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## RELIABILITY ENGINEERING SURVIVABILITY ENGINEERING

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**RELIABILITYENGINEERINGSURVIVABILITY**ENGINEERING قدرة الإستمرار في أداء المهام Prof. Dr. Ahmed Farouk Abdul Moneim**A System is considered SUCCESSFUL and**OPERATIONALLY FEASIBLE If and only if it is: RELIABLE 2) MAINTAINABLE 3) MANABLE 4) SUPPORTABLE Prof. Dr. Ahmed Farouk Abdul Moneim**What is RELIABILITY**RELIABILTY is the PROBABILITYthat a System performs Its FUNCTIONSas INTENDED for a SPECIFIED PERIOD OF TIME in a CERTAIN ENVIRONMENT ENVIRONMENT : Installation Weather, Altitude, Terrain PreventiveMaintenance Users Transportation Storage Time may be measured Chronically or in Distances, Revolutions,… Prof. Dr. Ahmed Farouk Abdul Moneim**WHY DO WE STUDY RELIABILITY ENGINEERING**1)To be able to evaluate RELIABILITY and hence we could CONTROL it 2) To Classify the system components as regards to their CRITICALITY 3) To decide, in the design stage, the size of REDUNDANCY 4) To be able to deploy RELIABILITY as a QUALITY characteristic into the different components of the system 5) To decide the Quantity of SPARE PARTS required by the system to SURVIVE a predefined period of time 6) To be able to design proper WARRANTY Policies 7) To decide proper MAINTENANCE Policies Prof. Dr. Ahmed Farouk Abdul Moneim**FUNDAMENTAL EXPRESSIONS**OF RELIABILITY Prof. Dr. Ahmed Farouk Abdul Moneim**Given Nelectric motors at the beginning of a survival test**(t = 0). number of failed motors up to time t The Ratio of number of FAILED Components NFto the TOTAL Number N is Is the FAILURE function or the PROBABILITY of Failure of a motor in the time period from 0 to time t Is Rate of Change of the FAILURE function with respect to time It is called FAILURE PROBABILITY DENSITY FUNCTION number of Survived motors up to time t The Ratio of number of SURVIVED Components NSto the TOTAL Number N is Prof. Dr. Ahmed Farouk Abdul Moneim**The Event of FAILURE and Event of SURVIVAL of a motor are**Mutually Exclusive EVENTS.Therefore,the SUM of their probabilities equals 1 is the SURVIVAL function (RELIABILITY) or the PROBABILITY of SURVIVAL of a motor in the time period from 0 to time t Consider again the Failure Probability Density Function: then This is the Unconditional Failure frequency related to the Initial Number of motors under test If on the other hand the Failure Frequency is related to the number of motors survived (not failed) at time t, we obtain the Conditional frequency as follows: Multiply denominator and numerator by N, we find Therefore, Is the FAILURE or HAZARD RATE Prof. Dr. Ahmed Farouk Abdul Moneim**A life test is conducted on 200 Transistors for a period of**20 months . The test results are given: Prof. Dr. Ahmed Farouk Abdul Moneim**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, … Prof. Dr. Ahmed Farouk Abdul Moneim**SUMMARY**Three Basic Functions: Failure Probability Density Function Failure PDF Failure Rate Failure Probability Function Survival Probability Function or RELIABILITY Prof. Dr. Ahmed Farouk Abdul Moneim**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, Prof. Dr. Ahmed Farouk Abdul Moneim**GENERAL FORMULA FOR MEAN RESIDUAL LIFE**f (t) t 0 TO MRL Prof. Dr. Ahmed Farouk Abdul Moneim**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 Prof. Dr. Ahmed Farouk Abdul Moneim**Case 2**COMPONENTS WITH INCREASING FAILURE RATE Example Prof. Dr. Ahmed Farouk Abdul Moneim**Case 3**COMPONENTS WITH DECREASING FAILURE RATE Example As a general rule Prof. Dr. Ahmed Farouk Abdul Moneim**BATH TUB DISTRIBUTIONandPRODUCT LIFECYCLE**Prof. Dr. Ahmed Farouk Abdul Moneim**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 Prof. Dr. Ahmed Farouk Abdul Moneim**WEIBULL DISTRIBUTION**• It is a Universal distribution • It fits all trends of Failure Rate by changing a single parameter β Prof. Dr. Ahmed Farouk Abdul Moneim**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) Prof. Dr. Ahmed Farouk Abdul Moneim**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 Prof. Dr. Ahmed Farouk Abdul Moneim**Mean Time To Failure**Prof. Dr. Ahmed Farouk Abdul Moneim**Variance of Time To Failure**Prof. Dr. Ahmed Farouk Abdul Moneim**SUMMARY OF WEIBULL Distribution**Prof. Dr. Ahmed Farouk Abdul Moneim**SOLVED EXAMPLES**Prof. Dr. Ahmed Farouk Abdul Moneim**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 Prof. Dr. Ahmed Farouk Abdul Moneim**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 Prof. Dr. Ahmed Farouk Abdul Moneim**(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 Prof. Dr. Ahmed Farouk Abdul Moneim**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 Prof. Dr. Ahmed Farouk Abdul Moneim**TRIAL & ERROR**1 2 2 Required Value =1.117 1.273 3 1.1321 1.1176 3.2 Then β = 3.2 Prof. Dr. Ahmed Farouk Abdul Moneim**Example 2 Continued**Then Notice the difference between Exponential and Weibull Distributions Prof. Dr. Ahmed Farouk Abdul Moneim**f(t)**F = 0.06 545.16 Time 72 F = 0 .06 Prof. Dr. Ahmed Farouk Abdul Moneim