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1

RELIABILITY AND MAINTAINABILITY ENGINEERING
EEE 571
2 UNITS

2

COURSE OUTLINE
1
...
1 The Importance of Reliability
1
...
2
...
2
...
2
...
3 Reliability Prediction Model: Laws of Probability Relevant to Reliability
Prediction
1
...
1

Multiplication Rule

1
...
2

Addition rule

1
...
3

Binomial distribution

1
...
5 The Reliability of a Parallel System
1
...
7 Reliability Measurement
1
...
1 Mean Time between Failures (MTBF)
1
...
1
...
7
...
2 MTBF of a Parallel System
1
...
1
...
7
...
8 Derivation of MTBF, MTTF and failure rate
1
...
9
...
9
...
9
...
10 Failure Pattern (The Bath-tub Curve) of Equipment
1
...
1 Early Failure Period
1
...
2 Constant Failure Period
1
...
3 Wearout Failure Period
1
...
0 Maintainability
2
...
2 Terminologies Relating To Maintainability
2
...
0 Availability
3
...
1
...
1
...
1
...
2 Availability of Items in Series and Parallel Combinations
4
...
1 Prototype Testing
4
...
3 Production Testing
4
...
0 Faults Analysis
5
...
1 Fault Tree Analysis
5
...
2
...
2
...




It is the probability of an item to perform a required function under specified
conditions for a stated period of time
...


It is worthy of note that other literature sources may define reliability by slightly different
statements, regardless of the approach, the three operative phrases “PERFORM A
REQUIRED FUNCTION”, “UNDER STATED CONDITION”, “FOR STATED
PERIOD OF TIME”, are always emphasized
...

Mathematically, reliability lies between 0 and 1
Reliability = 1
...

Reliability = 0
...

Reliability = 0
...
1 The Importance of Reliability
Unreliability has a number of unfortunate consequences and therefore for many
products and services is a serious threat
...
2 Configuration and Functionality of Systems
Functional ways in which componenets or systems are to be connected in order to work
are; series, parallel, and series-parallell connections
...

1
...
1 Series System
Simplest reliability model is a serial model where all the components must be working for the
system to be successful
...
Figure 2 represents the block diagram of a series system
...

A

B

Figure 2: Components Connected in Series
1
...
2 Parallel System
The parallel system is one which does not fail until the entire sub unit or components have
...

A

B

Figure 3: Components Connected in Parallel
1
...
3 Series-Parallel System
The series-parallel system is one which fails if the block in series connection fails or if both
blocks in parallel connection fail
...
Figure 4 represents a series-parallel connected system
...
3 Reliability Prediction Model: Laws of Probability Relevant to Reliability Prediction
Reliability prediction is the process of calculating the anticipated system reliability from
assumed component failure rates
...
It is necessary to represent
the system as a number of functional blocks
...
Usually, multiple components make up a system
and we often want to know the reliability of a system that uses more than one component
...
In calculating the reliability of a system made up of a number of sub-units each
having their own reliability, the type of system must first be defined
...
These laws are discussed below
...
3
...
If there are two sub-units A
and B connected in series, both sub-units can operate simultaneously
...
Hence
the probability of both sub units A and B operating successfully is
Pab = Pa x Pb

(1
...
1
...
e
...
,n) connected in series,
then we state the probability of successful operation of the new system as
Pan = Pa x Pb x … … … … … … x Pn

(1
...
1
...
3
...
2) is generally referred to as the multiplication rule of probabilities
...
85 and Pb = 0
...
85 x 0
...
765
1
...
2 Addition rule
From figure (3) there are only two sub-units A and B connected in parallel
...
3
...
1)

P(A or B) = 1 – [(1 – Pa)(1 – Pb)]

(1
...
2
...
or n) = 1- [(1-Pa)(1-Pb)……
...
3
...
3)

EXAMPLE
0
...
99
0
...
3
...
In n trials of such an
experiment, the distribution of the two classes of possible outcomes is discrete and of the
binomial type
...
95 for a single day, and it is expected to operate for 5 days (not continuously)
...


8

2) Find the possibility that the generating set will operate successfully for (i) 3 days; (ii)
at least 4 days
...



