1. AEM 336: Reliability & Sampling Prediction & Modeling

2. Outline System Reliability Series System Reliability Parallel System Reliability Series-Parallel System Reliability 2

3. Review Static Systems: Systems where failure of one component has NO effect on the probability of any other component failing Dynamic Systems: Components are dependent; failure of one component will affect the probability of failure of another component

4. Review Series System: A complex system of independent units connected together (interrelated) such that the entire system will fail if any one the units fail.

5. Review Parallel System: components are connected in such a way that a redundant, or standby, part can take over the function of a failed part to save the system.

6. Review The Product Rule: if a system has n components, each with a reliability P 1, P 2,…,P n, the reliability of the system (R s ) is R s = P 1 * P 2 * … * P n, where, R s = Prob. Of system functioning as intended P n = prob. Components functioning as intended Example: 3 components – A(.92), B(.95), & C(.96) R s = A * B * C = (.92) * (.95) * (.96) =.839

7. Review

8. Unreliability (U): defined as 1-Reliability U = 1 - P c for a component U = 1 - (P 1 * P 2 * … * P n ) Or U = 1 – (P c ) n Series Systems

9. Review Series System Reliability using Failure Rate (λ): R s = P 1 * P 2 * …* P n R s = e -λ 1 T * e -λ 2 T *… e -λ n T R s = e -T(λ 1 + λ 2 + … + λ n ) Where:λ = failure rate of component T = x-hour reliability of the system

10. Review Example: Failure Rates:λ 1 =.002 λ 2 =.001 λ 3 =.0025 λ 4 =.0005 ∑ =.0060/ T = 100 R s = e -T(λ1 + λ2 + … + λn) = e -100(.006) =.5488 or R s = e -100(.002) * e -100(.001) * e -100(.0025) * e -100(.0005) = R s =.8187 *.9048 *.7788 *.9512 =.5488 Calculator Tip: eˆ ((-100).002)

11. Parallel Reliability The reliability of a parallel (or redundant) system MUST be determined by 1 st calculating the probability that the system or part WILL fail (unreliability). R s = 1 – (U 1 * U 2 * …* U n ) Where: U x is the unreliability of a component AND R s = 1 – (U c ) n = 1 – (1 – P c ) n

12. Parallel Reliability Example: R A =.92; U A = 1 – P A =.08 R B =.95; U B = 1 – P B =.05 R C =.96; U C = 1 – P C =.04 R s = 1 – (1 – P c ) = 1 – (U A * U B * U C ) = = 1 – (.08 *.05 *.04) =.9998 SERIESvs.PARALLEL R A *R B *R C vs.1-(U A *U B *U C ) 83.9%vs.99.98%

13. Parallel vs. Series P c =.70 2 vs. 3 vs. 4 components at P c =.70 2 3 4.70 =.91 =.973 =.992 Parallel Series 2 3 4 (.70) 2 =.49 (.70) 3 =.34 (.70) 4 =.2401

14. Parallel vs. Series

15. Series-Parallel Systems U C =.914 (A) R A =.358.95 20.93 20.96 60 (B) R B =.234 (C) R C =.086 U B =.766 Series Part => (R A )(R BC ) Parallel Part =>(R BC ) Must find this 1 st ! ***Find unreliability of B & C ***

16. Series-Parallel Systems 1)R BC = 1-U BC = 1-U B U C = 1-(.766)(.914) = 1-.700 =.300 2)R A =.358 3)R S = (R A )(R BC ) = (.358)(.300) =.107.107

17. High vs. Low Level Redundancy Parallel Systems Parallel Components.7 High Level – Entire System in Parallel R S = 1 – {[1 – (.7*.7*.7)] * [1- (.7*.7*.7)]} =.5684 Series

18. High vs. Low Level Redundancy.7 Low Level – Component Level CAN BE REPLACED R S = [1 – (1 -.70)(1 -.70) 3 =.7536 Unreliability The Unreliability of each of the 3 Parallel Parts of the System

19. Dynamic Systems Series Dynamic Systems: Calculate the failure rate of system by summing the reciprocals of the means (MTBF) of each component Example – 100-hr Reliability failures/hr

20. Dynamic Systems failures/hr R S = e -λT = e -.00116(100) =.8905

21. Parallel Dynamic Systems 2 Types 1)Manual Switching 2)Electronic Switching

22. Assignment 1)Exam on Chapter VI Modeling & Prediction, Tuesday, November 2 2)Assignment: Worksheet on Prediction & Modeling Due November 2