B Weibull Reliability Analysis W - University of Washington

1 Weibull Reliability Analysis =) Scholz (425-865-3623, 7L-22)BoeingPhantomWorksMathematics & Computing TechnologyWeibull Reliability Analysis |FWS-5/1999|1 Wallodi WeibullWeibull Reliability Analysis |FWS-5/1999|2 Seminal PaperWeibull Reliability Analysis |FWS-5/1999|3 The Weibull Distribution Weibull distribution, usefuluncertainty modelfor{ wearout failure timeTwhen governed by wearout of weakest subpart{ material strengthTwhen governed by embedded aws or weaknesses, It has often been found useful based on empirical data ( Y2K) It is also theoretically founded on theweakest link principleT= min (X1;:::;Xn);withX1;:::;Xnstatistically independent random strengths orfailure times of then\links" comprising the have a natural nite lower endpoint, , link strength 0 or subpart time to failure 0.}}

2 ,! Weibull Reliability Analysis |FWS-5/1999|4 Theoretical Basis Under weak conditionsExtreme Value Theoryshows1that forlargenP(T t) 1 exp0BB@ 264t 375 1 CCAfort ; >0; >0 The above approximation has very much the same spirit as theCentral Limit Theoremwhich under some weak conditions ontheXiasserts that the distribution ofT=X1+:::+Xnisapproximately bell-shaped normal or Gaussian Assuming a Weibull model forT, material strength or cycle time tofailure, amounts to treating the above approximation as an equalityF(t)=P(T t)=1 exp0BB@ 264t 375 1 CCAfort ; >0; >01see: E. Castillo,Extreme Value Theory in Engineering, Academic Press, 1988 Weibull Reliability Analysis |FWS-5/1999|5 Weibull Reproductive PropertyIfX1;:::;Xnare statistically independentwithXi Weibull ( i; )thenP(min(X1;:::;Xn)>t)=P(X1>t) P(Xn>t)=nYi=1exp0BB@ 264t i375 1 CCA=exp0BB@ nXi=1264t i375 1 CCA=exp0BB@ 264t 375 1 CCAH enceT= min(X1;:::;Xn) Weibull ( ; ) with =0B@nXi=1 i1CA 1= Weibull Reliability Analysis |FWS-5/1999|6 Weibull ParametersThe Weibull distribution may be controlled by 2 or 3 parameters: thethreshold parameter T with probability 1 =0 =)2-parameter Weibull model.

3 Thecharacteristic lifeorscale parameter >0P(T + )=1 exp0B@ 24 35 1CA=1 exp( 1) =:632regardless of the value >0 theshape parameter >0,usually 1 Weibull Reliability Analysis |FWS-5/1999|72-Parameter Weibull Model We focus on Analysis using the2-parameter Weibull model Methods and software tools much better developed Estimation of in the 3-parameter Weibull modelleads to complications When a 3-parameter Weibull model is assumed,it will be stated explicitlyWeibull Reliability Analysis |FWS-5/1999|8 Relation of & to Statistical Parameters Theexpectationormean valueofT =E(T)=Z10tf(t)dt= (1 + 1= )with (t)=Z10exp( x)xt 1dx ThevarianceofT 2=E(T )2=Z10(t )2f(t)dt= 2 (1 + 2= ) 2(1 + 1= ) p-quantiletpofT, , by de nitionP(T tp)=ptp= [ log(1 p)]1= ;forp=1 exp( 1) =:632=)tp= Weibull Reliability Analysis |FWS-5/1999|9 Weibull Density Thecumulative distribution functionF(t)=P(T t)isjustone way to describe the distribution of the random quantityT Thedensity functionf(t) is another representation ( =0)f(t)=F0(t)=dF(t)dt= 0B@t 1CA 1exp0BB@ 264t 375 1 CCAt 0P(t T t+dt) f(t)dtF(t)=Zt0f(x)dxWeibull Reliability Analysis |FWS-5/1999|10 Weibull Density & Distribution Function05000100001500020000cyclesWeibul l density = 10000, = area under density = 1cumulative distribution functionpp01 Weibull Reliability Analysis |FWS-5/1999|11 Weibull Densities: E ect of cyclesprobability density020004000 = 0 = 1000 = 2000 = 1000, = Reliability Analysis |FWS-5/1999|12 Weibull Densities.

