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14: Analysis of Test Data Testing leads to failure, and failure leads to understanding."

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Definitions Failure rate = the number of failures per million hours of operation λ =1/MTBF Failure: the state of inability of an item to perform its required function Reliability = the probability that an item will perform a required function, under specified conditions, for a specified period of time, at a desired confidence level. (MTBF or Failure rate, λ) Minimum life = time to first failure

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Definitions, Continued Confidence level = probability that a given statement is correct Confidence limits = the extremes of a confidence interval within which the unknown has a designated probability of being included The minimum life of a device = the time of occurrence of the first failure. Pareto analysis = plot of individual failures versus the frequency of the failures.

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Example 1: MTBF An EEG machine has a MTBF of 4380 hours. What is the failure rate? λ = 1/MTBF λ = 1/4380 failures per hour λ = 0.000228 failures per hour λ = 228 failures per million hours

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Example 2: MTBF 10 power supplies are put on test, to be terminated after each has completed 1000 hours of operation. Two power supplies fail, one at 420 hours and the other at 665 hours. What is the failure rate of the power supplies? Eight units completed 1000 hours. Total test time = 8(1000) + 420 + 665 = 9085 hours λ = number of failures/total test time λ = 2/9085 = 0.000220 failures per hour λ = 220 failures per million hours

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General MTBF Tests in practice: Time terminated, failed parts replaced Time terminated, no replacement (last…) Failure terminated, failed parts replaced Failure terminated, no replacement (easy) No failures observed during the test …SOME EXAMPLES FOLLOW

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Time terminated, failed parts replaced: MTBF = N(td)/r where N = number of units tested td = test duration r = number of failures

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…continued The performance of ten pressure monitors is monitored while operating for a period of 1200 hours. The test results are listed below. Every failed unit is replaced immediately. What is the MTBF? Unit NumberTime of Failure (hours)#failures 1650 1 2420 1 3130 and 725 2 4585 1 5630 and 950 2 6390 1 7No Failure 0 8880 1 9No Failure 0 10220 and 675 2 11

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finished … N = 10 sets of tests r = 11 failues td = 1200 hours (test duration) MTBF = N(td)/r = 10(1200)/11 = 1091 hours

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Failure Terminated, Failed Parts Replaced Six TENS units were placed on test until all units failed, the last occurring at 850 hours. The test results are listed below. Every failed unit, except the last one, is replaced immediately. What is the MTBF? Unit NumberTime of Failure (hours) Failures 11301 28501 3120 and 6552 44401 57251 65801 MTBF = N(td)/r = 6(850)/7 = 729 hours

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No failures observed during the test For the case where no failures are observed, an MTBF value cannot be calculated. A lower one sided confidence limit must be calculated and the MTBF stated to be greater than that value. ml = 2(Ta)/Χ 2 α;2 where ml = lower one sided confidence limit Ta = total test time Χ 2 α;2 = the chi square value from the table in Appendix 1, where α is the risk level and 2 is the # of degrees of freedom

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continued … 10 ventilators are tested for 1000 hours without failure. What is the MTBF at a 90% confidence level? N = 10 test units td = 1000 = test duration r = 0 = failures 1 % = 0.90 = confidence level % = 0.10 = risk level Ta = N(td) = 10(1000) = 10000 = total test time ml = 2(Ta)/Χ 2 α;2 = 2(10000)/Χ 2 10;2 = 20000/4.605 = 4343 hours = lower 1 sided confidence limit… We can then state that the MTBF > 4343 hours, with 90% confidence.

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Reliability Reliability = exp( λt) = exp ( t/MTBF) Example: test time =3200 hours, λ = 220 failures/ million hours Reliability = exp(-3200*220/1000000) = exp(-.704) =.495 or ~ half will fail in 3200 hours….

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Confidence Limits Time Terminated Confidence Limits mL = 2(Ta)/Χ 2 α/2;2r+2 where mL = lower confidence limit Ta = total test time Χ 2 α/2;2r+2 = Chi square value from Appendix 1 for α risk level α and 2r+2 degrees of freedom mU = 2(Ta)/Χ 2 1-α/2;2r

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Confidence Limits, continued mL = 2(Ta)/Χ 2 α/2;2r and mU = 2(Ta)/Χ 2 1-α/2;2r … left to the reader -

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Pareto Analysis

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Field Data Plots:

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Final comments The data examples given are for estimate purposes only, most of the reliability estimates will likely be slightly on the low side, which can be a good thing…

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1 Math 10 Part 5 Slides Confidence Intervals © Maurice Geraghty, 2009.

1 Math 10 Part 5 Slides Confidence Intervals © Maurice Geraghty, 2009.

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