Transcript Reliability measures

2. Reliability measures
Objectives:
• Learn how to quantify reliability of a system
• Understand and learn how to compute the following measures
– Reliability function
– Expected life
– Failure rate and hazard function
• Learn some common probability density functions of time to failure
and learn when to apply them
– Exponential
– Normal
– Weibull
• Learn how to estimate hazard functions from data
• Learn how to select a reliability function for a given problem
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Reliability function
• Assumption: New equipment
• T=Failure time, random variable because we do
not know when a system will fail
• Probability density function of failure time, fT(t).
Units: # of failures per unit time
• Reliability function, R(t)= probability that system
will work properly at time t
• Failure distribution function, FT(t)= probability
that a system will fail by time t
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Notation
• Probability density function, fT(t)
• T random variable (in this case it is the
component life)
• t value that the random variable assumes
• fT(t)=limt0 P(t<T t+ t)/ t
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Expected life
• Expected life of a component or system,
E(T)
fT(t)
E(T)
t
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Hazard function
• Hazard function: h(t)=probability that, given that a
system has survived until time t, it will fail
between times t and t+t, divided by t. Units of
h(t): 1/unit time
• h(t)= fT(t)/R(t)
• Example start with N=1000 light bulbs, at T=1000
hrs, 300 light bulbs are still working. After 10 hrs
5 more bulbs fail. The hazard function is
approximately: h(1005)=5/(300*10)
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Shape of hazard function of most
real-life systems: bathtub
function
Aging
h(t)
Debugging,
or infant
mortality
t
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Relation between reliability measures
fT (t )
FT (t )
R (t )
h (t )
t
fT (t )
FT (t )
-
dFT (t )
dt
 fT (t ' )dt'
0
t
1   fT (t ' )dt'
0
fT (t )
t
1   fT (t ' )dt'
0
R (t )

dR (t )
dt
-
1  R (t )
1  FT (t )
-
dFT (t )
dt
1  FT (t )
h (t )
t
  h (t ')dt '
R ( 0) h ( t ) e 0
t
  h (t ')dt '
1  R ( 0) e 0

dR(t )
dt
R (t )
t
  h (t ')dt '
R ( 0) e 0
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Reliability and hazard functions
for well known distributions
• Exponential
– Good choice for systems or components whose
strength does not change with time and which
are subjected to extreme disturbances occurring
completely at random and independently.
– fT(t)=1/*exp(-t/ )
– R(t)= exp(-t/ )
– h(t)=1/
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Shape of exponential distribution
fT(t)
1/ 
E(T)=θ
t
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Normal distribution
Two parameter distribution
fT(t)
Standard deviation, 

t
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• No closed form analytical expression for
cumulative distribution
• Cumulative distribution of standard normal, (z),
has been tabulated. We also have excellent
polynomial approximations. Standard normal has
zero mean and unit standard deviation.
• Very easy to do reliability computations with
normal distributions
• Finding FT(t) if T is normal. Transform T into
standard normal.
• FT(t)=P(Tt)=P[(T-)/  (t-)/]= (z)
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Cumulative distribution of standard
normal variable
z
(z)
0
0.5
-1
0.16
-2
0.02
-3
0.001
-4
310-5
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f (t)
T
The area under the
curve to the left of zero
is the probability
of t being negative
0

t
If we model the time to failure using a normal distribution then
there is small probability of the time to failure being negative.
This does not make sense. Always check that the probability
of the time being negative is small compared to the
probabilities we are calculating in the problem at hand. For
example, if the we are working with systems whose failure
probabilities are about 10-3, then the probability of the time to
failure being negative should be about 10-5 or less.
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Weibull distribution
• Good choice for systems or components whose strength
deteriorates with time and which are subjected to extreme
disturbances occurring completely at random and
independently.
• Consider a building in Greece that is expected to be sustain
a very strong earthquake (say above 6.5 in the Richter
scale) once every ten years. Like any real life system, the
strength of the building deteriorates with time. A Weibull
distribution is a good candidate for modeling the time to
failure (or length of the life) of the building.
• Very popular for describing strength and life length
• Generalizes the exponential distribution
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Reliability function
t  
(
)
– R(t )  e  
for t greater than 
– Three parameter distribution:
•
•
•
•
use shape parameter, , to control shape
 is the scale parameter, affects dispersion
use location parameter, , to shift the mean value
shape parameter=1, Weibull reduces to the exponential
distribution
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Shape of Weibull probability density function
if shape parameter less than 3.6, density is skewed to the right
if shape parameter is greater than 4, density is skewed to the left
.
beta=0.5
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f(t)
6
4
2
0
0
2
4
6
t
theta=0.5
theta=1
theta=4
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.
Shape of Weibull probability density function
beta=1
f(t)
2
1
0
0
2
4
6
t
theta=0.5
theta=1
theta=4
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Shape of Weibull probability density function
beta=4
4
3
f(t)
.
2
1
0
0
2
4
6
t
theta=0.5
theta=1
theta=4
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Shape of Weibull probability density function
beta=10
8
.
f(t)
6
4
2
0
0
2
4
6
t
theta=0.5
theta=1
theta=4
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Effect of shape parameter
Consider building exposed to
earthquakes:
The larger the value of the shape
parameter, the larger the rate of
deterioration in strength
If the shape parameter is one then there
is no deterioration in the strength
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Statistics
E (T )  (   ) (1 
1

) 
2
1
 2  (   )2 [ (1  )  {(1  )}2 ]


median  Tˆ    (   )e

0.3665

Median: 50% probability lower than median,
50% higher than median
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Other common distributions
• Lognormal; If x is normal then exp(x) is
lognormal
• Gamma: quite similar to Weibull
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Estimating hazard function, failure
density function and reliability function
from data
Case 1: Large sample of data about failures (N
greater than 30)
• Start with N systems.
N (t ), number of systems that operate successful ly at time, t
N(t)
R(t) 
N
N (t )  N (t  t )
h (t ) 
N (t )  t
N (t )  N (t  t )
f (t ) 
N  t
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Case 2: Small samples
Study homework 3
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Selecting a probability distribution on
the basis of knowledge of the particular
physical situation causing failures
• In most real life problems, we do not have enough
data to estimate probability distributions.
Therefore, we rely on experience or on
analytically obtained associations of physical
situations causing failure and probability
distributions to select type of probability
distribution to failure.
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Weibull and exponential models
• Extreme disturbances occurring completely at
random and independently. Example: time of
occurrence and intensity of a strong earthquake
does not affect the time of occurrence and
intensity of the next.
• Probability of occurrence of one earthquake
during [t, t+dt] is  dt. Average rate of occurrence
of extreme disturbances is  disturbances/unit time
• Probability of a system failing because of a
disturbance, p(t)
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Earthquake intensity versus time
Severe earthquakes
Intensity
 severe earthquakes per year
return period, 1/ 
t
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Reliability
t
   p ( )d
R ( t )  e  P ( t )  e 0
f T ( t )   p ( t ) e  P ( t )
If p(t) is constant :
R (t )  e  pt
fT (t )  pe  pt
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Suggested reading
• Fox, E., “The Role of Statistical Testing in
NDA,” Engineering Design Reliability
Handbook, CRC press, 2004, p. 26-1.
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