignificantadvantageovertraditionaltoleranceanalysismethodsinthatitdoesnotrequireanexplicitfunctiontodescribetherelationshipbetweentheresultantassemblydimension(s)andmanufacturedcomponentdimensions.Suchanexplicitassemblyfunctionmaybedifficultorimpossibletoobtainforcomplex2-Dassemblies.TheDLMmethodisaconvenientdesigntool.Themodelsareconstructedofcommonengineeringelements:vectorchains,kinematicjoints,assemblydatums,dimensionaltolerances,geometricfeaturetolerancesandassemblytolerancelimits.ItiswellsuitedforintegrationwithacommercialCADsystemasagraphicalfrontend.Itisnotcomputationallyintensive,soitisideallysuitedforiterativedesign.Ageneralformulationisderived,detailedmodelingandanalysisproceduresareoutlinedandthemethodisappliedtotwoexampleproblems.21.IntroductionAnimportantconsiderationinproductdesignistheassignmentoftolerancestoindividualcomponentdimensionssotheproductcanbeproducedeconomicallyandfunctionproperly.Thedesignermayassignrelativelytighttolerancestoeachparttoensurethattheproductwillperformcorrectly,butthiswillgenerallydrivemanufacturingcosthigher.Relaxingtolerancesoneachcomponent,ontheotherhand,reducescosts,butcanresultinunacceptablelossofqualityandhighscraprate,leadingtocustomerdissatisfaction.Theseconflictinggoalspointouttheneedinindustryformethodstorationallyassigntolerancestoproductssothatcustomerscanbeprovidedwithhighqualityproductsatcompetitivemarketprices.Clearly,atooltoevaluatetolerancerequirementsandeffectswouldbemostusefulinthedesignstageofaproduct.Tobeusefulindesign,itshouldincludethefollowingcharacteristics:1.Bringmanufacturingconsiderationsintothedesignstagebypredictingtheeffectsofmanufacturingvariationsonengineeringrequirements.2.Providebuilt-instatisticaltoolsforpredictingtolerancestack-upandpercentrejectsinassemblies.3.Becapableofperforming2-Dand3-Dtolerancestack-upanalyses.4.Becomputationallyefficient,topermitdesigniterationanddesignoptimization.5.Useageneralizedandcomprehensiveapproach,similartofiniteelementanalysis,whereafewbasicelementsarecapableofdescribingawidevarietyofassemblyapplicationsandengineeringtolerancerequirements.6.Incorporateasystematicmodelingprocedurethatisreadilyacceptedbyengineeringdesigners.7.BeeasilyintegratedwithcommercialCADsystems,sogeometric,dimensionalandtolerancedatamaybeextracteddirectlyfromtheCADdatabase.8.Useagraphicalinterfaceforassemblytolerancemodelcreationandgraphicalpresentationofresults.3Toillustratetheproblemsassociatedwith2-Dtoleranceanalysis,considerthesimpleassemblyshowninfigure1,asdescribedbyFortini[1967].Itisadrawingofaone-waymechanicalclutch.Thisisacommondeviceusedtotransmitrotarymotioninonlyonedirection.Whentheouterringoftheclutchisrotatedclockwise,therollerswedgebetweentheringandhub,lockingthetwosotheyrotatetogether.Inthereversedirection,therollersjustslip,sothehubdoesnotturn.ThepressureangleΦΦ1betweenthetwocontactpointsiscriticaltotheproperoperationoftheclutch.IfΦΦ1istoolarge,theclutchwillnotlock;ifitistoosmalltheclutchwillnotunlock.HubSpringRingRollerONE-WAYCLUTCHASSEMBLYHubONE-WAYCLUTCHDIMENSIONSRingΦΦ11RollerΦΦ22abcceFigure1.One-wayclutchassemblyanditsrelevantdimensionsTheprimaryobjectiveofperformingatoleranceanalysisontheclutchistodeterminehowmuchtheangleΦΦ1isexpectedtovaryduetomanufacturingvariationsintheclutchcomponentdimensions.Theindependentmanufacturingvariablesarethehubdimensiona,thecylinderradiusc,andtheringradiuse.ThedistancebandangleΦΦ1arenotdimensioned.Theyareassemblyresultantswhicharedeterminedbythesizesofa,candewhenthepartsareassembled.Bytrigonometry,thedependentassemblyresultants,distancebandangleΦΦ1,canbeexpressedasexplicitfunctionsofa,cande.ΦΦ1=cos-1(a+ce-c)b=(e-c)2-(a+c)2(1)TheexpressionforangleΦΦ1maybeanalyzedstatisticallytoestimatequantitativelytheresultingvariationinΦΦ1intermsthespecifiedtolerancesfora,cande.Ifperformance4requirementsareusedtosetengineeringlimitsonthesizeofΦΦ1,thequalitylevelandpercentrejectsmayalsobepredicted.Whenanexplicitfunctionoftheassemblyresultantisavailable,suchasΦΦ1inequation(1),severalmethodsareavailableforperformingastatisticaltoleranceanalysis.Theseinclude:1.LinearizationoftheassemblyfunctionusingTaylorseriesexpansion,2.Methodofsystemmoments,3.Quadrature,4.MonteCarlosimulation,5.Reliabilityindex,6.Taguchimethod.Thenextsectionwillbrieflyreviewthesemethods.Establishingexplicitassemblyfunctions,suchasequation(1),todescrib