4 E ect of cyclesprobability density02000400060008000 = 1000 = 2000 = 3000 = 0, = Reliability Analysis |FWS-5/1999|13 Weibull Densities: E ect of cyclesprobability density01000200030004000 = .5 = 1 = 2 = 4 = 7 = 0, = 1000 Weibull Reliability Analysis |FWS-5/1999|14 Failure Rate or Hazard Function A third representation of the Weibull distribution is through thehazardorfailure rate function (t)=f(t)1 F(t)= 0B@t 1CA 1 (t) is increasingtfor >1 (wearout) (t) is decreasingtfor <1 (t) is constant for = 1 (exponential distribution)P(t T t+dtjT t)=P(t T t+dt)P(T t) f(t)dt1 F(t)= (t)dtF(t)=1 exp Zt0 (x)dx!andf(t)= (t)exp Zt0 (x)dx! Weibull Reliability Analysis |FWS-5/1999|15 Exponential Distribution Theexponential distributionis aspecial case: =1& =0F(t)=P(T t)=1 exp0B@ t 1 CAfort 0 This distribution is useful when parts fail due torandom external in uencesandnot due to wear out Characterized by thememoryless property,a part that has not failed by timetis as good as new,past stresses without failure are water under the bridge Good for describing lifetimes of electronic components,failures due to external voltage spikes or overloadsWeibull Reliability Analysis |FWS-5/1999|16 Unknown Parameters Typically will not know the Weibull distribution: ; unknown Will only have sample data=)estimatesc.

5 C getestimatedWeibull model for failure time distribution=)double uncertaintyuncertainty of failure time & uncertainty of estimated model Samples of failure times are sometimes very small,only 7 fuse pins or 8 ball bearings tested until failure,long lifetimes make destructive testing di cult Variability issues are often not su ciently appreciatedhow dosmall sample sizesa ect our con dence inestimates and predictions concerning future failure experiences? Weibull Reliability Analysis |FWS-5/1999|17 Estimation Weibull Population: Histogram for N = 10,000 & Density life = 30,000shape = modelestimated model from 9 data pointsWeibull Reliability Analysis |FWS-5/1999|18 Weibull Parameters & Sample Estimatest = tpp-quantilep=P(T < t ) characteristic life = 30, parameter = 4 25390 estimates from three samples of size n = 10 Weibull Reliability Analysis |FWS-5/1999|19 Generation of Weibull Samples Using the quantile relationshiptp= [ log(1 p)]1= one can generate a Weibull random sample of sizenby{generating a random sampleU1;:::;Unfrom a uniform [0;1] distribution{and computingTi= [ log(1 Ui)]1= ,i=1;:::;n.}}

6 {ThenT1;:::;Tncanbeviewedasarandomsample ofsizenfrom a Weibull population or Weibull distributionwith parameters & . Simulations are useful in gaining insight on estimation proceduresWeibull Reliability Analysis |FWS-5/1999|20 Graphical Methods Suppose we have a complete Weibull sample of sizen:T1;:::;Tn Sort these values from lowest to highest:T(1) T(2) ::: T(n) Recall that thep-quantile istp= [ log(1 p)]1= Computetp1<:::<tpnforpi=(i :5)=n,i=1;:::;n Plot the points (T(i);tpi),i=1;:::;nand expectthese points to cluster around main diagonalWeibull Reliability Analysis |FWS-5/1999|21 Weibull Quantile-Quantile Plot: Known Parameters Ttp050001000015000050001000015000 Weibull Reliability Analysis |FWS-5/1999|22 Weibull QQ-Plot: Unknown Parameters Previous plot requires knowledge of the unknown parameters & Note thatlog (tp)=log( )+wp= ;wherewp=log[ log(1 p)] Expect points log[T(i)];wpi ,i=1;:::;n, to cluster around linewith slope 1= and intercept log( ) This suggests estimating & from a tted least squares lineWeibull Reliability Analysis |FWS-5/1999|23 Maximum Likelihood Estimation Ift1.}

