Reliability Engineering - Part 1

1
Reliability Engineering- Part 1

2
Product Probability Law of Series Components

• If a system comprises a large number of
  components, the system reliability may be rather
  low, even though the individual components have
  high reliabilities, e.g. V-1 missile in WW II.

3
Transition of Component States
Normal state continues
Component fails
Failed state continues
N
F
Component is repaired
4
The Repair-to-Failure Process
5
Definitions of Reliability

• The probability that an item will adequately
  perform its specified purpose for a specified
  period of time under specified environmental
  conditions.
• The ability of an item to perform an required
  function, under given environmental and
  operational conditions and for a stated period of
  time. (ISO 8402)

6
Definition of Quality

• The totality of features and characteristics of a
  product or service that bear on its ability to
  satisfy or implies needs (ISO 8402).
• Quality denotes the conformity of the product to
  its specification as manufactured, while
  reliability denotes its ability to continue to
  comply with its specification over its useful
  life. Reliability is therefore an extension of
  quality into the time domain.

7
REPAIR -TO-FAILURE PROCESS
MORTALITY DATA tage in years L(t) number of
living at age t
t L(t) t L(t) t
L(t) t L(t)
0 1,023,102 15 962,270 50
810,900 85 78,221 1 1,000,000 20
951,483 55 754,191 90 21,577 2
994,230 25 939,197 60 677,771 95
3,011 3 990,114 30 924,609 65
577,822 99 125 4 986,767
35 906,554 70 454,548 5 983,817
40 883,342 75 315,982 10 971,804
45 852,554 80 181,765
After Bompas-Smith. J.H. Mechanical Survival
The Use of Reliability Data, McGraw-Hill Book
Company, New York , 1971.
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HUMAN RELIABILITY
t L(t), Number Living
at Age in Years Age t
R(t)L(t)/N F(t)1-R(t)
repair birth failure death Meaning of R(t)
(1) Prob. Of Survival (0.86) of an individual of
an individual to age t (40) (2) Proportion of a
population that is expected to Survive to a given
age t.
0 1,023,102
1. 0.
1 1,000,000
0.9774 0.0226 2
994,230
0.9718 0.0282 3
986,767 0.9645
0.0322 4
983,817 0.9616
0.0355 5
983,817 0.9616
0.0384 10
971,804 0.9499 0.0501
15 962,270
0.9405 0.0595
20 951,483
0.9300 0.0700 25
939,197
0.9180 0.0820 30
924,609 0.9037
0.0963 40
883,342 0.8634
0.1139 45
852,554 0.8333 0.1667
50 810,900
0.7926 0.2074
55 754,191
0.7372 0.2628 60
677,771
0.6625 0.3375 65
577,882 0.5648
0.4352 70
454,548 0.4443
0.5557 75
315,982 0.3088 0.6912
80 181,765
0.1777 0.8223
85 78,221
0.0765 0.9235 90
21,577
0.0211 0.9789 95
3,011 0.0029
0.9971 99
125 0.0001
0.9999 100
0 0.
1.
9
1.0
P
0.9
Survival distribution
0.8
0.7
0.6
0.5
Probability of Survival R(t) and Death F(t)
0.4
0.3
Failure distribution
0.2
0.1
0 10 20 30 40
50 60 70 80
90 100
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Time to Failure

• The time elapsing from when the unit is put into
  operation until it fails for the first time, i.e.
  a random variable T. It may also be measured by
  indirect time concepts
• The number of times a switch is operated
• The number of kilometers driven by a car
• The number of rotations of a bearing
• etc.

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State Variable

• The state of the unit at time t can be described
  by the state variable

12
Reliability, R(t) probability of survival to
(inclusive) age t the number of surviving at t
divided by the total sample Unreliability, F(t)
probability of death to age t (t is not
included) the total number of death before age t
divided by the total population
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Reliability - R(t)

• The probability that the component experiences no
  failure during the the time interval (0,t or,
  equivalently, the probability that the unit
  survives the time interval (0, t and is still
  functioning at time t.
• Example exponential distribution

14
Unreliability - F(t)

• The probability that the component experiences
  the first failure during (0,t.
• Example exponential distribution

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FALURE DENSITY FUNCTION f(t)
No. of Failures (death)
Age in Years
0.02260 0.00564 0.00402 0.00327 0.00288 0.00235
0.00186 0.00211 0.00240 0.00285 0.00353 0.00454 0.
00602 0.00814 0.01110 0.01500 0.01950 0.02410 0.02
710 0.02620 0.02020 0.01110 0.00363 0.00071 000012

