AEM 336: Reliability & Sampling

Prediction & Modeling

Outline

• System Reliability• Series System Reliability• Parallel System Reliability• Series-Parallel System Reliability

2

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

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.

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.

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

Review

Equivalent Component Reliability:

Rs = Pc * Pc * Pc = (Pc)n

Establishing Equivalency for non-equivalent reliabilities:

A(.92), B(.95), & C(.96)

= = .9432

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

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

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)

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

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%

Parallel vs. Series

2 vs. 3 vs. 4 components at Pc = .70

2

3

4

.70

.70

.70

.70

.70

.70

.70

.70

.70

= .91

= .973

= .992

Parallel Series

2

3

4

(.70)2 = .49

(.70)3 = .34

(.70)4 = .2401

Parallel vs. Series

1 2 3 40.00

0.20

0.40

0.60

0.80

1.00

1.20

0.70

0.91

0.97 0.99

0.49

0.34

0.24

Parallel

Series

Series-Parallel Systems

UC = .914

(A) RA = .358

.9520

.9320

.9660

(B) RB = .234

(C) RC = .086

UB = .766

Series Part => (RA)(RBC)Parallel Part => (RBC) Must find this 1st!

***Find unreliability of B & C ***

Series-Parallel Systems

1) RBC = 1-UBC = 1-UBUC = 1-(.766)(.914) = 1-.700 = .300

2) RA = .3583) RS = (RA)(RBC) = (.358)(.300) = .107

.107

High vs. Low Level Redundancy

Parallel Systems

Parallel Components

.7 .7 .7

.7 .7 .7

High Level – Entire System in Parallel

RS = 1 – {[1 – (.7*.7*.7)] * [1- (.7*.7*.7)]} = .5684

Series

High vs. Low Level Redundancy

.7 .7 .7

.7 .7 .7

Low Level – Component Level CAN BE REPLACED

RS = [1 – (1 - .70)(1 - .70)3 = .7536

The Unreliability of each of the 3 Parallel Parts of the System

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

Subsystem MTBF Reciprocals

1 5000 1/5000 = .0004

2 6000 1/6000 = .00016

3 4500 1/4500 = .0002

4 2200 1/2200 = .00045

5 8650 1/8650 = .000115

∑ .00116failures/hr

Dynamic Systems

Subsystem MTBF Reciprocals

1 5000 1/5000 = .0004

2 6000 1/6000 = .00016

3 4500 1/4500 = .0002

4 2200 1/2200 = .00045

5 8650 1/8650 = .000115

∑ .00116failures/hr

RS = e-λT = e-.00116(100) = .8905

Parallel Dynamic Systems

2 Types

1) Manual Switching

2) Electronic Switching

Assignment

1) Exam on Chapter VI Modeling & Prediction, Tuesday, November 2

2) Assignment: Worksheet on Prediction & Modeling Due November 2