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Fundamentals of Reliability               Fundamentals of Reliability             Engineering and Applications            ...
ASQ Reliability Division                  ASQ Reliability Division                  Short Course Series                  S...
Fundamentals of Reliability Engineering and              Applications                 E. A. Elsayed            elsayed@rci...
Outline        Part 1.   Reliability Definitions   Reliability Definition…Time dependent                Definition Time  ...
Outline  Part 2.    Reliability Calculations1. Use f f il1 U of failure d t                 data2. Density functions3.3 Re...
OutlinePart 3.     Failure Time Distributions1. Constant failure rate distributions1 C     t t f il       t di t ib ti2. I...
Outline  Part 2.      Reliability Calculations1. Use f f il1 U of failure d t               data  a)   Interval data (no c...
Basic Calculations  Suppose n0 identical units are subjected to a  test. During the interval (t, t +∆t ), we observed  nf ...
Basic Definitions Cont’dThe unreliability F(t) is                                 1                            F t   R t...
Calculations                            Time        Failure Density Hazard rate                            Interval       ...
Failure Density vs. Time×10-4           1   2       3     4     5   6   7   x 103                   Time i h              ...
Hazard Rate vs. Time×10-4   -         1   2      3     4      5   6   7   × 103                 Time in Hours             ...
Reliability CalculationsTime Interval   Failures  (Hours)         in the    Time Interval   Reliability R (t )            ...
Reliability vs. Time          y1   2      3     4      5   6   7   x 103        Time in hours                             ...
Exponential DistributionDefinition                          (t) h (t )        0, t  0                Time f (t )  ...
Exponential Model Cont’d                              Cont dStatistical Properties                                        ...
Exponential Model Cont’d                           Cont dStatistical Properties                                 5       1 ...
Empirical Estimate of F(t) and R(t)When the exact failure times of units is known, weuse an empirical approach to estimate...
Empirical Estimate of F(t) and R(t)            p                    ()       ()F (ti )    is obtained by several methods  ...
Empirical Estimate of F(t) and R(t)Assume that we use the mean rank estimator  ˆ (t )  i  F i           n 1  ˆ (t )  n ...
Empirical Estimate of F(t) and R(t)                   1   (t )   ˆ           ti .(n  1  i )     i                (  H...
Calculationsi   t (i)   t(i+1) F=i/10 R=(10-i)/10                   F i/10 R (10 i)/10   f 0.1/t                         ...
Reliability Function                     y           1.2             1ReliabilityR li bilit 0.8           0.6           0....
Probability Density Function           0.001600           0.001400           0.001200           0.001000Density           ...
Constant Failure Rate               0.001900               0.001700               0.001500               0 001500         ...
Exponential Distribution: Another ExampleGiven failure d tGi    f il    data:Plot the hazard rate, if constant then use th...
Input Data             25
Plot of the Data                   26
Exponential Fit                  27
Exponential Analysis
SummaryIn this part, we presented the three most importantrelationships in reliability engineering.We estimated obtained e...
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Fundamentals of reliability engineering and applications part2of3

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This is a three parts lecture series. The parts will cover the basics and fundamentals of reliability engineering. Part 1 begins with introduction of reliability definition and other reliability characteristics and measurements. It will be followed by reliability calculation, estimation of failure rates and understanding of the implications of failure rates on system maintenance and replacements in Part 2. Then Part 3 will cover the most important and practical failure time distributions and how to obtain the parameters of the distributions and interpretations of these parameters. Hands-on computations of the failure rates and the estimation of the failure time distribution parameters will be conducted using standard Microsoft Excel.
Part 2. Reliability Calculations
1.Use of failure data
2.Density functions
3.Reliability function
4.Hazard and failure rates

