statistical tolerance analysis
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Tolerance AnalysisTRANSCRIPT
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6/22/2015 statisticaltoleranceanalysisbasics:RootSumSquare(RSS)|ProductDevelopmentNotebook
http://www.pdnotebook.com/2010/06/statisticaltoleranceanalysisrootsumsquare/ 1/6
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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.
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6/22/2015 statisticaltoleranceanalysisbasics:RootSumSquare(RSS)|ProductDevelopmentNotebook
http://www.pdnotebook.com/2010/06/statisticaltoleranceanalysisrootsumsquare/ 2/6
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.
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6/22/2015 statisticaltoleranceanalysisbasics:RootSumSquare(RSS)|ProductDevelopmentNotebook
http://www.pdnotebook.com/2010/06/statisticaltoleranceanalysisrootsumsquare/ 3/6
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'[email protected],youneedtoaddupthefiveblockstoget,andtakethesquarerootofthesumofthesquaresofthestandarddeviationofthetolerances
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6/22/2015 statisticaltoleranceanalysisbasics:RootSumSquare(RSS)|ProductDevelopmentNotebook
http://www.pdnotebook.com/2010/06/statisticaltoleranceanalysisrootsumsquare/ 4/6
(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
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6/22/2015 statisticaltoleranceanalysisbasics:RootSumSquare(RSS)|ProductDevelopmentNotebook
http://www.pdnotebook.com/2010/06/statisticaltoleranceanalysisrootsumsquare/ 6/6
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