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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