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AEM 336: Reliability & Sampling

AEM 336: Reliability & Sampling. Prediction & Modeling. Outline. System Reliability Series System Reliability Parallel System Reliability Series-Parallel System Reliability. Review.

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AEM 336: Reliability & Sampling

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  1. AEM 336: Reliability & Sampling Prediction & Modeling

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

  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 P1, P2,…,Pn, the reliability of the system (Rs) is Rs = P1 * P2 * … * Pn, where, Rs = Prob. Of system functioning as intended Pn = prob. Components functioning as intended Example: 3 components – A(.92), B(.95), & C(.96) Rs = A * B * C = (.92) * (.95) * (.96) = .839

  7. Review Equivalent Component Reliability: Rs = Pc * Pc * Pc = (Pc)n Establishing Equivalency for non-equivalent reliabilities: A(.92), B(.95), & C(.96) = = .9432

  8. Review Unreliability (U): defined as 1-Reliability U = 1 - Pc for a component U = 1 - (P1 * P2 * … * Pn) Or U = 1 – (Pc)n Series Systems

  9. Review Series System Reliability using Failure Rate (λ): Rs = P1 * P2 * …* Pn Rs = e-λ1T * e-λ2T *… e-λnT Rs = e-T(λ1 + λ2 + … + λn) Where: λ = failure rate of component T = x-hour reliability of the system

  10. Review Example: Failure Rates: λ1 = .002 λ2= .001 λ3= .0025 λ4= .0005 ∑ = .0060/ T = 100 Rs= e-T(λ1 + λ2 + … + λn)= e-100(.006) = .5488 or Rs = e-100(.002) * e-100(.001) * e-100(.0025) * e-100(.0005) = Rs = .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 1st calculating the probability that the system or part WILL fail (unreliability). Rs = 1 – (U1 * U2 * …* Un) Where: Ux is the unreliability of a component AND Rs = 1 – (Uc)n = 1 – (1 – Pc)n

  12. Parallel Reliability Example: RA = .92; UA = 1 – PA = .08 RB= .95; UB= 1 – PB= .05 RC= .96; UC= 1 – PC= .04 Rs= 1 – (1 – Pc) = 1 – (UA * UB * UC) = = 1 – (.08 * .05 * .04) = .9998 SERIES vs. PARALLEL RA*RB*RC vs. 1-(UA*UB*UC) 83.9% vs. 99.98%

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

  14. Parallel vs. Series

  15. Series-Parallel Systems (B) RB = .234 .9320 (A) RA = .358 UB = .766 .9520 (C) RC = .086 .9660 UC = .914 Series Part => (RA)(RBC) Parallel Part => (RBC) Must find this 1st! ***Find unreliability of B & C ***

  16. Series-Parallel Systems .107 RBC = 1-UBC = 1-UBUC = 1-(.766)(.914) = 1-.700 = .300 RA = .358 RS = (RA)(RBC) = (.358)(.300) = .107

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

  18. High vs. Low Level Redundancy Low Level – Component Level CAN BE REPLACED .7 .7 .7 .7 .7 .7 RS = [1 – (1 - .70)(1 - .70)3 = .7536 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 RS = e-λT = e-.00116(100) = .8905

  21. Parallel Dynamic Systems 2 Types • Manual Switching • Electronic Switching

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

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