Transcript Reliability - Lyle School of Engineering

SMU
SYS 7340
NTU
SY-521-N
Logistics Systems Engineering
Reliability Fundamentals
Dr. Jerrell T. Stracener,
SAE Fellow
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Reliability - Basic Concepts
2
Reliability
• A product, service and system attribute as well
as an engineering function
• Reliability principles, methods and techniques
apply to:
products and services
and
Logistics systems
3
Reliability Concepts, Principles and Methodology
• Hardware
• Software
• Operator
• Service
• Product
• Production/Manufacturing Processes and
Equipment
• Product and Customer Support
• Systems
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What is Reliability
To the user of a product, reliability is problem free operation
Reliability is a function of stress
To understand reliability, understand stress on hardware
where its going to be used
how its going to be used
what environment it is going to be used in
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What is Reliability
– To efficiently achieve reliability, rely on
analytical understanding of reliability and less
on understanding reliability through testing
– Field Problems
Stress/Design
Parts and Workmanship
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What is Reliability
– Reliability affects market share:
During the 1970’s, Western color TV sets
were failing in service at a rate of five
times that prevailing in Japanese sets
Example 1:
 Prior to coming under Japanese
management, the U.S. Motorola factory
ran at a “fall-off” rate of 150 to 180 per
100 sets packed. This meant that 150
to 180 defects were found for every 100
sets packed.
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What is Reliability
– Reliability affects market share:
Example 1 (Continued):
 Three years later, after being taken over
by a Japanese company, the fall-off rate
at Quasar (the new name of the
factory) had gone down to a level of
about 3 or 4 per 100 sets.5
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What is Reliability
– Reliability affects market share:
Example 2:
Western automobiles have experienced a
similar problem as in example 1.
Consumer Reports annually published
frequency of repair statistics for
automobiles, taken from surveys of the
magazine’s many readers. In short, there
were almost no American car names
reported in vehicles with high reliability.
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What is Reliability
– Reliability affects market share:
Example 2 (Continued):
Consumers bought millions of imported
cars because they have the reputation of
reliability. Each million cars the US
imports represents abut $15 billion added
to the US trade deficit.6
– Reliability affects risk:
Example:
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What is Reliability
– Reliability affects risk:
Example:
The Challenger space shuttle solid rocket
motor was designed and qualified to
operate in the range of 50 to 90oF. On
January 27-28, the temperatures at the
launch site were predicted around 18oF.
The political decision to launch anyway
cost seven lives and a delay of over 30
months in the US space program.7
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Definitions
– Reliability is a characteristic of an item,
expressed by the probability that the item
will perform its required function under given
conditions for a stated time interval.1
– The probability that an item will perform a
required function without failure under stated
conditions for a stated period of time.2
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Definitions
– The probability that an item will perform its
intended function for a specified interval
under stated conditions.3
– The rigorous definition has four parts:4
1. Reliability is the probability that a system
2. will demonstrate specified performance
3. For a stated period of time
4. when operated under specified conditions.
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Definitions
– Reliability is a measure of the capability of a
system, equipment or component to operate
without failure when in service.
– Reliability provides a quantitative statement
of the chance that an item will operate
without failure for a given period of time in
the environment for which it was designed.
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Definitions
– In its simplest and most general form,
reliability is the probability of success.
– To perform reliability calculations, reliability
must first be defined explicitly. It is not
enough to say that reliability is a probability.
A probability of what?
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Definitions
– Succinctly put, reliability is a performance
attribute that is concerned with the
probability of success and frequency of
failures and is defined as:
The probability that an item will perform
its intended function under stated
conditions, for either a specified interval or
over its useful life.
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Definition of Reliability
The essential elements of a definition of reliability are:
System, subsystem, equipment or component
Satisfactory performance
Required period of operation
Conditions of operation
Environment
Operation
Maintenance
Support
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Systems
Definition &
Description
Definition of
Successful
Performance
Required
Period of
Operation
Reliability
Environment
Degree of
Customer
Satisfaction
Operation
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Reliability is
PERFORMANCE
OVER
TIME
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What Affects Reliability
–
–
–
–
–
–
–
Redundancy
Design Simplicity
Time
Learning Curve
Material Quality
Experience
Requirements
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Why is Reliability Modeling & Analysis Needed
Prediction of Product Performance
How many items will be required to meet
demand?
How much maintenance and support will
be required?
 Facilities
 Spares
 Maintenance Personnel
How many items will not meet warranty?
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Why is Reliability Modeling & Analysis Needed
– Basis for design, manufacturing and support
decisions
Evaluate Alternatives
Identify and rank drivers
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How is Reliability Used
– It is used to define the longevity of a product
and the associated cost it incurs.
– It helps identify risk of the product for both
the consumer and producer.
– It incorporates statistics to better identify
how much “give” or “take” can go into a
product or service. Usually, the higher the
reliability, the higher the initial cost.
– It predicts the likelihood of failure for a
product, service or system.
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How is Reliability Used
– Basic reliability and mission reliability
predictions are used through the item design
phase to perform
Design evaluations
 requirements assessment
 design comparisons
Trade-studies
 evaluation design alternatives
 rank design alternatives
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How is Reliability Used
Perform sensitivity analyses
 Mission effectiveness
 Supportability
 Life cycle costs
 Warranties
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Importance of Reliability
– Reliability is a measure of a product’s
performance that affects both product
function and operating and repair costs
– The reliability of a product is a primary factor
in determining operating and repair costs.
– Reliability determines whether or not a
product is available to perform its function.
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Reliability Goals
1.
2.
3.
4.
Increase competitive position
Increase customer satisfaction
Reduce customer support requirements
Decrease cost of ownership
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Reliability - Basic Metrics and Models
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Reliability Figures of Merit
• Basic Reliability
MTBF - Mean Time Between Failures
measure of product support requirements
• Mission Reliability
Ps or R(t) - Probability of mission success
measure of product effectiveness
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Basic Reliability
• Design and development
Basic reliability is a measure of serial reliability or
logistics reliability and reflects all elements in a system
• Measures
Air Force
Army
Navy
MFHBF - Mean Flight Hours Between Failures
MFHBUM - MFHB Unscheduled Maintenance
MFHBE - Mean Flight Hours Between Events
MFHBF - Mean Flight Hours Between Failures
MFHBMA - MFHB Maintenance Actions
Automotive Industry
Electronics Industry
Logistics
Number of defects per 100 vehicles
MTBF - Mean Time Between Failures
Mean Time Between System Failures
Percent On-Time Performance
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Mission Reliability
• Mission Reliability is defined as the probability that a system
will perform its mission essential functions during a
specified mission, given that all elements of the system
are in an operational state at the start of the mission.
• Measure
Ps or R(t) - Probability of mission success based on:
Mission Essential Functions
Mission Essential Equipment
Mission Operating Environment
Mission Length
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Reliability Life Characteristic Curve
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The Exponential Model:
Remarks
The Exponential Model is most often used in Reliability
applications, partly because of mathematical
convenience due to a constant failure rate.
The Exponential Model is often referred to as the
Constant Failure Rate Model.
The Exponential Model is used during the ‘Useful
Life’ period of an item’s life, i.e., after the ‘Infant
Mortality’ period before Wearout begins.
The Exponential Model is most often associated
with electronic equipment.
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Failure Density Function
Associated with a continuous random variable T,
the time to failure of an item, is a function f,
called the probability density function, or in
reliability, the failure density. The function f has
the following properties:
f (t)  0

