Reliability of Technical Systems

Transcription

1 eliabiliy of Technical Sysems

2 Main Topics Inroducion, Key erms, framing he problem eliabiliy parameers: Failure ae, Failure Probabiliy, Availabiliy, ec. Some imporan reliabiliy disribuions Componen reliabiliy Sofware reliabiliy Faul Tolerance Sysem reliabiliy: Srucure and Sae Modelling Dependen failure Human reliabiliy Saic and dynamic redundancy Advanced mehods for sysems modeling and simulaion eliabiliy of echnical Sysems

3 Mos relevan disribuions in eliabiliy Theorie: Coninuous ones: Discree ones: Exponeial, Weibull, Normal, Log Normal, Uniform ecangular. Poisson, Binomial Bernoulli. eliabiliy of echnical Sysems 3

4 Number Name Disribuion Binomial Exponenial 3 Log Normal 4 Normal 5 Poisson 6 Uniform 7 Weibull eliabiliy of echnical Sysems 4

5 Exponenial disribuion The exponenial disribuion is he only coninuous memoryless random disribuion. The exponenial disribuion is he mos common and simples disribuion funcion o model he reliabiliy of componens. The failure rae and he ime o failure are reciprocal. Applicaion: To esimae he reliabiliy of compones/sysems wih consan failure rae. eliabiliy of echnical Sysems 5

6 eliabiliy of echnical Sysems 6

7 eliabiliy of echnical Sysems 7

8 V E L Saionary availabiliy Only for repairable sysems he saionary availabiliy seady-sae availabiliy V is defined by EL + E MTTF MTTF + MTT The availabiliy V denoes he probabiliy, ha a sysem is properly funcioning a any poin in ime. The general ime dependen availabiliy V can only be calculaed afer he inroducion of a sae model. Mean ime o failure MTTF Mean ime o repair MTT Assumpion: Exponenial disribuion V / λ / λ + / µ / λ µ + λ λ µ µ λ + µ E L or E L E or E B E L /λ E /μ Where λ - failure rae, μ repaire rae eliabiliy of echnical Sysems 8

9 Saisical daa and saisical disribuion funcion A hisogram is he graphical version of a able ha shows wha proporion of cases relaive occurrence fall ino each of specified inervals. A generalizaion of he hisogram consruc a very smooh probabiliy densiy funcion from he supplied daa. Quesions he Hisogram Answers Wha is he mos common value? Wha disribuion cener, variaion and shape does he daa have? To check wih Kolmogorov`s or χ crieria. eliabiliy of echnical Sysems 9

10 Applicaion examples of disribuions Disribuion Exponenial Weibull Normal Logarihmic Normal Uniform Binomial Poisson Field of Applicaion for consan failure rae β> for monoone increasing raes β< for monoone decreasing raes In case of saisically independen random variables for ime o repair for wear-ou failures for n-ou-of-m sysems for saisical reliabiliy experimen eliabiliy of echnical Sysems 0

11 Exercise example: For a larger number of simulaneously implemened idenical devises i was discovered afer one year ha 5% of he devices failed, and were no repaired.. Wha is he mean lifeime E of he devices, if an exponenial disribuion wih parameer λ is assumed?. Wha is he mean lifeime E of he devices, if an uniform disribuion wih he parameers a and b is assumed and a 0? Wha assumpion leads o more opimisic reliabiliy forecas? eliabiliy of echnical Sysems

12 Soluion:. Exponenial disrbuion λ e year, F %, han 0.95 λ year e 0.95, λ ln 0.95 year / year Mean lifeime E exp /λ9,5 years eliabiliy of echnical Sysems

13 Soluion:. Uniform disrbuion b b a year, F %, han 0.95 b b 0.95, b year 0.95b b 0years b a b 0 Mean lifeime E uni a+b/0 years E exp E uni eliabiliy of echnical Sysems 3

14 4 eliabiliy of echnical Sysems Ineracion of differen disribued failure effecs I Assumpions: Failures are independen, Failure effecs inerac one afer anoher. In general: By failure effecs: *, F- *, N i i, d d f z z z + +, 0 d E Failure raes are added.

15 Ineracion of differen disribued failure effecs II Assumpions: Failures are independen, failure effecs, boh exponeial disribued wih λ and λ. e e e λ λ λ + λ also exponenial disribued wih λ λ + λ E λ + λ eliabiliy of echnical Sysems 5

16 Ineracion of differen disribued failure effecs III Assumpions: Failures are independen, failure effecs, boh weibull disribued wih α, β and α, β. e α β e α β e α β β + α only by β β weibull disribued wih / α /α +/α z β α β β + β α, fz* E? eliabiliy of echnical Sysems 6