L Berkley Davis Copyright 2009 MER035: Engineering Reliability Lecture 6 1 MER301: Engineering Reliability LECTURE 6: Chapter 3: 3.9, 3.11 and Reliability

L Berkley DavisCopyright 2009

MER035: Engineering ReliabilityLecture 6

1

MER301: Engineering Reliability

LECTURE 6:

Chapter 3: 3.9, 3.11 and ReliabilityExponential Distributions, Independence and Joint Distributions, Reliability



2

Summary

Exponential Distribution

Independence

Joint Distributions

Reliability



3


The Exponential Distribution predicts the probability of random failures

Frequently used in conjunction with a Poisson Process

For a Poisson process with parameter λ If x denotes the time until occurrence of the first

event after a specified time Then x has an Exponential Distribution with

mean=1/ λ



4

A Poisson Process Requirements for a Poisson Process

Random discrete events that occur in an interval that can be divided into subintervals

Probability of a single occurrence of the event is directly proportional to the size of a subinterval and is the same for all subintervals

If the sampling subinterval is sufficiently small, the probability of two or more occurrences of the event is negligible

Occurrences of the event in nonoverlapping subintervals are independent



5



Exponential and Poisson Distributions


6



7

Summary of Distribution Function Characteristics Poisson Distribution

Probability for a number of randomly occurring events to be found in an interval of of a specific size

Exponential Distribution Probability that next event of a Poisson process

occurs at a specified interval Weibull Distribution

Predicts probability of failure(failure rate) Exponential distribution a special case for

beta=1(random failures)



8

Example 6.1 Some strains of paramecia produce and secrete certain

particles that will destroy other paramecia, called sensitive, upon contact.

All paramecia unable to produce such particles are sensitive.

The number of particles emitted by any single non-sensitive paramecium is one every five hours.

In observing such a paramecium what is the probability that we must wait at most four hours before the first particle is emitted?

What is the probability that the time until the first particle is emitted is between two and three hours?



9

Example 6.1 Solution Let the measurement unit be one hour. This is then

a Poisson process with .The time at which the first particle is emitted has an exponential distribution with .The density function is then

and the probability we must wait at most 4 hours until the first particle is emitted is

5/1

5/1

5/)5/1()( tetf

55.0)5/1(44

0

5/ dtetP t



10

More than One Random Variable and Independence Most engineering design problems require that

multiple variables be dealt with in designing products Control variables “noise” variables

Independence implies that variation in one variable is not related to variation in another



11

Joint Distributions If X and Y are random

variables, a probability distribution defining their simultaneous behavior is a Joint Probability Distribution Applicable to both

continuous and discrete random variables

Applicable whether or not variables are independent

3-39

3-40



12

More than One Random Variable and Independence

Most engineering design problems require that multiple variables be dealt with in designing products Control variables “noise” variables

Independence implies that variation in one variable is not related to variation in another

Independent Variables X2 vs X2

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12

X1

X2 X2

X1 X28.29 8.277.49 8.948.65 7.916.83 8.664.58 9.29

10.73 8.447.22 7.999.75 11.79

11.35 9.441.91 9.289.17 6.858.24 5.8710.4 10.88.52 8.156.43 7.199.05 5.07

10.38 8.445.5 9.99

8.52 7.748.95 9.12

5.1,5.8

0.2,0.8

22

11

xx

xx

Salt Concentration in Watersheds

0

5

10

15

20

25

30

35

0 0.5 1 1.5 2

% area covered by roads

salt

co

nce

ntr

atio

n i

n r

un

off

Salt Conc

Salt Conc and %AreaNot Independent



13

Independent Random Variables


U n i o n C o l l e g eM e c h a n i c a l E n g i n e e r i n g

M E R 3 0 1 : E n g i n e e r i n g R e l i a b i l i t yL e c t u r e 1

4 1

P r o b a b i l i t y R u l e s

A d d i t i o n ( A o r B ) A a n d B a r e m u t u a l l y e x c l u s i v e

A a n d B a r e n o t m u t u a l l y e x c l u s i v e

M u l t i p l i c a t i o n ( A a n d B ) A a n d B a r e i n d e p e n d e n t

A a n d B a r e n o t i n d e p e n d e n t / c o n d i t i o n a l p r o b a b i l i t y

