6/22/2015 statisticaltoleranceanalysisbasics:RootSumSquare(RSS)|ProductDevelopmentNotebook

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statisticaltoleranceanalysisbasics:RootSumSquare(RSS)

Ifyou'reinterestedinlearningabouttoleranceanalysis(whyelsewouldyoubehere?),youwoulddowelltocheckoutmypostonwhatIconsidertobethebestGD&Ttrainingever.

...InmylastpostonworstcasetoleranceanalysisIconcludedwiththefactthattheworstcasemethod,althoughextremelysafe,isalsoextremelyexpensive.

Allowmetoelaborate,thenofferaresolutionintheformofstatisticaltoleranceanalysis.

cost

Aworstcasetoleranceanalysisisgreattomakesurethatyourpartswillalwaysfit,butifyou'reproducingmillionsofparts,ensuringeachandeveryoneworksisexpensiveand,undermostcircumstances,impractical.

Considerthesetwoscenarios.

6/22/2015 statisticaltoleranceanalysisbasics:RootSumSquare(RSS)|ProductDevelopmentNotebook

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1. Youmakeamillionparts,anditcostsyou$1.00perparttomakesurethateverysingleoneworks.2. Youmakeamillionparts,butdecidetogowithcheaper,lessaccurateparts.Nowyourcostis

$0.99perpart,but1,000partswon'tfit.

Inthefirst,scenario,yourcostis:

$1.00/part*1,000,000parts=$1,000,000

Inthesecondscenario,yourcostis:

$0.99/part*1,000,000parts=$990,000,

butyouhavetothrowawaythe1,000rejectswhichcost$0.99/part.Soyourtotalcostis:

$990,000+1,000*$0.99=$990,990.Whichmeansyousave$9,010.

Thoseactualnumbersaremakebelieve,butthelessonholdstrue:byproducinglessprecise(read:crappier)partsandthrowingsomeofthemaway,yousavemoney.

Soldyet?Good.Nowlet'stakealookatthetheory.

statisticaltoleranceanalysis:theory

Thefirstthingyou'llwanttothinkofisthebellcurve.Youmayrecallthebellcurvebeingusedtoexplainthatsomeofyourclassmatesweresmart,someweredumb,butmostwereaboutaverage.

Thesameprincipleholdstrueintoleranceanalysis.Thebellcurve(onlynowit'scalledthe"normaldistribution")statesthatwhenyoutakealotofmeasurements,beitoftestscoresorblockthicknesses,somemeasurementswillbelow,somehigh,andmostinthemiddle.

Ofcourse,"justabout"and"most"doesn'thelpyougetthingsdone.Mathdoes,andthat'swherethenormaldistribution(andexcel...attachmentbelow)comein.

sidebar:InitiallyIplannedondivingdeepintothemathofRSS,butHilemandoessuchagoodjobonthedetails,I'llstickwiththebroadstrokeshere.Ihighlysuggestprintingouthispostandsittingdowninaquietroom,it'stheonlywaytodigesttheheavystuff.

thenormaldistributionand"defectspermillion"

Usingthenormaldistribution,youcandeterminehowmanydefects(definedaspartsthatcomeinoutsideofallowabletolerances)willoccur.Thestandardunitofmeasureis"defectspermillion",sowe'llstickwiththat.

Therearetwonumbersyouneedtocreateanormaldistribution,andtheyarerepresentedby(pronounced"mew")and(pronounced("sigma")

isthemean,ameasureofthe"center"ofadistribution.isthestandarddeviation,ameasureofhowspreadoutadistributionis.Forexample,thenumbersets{0,10}and{5,5}bothhaveanaverageof5,butthe{0,10}setisspreadoutandthushasahigherstandarddeviation.

6/22/2015 statisticaltoleranceanalysisbasics:RootSumSquare(RSS)|ProductDevelopmentNotebook

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Usingoneofourblocks(rememberthose?)asanexample...