First, there are two possible outcomes, namely; success or failure
...
In other words, the probability of failure may be represented by (1-p)
...
e
...




Four, the probability distribution of the two possible outcomes, can be determined
through binomial expansion (p + q) 5
...

i) Probability that the set will operate successfully for 3 days = 10p3q2
...
95
and q = 1 - 0
...
05, therefore
10p3q2 = 10 x 0
...
052 = 0
...
e
P (at least 4 days) = p5 + 5p4q
= 0
...
954*0
...
978
1
...
If Rs is the reliability of a series system while R1, R2,……,
Rn are the respective reliabilities of the individual n sub-units of the system
...
R 2

(1
...
1)

If there are n components in series therefore
n

R s =R1
...
R n =  R i

(1
...
2)

i=1

If the series system is made up of exponential failing units, such that the sub-units reliabilities
are
R1 = e−λ1 t , R 2 = e−λ2 t , … … …
...
R2 …… Rn = e−λ1 t x e−λ2 t x … … …
...
+ λn

(1
...
3)

1
...
The reliability of the system Rp
(assume two sub-unit) is given as
Rp = 1 – [(1 – R1)(1 – R2)]

(1
...
1)

If there are n components in parallel, the reliability of a parallel system with n different subunits is equation as
Rp = 1 – [(1 – R1)(1 – R2)……
...
5
...
R n = e−λn t
Then the system reliability for a system with n sub-unit is given by
R P = 1 − [(1 − e−λ1 t )( 1 − e−λ2 t ) … (1 − e−λn t )]
For a system with two sub-units in parallel connection
RP2 = 1 – [(1 - R1)(1 - R2)] = R1 + R2 - R1R2
R P 2 = e−λ1 t + e−λ2 t − e−(λ1 +λ2 )t
For a system with three sub-units in parallel
Rp3 = 1 – [(1 - R1)(1 - R2)(1 - R3)]
Expanding and simplifying the expression on the RHS, we obtain
Rp3 = R1 + R2 + R3 - R1R2 – R2R3 - R1R3 - R1R2R3
Rp3 = e−λ1 t + e−λ2 t + e−λ3 t − e−(λ1 +λ2 )t − e−(λ3 +λ2 )t − e−(λ1 +λ3 )t − e−(λ1 +λ2 +λ3 +)t
Definitely, these equation are not of a simple exponential form, and therefore the
overall system reliability cannot be expressed in the form 𝑒 −𝜆𝑝 𝑡 , as we did in the of a series
system
...
6 The Reliability of a Series-Parallel System
Suppose we have a series-parallel system represented by the diagram shown below, we
need to find its reliability
Ra

R1

R2


...
Identify units in series within the system, and calculate the reliability of a single
equivalent unit, using the relationship
Rs = R1 x R2 x
...
x R∞ represents the reliability of each unit
ii
...
(1 – Rz)]

(ii)

iii
...
In this case, the series-parallel system reliability is
...
R∞)] x [1 – (1 – Ra)(1 – Rb)……
...
7 Reliability Measurement
Reliability can be quantified
...

1
...
1 Mean Time between Failures (MTBF)
Mean time between failures is applicable to items that can be repaired and returned to
use (repairable items)
...
It can be computed based
on two main types of test known as the non-replacement and replacement methods
...
T results in the observed MTBF are
given by;
T
n
(t1 − t 0 ) + (t 2 − t1 ) + (t n − t n−1 )
m=
n
(t n − t 0 )
m=
n
m=

Where,
n = number of failures
...
7
...
1)
(1
...
1
...
7
...
3)

12

EXAMPLE
Continuous tests were conducted on an electrical item and faults which were repaired
immediately occurred at the following times
...
Under this method it is assumed that the test time should be truncated (cut
off) before the items are subjected to failure possibly due to wear-out
...
Epstein demonstrated that the best estimate of MTBF, m,
for a truncated test is given by:
m=

test hours for failures + test hours for survivors
number of failures

(1
...
1
...
7
...
5)

