tolerance stack up

ToleranceStackAnalysisMethodsFritzScholz∗ResearchandTechnologyBoeingInformation&SupportServicesDecember1995AbstractThepurposeofthisreportistodescribevarioustolerancestackingmethodswithoutgoingintothetheoreticaldetailsandderivationsbehindthem.ForthosethereaderisreferredtoScholz(1995).Foreachmethodwepresenttheassumptionsandthengivethetolerancestackingformulas.Thiswillallowtheusertomakeaninformedchoiceamongthemanyavailablemethods.Themethodscoveredare:worstcaseorarithmetictolerancing,simplestatisticaltolerancingortheRSSmethod,RSSmethodswithinflationfactorswhichaccountfornonnormaldistributions,toleranc-ingwithmeanshifts,wherethelatterarestackedarithmeticallyorstatisticallyindifferentways,dependingonhowoneviewsthetrade-offbetweenparttopartvariationandmeanshifts.∗BoeingInformation&SupportServices,P.O.Box3707,MS7L-22,SeattleWA98124-2207,e-mail:[e-mail]ageofFirstOccurrencetermmeaningpageσ,σistandarddeviation,describesspreadofastatistical2,15distributionforparttopartvariationLiactualvalueofithdetailpartlengthdimension4Ggap,assemblycriterionofinterest,4usuallyafunction(sum)ofdetaildimensionsλinominalvalueofithdetailpartdimension4Titolerancevalueforithdetailpartdimension6γnominalgapvalue,assemblycriterionofinterest6idifferencebetweenactualandmean(nominal)value6ofithdetailpartdimension:i=Li−λiifmeanµi=nominalλi,andi=Li−µiifµi=λiaicoefficientfortheithterminthelinear7tolerancestack:G=a1L1+...+anLn,oftenwehaveai=±1Xiactualvalueofithinputtosensitivityanalysis;7inlengthstackingXiandLiareequivalentYoutputfromsensitivityanalysis;7inlengthstackingYandGareequivalentiGlossaryofNotationbyPageofFirstOccurrencetermmeaningpagefsmoothfunctionrelatingoutputtoinputs7insensitivityanalysis:Y=f(X1,...,Xn)Y=f(X1,...,Xn)≈a0+a1X1+...+anXnai=∂f(ν1,...,νn)/∂νi,i=1,...,na0=f(ν1,...,νn)−a1ν1−...−anνnνinominalvalueofithinputtosensitivityanalysis8inlengthstackingνiandλiareequivalentνnominaloutputvaluefromasensitivityanalysis8inlengthstackingνandγareequivalentTassygenericassemblytolerancederivedbyanymethod9Tarithassyassemblytolerancederivedbyarithmetic11tolerancestacking(worstcasemethod)Tarithassy=|a1|T1+...+|an|TnTdetailtolerancecommontoallparts11ρitoleranceratioρi=Ti/T111Tstatassyassemblytolerancederivedbystatistical14tolerancestacking(RSSmethod)Tstatassy=
a21T21+...+a2nT2niiGlossaryofNotationbyPageofFirstOccurrencetermmeaningpageTstatassy(Bender)assemblytolerancederivedbystatistical16tolerancestacking(RSSmethod)usingBender’sinflationfactorof1.5Tstatassy=1.5
a21T21+...+a2nT2nci,c,cinflationfactorforpartvariationdistribution17Tstatassy(c)assemblytolerancederivedbystatistical19tolerancestacking(RSSmethod)usingdistributionalinflationfactorsTstatassy(c)=Tstatassy(c1,...,cn)=
(c1a1T1)2+...+(cnanTn)2kdelimiterfortherectangularportionofthe21trapezoidaldensitypareaofmiddleboxofDIN-histogramdensity23ghalfwidthofmiddleboxofDIN-histogramdensity23µiactualprocessmeanforithdetailpartdimension25∆ishiftofprocessmeanfromnominal:∆i=µi−λi25ηi,ηfractionofabsolutemeanshiftinrelationtoTi25,26ηi=|∆i|/Ti,η=(η1,...,ηn)iiiGlossaryofNotationbyPageofFirstOccurrencetermmeaningpageLi,Uiloweranduppertolerance/specificationlimits:25Li=λi−Ti,Ui=λi+TiCpkaprocesscapabilityindexwhichaccountsfor25meanshiftsT∆,arith,1assy(η)assemblytolerancederivedbyarithmetic26stackingofmeanshiftsandRSSstackingofremainingnormalvariation;fixedTiwithtradeoffbetweenmeanshiftandpartvariationT∆,arith,1assy(η)=T∆,arith,1assy(η1,...,ηn)=η1|a1|T1+...+ηn|an|Tn+
[(1−η1)a1T1]2+...+[(1−ηn)anTn]2T
iparttolerancebasedonparttopartvariation,28,31eitherT
i=3σiorT
i=halfwidthofdistributionintervalT∆,arith,2assy(η)assemblytolerancederivedbyarithmetic28stackingofmeanshiftsandRSSstackingofremainingnormalvariation;inflatedTitoaccommodatemeanshifts(≤ηiTi)underfixedT
i=3σi=Ti/(1−ηi)partvariationT∆,arith,2assy(η)=T∆,arith,2assy(η1,...,ηn)=η1|a1|T
1/(1−η1)+...+ηn|an|T
n/(1−ηn)+
(a1T
1)2+...+(anT
n)2ivGlossaryofNotationbyPageofFirstOccurrencetermmeaningpageT∆,arithassy(η,c)assemblytolerancederivedbystatistical29stacking(RSSmethod)usingdistributionalinflationfactorsandarithmeticstackingofmeanshiftsT∆,arithassy(η,c)=η1|a1|T1+...+ηn|an|Tn+
[(1−η1)c1a1T1]2+...+[(1−ηn)cnanTn]2σµstandarddeviationformeanshiftdistribution33cµ,i,cµ,cµ,inflationfactorsformeanshiftdistributions33,33,35T∆,stat,1assy(η,c,cµ)assemblytolerancederivedbyRSSstacking34ofmeanshifts,RSSstackingofpartvariationandarithmeticallystackingthesetwo,assumingfixedpartvariationexpressedthroughT
iT∆,stat,1assy(η,c,cµ)=
c21a21T
21+...+c2na2n(1−ηn)2T
2n+
c2µ,1a21η21T
21/(1−η1)2+...+c2µ,na2nη2nT
2n/(1−ηn)2Ri,RrelativemeanshiftR=(R1,...,Rn)37Ri=∆i/(ηiTi),−1≤Ri≤1σ(Ri)standarddeviationofRi37vGlossaryofNotationbyPageofFirstOccurrencetermmeaningpagewiatoleranceweightfactor38wi=aiTi/
nj=1a2jT2j,ni=1w2i=1F(R)inflationfactorforgivenmeanshiftfactorR38T∆,stat,2assy(η)assemblytolerancederivedbyRSSstackingof39meanshiftsandRSSstackingofpartvariationwhichcanincreasewithdecreaseinmeanshifts;ηisthecommonboundonallpartmeanshiftfractionsT∆,stat,2assy(η)=
1−η+