ECE 510 Lecture 4 Reliability Plotting T&T Scott Johnson Glenn Shirley

Transcription

1 ECE 5 Lecure 4 Reliabiliy Ploing T&T 6.-6 Sco Johnson Glenn Shirley

2 Funcional Forms 6 Jan 23 ECE 5 S.C.Johnson, C.G.Shirley 2

3 Reliabiliy Funcional Forms Daa Model (funcional form) Choose funcional form for model o fi daa 6 Jan 23 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 23 ECE 5 S.C.Johnson, C.G.Shirley 4

5 Normal Disribuion μ = mean σ = sandard deviaion σ 2 = variance f F x x x x 2 e 2 dx x 2 e 2 rand normal NORMSINV ( CDF) where CDF is rand uniform 2 2 e x 2 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 23 ECE 5 S.C.Johnson, C.G.Shirley 5

6 CDF PDF Normal Disribuion Normal Disribuion x value μ = mean σ = sandard deviaion σ 2 = variance f F x x x x 2 e 2 dx x 2 e 2 rand normal NORMSINV ( CDF ) where CDF is rand uniform 2 2 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 23 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 23 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 23 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 23 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 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 23 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 23 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 2 6 Jan 23 ECE 5 S.C.Johnson, C.G.Shirley 2

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 23 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 2,5, hours 6 Jan 23 ECE 5 S.C.Johnson, C.G.Shirley 4

15 LogNormal Disribuion 5 = median ime o fail σ = sandard deviaion f F ln5 2 ln 2 e 2 d ln5 2 ln 2 e 2 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 23 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 2 ln 2 e 2 d ln5 2 ln 2 e 2 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 23 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 23 ECE 5 S.C.Johnson, C.G.Shirley 7

18 Use of Lognormal Disribuions I ~ SB e Le R R 6 Jan 23 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 23 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 23 ECE 5 S.C.Johnson, C.G.Shirley 2

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 23 ECE 5 S.C.Johnson, C.G.Shirley 2

22 Use of Weibull Disribuions When fail is caused by he wors of many iems When i fis he daa well Weibull 6 Jan 23 ECE 5 S.C.Johnson, C.G.Shirley 22

23 6 Jan 23 ECE 5 S.C.Johnson, C.G.Shirley 23 Main Reliabiliy Funcions e 2 e e 2 ln e

24 Muliple Mechanisms 6 Jan 23 ECE 5 S.C.Johnson, C.G.Shirley 24

25 Muliple Mechanisms Survivals muliply, hazard raes add: 6 Jan 23 ECE 5 S.C.Johnson, C.G.Shirley 25

26 Exercise 4.2 Hand fi 2 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 (pg. 94 in 3 rd ed): 6 Jan 23 ECE 5 S.C.Johnson, C.G.Shirley 26

27 Soluion 4.2 h h2 6 Jan 23 ECE 5 S.C.Johnson, C.G.Shirley 27

28 Reliabiliy Ploing 6 Jan 23 ECE 5 S.C.Johnson, C.G.Shirley 28

29 Reliabiliy Ploing Noe sraigh lines (doed, each Weibull) 6 Jan 23 ECE 5 S.C.Johnson, C.G.Shirley 29

30 Probi Plo NORMSINV(CDF) Our eyes deec sraigh lines 6 Jan 23 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 23 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 23 ECE 5 S.C.Johnson, C.G.Shirley 32

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 23 ECE 5 S.C.Johnson, C.G.Shirley 33

34 Uncerainies in Probi Plos 6 Jan 23 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 23 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 23 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 23 ECE 5 S.C.Johnson, C.G.Shirley 37

38 Percen Percen The Graph Paper Mehod Exponenial (semi-log) Normal 6 Jan 23 ECE 5 S.C.Johnson, C.G.Shirley 38

39 Percen Percen More Graph Paper Lognormal Weibull 6 Jan 23 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 23 ECE 5 S.C.Johnson, C.G.Shirley 4

41 The End 6 Jan 23 ECE 5 S.C.Johnson, C.G.Shirley 4