That is
m=

EXAMPLE
In order to determine the MTBF of a certain component, 50 were tested for a period lasting
200 hours
...
35 components
survived without failure
...
Total test hours before failure
2
...
Total survival hours
4
...
Total test hours before failures (for components which failed)
= (6 × 100) + (5 × 140) + (4 × 175) = 2000 hours
2
...
Total survival hours = 2000 + 7000 = 9000 hours
4
...
7
...
1 MTBF of a series system
The MTBF of a series system, Ms, is equal to the reciprocal of the system failure rate
Ms =

1
1
=
λs
λ1 + λ2 + ⋯ …
...
7
...
1
...
7
...
1
...
7
...
1
...
7
...
2 MTBF of a Parallel System
The MTBF of a parallel system with two sub-units is;
∞

∞

MP 2 = ∫ R P 2dt = ∫ (R1 + R 2 − R1 R 2 )dt
0

0
∞

= ∫ (e−λ1 t + e−λ2 t − e−(λ1 +λ2 )t )dt
0

MP 2 =

1
1
1
+
−
λ1 λ2 (λ1 + λ2 )

Similarly, for a three unit system in parallel connection

14

MP 3 =

1
1
1
1
1
1
1
+
+
−
−
−
−
λ1 λ2 λ3 (λ1 + λ2 ) (λ2 + λ3 ) (λ1 + λ3 ) (λ1 + λ2 + λ3 )

Where, λ1 , λ2 , λ3 are the unit failure rates respectively
...
+
λ 2λ 3λ
nλ

MPn =

1
...
1
...

As shown in equations below
...
R∞ ) [1 – (1 – Ra)(1 – Rb) … …
...
7
...
It is the
average time an item may be expected to function before failure or the average time that
elapses until a failure occurs
...
g by applying certain electrical, mechanical, heat, or humidity conditions)
until all have failed
...
, tn) then the observed MTTF is
given by
∑N
i=1(t i − t 0 )
N
(t1 − t 0 ) + (t 2 − t 0 ) + ⋯ + (t N − t 0 )
MTTF =
N
MTTF =

where
t0 = starting (reference) time
(t1 – t0) = period to 1st failure
(t2– t0) = period to 2nd failure
(tN – t0) = period to Nth failure
N = total number of failed components

(1
...
2
...
7
...
2)

15

EXAMPLE
Life testing is made on six (non-repairable) electrical lamps and the following results were
obtained
...
8 Derivation of MTBF, MTTF and failure rate
1

Derivation of MTBF, MTTF and failure rate from the first principle i
...
The general
expression for MTBF, m, is
∞

m = ∫ R(t)dt

(1
...
1)

0

for the case when λ is constant, R = e-λt and equation (1
...
1) becomes
∞

m = ∫ e−λt dt
0
∞
1
= − [ e−λt ]
λ
0

=
MTBF, m =

1 −∞
1
[e − e−0 ] =
λ
λ

1
λ

(1
...
2)

If failures are due to chance and if the failure rate is constant, then it follows that
λ=

1
for non − repairable items
MTTF
or

1
for repairable items
MTBF
Therefore, MTBF can be expressed as the integral of reliability, with the limits of integration
λ=

from 0 to ∞
...
e
...
Compare the reliability of a series system with a parallel system, if each system contain 3
sub-units having reliabilities 0
...
85 and 0
...

2
...
The link is operative if one channel is working and
the reliability of the switching unit is 0
...
calculate the reliability for one year operating
period using
i
...
Two parallel channels
iii
...
An electrical power system consists of three sections connected in series
...
Calculate the MTBF of the system
4
...
Calculate the reliability (Rs) of the system shown in figure 2
ITEM C
R(t) = 0
...
9
ITEM B
R(t) = 0
...
95

ITEM A
R(t) = 0
...
95

Figure 2: A Series-Parallel System
1
...
It is a statistical event
...


