ECE 510 Lecture 4 Reliability Plotting T&T Scott Johnson Glenn Shirley
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1 ECE 5 Lecure 4 Reliabiliy Ploing T&T 6.-6 Sco Johnson Glenn Shirley
2 Funcional Forms 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley
3 Reliabiliy Funcional Forms Daa Model (funcional form) Choose funcional form for model o fi daa 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 3
4 A Funcion Besiary Besiary: A medieval collecion of sories providing physical and allegorical descripions of real or imaginary animals Coninuous disribuions Normal Exponenial Lognormal Weibull Gamma Bea Discree disribuions Hypergeomeric Binomial Poisson Saisical disribuions Chi-square Suden s F Mos common for reliabiliy 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 4
5 Normal Disribuion μ = mean σ = sandard deviaion σ = variance f F x x x x e dx x e rand normal NORMSINV ( CDF) where CDF is rand uniform e x Using Excel: PDF = NORMDIST(x,μ,σ,FALSE) CDF = NORMDIST(x,μ,σ,TRUE) Plo using: y-axis = probi = NORMSINV(CDF) x-axis = x σ = /slope μ = x-inercep = (y-inercep) / slope 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 5
6 CDF PDF Normal Disribuion Normal Disribuion x value μ = mean σ = sandard deviaion σ = variance f F x x x x e dx x e rand normal NORMSINV ( CDF ) where CDF is rand uniform Normal Disribuion x value Using Excel: PDF = NORMDIST(x,μ,σ,FALSE) CDF = NORMDIST(x,μ,σ,TRUE) Plo using: y-axis = probi = NORMSINV(CDF) x-axis = x σ = /slope μ = x-inercep 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 6
7 f() H() F() S() h() Normal Disribuion Reliabiliy Plos Reliabily Funcion F() = CDF PDF f() = d/d CDF Survival Funcion S() = -F() Failure rae = h() = f()/s() Cumulaive Hazard Funcion H() Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 7
8 Use of Normal Disribuions Mos measuremen error Sum of random hings is normal 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 8
9 Exponenial Disribuion λ = scale facor f F x x x e e x e ln( CDF) rand exponenial where CDF is rand uniform Using Excel: PDF = λ*exp(-λx) CDF = -EXP(-λx) Plo using: y-axis = exbi = -LN(-CDF) x-axis = x λ = slope 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 9
10 CDF PDF Exponenial Disribuion Exponenial Disribuion x value λ = scale facor f F x x x e e x ln( CDF) rand exponenial where CDF is rand uniform Exponenial Disribuion 3 4 x value Using Excel: PDF = λ*exp(-λx) CDF = -EXP(-λx) Plo using: y-axis = exbi = -LN(-CDF) x-axis = x λ = slope 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley
11 f() H() F() S() h() Exponenial Reliabiliy Plos Reliabily Funcion F() = CDF Survival Funcion S() = -F() Failure rae = h() = f()/s() PDF f() = d/d CDF Cumulaive Hazard Funcion H() Jan 3 ECE 5 S.C.Johnson, C.G.Shirley
12 Use of Exponenial Disribuions Consan fail rae No memory of he pas; no age Radioacive decay Sof errors, exernal environmen Easy o calculae MTTF = /λ 5 e. 5 Median ime o fail from so F 5 5 ln 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley
13 Exercise 4. Given an exponenial fail disribuion wih.4% khr wha is he probabiliy of failure wihin 5, hours of use? Wha is he MTTF? 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 3
14 Soluion 4. Conver o pure unis.4% khr. 4 fails hour hen evaluae he fail funcion a 5, hours F. 4 5, e e.6.6% The MTTF is even easier MTTF,5, hours 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 4
15 LogNormal Disribuion 5 = median ime o fail σ = sandard deviaion f F ln5 ln e d ln5 ln e Using Excel: PDF = NORMDIST(ln(),ln(5),σ,FALSE)/ CDF = NORMDIST(ln(),ln(5),σ,TRUE) Plo using: y-axis = probi = NORMSINV(CDF) x-axis = ln() σ = /slope ln(5) = x-inercep rand normal exp NORMSINV where CDF is rand uniform CDF 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 5
16 CDF PDF LogNormal Disribuion LogNormal Disribuion x value 5 = median ime o fail σ = sandard deviaion f F rand normal exp NORMSINV where CDF is rand uniform ln5 ln e d ln5 ln e CDF LogNormal Disribuion x value Using Excel: PDF = NORMDIST(ln(),ln(5),σ,FALSE)/ CDF = NORMDIST(ln(),ln(5),σ,TRUE) Plo using: y-axis = probi = NORMSINV(CDF) x-axis = ln() σ = /slope ln(5) = x-inercep 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 6
17 f() H() F() S() h() Lognormal Reliabiliy Plos Reliabily Funcion F() = CDF Survival Funcion S() = -F() Failure rae = h() = f()/s() PDF f() = d/d CDF Cumulaive Hazard Funcion H() Mosly decreasing failure rae: IM-ype mechanism 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 7
18 Use of Lognormal Disribuions I ~ SB e Le R R 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 8
19 Weibull Disribuion β = shape parameer α = scale parameer γ = locaion parameer f x x F x e x exp x exp Noe: α and β are ofen swapped in meaning! Excel swaps hem (below). T&T use βm and αc. rand Weibull ln CDF where CDF is rand uniform Using Excel: PDF = WEIBULL(x,β,α,FALSE) CDF = WEIBULL(x,β,α,TRUE) =-EXP(-((x/α)^β)) Noe γ= in Excel Plo using: y-axis = weibi = ln(-ln(-cdf)) x-axis = ln(x) β = slope α = exp(-inercep/slope) 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 9