0 1 2 3 4 5 10 15 20 25 30 35 40 45 50 60 65 70 75
80 85 90 95 99 100
23,102 5,770 4,116 3,347 2,950
12,013 9,543 10,787 12,286 14,588
18,055 23,212 30,788 41,654 56,709
99,889 123,334 138,566 134,217 103,554 56,634
18,566 2,886 125 0
0.00540 0.00454 0.00284 0.00330 0.00287 0.00192 0.
00198 0.00224 0.00259 0.00364 0.00393 0.00436 0.00
637 0.00962 0.01367 0.01800 0.02200 0.02490 0.0261
0 0.02460 0.01950 0.00970 0.00210 -
- -
16
140
120
100
Number of Deaths (thousands)
80
60
40
20
Age in Years (t)
20
40
60
80
100
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0.14
0.12
0.10
Failure Density f (t)
0.8
0.6
0.4
0.2
Age in Years (t)
20
40
60
80
100
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Failure Density - f(t)
(exponential distribution)
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CALCULATION OF FAILURE RATE r(t)
Age in Years
No. of Failures (death)
Age in Years
No. of Failures (death)
r(t)
r(t)
0 1 2 3 4 5 10 15 20 25 30 35
23,102 5,770 4,116 3,347 2,950 12,013
9,534 10,787 12,286 14,588 18,055 23,212
0.02260 0.00570 0.00414 0.00338 0.00299 0.00244 0.
00196 0.00224 0.00258 0.00311 0.00391 0.0512
40 45 50 55 60 65 70 75 80 85 90 95 99
30,788 41,654 56,709 76,420
99,889 123,334 138,566 134,217 103,554 56,634
18,566 2,886 125
0.00697 0.00977 0.01400 0.02030 0.02950 0.04270 0.
06100 0.08500 0.11400 0.14480 0.17200 0.24000 1.20
000
20
Random failures
Early failures
Wearout failures
0.2
0.15
Failure Rate r(t)
0.1
0.05
20
40
60
80
100
Failure rate r(t) versus t.
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Failure Rate, (faults/time)
Period of Approximately Constant failure rate
Infant Mortality
Old Age
Time
Figure 11-2 A typical bathtub failure rate
curve for process hardware. The failure rate is
approximately constant over the mid-life of the
component.
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Comments on Bathtub Curve

• Often units are tested before they are
  distributed to the users. Thus, much of the
  infant mortality will be removed before the units
  are delivered for use.
• For the majority of mechanical units, the failure
  rate will usually show a slightly increasing
  tendency in the useful life period.

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Failure Rate - r(t)

• The probability that the component fails per unit
  time at time t, given that the component has
  survived to time t.
• Example

The component with a constant failure rate is
considered as good as new, if it is functioning.
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As Good As New?

• This implies that the probability that a unit
  will be functioning at time tx, given that it is
  functioning at time t, is equal to the
  probability that a new unit has a time to failure
  longer than x. Hence the remaining life of a
  unit, functioning at time t, is independent of t.
  The exponential distribution has no memory.

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Relation Between Reliability and Failure Rate
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Interpretation of Failure Rate

• In actuarial statistics, the failure rate is
  called the force of mortality (FOM). The failure
  rate or FOM is a function of the life
  distribution of a single unit and an indication
  of the proneness of failure of the unit after
  time t has elapsed.

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Failure-Rate Experiment

• Split the time interval (0, t) into disjoint
  intervals of equal length dt. Then put n
  identical units into operation at time t0. When
  a unit fails, record the time and leave that unit
  out. For each interval, determine
• The number of units n(i) that fail in interval i.
• The functioning times of the individual units in
  interval i. If a unit has failed before interval
  i, its functioning time is zero.

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Failure-Rate Experiment
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Mean Time to Failure - MTTF

• Variance of Time to Failure

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Failure Rate
Failure Density
Unreliability
Reliability
1
1
Area 1
1 - F (t)
f (t)
F (t)
R (t)
0
0
t (a)
t (b)
t (c)
t (d)
Figure 11-1 Typical plots of (a) the failure
rate (b) the failure density f (t), (c) the
unreliability F(t), and (d) the reliability R (t).
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TABLE 11-1 FAILURE RATE DATA FOR VARIOUS
SELECTED PROCESS COMPONENTS1
Instrument

Fault/year Controller

0.29 Control valve

0.60 Flow measurement (fluids)

1.14 Flow measurement (solids)
3.75 Flow
switch
1.12 Gas -
liquid chromatograph
30.6 Hand valve

0.13 Indicator lamp

0.044 Level measurement
(liquids)
1.70 Level measurement (solids)

6.86 Oxygen analyzer

5.65 pH meter

5.88 Pressure measurement

1.41 Pressure relief valve

0.022 Pressure switch

0.14 Solenoid valve

0.42 Stepper motor

0.044 Strip chart recorder

0.22 Thermocouple temperature measurement
0.52 Thermometer
temperature measurement
0.027 Valve positioner

0.44 1Selected from Frank P. Lees, Loss
Prevention in the Process Industries (London
Butterworths, 1986), p. 343.
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Example

