Reliability of Technical Systems

Transcription

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2 Main Topics 1. Introduction, Key Terms, Framing the Problem 2. Reliability Parameters: Failure Rate, Failure Probability, etc. 3. Some Important Reliability Distributions 4. Component Reliability 5. Software Reliability 6. Fault Tolerance 7. System Reliability I 8. Dependent Failures 9. System Reliability II 10. Static and Dynamic Redundancy 11. Advanced Methods for System Modelling and Simulation I 12. Advanced Methods for System Modelling and Simulation II 13. Human Reliability 2

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5 Comparison Between Failure and Mortality Rate failure rate z(t) mortality rate time t age Failure rate: Probability per dt that a unit that survived until time t will fail in the next time interval. Mortality rate: Probability at a given age to die within the next year. 5

6 Bathtub Curve (I) The bathtub curve describes the live time of a component. failure rate z(t) Phase I Phase II Phase III λ T F T A time t Phase Failure Rate Meaning 1 decreasing Phase of early failures ( optimising phase ) 2 constant Phase of random failures (no systematic failures) 3 increasing Phase of late failures (e.g. wear-out, aging, etc.) 6

7 Failure (Fault): Wrong or "missing" function of a component. Failure causes: Design failure Manufacture failure Operation failures Failures due to disturbances Random physical failures Handling failures Maintenance failures Wearing failures 7

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9 failure rate Bathtub Curve (II) time Phase I Phase II Phase III At time T F : Weibull distributed with β < 1, T F = some weeks. Exponential distributed with β = 1, z(t) = λ (constant). Normal distributed (WD with β 3,44; or Log Normal), T A = years. 9

10 Bathtub Curve and Reliability wear-out 10

11 For some type of components the late phase typically begins after many decades, when the system is out of use anyway. Therefore the exact modelling of this phase is not of interest for such components. 11

12 Failure Rates of Various Unit Types bathtub curve Hardware / Mechanic Components Software Comments: The hypothesis of a constant failure rate has to be verified for each case by statistical method. It holds true if, and only if, the probability model of an exponential distribution is correct. Non constant failure rates implicate another probability model, e.g. the Weibull distribution. 12

13 How Phase I and Phase III can be reduced? Phase I: Pre-aging ( Burn-in ) Phase III: Preventive Replacement 13

14 Pre-aging 14

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16 Reliability or failure data can be obtained from a number of sources. The most useful are: Reliability Data Field data Reliability databases (OREDA, GIDEP, etc) Reliability test (experiment) data The other sources are in most cases incomplete. 16

17 Realistic stress of several components Measurement of the time to failure 17

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20 Malfunctions of an unit (failure modes) Functions Closing Opening Remain closed Remain opened Types of failure Fails open Only partly closed Fails closed Only partly opened Opens completely Partly opens Closes completely Partly closes 20

21 Techniques to speed up the experiment Sequential test Accelerated test Extrapolation 21

22 Sequential Test (I) If the actual failure rate λ does not exceed the limit λ 1 (λ < λ 1 ) with high probability w 1, the components are to be accepted. On the other hand, if λ does not under-run the limit λ 2 (λ > λ 2 ) with high probability w 2, the components are to be rejected. Given: λ 1, sample size n, acceptance threshold k, time of experiment t. Let X be the number of failures (Poisson distributed) within the time interval [0, t]. 22

23 Sequential Test (II) For given probabilities α and β is true: Probability of acceptance Probability of rejection For λ= λ1 1- α α For λ= λ2 β 1- β Accept decision if: i β ln 1 λ2 λ1 + λ α 2 λ2 ln ln λ λ 1 1 n t Reject decision if: i 1 β ln λ2 λ1 + λ α 2 λ2 ln ln λ λ 1 1 n t 23

24 Sequential Test (III): Illustration number of failures reject decision accept decision cumulative testing time The minimum testing time (nt) min and the minimum number of failure i min amount to: where α and β are probabilities 24

25 Sequential Test (IV): Algorithm failure rate is high reject failure rate is high reject test until first failure failure rate is neither high nor low test until next failure failure rate is neither high nor low... failure rate is low accept failure rate is low accept 25

26 Extrapolation (I) Failure prediction by extrapolation: Shorter testing time Test is non-destructive Conditions: Just drift failures Failure criteria are known 26

27 Extrapolation (II): Illustration parameter y failure criteria t t exp t pred t exp t pred testing time predicted life time (forecast through extrapolation) 27

28 Accelerated Test (I) 28

29 Accelerated Test (II): Algorithm Acceleration factor F 1,2 = L L 1 2 = e E K 1 1 ( T 1 T 2 ) Measure mean life time L 2 of component by T 2 > T 1, where T 1 is a low temperature. If the quotient E/K is unknown, repeat the test by T 3 > T 1 and find it from L L 2 3 = e E 1 1 ( ) K T T 2 3. E 1 1 ( ) K T T 1 = 2. L L e Then calculate

30 Accelerated Test (III): Illustration 30