)()()()()( BandAPBPAPBAPBorAP

0)()(

)()()()(

BAPBandAP

BPAPBAPBorAP

)()()( BPAPBandAP

)()()( APABPBPBAPBandAP


14

Independent Random Variables

Independent Variables X2 vs X2

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12

X1

X2 X2

)8.95.6( 2 XP

)106( 1 XP

)8.95.6()106()8.95.6106( 2121 XPXPXandXP

X1 &x2

5.1,5.8

0.2,0.8

22

11

xx

xx



15

Union Coll egeMechanical Engi neeri ng

MER301: Engi neering ReliabilityLecture 16

Actual Defects

LSL USL

= Product Std. Dev.= Product Mean

LSL USL

Observed Defects

Measurement system variance

Product variance

Impact of Measurement System Variation on Variation in Experimental Data

22mactualobs

actual

actual

actual

obs observedobs

tmeasuremenm

actualact



Actual Defects

LSL USL


LSL USL

Observed Defects


Product variance


22mactualobs

actual

actual

actual

obs observedobs

tmeasuremenm

actualact



Actual Defects

LSL USL


LSL USL

Observed Defects


Product variance


22mactualobs

actual

actual

actual

obs observedobs

tmeasuremenm

actualact



Actual Defects

LSL USL


LSL USL

Observed Defects


Product variance


22mactualobs

actual

actual

actual

obs observedobs

tmeasuremenm

actualact



Actual Defects

LSL USL


LSL USL

Observed Defects


Product variance


22mactualobs

actual

actual

actual

obs observedobs

tmeasuremenm

actualact



Actual Defects

LSL USL


LSL USL

Observed Defects


Product variance


22mactualobs

actual

actual

actual

obs observedobs

tmeasuremenm

actualact



Actual Defects

LSL USL


LSL USL

Observed Defects


Product variance


22mactualobs

actual

actual

actual

obs observedobs

tmeasuremenm

actualact



Actual Defects

LSL USL


LSL USL

Observed Defects


Product variance


22mactualobs

actual

actual

actual

obs observedobs

tmeasuremenm

actualact



Actual Defects

LSL USL


LSL USL

Observed Defects


Product variance


22mactualobs

actual

actual

actual

obs observedobs

tmeasuremenm

actualact



Actual Defects

LSL USL


LSL USL

Observed Defects


Product variance


22mactualobs

actual

actual

actual

obs observedobs

tmeasuremenm

actualact

)(...)()()...( 10211021 XPXPXPXXandXP

diameterX

Unio n Col l eg eMec ha nic al E ngi ne eri ng

ME R30 1: E ngi ne erin g R el iabi l i tyLe ct ur e 1 6

Actual Defects

LSL USL


LSL USL

Observed Defects


Product variance


22mactualobs

actual

actual

actual

obs observedobs

tmeasuremenm

actualact

919.0)2515.02485.0( XP

0005.0

2508.0

0005.0

2508.0

Binomial ProblemN=10,x=10.p=0.919


U n i o n C o l l e g eM e c h a n i c a l E n g i n e e r i n g

M E R 3 0 1 : E n g i n e e r i n g R e l i a b i l i t yL e c t u r e 1

4 1

P r o b a b i l i t y R u l e s

A d d i t i o n ( A o r B ) A a n d B a r e m u t u a l l y e x c l u s i v e

A a n d B a r e n o t m u t u a l l y e x c l u s i v e

M u l t i p l i c a t i o n ( A a n d B ) A a n d B a r e i n d e p e n d e n t

A a n d B a r e n o t i n d e p e n d e n t / c o n d i t i o n a l p r o b a b i l i t y

)()()()()( BandAPBPAPBAPBorAP

0)()(

)()()()(

BAPBandAP

BPAPBAPBorAP

)()()( BPAPBandAP

)()()( APABPBPBAPBandAP


16

Joint Distributions If X and Y are random

variables, a probability distribution defining their simultaneous behavior is a Joint Probability Distribution

Applicable to both continuous and discrete random variables Applicable whether or not variables are independent

Salt Concentration in Watersheds

0

5

10

15

20

25

30

35

0 0.5 1 1.5 2

% area covered by roads

salt

co

nce

ntr

atio

n i

n r

un

off

Salt Conc



17

Independence of Random Variables or Events

Concept of independent random variables/events underlies many statistical analysis methods

If Y=f(x1,x2,…xn) then the values y1,…yk obtained from k repetitions of an experiment are independent if and only if x1,…xn are random independent variables



18

Reliability Reliability Analysis quantifies whether or not a design

functions adequately for the required life when operating under the conditions for which it is designed- whether it… Meets Functional Requirements over Operating Life Achieves Design Life before Replacement/Repair Tolerates Needed range of Environmental Conditions