Veröffentlicht in: Technologie, Business

Fundamentals of reliability engineering and applications part2of3

  1. 1. Fundamentals of Reliability  Fundamentals of Reliability Engineering and Applications Part 2 of 3 E. A. Elsayed ©2011 ASQ & Presentation Elsayed Presented live on Dec 07th, 2010http://reliabilitycalendar.org/The_Reliability_Calendar/Short_Courses/Shliability Calendar/Short Courses/Short_Courses.html
  2. 2. ASQ Reliability Division  ASQ Reliability Division Short Course Series Short Course Series The ASQ Reliability Division is pleased to  present a regular series of short courses  featuring leading international practitioners,  academics, and consultants. academics and consultants The goal is to provide a forum for the basic and  The goal is to provide a forum for the basic and continuing education of reliability  professionals.http://reliabilitycalendar.org/The_Reliability_Calendar/Short_Courses/Shliability Calendar/Short Courses/Short_Courses.html
  3. 3. Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu elsayed@rci rutgers edu Rutgers University December 7, 2010 , 1
  4. 4. Outline Part 1. Reliability Definitions Reliability Definition…Time dependent Definition Time characteristics Failure Rate Availability MTTF and MTBF Time to First Failure Mean Residual Life Conclusions C l i 2
  5. 5. Outline Part 2. Reliability Calculations1. Use f f il1 U of failure d t data2. Density functions3.3 Reliability function4. Hazard and failure rates 3
  6. 6. OutlinePart 3. Failure Time Distributions1. Constant failure rate distributions1 C t t f il t di t ib ti2. Increasing failure rate distributions3.3 Decreasing failure rate distributions4. Weibull Analysis – Why use Weibull? 4
  7. 7. Outline Part 2. Reliability Calculations1. Use f f il1 U of failure d t data a) Interval data (no censoring) b) Exact failure times are known2. Density functions3. Reliability function y4. Hazard and failure rates 5
  8. 8. Basic Calculations Suppose n0 identical units are subjected to a test. During the interval (t, t +∆t ), we observed nf (t ) failed components Let ns (t ) be the components. surviving components at time t , then we define:Failure density function f ˆ (t )  n f (t ) n 0 t ˆ (t )  nf (t ) , hFailure rate function ns (t )tReliability function Rˆ (t )  P (T  t )  n s (t ) r n0 6
  9. 9. Basic Definitions Cont’dThe unreliability F(t) is 1 F t   R t ︵ ︶ ︵ ︶Example: 200 light bulbs were tested and the failures in1000-hour intervals are Time Interval (Hours) Failures in the interval 0-1000 100 1001-2000 40 2001-3000 2001 3000 20 3001-4000 15 4001-5000 10 5001-6000 8 6001-7000 7 Total 200 7
  10. 10. Calculations Time Failure Density Hazard rate Interval I t l f (t ) x 104 h(t ) x 104 100 100Time Interval Failures 0-1000 200  103  5 .0 200  103  5 .0 (Hours) in the interval 0-1000 100 1001-2000 1001 2000 40 40  2 .0 0  4 .0 01001-2000 40 200  103 100  1032001-3000 203001-4000 15 2001 3000 2001-3000 20  1 .0 20  3 .3 34001-5000 10 200  103 60  1035001-6000 86001-7000 7 …… …….. ……Total 200 7 7 6001-7000 200  103  0 .3 5 7  103  10 8
  11. 11. Failure Density vs. Time×10-4 1 2 3 4 5 6 7 x 103 Time i h Ti in hours 9
  12. 12. Hazard Rate vs. Time×10-4 - 1 2 3 4 5 6 7 × 103 Time in Hours 10
  13. 13. Reliability CalculationsTime Interval Failures (Hours) in the Time Interval Reliability R (t ) interval 0-1000 0 1000 5/5=1.0 5/5=1 0 0-1000 100 1001-2000 40 1001-2000 / 2.0/4.0=0.5 2001-3000 2001 3000 20 3001-4000 15 4001-5000 10 2001-3000 1/3.33=0.33 5001-6000 5001 6000 8 6001-7000 7 …… ……Total 200 6001-7000 0.35/10=.035 11
  14. 14. Reliability vs. Time y1 2 3 4 5 6 7 x 103 Time in hours 12