17

1
...
1 Classification of Failures
Failure may be classified according to
i
...
Timing of failure
iii
...
A combination of (i) – (iii)
i
...
Misuse Failure: Failure attributable to the application of stresses beyond the stated
capabilities of the item
...
c mains to an equipment
specified for use with 110V a
...

b
...

ii
...
Sudden failure: Failure that could not be anticipated by prior examination
b
...
By Degree of Failures
a
...
By a Combination of Failure
a
...
Examples are (i)
blowing of fuse (ii) open-ciruit in wire-wound resistors or relay coils, or short-circuitcircuit failure in capacitors
b
...
An example is a
change in the value of the resistance of a resistor due to over-operational stress
...
9
...
The number of failures occuring per unit time is known as the failure rate
...
Strictly speaking, failure rate λ(t) is normally defined by the mathematical
relation,

λ(t) = lim
Δt  0

1
ΔNf
1
dNf
x
=
x
Ns
Δt
Ns
dt

where, Ns = number of surviving items after a life test
...
9
...
1)

18

Nf = number of failed items during the time interval, t

EXAMPLE
10 items have failed out of 1010 put on test during a period of 5000 hours
...

SOLUTION
λ=

1
Nf
x
Ns
t

λ=

1
10
x
1000 5000

i
...
0002 percent/hour

Note: Failure rate is most commonly expressed as a percentage per 1000 hours
...
2 percent/103 hrs
1
...
3 Reliability and Unreliability Equation and Curves
If failure rate is contant, the probability of no failure occurring in a given time is
R = e-λt

Where, R is the probability of no failure in time t i
...
The unreliability, Q is
defined as the probability of total failure
...
9
...
1)

Q = 1 - R = 1 - e-λt

(1
...
3
...
9
...
1
ideal

R, Q

Q = 1 - e-λt

R = e-λt
0

t

Figure 1
...
3
...
10 Failure Pattern (The Bath-tub Curve) of Equipment
The bath-tub curve is a representation of the reliability performance of components or nonrepaired items
...
If we observe the items over their lifetime
without replacement then we can observe three distinct shapes or periods
...
The bath-tub
curve is obtained from the addition of three curves
...
It reflects failure due to weak parts that escape final testing
...
Constant failure
curve represents a constant and relatively low failure that would occur if no weak parts of
equipment were present in the equipment i
...
, it occurs by chance
...
The addition of the three

Failure Rate, λ(t)

separate curves gives a composite curve which takes on the shape of a bathtub

Early
Failure
Period

Wearout
period

i

Useful life period

Early
Failure
curve

Constant failure curve

iii

ii

Wearout
curve
ii
iii

i

0

A

B

Time

Figure 1
...
1: The Bath-tub Curve
The bath-tub curves has three distinct phases in the life of equipment
...
Other names by which this
period is called are infant mortality’, and ‘burn in’, the second phase is called the ‘constant
failure’ period or ‘useful life’ period
...
The third phase, called the wear-out failure period
lasts for very few years or few thousands of continuous working hours
...


20

1
...
1 Early Failure Period
The most common causes of early failure are
Manufacturing Fault
Manufacturing faults are those which were not detected before the dispatch of an item to the
customer
...

Design Fault
Design faults are those which are caused by wrong or inaccurate design and which may be
undetected until equipment is sold out to a customer
...

Misuse fault
Misuse fault may be due to incompetent operation of an equipment, most probably by the
customer
...
Misuse fault may also occur by operating an equipment in a
hazardous environment for which it was not designed
...
e
...
g
...
The
drawing of an umbrella is most often used
...


Installation Fault
Incorrect or poor installation of an equipment or system may also cause an early failure
...
For the installation of every equipment there are specific precautions to be
taken in order to avoid equipment failure later
...
It is assumed that during this period the

21

customer should be free of repair cost in case of any failure except if it is due to misuse or
mechanical damage
...
10
...
This is the
period during which equipment is most usefully employed and failures which occur are
usually assumed to be stress related
...

1
...
3 Wearout Failure Period
Wear out failure is a period when an equipment or system comes to the end of it its useful life
period its failure may increase because, in addition to chance failure, parts starts to
deteriorates and wear out
...
The time interval that elapses before this
period starts may of course, be extended by planned maintenance and repairs
...

NOTE: The early failure period may be represented by a gamma distribution, the useful
period by an exponential (poisson distribution) or Weibull distribution and the wear out
period by a normal distribution
...
11 Some of the Failures Which the Designer/Manufacturer Will Not Be Held
Responsible


Faults due to lack of adherence to maintenance procedures and instructions
...




Faults caused by human intervention
...




Replacement item from the customer, immediately faulty when used
...