The objective of Reliability Analysis is to predict the probability of failure at a specific time t, number of cycles, etc Failure density function, f(t) Cumulative density function, F(t)=probability of failure at

time t Reliability function, R(t) = I-F(t)=probability of no failure at

time t



19

Reliability Functions

Four Reliability Functions Failure density function, f(t) may be a Weibull,

exponential, or other distribution Cumulative density function, F(t) Reliability function, R(t) = I-F(t) Hazard rate of distribution, h(t)

t

t

dttftFtR

dttftF

0

0

)(1)(1)(

)()(



20

Reliability Definitions Probability of failure per unit time or cycle

Mean Time between Failures(MTBF)

Types of Failures during Product Life Infant Mortality Random Failure Old Age

For the Exponential Distribution

)/()( hoursoperatingfailuresofnumber

)/()(/1 failuresofnumberhoursoperatingMTBF

gincrea

tcons

gdecrea

sin,1

tan,1

sin,1

t

t

t edtetF 1)(0

tcons tan,1

tetFtR )(1)(

)(

/)()(

tR

dttdRth


failure rate for delta=1000X beta=0.5 beta=1 beta=2 beta=3 beta=40 0 0 0 0 0

10 0.095 0.01 0 0 050 0.2 0.049 0.002 0 0

100 0.271 0.095 0.01 0.001 0200 0.361 0.181 0.039 0.008 0.002500 0.507 0.393 0.221 0.118 0.061

1000 0.632 0.632 0.632 0.632 0.6322000 0.757 0.865 0.982 1 15000 0.893 0.993 1 1 1

10000 0.958 1 1 1 1


21

The Weibull Distribution

1)()(

x

th

))(exp()(1)(

x

tFtR

reliability function for delta=1000X beta=0.5 beta=1 beta=2 beta=3 beta=40 1 1 1 1 1

10 0.905 0.99 1 1 150 0.8 0.951 0.998 1 1

100 0.729 0.905 0.99 0.999 1200 0.639 0.819 0.961 0.992 0.998500 0.493 0.607 0.779 0.882 0.939

1000 0.368 0.368 0.368 0.368 0.3682000 0.243 0.135 0.018 0 05000 0.107 0.007 0 0 0

10000 0.042 0 0 0 0

hazard rate for delta=1000X beta=0.5 beta=1 beta=2 beta=3 beta=40

10 0.005 0.001 0.00002 3E-07 4E-0950 0.002236 0.001 0.0001 7.5E-06 5E-07

100 0.001581 0.001 0.0002 0.00003 0.000004200 0.001118 0.001 0.0004 0.00012 0.000032500 0.000707 0.001 0.001 0.00075 0.0005

1000 0.0005 0.001 0.002 0.003 0.0042000 0.000354 0.001 0.004 0.012 0.0325000 0.000224 0.001 0.01 0.075 0.5

10000 0.000158 0.001 0.02 0.3 4



22

The Weibull Distribution Hazard Rate

h(x) for x=delta

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0 1 2 3 4 5

beta

haz

ard

rat

e

h(x) h(x) for x=2*delta

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 1 2 3 4 5

beta

haz

ard

rat

e

h(x)

11 21000

)()(

xth

1000)()( 1

x

th

delta=1000

delta=1000



23

Reliability What kinds of Systems require reliability analysis?

Individual components(gas turbine compressor blades) or Assemblies(motors, pumps,valves, instruments)

Software (firmware, distributed controls code,plant level controls code, applications code)

Systems comprising multiple components or assemblies or software(gas turbine gas fuel system)

What are the methods used in reliability analysis? Testing to establish capability of components, assemblies,

systems, software Reliability Block Diagrams to model entire Systems Analysis of Reliability Models to assess performance Field Data gathering to validate models



24

Reliability Testing Reliability Tests are conducted to establish the tolerance

of components,assemblies, and systems to environmental factors Dust, dirt,chemical contaminants Vibration Operating Temperature range Ambient Temperature range Moisture,humidity

Reliability Tests are typically conducted by exercising the system Subjecting systems to environmental stress testing Exercising the system through a number of operational

cycles equivalent to product life requirements Conducting HALT tests to get data more quickly



25

Reliability Analysis Most designs consist of multiple subsystems,

assemblies,and components each with it own probability of failing or not Probability of overall system functioning is a

function of subsystem characteristics The way subsystems are arranged in a system

determines overall reliability



26

Reliability of a System


Venn Diagram


27

Reliability of a System)()()( 2121 CPCPCandCP

)()()()( 212121 CandCPCPCPCorCP

A not A SumB A and B not A and B P(B)

not B A and not B not A and not B P(not B)P(A) P(not A) 1

AB

BA

)()()()( BAPBPAPBorAP

A=A & B+

A & not BnotA &B notA

&notB

“coin flip”

“A or NotA” Diagram

“Probability A or B”