  15. 15. Exponential DistributionDefinition (t) h (t )     0, t  0 Time f (t )   exp( t ) R (t )  exp( t )  1  F (t ) p( 13
  16. 16. Exponential Model Cont’d Cont dStatistical Properties 5 1 0 6 F a il u r e s / h r 1    M TTF  MTTF=200,000 hrs or 20 years  5 1 0  6 F a il u r e s / h r 1  V a ri a n c e  2 Standard deviation of MTTF is 200,000 hrs or 20 years M e d i a n lif e ln 1 Median life =138,626 hrs or 14  2  ︵ ︶ years 14
  17. 17. Exponential Model Cont’d Cont dStatistical Properties 5 1 0 6 F a il u r e s / h r 1    M TTF  MTTF=200,000 hrs or 20 years It is important to note that the MTTF= 1/failure MTTFis only applicable for the constant failure ratecase (failure time follow exponential distribution. ( p 15
  18. 18. Empirical Estimate of F(t) and R(t)When the exact failure times of units is known, weuse an empirical approach to estimate the reliabilitymetrics. The most common approach is the Rank ppEstimator. Order the failure time observations (failuretimes) in an ascending order: ... ... t1  t2   ti 1  ti  ti  1   t n 1  t n 16
  19. 19. Empirical Estimate of F(t) and R(t) p () ()F (ti ) is obtained by several methods y i1.1 Uniform “naive” estimator naive n i2.2 Mean rank estimator n1 . i  0 .33. Median3 M di rank estimator (B k ti t (Bernard) d) n 0 4 / i 3 / 84. Median rank estimator (Blom) n 1 4 17
  20. 20. Empirical Estimate of F(t) and R(t)Assume that we use the mean rank estimator ˆ (t )  i F i n 1 ˆ (t )  n  1  i R i ti  t  ti 1 i  0,1, 2,..., n n 1Since f (t ) is the derivative of F(t ), then F (ti 1 )  F (ti ) ˆ ˆ f (ti )  ˆ ti  ti 1  ti ti .(n  1) ( 1 f (ti )  ˆ ti .(n  1) 18
  21. 21. Empirical Estimate of F(t) and R(t) 1  (t )  ˆ ti .(n  1  i ) i ( H (ti )   ln ( R(ti ) ˆ ˆExample:Recorded failure times for a sample of 9 units areobserved at t=70 150 250 360 485 650 855 =70, 150, 250, 360, 485, 650, 855,1130, 1540. Determine F(t), R(t), f(t),  t ,H(t) ︵ ︶ 19
  22. 22. Calculationsi t (i) t(i+1) F=i/10 R=(10-i)/10 F i/10 R (10 i)/10 f 0.1/t f=0.1/t  1/(t.(10 i)) =1/(t.(10‐i)) H(t)0 0 70 0 1 0.001429 0.001429 01 70 150 0.1 0.9 0.001250 0.001389 0.105360522 150 250 0.2 0.8 0.001000 0.001250 0.223143553 250 360 0.3 0.7 0.000909 0.001299 0.356674944 360 485 0.4 0.6 0.000800 0.001333 0.510825625 485 650 0.5 0.5 0.000606 0.001212 0.693147186 650 855 0.6 0.4 0.000488 0.001220 0.916290737 855 1130 0.7 0.3 0.000364 0.001212 1.20397288 1130 1540 0.8 0.2 0.000244 0.001220 1.609437919 1540 - 0.9 0.1 2.30258509 20
  23. 23. Reliability Function y 1.2 1ReliabilityR li bilit 0.8 0.6 0.4 0.2 0 0 200 400 600 800 1000 1200 Time 21
  24. 24. Probability Density Function 0.001600 0.001400 0.001200 0.001000Density 0.000800Function 0.000600 0.000400 0.000200 0.000000 0 000000 0 200 400 600 800 1000 1200 Time 22
  25. 25. Constant Failure Rate 0.001900 0.001700 0.001500 0 001500 0.001300 0.001100Failure Rate 0.000900 0.000700 0.000500 0.000300 0.000100 0 200 400 600 800 1000 1200 Time 23
  26. 26. Exponential Distribution: Another ExampleGiven failure d tGi f il data:Plot the hazard rate, if constant then use theexponential distribution with f (t), R (t) and h (t) asdefined before.We use a software to demonstrate these steps. 24
  27. 27. Input Data 25
  28. 28. Plot of the Data 26
  29. 29. Exponential Fit 27
  30. 30. Exponential Analysis
  31. 31. SummaryIn this part, we presented the three most importantrelationships in reliability engineering.We estimated obtained estimate functions for failurerate, reliability and failure time. We obtained thesefunction for interval time and exact failure times times. 29

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