Tolerance Stack Analysis Methods

1 Tolerance Stack Analysis MethodsFritz Scholz Research and TechnologyBoeing Information & Support ServicesDecember 1995 AbstractThe purpose of this report is to describe various Tolerance stackingmethods without going into the theoretical details and derivationsbehind them. For those the reader is referred to Scholz (1995). Foreach method we present the assumptions and then give the tolerancestacking formulas. This will allow the user to make an informed choiceamong the man yavailable Methods covered are: worst case or arithmetic tolerancing,simple statistical tolerancing or the RSS method, RSS Methods withinflation factors which account for nonnormal distributions, toleranc-ing with mean shifts, where the latter are stacked arithmeticall yorstatisticall yin different wa ys, depending on how one views the trade-off between part to part variation and mean shifts. Boeing Information & Support Services, Box 3707, MS 7L-22, Seattle WA 98124-2207,e-mail: of Notation by Page of First Occurrencetermmeaningpage , istandard deviation, describes spread of a statistical2, 15distribution for part to part variationLiactual value of ithdetail part length dimension4 Ggap, assembly criterion of interest,4usually a function (sum) of detail dimensions inominal value of ithdetail part dimension4 Titolerance value for ithdetail part dimension6 nominal gap value, assembly criterion of interest6 idifference between actual and mean (nominal) value6of ithdetail part dimension: i=Li iifmean i= nominal i,and i=Li iif i = iaicoefficient for the ithterm in the linear7tolerance Stack :G=a1L1+.

2 +anLn,often we haveai= 1 Xiactual value of ithinput to sensitivity Analysis ;7in length stackingXiandLiare equivalentYoutput from sensitivity Analysis ;7in length stackingYandGare equivalentiGlossary of Notation by Page of First Occurrencetermmeaningpagefsmooth function relating output to inputs7in sensitivity Analysis :Y=f(X1,..,Xn)Y=f(X1,..,Xn) a0+a1X1+..+anXnai= f( 1,.., n)/ i,i=1,..,na0=f( 1,.., n) a1 1 .. an n inominal value of ithinput to sensitivity analysis8in length stacking iand iare equivalent nominal output value from a sensitivity analysis8in length stacking and are equivalentTassygeneric assembly Tolerance derived by any method9 Tarithassyassembly Tolerance derived by arithmetic11tolerance stacking (worst case method)Tarithassy=|a1|T1+..+|an|TnTdetai ltolerance common to all parts11 itolerance ratio i=Ti/T111 Tstatassyassembly Tolerance derived by statistical14tolerance stacking (RSS method)Tstatassy= a21T21+.

3 +a2nT2niiGlossary of Notation by Page of First OccurrencetermmeaningpageTstatassy(Bende r)assembly Tolerance derived by statistical16tolerance stacking (RSS method)using Bender s inflation factor of a21T21+..+a2nT2nci,c,cinflation factor for part variation distribution17 Tstatassy(c)assembly Tolerance derived by statistical19tolerance stacking (RSS method) usingdistributional inflation factorsTstatassy(c)=Tstatassy(c1,..,cn)= (c1a1T1)2+..+(cnanTn)2kdelimiter for the rectangular portion of the21trapezoidal densityparea of middle box of DIN-histogram density23ghalf width of middle box of DIN-histogram density23 iactual process mean for ithdetail part dimension25 ishift of process mean from nominal: i= i i25 i, fraction of absolute mean shift in relation toTi25, 26 i=| i|/Ti, =( 1,.., n)iiiGlossary of Notation by Page of First OccurrencetermmeaningpageLi,Uilower and upper Tolerance /specification limits:25Li= i Ti,Ui= i+TiCpka process capability index which accounts for25mean shiftsT ,arith,1assy( )assembly Tolerance derived by arithmetic26stacking of mean shifts and RSS stacking ofremaining normal variation; fixedTiwithtradeoff between mean shift and part variationT ,arith,1assy( )=T ,arith,1assy( 1.)

4 , n)= 1|a1|T1+..+ n|an|Tn+ [(1 1)a1T1]2+..+[(1 n)anTn]2T ipart Tolerance based on part to part variation,28, 31eitherT i=3 iorT i=halfwidthof distribution intervalT ,arith,2assy( )assembly Tolerance derived by arithmetic28stacking of mean shifts and RSS stacking ofremaining normal variation; inflatedTito accommodate mean shifts ( iTi) underfixedT i=3 i=Ti/(1 i) part variationT ,arith,2assy( )=T ,arith,2assy( 1,.., n)= 1|a1|T 1/(1 1)+..+ n|an|T n/(1 n)+ (a1T 1)2+..+(anT n)2ivGlossary of Notation by Page of First OccurrencetermmeaningpageT ,arithassy( ,c)assembly Tolerance derived by statistical29stacking (RSS method) using distributionalinflation factors and arithmetic stacking ofmean shiftsT ,arithassy( ,c)= 1|a1|T1+..+ n|an|Tn+ [(1 1)c1a1T1]2+..+[(1 n)cnanTn]2 standard deviation for mean shift distribution33c ,i,c ,c ,inflation factors for mean shift distributions33, 33, 35T ,stat,1assy( ,c,c )assembly Tolerance derived by RSS stacking34of mean shifts, RSS stacking of part variationand arithmetically stacking these two,assuming fixed part variation expressed throughT iT ,stat,1assy( ,c,c )= c21a21T 21+.