• Consider two independent components with failure
  rates?1and?2, respectively. Determine the
  probability that component 1 fails before
  component 2.
• Similarly,

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A System with n Components in Parallel

• Unreliability
• Reliability

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A System with n Components in Series

• Reliability
• Unreliability

35
Upper Bound of Unreliability for Systems with n
Components in Series
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The Poisson Process
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Assumptions of Homogeneous Poisson Process (HPP)

• Suppose we are studying the occurrence of a
  certain event A in the course of a given time
  period. Let us assume
• A can occur at any time in the interval. The
  probability of A occurring in the interval
  is independent of t and may be written
  as where is a positive
  constant.
• The probability of more than one event A in this
  interval is , which is a function with
  property
• Let (t11,t12, (t21,t22,be any sequence of
  disjoint intervals in the time period in
  question. Then the events A occurs in
  (tj1,tj2, j 1, 2, , are independent.

The process is said to have intensity?
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Probability of No Event Occurring in (0, t

• Let N(t) denotes the number of times the event A
  occurs during the period (0, t. Let

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Exponential Distribution

• Let T1 denotes the time point when A occurs for
  the first time. T1 is a random variable and

Thus, thee waiting time T between consecutive
occurrences in a HPP is exponentially distributed.
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Probability of n Events Occurring in (0, t
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Poisson Distribution

• Since
• This distribution is called the Poisson
  distribution with parameter and random
  variable n
• Notice that, since the expected number of
  occurrences of event A per unit time (t1) is
  , expresses the intensity of the process.

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Example 1

• Suppose that exactly one event (failure) of a HPP
  with intensity?is known to have occurred in the
  interval (0, t0. Determine the distribution of
  the time T1 at which this event occurred.

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Example 1
In other words, the time at which the first
failure occurs in uniformly distributed over
(0,t0. The expected time is thus
44
Example 2

• Suppose that the failure of a system are
  occurring in accordance with a HPP. Some failures
  develop into a consequence C, and others do not.
  The probability of this development is p and is
  assumed to be constant for each failure. The
  failure consequences are further assumed to be
  independent of each other. Determine the
  distribution of the consequences.

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Example 2
Thus, M(t) is also a HPP with intensity (p?)
. The mean number of C consequences in (0,t is
(p?t).
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Gamma Distribution

• Consider a unit that is exposed to a series of
  shocks which occur a HPP with intensity ?. The
  time intervals T1, T2, T3, , between consecutive
  shocks are then independent and exponentially
  distributed with parameter ?. Assume that the
  unit fails exactly at the kth shock, and not
  earlier. The time to failure of the unit
• is then gamma distributed (k, ?).

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Gamma Distribution

• Consider a homogeneous Poisson distribution,

The waiting time until the kth occurrence of
event A in a HPP with intensity ?is gamma
distributed. The gamma distribution (1, ?) is
an exponential distribution with parameter ?.
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Weibull Distribution

• For majority of mechanical units, the failure
  rates are slightly increasing in the useful life
  period (not constant). A distribution often used
  when r(t) is monotonic is the Weibull
  distribution. The time to failure T of a unit is
  said to be Weibull distributed with scale
  parameter?and shape parametera.

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Weibull Distribution
50
Weibull Distribution

1. a1 The Weibull distribution reduces to
   exponential distribution.
2. agt1 The failure rate is increasing.
3. alt1 The failure rate is decreasing.

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Weibull Distribution
Hence, the probability of time to failure larger
than 1/? is independent of a. This quantity is
called the Characteristic Lifetime.
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Three-Parameter Weibull Distribution
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Normal Distribution

• The normal distribution is sometimes used as a
  lifetime distribution, even though it allows
  negative TTF values with positive probability.
• The probability density of standard normal
  distribution is

54
Normal Distribution

• The unreliability, reliability and failure rate
  can be written as

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Normal Distribution, Left Truncated at 0
56
Log Normal Distribution

• The time to failure T of a unit is said to be
  lognormally distributed if Yln(T) is normally
  distributed.
• The lognormal distribution is commonly used as a
  distribution for repair time. When modeling the
  repair time, it is natural to assume that the
  repair rate in increasing in a first phase. When
  the repair has been going on for a rather long
  time, this indicates serious problems. It is
  thus natural to believe that the repair rate is
  decreasing after a certain period of time.

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Log Normal Distribution