1 / 43

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

astroman
Télécharger la présentation

MER301: Engineering Reliability

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related