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MER301: Engineering Reliability

MER301: Engineering Reliability. LECTURE 4: Chapter 3: Lognormal Distribution, The Weibull Distribution, and Discrete Variables. Summary of Topics. Lognormal Distribution Weibull Distribution Probability Density and Cumulative Distribution Functions of Discrete Variables

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MER301: Engineering Reliability

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  1. MER301: Engineering Reliability LECTURE 4: Chapter 3: Lognormal Distribution, The Weibull Distribution, and Discrete Variables MER301: Engineering Reliability Lecture 4

  2. Summary of Topics • Lognormal Distribution • Weibull Distribution • Probability Density and Cumulative Distribution Functions of Discrete Variables • Mean and Variance of Discrete Variables MER301: Engineering Reliability Lecture 4

  3. Normal Distribution • Many Physical Phenomena are characterized by normally distributed variables • Engineering Examples include variation in such areas as: • Dimensions of parts • Experimental measurements • Power output of turbines MER301: Engineering Reliability Lecture 4

  4. Lognormal Distribution • Special case of the normal distribution where and the variable w is normally distributed • Chemical processes and material properties are often characterized by lognormal distributions • Parameters and are the mean and variance of W, respectively MER301: Engineering Reliability Lecture 4

  5. Lognormal Distribution MER301: Engineering Reliability Lecture 4

  6. Lognormal Distribution MER301: Engineering Reliability Lecture 4

  7. Lognormal Example 4.1 • Gas Turbine CO Emissions • is a normally distributed function of combustor fuel/air ratio • Mean value of CO will need to be 9ppm or less MER301: Engineering Reliability Lecture 4

  8. Lognormal Example 4.1(cont) MER301: Engineering Reliability Lecture 4

  9. Lognormal Example 4.1 • excel spreadsheet for the CO example MER301: Engineering Reliability Lecture 4

  10. Skewness and Kurtosis: Tflame Example • Skewness • Skewness characterizes the degree of asymmetry of a distribution around its mean. Positive skewness indicates a distribution with an asymmetric tail extending towards more positive values. Negative skewness indicates a distribution with an asymmetric tail extending towards more negative values" (Microsoft, 1996). Samples from Normal distributions produce a skewness statistic of about zero. • ses can be estimated roughly using a formula from Tabachnick & Fidell,1996 • Kurtosis • kurtosis characterizes the relative peakedness or flatness of a distribution compared to the normal distribution. Positive kurtosis indicates a relatively peaked distribution. Negative kurtosis indicates a relatively flat distribution. Samples from Normal distributions produce a kurtosis statistic of about zero • sek can be estimated roughly using a formula from Tabachnick & Fidell, 1996

  11. Lognormal Example 4.1(cont)

  12. Lognormal Example 4.1(cont) CO is given by the equation Let and So that or MER301: Engineering Reliability Lecture 4 12

  13. Lognormal Example 4.1(cont) MER301: Engineering Reliability Lecture 4

  14. Lognormal Example 4.1(cont) • CO is given by the equation • Test Ln(CO/4.5) and CO for normality …. • Ln(CO/4.5) is normally distributed and CO is not MER301: Engineering Reliability Lecture 4

  15. Lognormal Example 4.1(cont) • CO is given by the equation • Let and So that or Now we want MER301: Engineering Reliability Lecture 4

  16. Lognormal Example 4.1(cont) • From the analysis of the data for flame temperature and W • Then MER301: Engineering Reliability Lecture 4

  17. Lognormal Example 4.1(cont) • Summary of the CO Lognormal Distribution Mean CO Standard Deviation of CO System Does not meet CO Requirements -combustor needs a factor of 5 improvement in CO performance MER301: Engineering Reliability Lecture 4

  18. Weibull Distribution • Widely used to analyze and predict failure for physical systems • failure may be a function of time, cycles, starts, landings, etc • Can provide reasonably accurate failure predictions with small samples • Important in safety critical systems MER301: Engineering Reliability Lecture 4

  19. The Weibull Distribution MER301: Engineering Reliability Lecture 4

  20. The Weibull Failure Function…. MER301: Engineering Reliability Lecture 4

  21. The Weibull Distribution Two parameters define the Weibull distribution: b, the shape parameter, is a measure of the time dependency of the probability of failure. Completely random failures(random errors, external shocks) have a b = 1. Failures which increase in probability over time(wearout, old age) have b > 1, and failures whose probability decreases over time(manufacturing errors) have 0 < b < 1. ,the scale parameter, is the time at which a cumulative 63.2% of the population is expected to have failed MER301: Engineering Reliability Lecture 4

  22. MER301: Engineering Reliability Lecture 4

  23. 63.2% Failure Rate at x=delta =1000 Infant Mortality Old Age Useful Life X=1000

  24. Weibull Plots-Cumulative Density Function Many kinds of failure data plot as a straight line with slope b . The x- axis is time and the y-axis is the cumulative failure density function F(t), Weibull plots are used to predict cumulative failures at any time. For instance, with b = 1.66 and = 177051, after 30000 time units 5% of the population will have failed. ln(ln(1/(1-63.2%)) = 0. So, h is the y-intercept of the straight line plot. t=30000 MER301: Engineering Reliability Lecture 4