28

Reliability of a System

)3()()()3( 2121 BlockPBlockPBlockPBlockandBlockandBlockP

)()()()()()()()( ABCPBCPACPABPCPBPAPCorBorAP

Block 3A

B

C



29

Aircraft Reliability Four Engine Aircraft-two engines per wing and at least one engine per wing must function

For a wing, either engine must function so that probability that a wing is OK is an “or” problem For the aircraft, one engine on each wing must function at the same time so that the probability that both wings are OK is an

“and” problem Addition ( A or B)

A and B are mutually exclusive

A and B are not mutually exclusive

Multiplication( A and B) A and B are independent

A and B are not independent/conditional probability

)()()()( BandAPBPAPBorAP

)()()( BPAPBorAP

)()()( BPAPBandAP

)()( BPBAPBandAP



30

Aircraft Reliability

A

B D

C

Let Engines A and B be on Wing 1 and Engines C and D be on Wing 2 Assume the reliability (probability of not failing) of all engines is the

same RA=RB=RC=RD

The probability a wing is functional is then(addition rule)

Both wings must be functional for the plane to fly so the probability of flight is then (multiplication rule)

DCDCBABA RRRRRRRROKwingPOKwingP )2()1(

)()()2()1( DCDCBABA RRRRRRRROKwingPOKwingP

Four Engine Plane Reliability

0.95

0.96

0.97

0.98

0.99

1

1.01

0.8 0.85 0.9 0.95 1 1.05

Engine reliability

Air

pla

ne

Re

liab

ility

Series1



31

Aircraft ReliabilityEngine Airplane Failure/Flight

0.85 0.955506 0.044490.9 0.9801 0.0199

0.95 0.995006 0.004990.98 0.9992 0.00080.99 0.9998 0.00020.997 0.999982 0.0000180.999 0.999998 0.000002

0.9999 1 0.000000020.999978 1 0.000000001

Four Engine Plane Reliability

0.95

0.96

0.97

0.98

0.99

1

1.01

0.8 0.85 0.9 0.95 1 1.05

Engine reliability

Air

pla

ne

Re

liab

ility

Series1

Four Engine Aircraft-two engines per wing and at least one engine per wing must function for a successful flight 0.999 engine reliability gives two chances/million of a flight

failure….. Is this good enough?



32

Aircraft Reliability

Engine Airplane Failure/Flight Failures/1000 Flights Failures/Year Failures/Fleet Life Failures/Billion0.85 0.955506 0.04449 44.49375 16240.21875 487206.6 444937500.9 0.9801 0.0199 19.9 7263.5 217905 19900000

+/- 2 sigma 0.95 0.995006 0.00499 4.99375 1822.71875 54681.6 49937500.98 0.9992 0.0008 0.79984 291.9416 8758.2 7998400.99 0.9998 0.0002 0.19999 72.99635 2189.9 199990

+/- 3 sigma 0.997 0.999982 0.000018 0.018 6.57 197.1 180000.999 0.999998 0.000002 0.002 0.73 21.9 2000

0.9999 1 0.00000002 0.00002 0.0073 0.2 20one/billion 0.999978 1 0.000000001 0.000001 0.00036 0.01 1

We must deal with the entire fleet over its lifetime Assume a fleet of 1000 planes with one flight per plane per day

and a life of 30 years

To reach these high levels of reliability, scheduled engine maintenance is performed to “re-start”the clock



33

Triple Modular Redundant System

Any two of three components must operate for the system to function Assume that RA=RB=RC

On-Line Repair often incorporated into Triple Modular Redundant systems

B

C

C

A

B

A

)1()1()1(1Re CBCABA RRRRRRliabilitySystem



34

Triple Modular Redundant System

Triple Modular redundant System ReliabilityComponent Path System Reliability Failures/Million Failures/Billion

0.5 0.25 0.578125 421875 4218750000.6 0.36 0.737856 262144 2621440000.7 0.49 0.867349 132651 1326510000.8 0.64 0.953344 46656 46656000

0.85 0.7225 0.978630766 21369.234 213692340.9 0.81 0.993141 6859 6859000

0.95 0.9025 0.999073141 926.859 9268590.975 0.9506 0.999879446 120.554 1205540.99 0.9801 0.999992119 7.881 78810.997 0.994 0.999999784 0.216 2160.999 0.998 0.999999992 0.008 8

0.9995 0.999 0.999999999 0.001 10.9999 0.9998 1 8.00005E-06 0.008



35

Summary


Independence

Joint Distributions

Reliability

Documents

L Berkley Davis Copyright 2009 MER035: Engineering Reliability Lecture 6 1 MER301: Engineering Reliability LECTURE 6: Chapter 3: 3.9, 3.11 and Reliability