Faults caused by external influences outside the designed systems specifications
...




Fault analysis pending due to insufficient information
...


22

LECTURE TWO
MAINTAINABILITY
2
...
It can also be defined as the probability
that the item will regain its operational status after repair, under normal condition
...
Maintainability is embodied in the design of the product
...
Maintainability can be quantified and can assume a
dimensionless numerical figure between 0 and 1, except if expressed in percentage form
...
This is aimed at reducing the chances of a catastrophic failure
...
In all the sections every item
is put on maintenance schedule
...

Emergency maintenance: This is the type of maintenance applied when the system has
completely broken down
2
...
System location
...
Efficiency and skill of the maintenance crew
...
Provision of spare parts
2
...
A typical sequence of maintenance tasks comprises fault
localization, fault isolation, fault correction involving removal, replacement or

23

reassembly and alignment or adjustment, and checkout
...
Mathematically, MTTR is given as;
∑ ni λi t mi
ni λi

MTTR =

(2
...
1)

Where,
ni = quantity of similar parts
λi = parts failure rate
t mi = predicted maintenance operation time


Maintenance action (repair) rate (µ): is the number of maintenance actions that
can be carried out on a particular item per hour or average number of items
(equipments) which can be restored to normal working condition per hour,
assuming that there is no waiting between repair jobs
...
Determine maintainability of
1

the system for a time of 22 hours
...
5
1

= 1 − 0
...
918
Comment: The probability, M of the system being returned to a working condition within
1

22 hours is 0
...
8%)
...
3 Design and Methods for Improving Maintainability
The following cogent points are summarized below as contributory factors towards
improving maintainability:


Provision of maintenance check points which are easily accessible and hazard free
...
Keep it
simple
...




Provision of a well written maintenance and repair manual which can be easily
followed and understood
...




Provision of adequate spares
...




Consult the maintenance engineer during the design phase and agree upon a set of
documents to be handed over to the maintenance people
...




Reduce maintenance frequency overall by ruggedizing and over-specifying
components to withstand occasional overload
...
For
example on hot or heavy items or where there is stored mechanical or electrical
energy
...
Software can also be
made modular
...


25

LECTURE THREE
AVAILABILITY
3
...
Availability is affected by the failure rate and by maintenance time
...
1

The equation of (3
...

3
...

3
...
1

Steady- State Availability (Ass)

Steady state availability is the proportion of time that a system is available for use when the
overall period is of considerable duration
...
2)

(3
...
4)

On the other hand, system unavailability can be expressed as;
̅ ss = 1 − Ass
Steady state unavailability , A
̅ ss =
A
3
...
2

λ
λ+ µ

(3
...
Instantaneous availability can be expressed mathematically as;

26

A=

µ
λ
−
exp(λ+ µ)t
λ+ µ
λ+ µ

(3
...
6) and (3
...
7)
µ
λ+ µ

should be noted that if t is large, then equation (3
...
It

, which is the same as

equation (3
...
In other words, instantaneous availability becomes equal to steay-state
availability when time value is large (i
...
1
...
Here, one is concerned with the
availability of an system to function when it is requied for a particular military mission
...
2 Availability of Items in Series and Parallel Combinations
Availabilities of sub units can be combined in the same way as reliabilities
...
, An
respectively is given by
As = A1, A2, …
...
2
...
(1-An)]

(3
...
2)

EXAMPLE
An electrical generating set designed for continuous operation fails twice in a period of 123
days
...
Determine the following
parameters:
1
...
MTTR (days)
3
...
Total operating time = (123-3) = 120 days
Total number of failure = 2
Therefore MTBF =

120
2

= 60 days

27

2
...
Availability =
=

60
3
60 + 2

1

day or 1 2 day
MTBF
MTBF+MTTR

× 100%

= 97
...
Given that
UNIT

MTBF (Hours)

MTTR (Hours)

A

250

5

B

200

10

C

650

6

D

800

8

i
...
Calculate the availability of the system if two of the C modules are connected in
parallel
SOLUTION
i
...
95
= 0
...
99

Since the units are connected in series, the overall system availability is the product of
individual availability
Overall system availability = 0
...
95 * 0
...
99 = 0
...
922

28

ii) If two of the unit C modules are connected in parallel, then the whole system
arrangement will be configure as shown below
C
A

D

B
C

The resultant availability Acp of two C modulus connected in parallel is given by
A cp = 1 - 1-A c 

2

Where Ac = availability of one unit C
From the above calculation of (i),
Hence

Ac = 0
...
99   0
...
98 x 0
...
999 x 0
...
922

29

ASSIGNMENT
1
...