Let'ssayyoumeasurefiveblocksliketheoneabove(inpracticeit'sbesttomeasure30attheveryleast,butwe'llkeepitat5

fortheexample)andgetthefollowingresults:

x1=1.001"x2=0.995"x3=1.000"x4=1.001"x5=1.003"

Theaverage()is1.000(andthestandarddeviation()is.003.Plugthoseintoanormaldistribution,andyourtolerancesbreakdownlikethis.(seethe'afterproduction'tabintheattachedexcelfileforformulas)

Ifyourequiretheblockstobe1.000.003(1),theblockswillpassinspection68.27%ofthetime...317,311defectspermillion.

Ifyourequiretheblockstobe1.000.006(2),theblockswillpassinspection95.45%ofthetime...45,500defectspermillion

Ifyourequiretheblockstobe1.000.009(3),theblockswillpassinspection99.73%ofthetime...2,700defectspermillion

andsoon.

Usingthedataaboveyoucansaywithconfidence(assumingyoumeasuredenoughblocks!)thatifyouweretouseamillionblocks,allbut2700ofthemwouldcomeinbetween0.991and1.009.

rootsumsquareandthestandarddeviation

Ifyou'vefollowedthelogiccloselyyoumaynoticeacatch22.Ideally,youwanttodoatoleranceanalysisbeforeyougotoproduction,buthowcanyoudetermineorwithouthavingsamplestotest...whichyouwillonlygetafterproduction?

Youmake(andstate...repeatedly)assumptions

Thepartiseasy.Youjustassumethatthemeanwillbeequaltothenominal(inourcase,1.000).Thisisusuallyasolidassumptionandonlybeginstogetdiceywhenyoutalkaboutthenominalshifting(someliketoplanforupto1.5!)overthecourseofmillionsofcycles(perhapsduetotoolwear),butthatisanothertopic.

For,aconservativeestimateisthatyourtolerancecanbeheldtoaqualityof3,meaningthatatoleranceof.005willyieldyouaof0.005/3=0.00167.

Let'splaythisout...Ifyouarestackingfiveblocks@1.000.005,youneedtoaddupthefiveblockstoget,andtakethesquarerootofthesumofthesquaresofthestandarddeviationofthetolerances

6/22/2015 statisticaltoleranceanalysisbasics:RootSumSquare(RSS)|ProductDevelopmentNotebook

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(wordyIknow),whichlookslikethis...SQRT([.005/3]^2+[.005/3]^2+[.005/3]^2+[.005/3]^2+[.005/3]^2)...(youdivideby3becauseyouareassumingthatyourtolerancesrepresent3standarddeviations)

That'saswordyasI'mgoingtogetonthemath(thepostisalreadylongerthani'dlike),youcanseeitworkingforyourselfinthe'beforeproduction'tabintheattachedexcelfileforformulas)

Justremembertotreatthosenumberswiththerespectthattheydeserveandthatindustryacceptedassumptionsarenoreplacementforahearttoheart(andemailtrail)withyourmanufacturer.Tryingtopushamanufacturertoholdtolerancestheyaren'tcomfortablewithusadrainingandoftenfutileexercise.

Thetolerancesdictatethedesign,nottheotherwayaround.

update:Myseriesofpostsonworstcase,rootsumsquare,andmontecarlotoleranceanalysisstartedoffasjustabriefintroductiontothebasics.SincethenIhaveheardfromanumberofyouaskingforaclear,concise(everythingelseoutthereissoheavy),usableguidetoboththemathbehindtoleranceanalysisandrealworldexamplesofwhentouseit.I'mcurrentlyworkingonit,butwouldlovetohearwhatYOUwouldlikeoutofit.Letmeknowinthecommentsorcontactmethroughthesite.

Relatedposts:

1. statisticaltoleranceanalysisbasics:worstcasetoleranceanalysis2. statisticaltoleranceanalysisbasics:MonteCarloSimulation3. whentousenonlinearfiniteelementanalysis

articlebychrisloughnane

chrisisamechanicalengineer,patentagent,andprogrammerinaBostonareadesignconsultancyandhaswritten190articlesfortheProductDevelopmentNotebook.

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