5 +c2na2n(1 n)2T 2n+ c2 ,1a21 21T 21/(1 1)2+..+c2 ,na2n 2nT 2n/(1 n)2Ri,Rrelative mean shiftR=(R1,..,Rn)37Ri= i/( iTi), 1 Ri 1 (Ri)standard deviation ofRi37vGlossary of Notation by Page of First Occurrencetermmeaningpagewia Tolerance weight factor38wi=aiTi/ nj=1a2jT2j, ni=1w2i=1F(R)inflation factor for given mean shift factorR38T ,stat,2assy( )assembly Tolerance derived by RSS stacking of39mean shifts and RSS stacking of part variationwhich can increase with decrease in mean shifts; is the common bound on all partmean shift fractionsT ,stat,2assy( )= 1 + 2/2+ 3 a21T21+..+a2nT2nT ,arith,rassy( ,c)reduced assembly Tolerance using the the RSS part ofT ,arithassy( ,c)T ,arith,rassy( ,c)= 1|a1|T1+..+ n|an|Tn+.927 [(1 1)c1a1T1]2+..+[(1 n)cnanTn]2vi1 Introduction and OverviewTolerance Stack Analysis Methods are described in various books and pa-pers, see for example Gilson (1951), Mansoor (1963), Fortini (1967), Wade(1967), Evans (1975), Cox (1986), Greenwood and Chase (1987), Kirschling(1988), Bj rke (1989), Henzold (1995), and Nigam and Turner (1995).

6 Un-fortunately, the notation is often not standard and not uniform, making theunderstanding of the material at times difficult. For a critical review of theseand some new Methods and the mathematical derivation behind them seeScholz (1995).Although the above cited references date back as far as Gilson s 1951monograph, he provides several older references, namely Gramenz (1925),Ettinger and Bartky (1936), Rice (1944), Epstein (1946), Bates (1947, 1949),Nielson (1948), Gladman (1945), Loxham (1947) and some not associatedwith a person and thus omitted here. So far we have not been able toobtain any of these references, but it appears doubtful that anything beyondstraight arithmetic or statistical tolerancing is contained in these. However,it would be of interest to find out who first proposed these two cornerstonesof tolerancing and the various nuances that have are no doubt many other sources which are internal to variouscompanies and thus not very accessible to most people.

7 For example, Wade(1967) mentions an article on statistical tolerancing by Backhaus and Fieldenthat appeared in an Corporation in-house publication. So far wehave not been able to get a copy of this article. Other in-house writings onthe subject are protected, such as Boeing s proprietary Tolerancing-DesignGuide (1990) by Griess. Other companies have made their tolerancing guideswidely available. As an example we cite the Motorola guide, authored byHarry and Stewart (1988). Unfortunately we have not been able to come upwith sound, theoretical underpinnings for their proposed Methods for dealingwith mean shifts and thus we will omit them here. See Scholz (1995) for is of interest to examine how the ASME standard andits companion ASME treat this subject. The former containsa very short Section , pp 38-39, which briefly mentions the basic formsof arithmetic and statistical tolerancing in connection with a new drawingsymbol indicating a statistical Tolerance , namely ST.

8 This symbol is intro-1duced there for the first time and it is to be expected that future editionsof this standard will move toward taking advantage of statistical tolerancestacking. At this point the above symbol indicates that tolerances set withthis symbol are to be monitored by statistical process control Methods . Howthat is done is still left up to the user. Other symbols with similar intent arealready in use in various any exposition on tolerancing will include the two cornerstones,arithmetic and statistical tolerancing. We will make no exception, since thesetwo Methods provide conservative and optimistic benchmarks, arithmetic tolerancing it is assumed that the detail part dimen-sions can haveanyvalue within the Tolerance range and the arithmeticallystacked tolerances describe the range ofallpossible variations for the assem-bly criterion of the basic statistical tolerancing scheme it is assumed that detail partdimensions vary randomly according to a normal distribution, centered atthe midpoint of the Tolerance range and with its 3 spread covering thetolerance interval.

9 For given part dimension tolerances this kind of statisticalanalysis typically leads to much tighter assembly tolerances, or for givenassembly Tolerance it requires considerably less stringent tolerances for detailpart dimensions, resulting in significant savings in cost or even making thedifference between feasibility or infeasibility of a proposed has shown that the results are usually not quite as good as ad-vertised. Assemblies often show more variation in the toleranced dimensionthan predicted by the statistical tolerancing method. The causes for thislie mainly in the violation of various distributional assumptions, but some-times also in the misapplication of the method by not understanding theassumptions. Not wanting to give up on the intrinsic gains of the statisticaltolerancing method one has tried to relax these distributional assumptionsin a variety of ways. As a consequence such assumptions are more likely tobe met in such relaxation is to allow other than normal distributions.

10 Such dis-tributions essentially cover the Tolerance interval with a wider spread, but arestill centered on the Tolerance interval midpoint. This results in somewhatless optimistic gains than those obtained under the normality assumption,but it usually still yields better results than those given by arithmetic toler-2ancing, especially for Tolerance chains involving many detail relaxation of assumptions concerns the centering of the distribu-tion on the Tolerance interval midpoint. The realization that it is difficult tocenter any process exactly where one wants it to be has led to several meanshift models. In these the distribution may be centered anywhere within acertain small neighborhood around the nominal Tolerance interval midpoint,but usually it is still assumed that the distribution is normal and its 3 spread is within the Tolerance limits. This means that while we allow someshift in the detail process mean we either require a simultaneous reduction inpart variability or we have to widen the Tolerance interval.