and
for all values of t
 f (t )dt  1
0
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The Exponential Model:
A random variable T is said to have the Exponential
Distribution with parameters , where  > 0, if the
failure density of T is:
1
f (t)  e

0

t

,
for t  0
,
elsewhere
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Failure Distribution Function
The failure distribution function or, the probability
distribution function is the cumulative proportion
of the population failing in time t, i.e.,
t
Ft   P(T  t )   f ( y)dy
0
 1 e

t

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The Reliability Function
The Reliability of an item is the probability that
the item will survive time t, given that it had not
failed at time zero, when used within specified
conditions, i.e.,

Rt   P(T  t )   f ( t )dt  1  F( t )
t
e

t

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Failure Rate
1 d
f (t)
h(t)  
R (t) 
R ( t ) dt
R (t)

1


Remark: The failure rate h(t) is a measure of
proneness to failure as a function of age, t.
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Cumulative Failure Rate
The cumulative failure rate at time t, H(t), is the
cumulative number of failures at time t, divided
by the cumulative time, t, i.e.,
t
1
H( t )   h ( y)dy
t0

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The Reliability Function
The reliability of an item at time t may be
expressed in terms of its failure rate at time t
as follows:

 0 h ( y ) dy
R ( t )  exp    h ( y)dy   e
 0

t
t
where h(y) is the failure rate
R(t )  e
 t
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MTTF and MTBF
Mean Time to Failure (or Between Failures) MTTF
(or MTBF) is the expected Time to Failure (or
Between Failures)


0
0
MTBF   tf (t )dt   R(t )dt

Remarks:
MTBF provides a reliability figure of merit for expected failure
free operation MTBF provides the basis for estimating the
number of failures in a given period of time Even though an
item may be discarded after failure and its mean life
characterized by MTTF, it may be meaningful to characterize
the system reliability in terms of MTBF if the system is
restored after item failure.
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The Weibull Model:
• Definition
A random variable T is said to have the Weibull
Probability Distribution with parameters  and ,
where  > 0 and  > 0, if the failure density of T is:

t
 
 1    ,

f (t)   t e

0
,
for t  0
elsewhere
• Remarks
 is the Shape Parameter
 is the Scale Parameter (Characteristic Life)
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Properties of The Weibull Model:
• Probability Distribution Function
F( t )  1 - e
t
 


