Life Data Analysis using the Weibull distribution

Transcription

1 RELIABILITY ENGINEERING Life Data Analysis using the Weibull distribution PLOT Seminar October 2008 Ing. Ronald Schop Weibull: Reliability Engineering

2 Content Why Reliability Weibull Statistics Weibull Shape Parameter β Cases: 1 - Steering Link Failures 2 - Is the new Design better? 2

3 Customer & Supplier Safety Pricing Availability Reliability Effects: Warranty Costs R&M Costs Cost of Ownership Residual Value Business Performance BOTH CUSTOMER & SUPPLIER Image Re-buying 3

4 Customer Costs Life Cycle Costs Customer Costs Current Warranty Repair & Maintenance Total Costs Corrective New Price Costs Life Cycle Costs (EURO) Preventive Current Re-sale Price 4

5 Statistics Real Life Bath Tub Curve 2500 USA 1990 Deaths Per D eaths/ Age Intervals (years) 5

6 Statistics Simplified Model: Bath Tub curve Failures / System Early Life Useful Life Late Life Infant Mortality Random Wear out System Life (years) 6

7 Statistics Worldwide WEIBULL Distribution is used for: Life Data (Warranty, Field-test & Rig Testing data) Engineering Analyses 1 - Extremely small sample size. 2 - Dirty data sets. 3 - All failure modes fit in Weibull. 4 - Focus on low % failure rate. 5 - Both failures and suspensions. 6 - Managers can read the plot! 7

8 WEIBULL 2-parameter 2 formula: Statistics F( t) = 1 exp η t β Shape Parameter : β Characteristic Life : η Failure Percentage : F(t) NOTE: Only one failuremode per Weibull 8

9 Regression Method Weibull Plot Regression Line Correlation Shape Parameter Sample Number Cumulative % Failures (log-log) Characteristic Life Mileage / Time (Log scale) 9

10 Statistics 10

11 Statistics B(10) on Weibull plot 11

12 The Weibull SHAPE Parameter: β Clues to Failure - Modes η rise run β = rise/run 12

13 Interpreting the Plot Infant Mortality - Beta < 1 Solid State Electronics Beta = good stress screening Beta = bad stress screening Production Problems, quality control, misassemble,... Overhaul Problems Incorrect Maintenance Changes Improves with age like Whisky Overhauling is dumb...inappropriate... 13

14 Weibull Distribution 0,5 0,4 0,3 Beta = 0,5 Infant Mortality Beta < 1 0,2 0,

15 Random Failures Beta = Exponential Maintenance errors, human errors Random failures are independent of age. An old part is as good as a new part. Ageless... Mother Nature: lightning strikes, foreign object damage, woodpecker attacks on power poles Mixtures of more than 3 failure modes (assuming different betas) 15

16 Weibull Distribution 0,5 0,4 Random Failures Beta = 1...Exponential 0,3 Beta = 1 0,2 0,

17 Early Wearout 1.0 <Beta<4.0 Most Mechanical Failure Modes... Low Cycle Fatigue, Bearing Failures Corrosion, Erosion Overhauls, parts replacement may be cost effective. Approximate replacement period may be read off plot. These are often Design Deficiencies......Unpleasant surprises... 17

18 Weibull Distribution 0,5 0,4 0,3 Beta = 3,25 Early Wearout 1.0 <Beta<4.0 0,2 0,

19 Old Age Rapid Wearout, Beta > 4.0 Steep Betas within the Design Life cause great concern, & panic... Panic because the entire fleet will fail quickly as they age. Beyond the Design Life, they are joy! Most steep Weibulls have a safe, benign period within which the risk is very low. The steeper beta, the smaller the variation in the times-to to-failure, and the more predictable the results. 19

20 Weibull Distribution 0,5 0,4 0,3 Beta = 5 Old Age Rapid Wearout Beta > 4.0 0,2 0,

21 Weibull Fits a Wide Range of Shapes Probability Density 0,5 0,4 0,3 0,2 0,1 Bath Tub Can be modelled Weibull: Probability Density f(t) / β Beta = 0,5 Beta = 1 Beta = 3,25 Beta = Time 21

22 Weibull Distributions for a Complex System Design Life Failures / System Infant Mortality Random Early Wear out Old age Rapid Wear out β < 1 β = 1 1 < β < 4 β > 4 22

