reliability engineering.pptx
TRANSCRIPT
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Reliability Engineering
Ashraf Khalil
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Billington and Allen, “Reliability Evaluation of Engineering Systems”, 1983.
David J Smith, “Reliability, Maintainability and Risk”, Butterworth-Heinemann, 1997.
Walpole, Raymond, Sharon and Keying, “Probability and Statistics for Engineers and Scientists”, Prentice-Hall, 2002.
References
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Reliability engineering is an engineering field that deals with the study, evaluation, and life-cycle management of reliability
The ability of a system or component to perform its required functions under stated conditions for a specified period of time.
Reliability is theoretically defined as the probability of failure, the frequency of failures, or in terms of availability
What is Reliability?
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The reliability is the probability of success, Mean Time Between Failure (MTBF).
The prevention of system loose function. OR Reliability is the probability of a device
performing its purpose adequately for the period of time intended under the operating conditions encountered." Billington and Allen (1983).
What is Reliability?
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Used to estimate the reliabilities of individual devices, such as electronic components,
and the reliabilities of systems constructed of components.
Mathematical – based on probability theory.
Reliability Theory
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Maintainability: The ability of an item, under stated conditions of use, to be retained in, or restored to, a state in which it can perform its required function(s), when maintenance is performed under stated conditions and using prescribed procedures and resources. Expressed as Mean Time To Repair (MTTR).
Definitions
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Availability: Is the probability that a system is available for use at a given time- a function of reliability and maintainability.
A=Up time/(Up time+ Down time) =MTBF/(MTBF+MDT). Failure: The termination of the ability of an
item to perform its required function. A fault: is "An accidental condition that
causes a functional unit to fail to perform its required function"
Definitions
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expected number of failures in a given time period
average time between failures average down time expected revenue loss due to failure expected loss of output due to failure
Reliability Indicators
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MTBF = MTTF + MTTR Where MTTF is mean time to failure MTTR is mean time to repair
Mean Time Between Failures
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The probability that an item will fail in the interval from 0 to time t is F(t), the reliability is then:
R(t)=1-F(t)
Basic Theory
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Assumes that items in the series are independent
All items must work for the system to work
Reliability of a Series System
R1 R3R2
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The probability of failure in the interval t to t+dt which is
λ(t) dt where λ(t) is the failure rate
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The variation of failure rate of electrical or electronic components.
The Bathtub Distribution
Early Failure
Useful
Life
Wearout
Failure
Failure Rate
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The set of all possible outcomes of a statistical experiment is called the Sample Space and is represented by the symbol S.
Example 1 The possible outcomes when a coin is tossed? S = {H, T }
Example 2 The experiment of tossing a die? S={1,2,3,4,5,6}
Probability
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Example 3: Tree Diagram Tossing a coin then a die! S = {HH, HT, T1, T2, T3, T4, T5, T6 }
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Example 4: Suppose that three items are selected at
random from a manufacturing process. Each item is inspected and classified (D=Defective or N=Nondefective).
S={DDD,DDN,DND,DNN,NDD,NDN,NND,NNN}
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Event: An event is a subset of a sample space. Example: The outcome is dividable by 3 (toss a die) A= {3, 6}
Example: The number of defective parts is more than
1. B = {DDN, DND, NDD, DDD}
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The complement of an event A with respect to S is the subset of all elements of S that are not in A. We denote the complement of A by the symbol A’.
A={DDD, DDN, DND, DNN} A’={NDD, NDN, NND, NNN}
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The intersection of two events A and B, denoted by the symbol A⋂B, is the event containing all elements that are common to A and B.
Example M={a,e,i,o,u} N={r,s,t} M ⋂ N= ∅
Two events are called mutually exclusive if :
A⋂B = ∅
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The union of the two events A and B, denoted by the symbol A∪ B, is the event containing all the elements that belong to A or B or Both.
Example A={a,b,c} B={b,c,d,e} A∪ B={a,b,c,d,e}
Example M={x|3<x<9} and N={y|5<y<12}, then M∪N = {z|3<z<12}
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A∩B = regions a and 2. B ∩C= regions 1 and 3 A∪C=regions 1,2,3,4,5, and 7, B’∩A= regions 4 and 7, A∩B∩C= region 1, (A∪B) ∩ C’=regions 2,6, and 7.
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If A and B are any two events, then P(A∪B)=P(A)+P(B)-P(A∩B)
If A and B are mutually exclusive: P(A∪B)=P(A)+P(B) Why P(A∩B)=0
Additive Rules
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Problem 1:The probability that a man will be alive
in 10 years is 0.8 and the probability that his wife will be alive in 10 years is 0.9. Find the probability that in 10 years
A- Both will be alive.B- Only the man will be alive.C- Only the wife will be alive.D- At least one will be alive.
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)(72.08.09.0)().()(
)(9.0)().(8.0)(
eventstindependenBPAPBAP
alivebewillwifehisBPalivebewillmanaAP
08.072.08.0
)()()(
BAPAPalivebewillmantheOnlyP
Solution:
18.072.09.0
)()()(
BAPBPalivebewillwifetheOnlyP
A- Both will be alive
B- Only the man will be alive:
C- Only the wife will be alive:
08.072.08.0
)()()(
BAPAPalivebewillmantheOnlyP
D- At least one will be alive:
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Two dice are tossed together. Let A be the event that the sum of the faces are odd, B the event that at least one is a one. What is the probability that:(i) Both A and B occur?(ii) Either A or B or both occur?(iii) A and not B occur?(iv) B and not A occur?
Problem 5:
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A = The sum is odd B = at least one face is a one.
36;6)(65)4)(63)(62)(61)(6(6
6)5)(54)(53)(52)(51)(5(5
6)5)(44)(43)(42)(41)(4(4
6)5)(34)(33)(32)(31)(3(3
6)5)(24)(23)(22)(21)(2(2
6)(15)(14)(13)(12)(11)(1S
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111)(61)1)(51)(41)(3(2
6)(15)(14)(13)(12)(11)(1B
18;5)(63)1)(6(6
6)4)(52)(5(5
5)3)(41)(4(4
6)4)(32)(3(3
5)3)(21)(2(2
6)(14)(12)(1A
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Both A and B occur:
1)(61)1)(46)(24)(12)(1(1BA
23
6B)(AP
Either A or B or both occur:
36
23B)(AP
2361118BA
I
II
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A and not B occurs:
36
12)BP(A12;618BA
B and not A occurs:
36
5)BP(A5;AB 611
IV
III
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P(B|A) = the probability that B occurs given that A occurs.
Conditional Probaility