,
for t  0
where F(t) is the Fraction of Units Failing in Time t
• Reliability Function
R (t)  e
t
 


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The Weibull Model - Weibull Probability Paper (WPP):
Weibull Probability Paper links
http://perso.easynet.fr/~philimar/graphpapeng.htm
http://www.weibull.com/GPaper/index.htm
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Use of Weibull Probability Paper:
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Properties of the Weibull Model:
• 100th Percentile
t P   - ln(1 - p)
1

and, in particular
t 0.632  
• MTBF (Mean Time Between Failure)
1 
MTBF     1
 
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The Gamma Function 

(a )   e x dx
x
a 1
0
(a  1)  a(a )
Values of the
Gamma Function
y=a
1
1.01
1.02
1.03
1.04
1.05
1.06
1.07
1.08
1.09
1.1
1.11
1.12
1.13
1.14
1.15
1.16
1.17
1.18
1.19
1.2
1.21
1.22
1.23
1.24
 (a)
1
0.9943
0.9888
0.9836
0.9784
0.9735
0.9687
0.9642
0.9597
0.9555
0.9514
0.9474
0.9436
0.9399
0.9364
0.933
0.9298
0.9267
0.9237
0.9209
0.9182
0.9156
0.9131
0.9108
0.9085
a
1.25
1.26
1.27
1.28
1.29
1.3
1.31
1.32
1.33
1.34
1.35
1.36
1.37
1.38
1.39
1.4
1.41
1.42
1.43
1.44
1.45
1.46
1.47
1.48
1.49
 (a)
0.9064
0.9044
0.9025
0.9007
0.899
0.8975
0.896
0.8946
0.8934
0.8922
0.8912
0.8902
0.8893
0.8885
0.8879
0.8873
0.8868
0.8864
0.886
0.8858
0.8857
0.8856
0.8856
0.8858
0.886
a
1.5
1.51
1.52
1.53
1.54
1.55
1.56
1.57
1.58
1.59
1.6
1.61
1.62
1.63
1.64
1.65
1.66
1.67
1.68
1.69
1.7
1.71
1.72
1.73
1.74
 (a)
0.8862
0.8866
0.887
0.8876
0.8882
0.8889
0.8896
0.8905
0.8914
0.8924
0.8935
0.8947
0.8959
0.8972
0.8986
0.9001
0.9017
0.9033
0.905
0.9068
0.9086
0.9106
0.9126
0.9147
0.9168
a
 (a)
1.75 0.9191
1.76 0.9214
1.77 0.9238
1.78 0.9262
1.79 0.9288
1.8 0.9314
1.81 0.9341
1.82 0.9369
1.83 0.9397
1.84 0.9426
1.85 0.9456
1.86 0.9487
1.87 0.9518
1.88 0.9551
1.89 0.9584
1.9 0.9618
1.91 0.9652
1.92 0.9688
1.93 0.9724
1.94 0.9761
1.95 0.9799
1.96 0.9837
1.97 0.9877
1.98 0.9917
1.99 0.9958
2 49
1
Properties of the Weibull Model:
• Variance of T
  2  2  1 
     1     1
  
  
2
2
• Failure Rate
 -1
h(t)   t

Notice that h(t) isa decreasing function of t if  < 1
a constant if  = 1
an increasing function of t if  > 1
50
Properties of the Weibull Model:
• Cumulative Failure Rate
-1
t
h(t)
H( t )   


• The Instantaneous and Cumulative Failure Rates,
h(t) and H(t), are straight lines on log-log paper.
• The Weibull Model with  = 1 reduces to the
Exponential Model.
• Any straight line on Weibull Probability paper is a
Weibull Distribution with slope,  and intercept,
- ln , where the ordinate is ln{ln(1/[1-F(t)])}
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the abscissa is ln t.
Properties of the Weibull Model:
• Conditional Probability of Surviving Time t2, given
survival to time t1, where t1 < t2,


1  t
R t 2 t1   e




t
2 1
 ,
if  > 1
• Mode - The value of time (age) that maximizes the
failure density function.
t mod e  1  1 
1
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The Weibull Model - Distributions:
Probability Density Function
f(t)
t
t is in multiples of 
53
The Weibull Model - Distributions:
Reliability Functions
R(t)
t
t is in multiples of 
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The Weibull Model - Distributions:
Failure Rates
h(t)
t
t is in multiples of 
h(t) is in multiples of 1/ 
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The Weibull Model
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