  25. Weibull Plots-Cumulative Density Function (Two Cycle Weibull Paper) X=10000 X=100 X=1000

  26. Discrete Distribution Probability Mass Function • Describes how the total probability of 1 is distributed among various possible values of the variable X MER301: Engineering Reliability Lecture 4

  27. Define the following probabilities Let A= probability of 2=1/36 Let B= probability of 3=2/36 Let C= probability of 4=3/36 Let D= probability of 5=4/36 Let E= probability of 6=5/36 Let F= probability of 7=6/36 Let H= probability of 9=4/36 Let I= probability of 10=3/36 Let J= probability of 11=2/36 Let K= probability of 12=1/36 The Sum of Two Dice…

  28. Define the following probabilities Let A= probability of 2=1/36 Let B= probability of 3=2/36 Let C= probability of 4=3/36 Let D= probability of 5=4/36 Let E= probability of 6=5/36 Let F= probability of 7=6/36 Let H= probability of 9=4/36 Let I= probability of 10=3/36 Let J= probability of 11=2/36 Let K= probability of 12=1/36 The Sum of Two Dice…The Probability Mass Function For a Probability Mass Function

  29. The Sum of Two Dice…TheCumulative Distribution i • Define the following probabilities • Let A= probability of 2=1/36 • Let B= probability of 3=2/36 • Let C= probability of 4=3/36 • Let D= probability of 5=4/36 • Let E= probability of 6=5/36 • Let F= probability of 7=6/36 • Let G= probability of 8=5/36 • Let H= probability of 9=4/36 • Let I= probability of 10=3/36 • Let J= probability of 11=2/36 • Let K= probability of 12=1/36 MER301: Engineering Reliability Lecture 1 30

  30. The Sum of Two Dice…The Mean…. • Define the following probabilities • Let A= probability of 2=1/36 • Let B= probability of 3=2/36 • Let C= probability of 4=3/36 • Let D= probability of 5=4/36 • Let E= probability of 6=5/36 • Let F= probability of 7=6/36 • Let G= probability of 8=5/36 • Let H= probability of 9=4/36 • Let I= probability of 10=3/36 • Let J= probability of 11=2/36 • Let K= probability of 12=1/36 MER301: Engineering Reliability Lecture 1 31

  31. The Sum of Two Dice…The Variance…. • Define the following probabilities • Let A= probability of 2=1/36 • Let B= probability of 3=2/36 • Let C= probability of 4=3/36 • Let D= probability of 5=4/36 • Let E= probability of 6=5/36 • Let F= probability of 7=6/36 • Let G= probability of 8=5/36 • Let H= probability of 9=4/36 • Let I= probability of 10=3/36 • Let J= probability of 11=2/36 • Let K= probability of 12=1/36 MER301: Engineering Reliability Lecture 1 32

  32. Probability Mass Function 3-29 MER301: Engineering Reliability Lecture 4

  33. Example 4.2 • Consider a group of five potential blood donors – A, B, C, D, and E – of whom only A and B have type O+ blood. Five blood samples, one from each individual, will be typed in random order until an O+ individual is identified. • Let X=the number of typings necessary to identify an O+ individual • Determine the probability mass function of X MER301: Engineering Reliability Lecture 4

  34. Cumulative Distribution Function 3-31 MER301: Engineering Reliability Lecture 4

  35. Example 4.3 • For the previous example (4.2), determine F(x) for each value of x in the set of possible values x=1 to x=4 MER301: Engineering Reliability Lecture 4

  36. Expected Value or Mean of theDiscrete Distribution MER301: Engineering Reliability Lecture 4

  37. Example 4.4 • Consider a university having 15,000 students • X= number of courses for which a randomly selected student is registered. • The probability mass function can be found by knowing how many students signed up for any specific number of classes • Determine the probability mass function f(x). • Calculate the mean/expected number of courses per student. MER301: Engineering Reliability Lecture 4

  38. Variance of Discrete Distributions MER301: Engineering Reliability Lecture 4

  39. Example 4.5 • In example 4.4 the density function is given as shown • Determine the variance and the standard deviation. MER301: Engineering Reliability Lecture 4

  40. Expected Value of a Function • If the random variable X has a set of possible values x1,x2,…,xn and a probability mass function f(x), the the expected value of a function h(X) can be estimated as where MER301: Engineering Reliability Lecture 4

  41. Example 4.6 • Let X be the number of cylinders in the engine of the next car to be tuned up at a certain facility. • The cost of the tune up • h(x)=20+3x+0.5x2 • Assume 50%,30%,and 20% of cars have four, six, and eight cylinders, respectively • Since x is a random variable, so is h(x) • Write the density function f(y) for y=h(x) • Determine the expected value for Y=h(X) MER301: Engineering Reliability Lecture 4

  42. Summary of Topics • Lognormal Distribution • Weibull Distribution • Probability Density and Cumulative Distribution Functions of Discrete Variables • Mean and Variance of Discrete Variables MER301: Engineering Reliability Lecture 4

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