𝛍𝟏 𝛍𝟐
𝐀𝐧𝐬: 𝐀 𝐬𝐬(𝐒𝐞𝐫𝐢𝐞𝐬) =
(𝛍𝟏 + 𝛌𝟏 )(𝛍𝟐 + 𝛌𝟐 )
2
...

𝐀𝐧𝐬: 𝐀 𝐬𝐬(𝐒𝐞𝐫𝐢𝐞𝐬) =

𝛍𝟐
(𝛍 + 𝛌)𝟐

3
...
Find the steady- state availability of the
system
𝐀𝐧𝐬: 𝐀 𝐬𝐬 =

µ𝟐 + 𝟐𝛍𝛌
(𝛌 + µ)𝟐

30

LECTURE FOUR
TEST CHARACTERISTICS OF ELECTRICAL AND ELECTRONIC
COMPONENTS
4
...
They are; prototype testing, pre-production testing and production
testing
...
Both reliability
demonstration and acceptance testing are based on an agreement between the manufacturer
and the customer
...
1 Prototype Testing
Before a design goes into production, a prototype or few prototypes are first produced
and are thoroughly tested to prove that a design will meet the specification
...

It is very desirable that prototypes are tested under the correct environmental conditions
by competent staff who are independent of the designers
...

Since the designer should be free from the responsibility of testing the prototype of his own
design, therefore, the designer must be completely answerable for its performance and
reliability
...
As a result of prototype testing three types of errors may be revealed
...




Manufacturing errors: Where the prototype has not been made correctly, in line with
the design



Part faults: Where individual components parts of the prototype are found to be
defective
...
2 Pre-production Testing
Pre-production testing is normally done before bulk manufacture of the item is
embarked upon in order to ensure that the specifications can be met under normal

31

manufacturing conditions
...
Pre-production testing encompasses performance test,
environmental test, reliability test, maintainability test, packaging and transportation test
(involving vibration and shock tests), physical characteristic test (to consider the positioning
of operating knobs and switches etc
...
It will be
quite useful to examine in some detail some of the important tests already mention under
preproduction testing
...
The test can, of course be done in
actual working environment if it is so available
...
In practice, test equipment may include ovens
which can simulate various combination of temperature and humidity, refrigeration
plant and equipment to simulate various combinations of vibration, shock etc
...




Maintainability Test: Maintainability test is used in checking the relative speed
which it takes the same repair to be carried out on two alternative designs
...
This
parameter is particularly important in view of the desired availability of the product
...
This explains
why packaging and transport test is essential
...
This
test may involve vibration and shock tests and also storage test
...
c
...
Some devices

32

are designed to operate on a
...
100 - 110V, 115 - 127V, 200 – 220V, 50/60 Hz, etc
...

 Ergonomics Test
This test is aimed at determining the effect of interface of a pre-production model
with operators and maintenance personnel
...
It should be mentioned that the complexity of an equipment design,
number and location of operating switches and knobs constitute an important factor in
determining an operator’s convenience and accuracy in carrying out his job
...
3 Production Testing
Production testing is designed to verify the conformity of a finished product/equipment
with specification
...
At this stage, failures may be attributed to
component defect, production methods etc
...
Tests carried out during the production stage embrace virtually all parts of the
engineering inspection and/or quality departments
...
4 Reliability Demonstration and Acceptance Testing
Reliability demonstration and acceptance testing is associated with the post-production
stage
...
Reliability demonstration test is a test which is normally
carried out when a manufacturer, having produced a certain equipment, wishes to
demonstrate to customer that its reliability is at least as good as he claims it to be
...
In this case the customer will be said to be doing what is called a reliability
acceptance test
...
0 Fault Test in Electrical and Electronic Components
Fault test in electrical and electronic components has to do with

fault identification,

correction and tolerance to improve design
...

