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HW9_reliability_soln2

# HW9_reliability_soln2 - 16.881 Robust System Design...

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16.881 – Robust System Design Solution to Homework #9 Robust Design and Reliabilty Due Date: Monday, 20 July Objectives: Understand basic concepts of reliability engineering Understand the relationship of robust design to reliability Assignment 1) Given that for small time intervals t λ ( t ) t = Pr(The system will fail within the period t to t + ∆ t The product survives to time t ) R ( t ) = Pr(The product survives to time t ) dF ( t ) = Pr( The s ystem will fail within the period t to t + dt ) R ( t ) + F ( t ) = 1 Prove that t R ( t ) = e 0 ( ξ ) d 1 of 2

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2 of 2 Solution to #1 Consider the definition of conditional probability The first two equations given in the problem statement fill out the right hand side of the equation. lso, in order to fail within the time interval t + t , a product must survive to time t , the third equation in the problem statement completes the left hand side of the definition. The fourth equation given in the problem statement is Taking the derivative and rearranging yields Combining this with (1) gives For small time intervals t is equivalent to dt , so Changing t into ξ gives Integrating both sides from 0 to t gives Since R(0) is always 1 and ln(1) is zero Which is the result we wanted to prove. ) Pr( ) Pr( ) & Pr( B B A B A = = t d e t R 0 ) ( ) ( λ = t d R t R 0 ) ( )) 0 ( ln( )) ( ln( = t t d R dR 0 0 ) ( ) ( ) ( t t t R t dR = ) ( ) ( ) ( 1 ) ( ) ( = + t F t R dt t t R t dR ) ( ) ( ) ( = d R dR ) ( ) ( ) ( = A Therefore
3 of 2 2) hat form will the reliability curve R ( t ) take if the early failure region is removed through “burn-in” and the wear-out region is avoided by retiring the product before the wear-out period? How is this form of R ( t ) related to a Weibull distribution? Solution to #2 The following slide appeared in the class notes If the early failure period and the wear out period are removed, then λ ( t ) is constant.

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HW9_reliability_soln2 - 16.881 Robust System Design...

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