1 / 26

TIME TO FAILURE AND ITS PROBABILITY DISTRIBUTIONS

TIME TO FAILURE AND ITS PROBABILITY DISTRIBUTIONS. Lectures 3 and 4. Failure Data and Related Functions. 200 bulbs have been subjected to a Life Test for 700 hours:. f(t ). F(t) Failure Probability = N F / N.

aaralyn
Download Presentation

TIME TO FAILURE AND ITS PROBABILITY DISTRIBUTIONS

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. TIME TO FAILURE AND ITS PROBABILITY DISTRIBUTIONS Lectures 3 and 4

  2. Failure Data and Related Functions 200 bulbs have been subjected to a Life Test for 700 hours: f(t) F(t) Failure Probability = NF / N f (t) Failure Probability Density Function = Д F/Д t = Д NF / (N Д t) R(t) Reliability = 1 – F(t) = 1 - NF / N =(N – NF) / N = NS / N h (t) Failure Rate = Д NF / (NSД t) 0.35 0..2 0.15 0.125 0.1 0.04 t

  3. F(t) Failure Probability = NF / N f (t) Failure Probability Density Function = Д F/Д t = Д NF / (N Д t) R(t) Reliability = 1 – F(t) = 1 - NF / N =(N – NF) / N = NS / N h (t) Failure Rate = Д NF / (NSД t) THEN AN IMPORTANT SPECIAL CASE THE FAILURE RATE IS CONSTANT As examples: All transistors, Diodes, Micro Switches, …

  4. SUMMARY Three Basic Functions: Failure Probability Density Function Failure PDF Failure Rate Failure Probability Function Survival Probability Function or RELIABILITY

  5. GENERAL FORMULA FOR MEAN TIME TO FAILURE The Time To Failure TTF of any component or system is a Random Variable with probability density function f (t) Accordingly, the Mean Time To Failure MTTF can be evaluated as follows: f (t) But t Substituting, MTTF Integrating by parts we get, Then finally,

  6. GENERAL FORMULA FOR MEAN RESIDUAL LIFE f (t) t 0 TO MRL

  7. COMPONENTS WITH CONSTANT FAILURE RATE Case 1 We have already shown that In case of CONSTANT Failure Rate Therefore, the probability density function f (t) in case of Constant Failure Rate will be Is known asEXPONENTIAL Distribution This distribution f(t) In this case t

  8. Case 2 COMPONENTS WITH INCREASING FAILURE RATE Example

  9. Case 3 COMPONENTS WITH DECREASING FAILURE RATE Example As a general rule

  10. BATH TUB DISTRIBUTIONandPRODUCT LIFECYCLE

  11. Hazard rate h (t) Infant Mortality Useful Life Wear out Burn-in IFR DFR CFR time Start of Life Start of commissioning Start of Deterioration End of Life

  12. WEIBULL DISTRIBUTION • It is a Universal distribution • It fits all trends of Failure Rate by changing a single parameter β

  13. Derivation of WEIBULL distribution Consider the following Failure Rate as function of time Failures / time As m=0, we get the case of Constant Failure Rate (CFR) As m>0, we get case of Increasing Failure Rate (IFR) As m<0, we get case of Decreasing failure rate (IFR) k is Constant Consider a Characteristic Time η as the time interval during which the mean number of failures is ONE Put m+1 = β, then we find h(t) as follows: As β=1, we get the case of Constant Failure Rate (CFR) As β> 1, we get case of Increasing Failure Rate (IFR) As β< 1, we get case of Decreasing failure rate (DFR)

  14. Failure Rate h (t) β is the shape factor η is the Characteristic Time When β=1 the failure rate becomes constant and Weibull distribution turns to be Exponential Reliability Probability Density Function PDF f (t) PDF = t

  15. Mean Time To Failure

  16. Variance of Time To Failure

  17. SUMMARY OF WEIBULL Distribution

  18. SOLVED EXAMPLES

  19. Example 1 The pdf of time to failure of a product in years is given • Find the hazard rate as a function of time. • Find MTTF • If the product survived up to 3 years, find the mean residual life MRL • Find the design life for a reliability of 0.9 ___________________________________________________ 1) Find the Reliability R(t) MRL f(t) 0 3 10 t 2) Find the Hazard rate h(t) 3) Find MTTF Since h(t) is Increasing function with time Then the product is in deteriorating h(t) 4) Find MRL t 10 5) Find Design Life at R=0.9

  20. Example 2 Ten V belts are put on a life test and time to failure are recorded. After 1600 hours six of them failed at the following times: 476, 529, 550, 921, 1247 and 1522. Calculate MTTF, reliability of a belt to survive to 900 hours. Design life corresponding to Reliability of 0.94 based on: a) CFR b) Weibull distribution MTTF=(476+529+550+921+1247+1522+4*1600)/10 =1164.495 hrs Based on CFR CONTINUED

  21. (1) Based on Weibull Variance=σ2=[(476-1164.945)2+(529-1164.945)2+(550-1164.945) 2+(921-1164.945)2 +(1247-1164.945)2+(1522-1164.945)2+4*(1600-1164.945)2)/(10-1) =158719.1 (2) Square (1) and divide (2)/(1), we find the following equation in ONE Unknownβ. CONTINUED

  22. The above equation is TRANSCENDENTAL and could be solved by Trial and Error Method or by Excel GOAL SEEK Using Excel GOAL SEEK, we find: β = 3.2 η will be found from (1) as follows: CONTINUED

  23. TRIAL & ERROR 1 2 2 Required Value =1.117 1.273 3 1.1321 1.1176 3.2 Then β = 3.2

  24. Example 2 Continued Then Notice the difference between Exponential and Weibull Distributions

  25. f(t) F = 0.06 545.16 Time 72 F = 0 .06

More Related