1
...
Fault tree analysis is a systematic way of identifying all
possible faults that could lead to system failure
...
The fault tree diagram for conducting a failure analysis for the system is
a diagram illustrating all the possible connection in which the system fails
...
The two most commonly used
gates in a fault tree are the AND and OR gates
...
If the occurrence of
either input event causes the output event to occur, then these input events are connected
using an OR gate
...
This can be illustrated using the diagram
below

Cooling system
overflow
Either event
can occur

Failure

OR

Fill mode
remains on

Valve
Stuck in
Open position

AND

Both events
Must occur

Basic events
Timeout
Control
fails

Sensor
fails

34

Fault Tree vs Reliability Block Diagram
TOP

1

(i)

1

3

2

3

2

TOP

1

(ii)
2
1

3

2

3

TOP

2

(iii)

1
3

G1
G1

1

3

2

The main element of a fault tree are:
1
...
Basic gates, they are the lowest level of identified causes
3
...
The Inpection Method
Consider the system shown below consisting of block boxes A, B, C
...
When block A, for instance operates satisfactorily, it is in
state a, when it has failed it is in state a , etc
...
Sn-1  the possible no of states the system below can assume
...

(i)

A

B

C

35

A

(ii)

B

C

(iii)

C

A

B

(iv)

A

B

C

Recall
R A (t) = e
R B (t) = e
R C (t) = e

-αt

-βt

-γt

SOLUTION
Since the system is always in one and only one of the m = 3 finite states, which are mutually
exclusive and which together exhaust all possibilities, then the possible no of states the
system described above can assume if each block were to be in one of two states are listed
below i
...
e-βt
...
 a + b 

System reliability, R s = P(s) = P(c)
...
b  + c

System reliability, R s = P(s) =  Pa
...
Rb(t)  + Rc(t)

=  e-αt
...
Solving By Event Space Method
Configuration (i)
The system functions satisfactorily if all the components operate satisfactorily
...
Rb(t)
...
e-βt
...
e a, b, c, ab, ac, bc, abc
...

Rs = P S4  + P S2  + P S1  + P S6  + P S5  + P S3  + P S7 

  

= 1 - P S0  = 1 - P a P b P c



but = P(a) = Ra(t); P a = 1 - Ra(t)
Rs = 1 - 1 - Ra(t) 1 - Rb(t) 1 - Rc(t) 

= 1 - 1 - e-αt 1 - e-βt 1 - e -γt 

Configuration (iii)
For the system to function satisfactorily, either of the blocks must function
satisfactorily CA, CB or CAB
Rs = P S3  + P S5  + P S7 

   
= P  a 
...
P  c  + P  a 
...
P  c  + P  a 
...
P  c 

= P abc + P abc + P  abc 

= 1- R a  t    R b  t    R c  t   +  R a  t   1- R b  t    R c  t   +  R a  t    R b  t    R c  t  
= 1- e-αt  e-βt  e-γt  +  e-αt 1- e-βt  e-γt  +  e -αt  e -βt  e -γt 



= 1- e-αt  e-β + γ

 + 1- e  e    + e 
-βt

- α+γ

- α + β + γ t



For Configuration (iv)
In this case, for the system to function satisfactorily, either of blocks AB&C function
satisfactorily
...

i
...
Give the logical expression for the malfunctioning of the same system
iii
...

iv
...

SOLUTION
Let s = reliability of the system
s = ae + bde + cde

s = e  a + bd + cd 
s = e a + d  b + c

Hence, using the parallel and series configuration approach, the block diagram is as shown
below
...
Either blocks AE, BDE, CDE of ABCDE
must be present
...
1 Methods of fault Analysis
The two methods that can be use in analyzing faults are Cut-Set Method and Tie-Set
Method
5
...
1 Cut-Set Method
All possible combination (SETs) under which the system will not function
Recall that by inspection, we got
Rs = e  a + d  b + c  
R f =R s = e  a + d  b + c   = e + ad + abc

41

Any of these combination when removed will cause the system to malfunction
...
2
...
The tie-set
method also known as success path is the opposite of the cut set method
...


R s = ae + edb + edc
If ae is working, then the system must work and vice versa

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