1. L Berkley Davis Copyright 2009 MER035: Engineering Reliability Lecture 6 1 MER301: Engineering Reliability LECTURE 6: Chapter 3: 3.9, 3.11 and Reliability Exponential Distributions, Independence and Joint Distributions, Reliability

2. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 2 Summary  Exponential Distribution  Independence  Joint Distributions  Reliability

3. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 3 Exponential Distribution  The Exponential Distribution predicts the probability of random failures  Frequently used in conjunction with a Poisson Process  For a Poisson process with parameter λ If x denotes the time until occurrence of the first event after a specified time Then x has an Exponential Distribution with mean=1/ λ

4. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 4 A Poisson Process  Requirements for a Poisson Process Random discrete events that occur in an interval that can be divided into subintervals Probability of a single occurrence of the event is directly proportional to the size of a subinterval and is the same for all subintervals If the sampling subinterval is sufficiently small, the probability of two or more occurrences of the event is negligible Occurrences of the event in nonoverlapping subintervals are independent

5. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 5 Exponential Distribution

6. L Berkley Davis Copyright 2009 Exponential and Poisson Distributions MER301: Engineering Reliability Lecture 6 6

7. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 7 Summary of Distribution Function Characteristics  Poisson Distribution Probability for a number of randomly occurring events to be found in an interval of of a specific size  Exponential Distribution Probability that next event of a Poisson process occurs at a specified interval  Weibull Distribution Predicts probability of failure(failure rate) Exponential distribution a special case for beta=1(random failures)

8. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 8 Example 6.1  Some strains of paramecia produce and secrete certain particles that will destroy other paramecia, called sensitive, upon contact.  All paramecia unable to produce such particles are sensitive.  The number of particles emitted by any single non- sensitive paramecium is one every five hours.  In observing such a paramecium what is the probability that we must wait at most four hours before the first particle is emitted?  What is the probability that the time until the first particle is emitted is between two and three hours?

9. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 9 Example 6.1 Solution  Let the measurement unit be one hour. This is then a Poisson process with.The time at which the first particle is emitted has an exponential distribution with.The density function is then and the probability we must wait at most 4 hours until the first particle is emitted is

10. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 10 More than One Random Variable and Independence  Most engineering design problems require that multiple variables be dealt with in designing products Control variables “noise” variables  Independence implies that variation in one variable is not related to variation in another

11. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 11 Joint Distributions  If X and Y are random variables, a probability distribution defining their simultaneous behavior is a Joint Probability Distribution Applicable to both continuous and discrete random variables Applicable whether or not variables are independent 3-39 3-40

12. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 12 More than One Random Variable and Independence  Most engineering design problems require that multiple variables be dealt with in designing products Control variables “noise” variables  Independence implies that variation in one variable is not related to variation in another Salt Conc and %Area Not Independent

13. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 13 Independent Random Variables

14. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 14 Independent Random Variables X1 &x2

15. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 15 Binomial Problem N=10,x=10.p=0.919

16. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 16 Joint Distributions  If X and Y are random variables, a probability distribution defining their simultaneous behavior is a Joint Probability Distribution  Applicable to both continuous and discrete random variables Applicable whether or not variables are independent

17. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 17 Independence of Random Variables or Events  Concept of independent random variables/events underlies many statistical analysis methods  If Y=f(x1,x2,…xn) then the values y1,…yk obtained from k repetitions of an experiment are independent if and only if x1,…xn are random independent variables

18. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 18 Reliability  Reliability Analysis quantifies whether or not a design functions adequately for the required life when operating under the conditions for which it is designed- whether it… Meets Functional Requirements over Operating Life Achieves Design Life before Replacement/Repair Tolerates Needed range of Environmental Conditions  The objective of Reliability Analysis is to predict the probability of failure at a specific time t, number of cycles, etc Failure density function, f(t) Cumulative density function, F(t)=probability of failure at time t Reliability function, R(t) = I-F(t)=probability of no failure at time t

19. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 19 Reliability Functions  Four Reliability Functions Failure density function, f(t) may be a Weibull, exponential, or other distribution Cumulative density function, F(t) Reliability function, R(t) = I-F(t) Hazard rate of distribution, h(t)

20. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 20 Reliability Definitions  Probability of failure per unit time or cycle  Mean Time between Failures(MTBF)  Types of Failures during Product Life Infant Mortality Random Failure Old Age  For the Exponential Distribution

21. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 21 The Weibull Distribution

22. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 22 The Weibull Distribution Hazard Rate delta=1000

23. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 23 Reliability  What kinds of Systems require reliability analysis? Individual components(gas turbine compressor blades) or Assemblies(motors, pumps,valves, instruments) Software (firmware, distributed controls code,plant level controls code, applications code) Systems comprising multiple components or assemblies or software(gas turbine gas fuel system)  What are the methods used in reliability analysis? Testing to establish capability of components, assemblies, systems, software Reliability Block Diagrams to model entire Systems Analysis of Reliability Models to assess performance Field Data gathering to validate models

24. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 24 Reliability Testing  Reliability Tests are conducted to establish the tolerance of components,assemblies, and systems to environmental factors Dust, dirt,chemical contaminants Vibration Operating Temperature range Ambient Temperature range Moisture,humidity  Reliability Tests are typically conducted by exercising the system Subjecting systems to environmental stress testing Exercising the system through a number of operational cycles equivalent to product life requirements Conducting HALT tests to get data more quickly

25. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 25 Reliability Analysis  Most designs consist of multiple subsystems, assemblies,and components each with it own probability of failing or not Probability of overall system functioning is a function of subsystem characteristics The way subsystems are arranged in a system determines overall reliability

26. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 26 Reliability of a System

27. L Berkley Davis Copyright 2009 Venn Diagram MER301: Engineering Reliability Lecture 6 27 Reliability of a System A B A= A & B+ A & not B notA &B notA &notB “ coin flip ” “A or NotA” Diagram “Probability A or B”

28. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 28 Reliability of a System Block 3 ABCABC

29. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 29 Aircraft Reliability  Four Engine Aircraft-two engines per wing and at least one engine per wing must function For a wing, either engine must function so that probability that a wing is OK is an “or” problem For the aircraft, one engine on each wing must function at the same time so that the probability that both wings are OK is an “and” problem  Addition ( A or B) A and B are mutually exclusive A and B are not mutually exclusive  Multiplication( A and B) A and B are independent A and B are not independent/conditional probability

30. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 30 Aircraft Reliability  Let Engines A and B be on Wing 1 and Engines C and D be on Wing 2 Assume the reliability (probability of not failing) of all engines is the same R A =R B =R C =R D The probability a wing is functional is then(addition rule)  Both wings must be functional for the plane to fly so the probability of flight is then (multiplication rule)

31. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 31 Aircraft Reliability  Four Engine Aircraft-two engines per wing and at least one engine per wing must function for a successful flight 0.999 engine reliability gives two chances/million of a flight failure….. Is this good enough?

32. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 32 Aircraft Reliability  We must deal with the entire fleet over its lifetime Assume a fleet of 1000 planes with one flight per plane per day and a life of 30 years  To reach these high levels of reliability, scheduled engine maintenance is performed to “re-start”the clock

33. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 33 Triple Modular Redundant System  Any two of three components must operate for the system to function Assume that R A =R B =R C  On-Line Repair often incorporated into Triple Modular Redundant systems B C C A B A

34. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 34 Triple Modular Redundant System

35. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 35 Summary  Exponential Distribution  Independence  Joint Distributions  Reliability