20 CDF PDF Weibull Disribuion Weibull Disribuion x value Weibull Disribuion bea alpha β = shape parameer α = scale parameer γ = locaion parameer Noe: α and β are ofen swapped in meaning! Excel swaps hem (below). T&T use βm and αc. f x x F x x exp x exp where CDF isrand uniform rand Weibull ln CDF x value Using Excel: PDF = WEIBULL(x,β,α,FALSE) CDF = WEIBULL(x,β,α,TRUE) =-EXP(-((x/α)^β)) Noe γ= in Excel Plo using: y-axis = weibi = ln(-ln(-cdf)) x-axis = ln(x) β = slope α = exp(-inercep/slope) 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley
21 f() f() S() H() S() H() F() h() F() h() Weibull Reliabiliy Plos Weibull, β=.5 (<) Weibull, β=.5 (>) Reliabily Funcion F() = CDF Failure rae = h() = f()/s() Reliabily Funcion F() = CDF Failure rae = h() = f()/s() Survival Funcion S() = -F() Cumulaive Hazard Funcion H() Survival Funcion S() = -F() Cumulaive Hazard Funcion H() PDF f() = d/d CDF Decreasing failure rae: Infan Moraliy (IM) ype mechanism PDF f() = d/d CDF Increasing failure rae: Wearou (WO) ype mechanism 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley
22 Use of Weibull Disribuions When fail is caused by he wors of many iems When i fis he daa well Weibull 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley
23 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 3 Main Reliabiliy Funcions e e e ln e
24 Muliple Mechanisms 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 4
25 Muliple Mechanisms 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 5 h h h F F S S F S S S o o o Survivals muliply, hazard raes add:
26 Exercise 4. Hand fi Weibull disribuions o he human moraliy daa like his: Plo boh he hazard rae h() (like above) and he fail funcion F(). Useful: for he Weibull, from T&T able 4.3: h 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 6
27 Soluion 4. 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 7 h e e F h h
28 Reliabiliy Ploing 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 8
29 Reliabiliy Ploing Noe sraigh lines (doed, each Weibull) 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 9
30 Probi Plo NORMSINV(CDF) Our eyes deec sraigh lines 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 3
31 Excel NORMxxx Funcions.8.8 CDF Probi Probi = NORMSINV(CDF) CDF = NORMSDIST(Probi) 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 3
32 Probi Plos in Excel Plo using: y-axis = probi = NORMSINV(CDF) x-axis = x σ = /slope μ = x-inercep = (y-inercep) / slope 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 3
33 Probi Plos in Excel probis daa Plo using: y-axis = probi = NORMSINV(CDF) x-axis = x σ = /slope μ = x-inercep = (y-inercep) / slope 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 33
34 Uncerainies in Probi Plos 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 34
35 Exbi Plos exbis daa Plo using: y-axis = exbi = -LN(-CDF) x-axis = x λ = slope Noe ha exbi is no a sandard name 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 35
36 Weibi Plos Weibi ln daa Plo using: y-axis = Weibi = ln(-ln(-cdf)) x-axis = ln(x) β = slope α = exp(-inercep/slope) Noe ha Weibi is a sandard name 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 36
37 Lognormal Probi Plo probis ln daa Plo using: y-axis = probi = NORMSINV(CDF) x-axis = ln() σ = /slope ln(5) = x-inercep 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 37
38 Percen Percen The Graph Paper Mehod Exponenial (semi-log) Normal 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 38
39 Percen Percen More Graph Paper Lognormal Weibull 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 39
40 Exercise 4.3 Make probi, exbi, Weibi, and lognormal probi plos Deermine parameers for each plo Look a all 4 daa ses ( 3) Deermine which ype each disribuion is Give he parameers for each correc disribuion 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 4
41 Daa exponenial λ = 3. Soluion 4.3 Daa normal µ =.6 σ =.88 Daa Weibull α = 3. β =.97 Daa3 lognormal µ =.88 σ =.67 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 4
42 JMP Plos JMP versions of probi, exbi, and Weibi plos 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 4
43 Number of Occurances CDF Probi Truncaed Disribuions Hisogram Value (Min of Range) Cumulaive Disribuion Funcion (CDF) Value CDF, Probi Scale Value 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 43
44 Top-Truncaed Disribuions Rank.3 Coun.4 Rank.3 Coun Missing.4 Noe Adj CDF doesn reach 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 44
45 Exercise 4.4 Make a runcaed probi plo of he daa on ab Ex8. Find he mean and sandard deviaion of he original disribuion as bes you can. 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 45
46 Soluion Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 46
47 Daa Censoring Missing daa is called censored Type I, ime censored Exac imes o fail up o ime ; no daa afer Type II, fail coun censored Exac imes o fail for he firs r unis o fail; no daa afer Mulicensored or readou Have a ime inerval wihin which each uni failed up o max; no daa afer 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 47
48 The End 6 Jan 3 ECE 5 S.C.Johnson, C.G.Shirley 48
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