7 Tnare the observed sample values one can contemplatethe probability of obtaining such a sample or of values nearby, ,P(T12[t1 dt=2;t1+dt=2];:::;Tn2[tn dt=2;tn+dt=2])=P(T12[t1 dt=2;t1+dt=2]) P(Tn2[tn dt=2;tn+dt=2]) f ; (t1)dt f ; (tn)dtwheref(t)=f ; (t) is the Weibull density with parameters ( ; ) Maximum likelihood estimation maximizes this probabilityover & =)maximum likelihood estimates ( )c andc Weibull Reliability Analysis |FWS-5/1999|24 General Remarks on Estimation MLEs tend to be optimal in large samples (lots of theory) Method is very versatile in extending to may other data scenarioscensoring and covariates Least squares method applied to QQ-plot is not entirely appropriatetends to be unduly a ected by stray observationsnot as versatile to extend to other situationsWeibull Reliability Analysis |FWS-5/1999|25 Weibull Plot:n=20cycles/hoursprobability model, Weibull ( 100 , 3 ) model, Weibull ( 108 , )least squares model, Weibull ( , ) Weibull Reliability Analysis |FWS-5/1999|26 Weibull Plot.

8 N= 100cycles/hoursprobability model, Weibull ( 100 , 3 ) model, Weibull ( , )least squares model, Weibull ( , ) Weibull Reliability Analysis |FWS-5/1999|27 Tests of Fit (Graphical) The Weibull plots provide an informal diagnosticfor checking the Weibull model assumption The anticipated linearity is based on the Weibull model properties Strong nonlinearity indicates that the model is not Weibull Sorting out nonlinearity from normal statistical point scattertakes a lot of practice and a good sense for the e ectof sample size on the variation in point scatter Formal tests of t are available for complete samples2and also for some other censored data D'Agostino and Stephens,Goodness-of-Fit Techniques, Marcel Dekker 1986 Weibull Reliability Analysis |FWS-5/1999|28 Formal Goodness-of-Fit Tests LetFb ;b (t) be the tted Weibull distribution function LetdFn(t)=#fTi t;i=1.

9 Ngnbe the empirical distribution function Compute a discrepancy metricDbetweenFb ;b anddFn,DKS(Fb ;b ;dFn)=supt Fb ;b (t) dFn(t) Kolmogorov-SmirnovDCvM(Fb ;b ;dFn)=Z10 Fb ;b (t) dFn(t)!2fb ;b (t)dtCramer-von MisesDAD(Fb ;b ;dFn)=Z10 Fb ;b (t) dFn(t)!2Fb ;b (t)(1 Fb ;b (t))fb ;b (t)dtAnderson-Darling The distributions ofD, when sampling from a Weibull population,are known andp-values of observed valuesdofDcan be calculatedp=P(D d)=)BCSLIB: HSPFITW eibull Reliability Analysis |FWS-5/1999|29 Kolmogorov-Smirnov distance n = 10 Weibull Reliability Analysis |FWS-5/1999|30 Weibull Plots:n=10 (KS) = (CvM) = (AD) = = 10 (KS) = (CvM) = (AD) = (KS) = (CvM) = (AD) = (KS) = (CvM) = (AD) = (KS) = (CvM) = (AD) = (KS) = (CvM) = (AD) = (KS) = (CvM) = (AD) = (KS) = (CvM) = (AD) = Reliability Analysis |FWS-5/1999|31 Weibull Plots.

10 N=20 (KS) = (CvM) = (AD) = = 20 (KS) = (CvM) = (AD) = (KS) = (CvM) = (AD) = (KS) = (CvM) = (AD) = (KS) = (CvM) = (AD) = (KS) = (CvM) = (AD) = (KS) = (CvM) = (AD) = (KS) = (CvM) = (AD) = Reliability Analysis |FWS-5/1999|32 Weibull Plots:n=50 (KS) = (CvM) = (AD) = = 50 (KS) = (CvM) = (AD) = (KS) = (CvM) = (AD) = (KS) = (CvM) = (AD) = (KS) = (CvM) = (AD) = (KS) = (CvM) = (AD) = (KS) = (CvM) = (AD) = (KS) = (CvM) = (AD) = Reliability Analysis |FWS-5/1999|33 Weibull Plots.