23 Cases 1 Steering Link Failures 2 Is the new Design better? With Permission from Dr. R. B. Abernethy 23

24 Forecasting Steering Link Failures 10 steering links on 18 Wheelers have suddenly failed across the USA. Safety, high losses, and adverse pr has management concerned. There are 10,010 units installed. Management wants to know: 1. How many failures will there be? When? 2. What will this problem cost the company? 3. What corrective action can be taken? 24

25 With 10 failures the Weibull B1 life is 18,000 miles; this is unacceptable. 25

26 How many failures? When? Next year we predict 576 failures with an upper 90% bound of 872. The next failure is expected in six days. 26

27 2. What will this problem cost? First year loss is predicted to be just over a million dollars but b might be as high as $1.6M. 27

28 3. What is the best corrective action? Engineering says the root cause is the recent addition of power steering kits overstressing the links. It will take 2 years to redesign, test and produce 10,000 stronger links for retrofit. An immediate forced retrofit would be too painful and embarrassing as we do not have enough spare steering links to do it. Many trucks would be down for months. Opportunistic maintenance may be the answer. 28

29 3. What is the best corrective action? The optimal replacement interval for minimum cost is at 22, 000 miles, based on the planned replacement cost of $95 and the unplanned failure cost of $

30 3. What is the best corrective action? A convenience retrofit at the 20,000 mile oil change would be close to optimal, less painful for our customers, and it would give us time to produce new links and solve the problem. Risk Reduction: 12 months from 576 to 187 failures, 24 months from 3470 to 397 failures. 2 Year Cost Analysis: No correction action = 3470 x $1850= $6.4M 20K replace = 397 x $1850 +( ) x $95 = $1.6M..($4.8M Cost Avoidance) The sooner the new links are available the sooner the problem is fixed. 3 shift overtime is required. 30

31 3. What is the best corrective action? Answer: Retrofit at 20,000 miles. Failures Avoided 31

32 4. Liabilities? Engineering predicts that the links fail at low speed, usually in reverse in a turn. So far this has been true. No one has been injured to date. Our liability estimates are based on these assumed low cost failures. If the assumption is wrong and the links fail at high speed, liabilities would be increased. 32

33 Summary / Conclusions Installation of the redesigned replacement links will end the problem. The redesign should be initiated with priority and 3 shift overtime. There is some indication that only one type of power steering kit produces the failures. The power steering vendor should be contacted for a coordinated response to this problem. This case study is based on actual data from two manufacturers that experienced this problem, one with trucks, one with cars. In both cases Weibull analysis provided accurate predictions and analysis for management action. 33

34 Case 2 - Is The New Design Better? Customers say our product is not perfect. perfect. It fails rather often! We have tested 8 of the new design and 8 of the old units. Is the new design significantly better than the old design or are the differences just statistical noise, and the new product is no better? Here are the test results: 34

35 Is There Real Improvement? 35

36 The Likelihood Ratio and Contour Tests Provide the Answer The Likelihood Ratio Test says the New Design is not significantly better at 90% but would be at a lower confidence level. The Likelihood Contour Test compliments the ratio test and provides a picture, a plot, of the likelihood contours on the next slide. The contour test confirms the Likelihood Ratio Tests results. 36

37 Likelihood Contours Overlap 37

38 Summary- Is There Real Improvement? We do not have enough data to show the new design is better. More testing may prove it is significantly better. More then two sets may be compared with the likelihood methods. There are many applications for this technology, comparing vendor A with B, comparing different materials, different environments as well as new design versus old design. Statistical confidence is provided. 38

39 Summary Weibull distribution will fit all failure-modes. Simple plot managers can read and understand. Even with very little data analysis can be done. Focus on Engineering s s interesting area. (B1 B5) New Methods are developed for extensive analysis like: Batch Analysis. Failure forecasting. Optimal replacement point. Test requirements; ZERO failures, Sudden Death, and more. Combining different test-lengths and sample seizes. Confidence intervals. Crow-AMSAA System analysis (multiple modes). 39

40 Handbooks and Software All methods presented, and much more is documented in The New Weibull Handbook by Dr. Robert Abernethy All analysis are created with the WinSmithWeibull / Visual Software Handbook + tutorial + Software is available as a self-study study packet. Public Weibull Reliability Engineering Workshops are given in the Netherlands (PATO) and the USA. Private Workshops are delivered in the EU by Weibull.nl The workshop can be tailored to the customers needs, are less expensive and are delivered on site. All workshops INCLUDE the Handbook and the Software plus full personal license for every student, and will make a novice to a full Weibull Engineer. Detailed info on 40

41 Thank you for your attention Ing. Ronald Schop Weibull: Reliability Engineering 41