Identification of Transverse Crack in a Cracked Cantilever Beam by

Identification of Transverse Crack in a Cracked Cantilever Beam  by
Identification of Transverse
Crack in a Cracked Cantilever Beam
Using Fuzzy Logic and Kohonen Network
A Thesis Submitted in Partial Fulfillment of the Requirements for the
Award of the Degree of
Master of Technology
in
Mechanical Engineering
(Machine Design & Analysis)
by
Sasanka Choudhury
Roll No.209ME1259
NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA
राष्ट्रीय प्रौधोगिकी संस्थान राउरकेऱा
PIN-769008
ODISHA, INDIA
June 2011
Identification of Transverse
Crack in a Cracked Cantilever Beam
Using Fuzzy Logic and Kohonen Network
A Thesis Submitted in Partial Fulfillment of the Requirements for the
Award of the Degree of
Master of Technology
in
Mechanical Engineering
(Machine Design & Analysis)
by
Sasanka Choudhury
Roll No.209ME1259
Under the Supervision of
Dr. Dayal Ramakrushna Parhi
NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA
राष्ट्रीय प्रौधोगिकी संस्थान राउरकेऱा
PIN-769008
ODISHA, INDIA
June 2011
To Bhagawan Shri Sathyasai Baba & My beloved Parents
National Institute of Technology Rourkela
Certificate
This is to certify that the thesis entitled, “Identification of Transverse Crack in a
Cracked Cantilever Beam using Fuzzy Logic and Kohonen Network” submitted by Mr.
Sasanka Choudhury to National Institute of Technology Rourkela, is a record of bonafide
research work carried out by him under my supervision and is worthy of consideration for
the award of the degree of Masters of Technology in Mechanical Engineering with
specialization in Machine Design and Analysis. The embodiment of this thesis has not
been submitted to any other University and/or Institute for the award of any degree or
diploma.
Date:
Dr. Dayal Ramakrushna Parhi
Professor
Dept. of Mechanical Engineering
National Institute of Technology
Rourkela-769008
ACKNOWLEDGEMENT
First, I offer my humble pranam on the lotus feet of Bhagawan Sri Sathya Sai Baba for his
subtle support in all difficulties through out my life. Its my wishes to express my gratitude
towards my parents for their eternal effort.
I would like to express my greatest appreciation to my supervisor Dr. Dayal Ramakrushna
Parhi for his encouragement and guidance, timely suggestions during the completion of the
project. The discussions held with him during the research work, which helped me a lot in
materializing this dissertation work and also to grow up my knowledge. My sincere thank
also to Prof. R. K. Sahoo and Prof. N. Kavi, for their valuable suggestions and comments. I
mention the help and support received from my dearest friends. The guidance and support
received from all the members who contributed and who else part of this project, whose
names could have not been mentioned here, is highly acknowledged.
I owe my indebtedness to my younger brother Satish and elder brother Sanjib for their
moral support and encouragement. Last but not least, I deeply appreciate the help and
guidance of Prof. Sarat Kumar Patra in every time I need during my stay at NIT Rourkela.
Sasanka Choudhury
i
ABSTRACT
The issue of crack detection and diagnosis has gained wide spread industrial interest.
Crack/damage affects the industrial economic growth. Generally damage in a structural
element may occur due to normal operations, accidents, deterioration or severe natural
events such as earth quake or storms. Damage can be analyzed through visual inspection or
by the method of measuring frequency, mode shape and structural damping. Damage
detection by visual inspection is a time consuming method and measuring of mode shape as
well as structural deflection is difficult rather than measuring frequency. As Nondestructive method for the detection of crack is favorable as compared to destructive
methods. So, our analysis has been made on the basis of non-destructive methods with the
consideration of natural frequency. Here the crack is transverse surface crack. In the current
analysis, methodologies have been developed for damage detection of a cracked cantilever
beam using analytical, fuzzy logic, kohonen network as well as experimental. Theoretical
analysis has been carried out to calculate the natural frequency with the consideration of
mass and stiffness matrices. The data obtained from theoretical analysis has been fed to
fuzzy controller as well as the kohonen competitive learning network.
The Fuzzy Controller uses the different membership functions as input as well as output.
The input parameters to the Fuzzy Controller are the first three natural frequencies. The
output parameters of the fuzzy controller are the relative crack depth and relative crack
location. Several Fuzzy rules have been trained to obtain the results for relative crack depth
and relative crack location.
Kohonen network is nothing but a competitive learning network is used here for the
detection of crack depth and location. It is processed through a vector quantization
algorithm.
A comparative study has been made between fuzzy logic technique and Kohonen network
technique after experimental verification. It has been observed that the process of kohonen
network can predict the depth and location accurately as close to fuzzy logic technique.
ii
CONTENTS
Acknowledgement................................................................................................................... i
Abstract .................................................................................................................................. ii
Contents ................................................................................................................................. iii
List of Tables ......................................................................................................................... vi
List of Figures ...................................................................................................................... vii
Nomenclature ........................................................................................................................ ix
Chapter 1 Introduction .......................................................................................................... 1
1.1. Theme of Thesis .......................................................................................................... 2
1.2. Motivation of Work .................................................................................................... 4
1.3. Thesis Layout ............................................................................................................. 5
Chapter 2 Literature Survey ................................................................................................. 7
2.1. Introduction ................................................................................................................. 8
2.2. Overview .................................................................................................................... 8
2.3. Methodologies uses in the area of research on crack detection .................................. 9
Chapter 3 Theoretical vibration analysis for identification of crack ............................. 21
3.1. Introduction ............................................................................................................... 22
3.2. Local flexibility of a cracked cantilever beam under axial load and bending moment
......................................................................................................................................... 22
3.3. Analysis of vibration characteristics of a cracked cantilever beam........................... 25
3.3.1. Analysis of free vibration .............................................................................. 25
3.3.2. Analysis of forced vibration .......................................................................... 27
3.4. Finite element formulation ........................................................................................ 27
3.4.1. Theory............................................................................................................ 27
3.4.2. Governing equations ...................................................................................... 28
3.4.3. Process of crack detection ............................................................................. 30
Chapter 4 Analysis of Fuzzy Inference System for Identification of Crack .................. 31
4.1. Introduction ............................................................................................................... 32
iii
4.2. Fuzzy sets and membership functions ....................................................................... 33
4.3. Fuzzy inference system ............................................................................................. 33
4.3.1. Fuzzy linguistic variables .............................................................................. 34
4.3.2. Fuzzy controller/ Fuzzy If-then rule .............................................................. 35
4.3.3. Creating Fuzzy rules ...................................................................................... 35
4.3.4. Fuzzification .................................................................................................. 36
4.3.5. Defuzzification of output distribution ........................................................... 36
4.4. Fuzzy mechanism used for localization and identification of crack ......................... 37
4.5. Function of fuzzy controller for localization and identification of crack .................. 39
4.5.1. Discussion...................................................................................................... 45
4.5.2. Comparison of Results .................................................................................. 52
4.5.3. Summary........................................................................................................ 52
4.6. Why fuzzy logic is used ............................................................................................ 52
Chapter 5 Analysis of Kohonen Network for identification of crack .............................. 54
5.1. Introduction ............................................................................................................... 55
5.2. Essential processes of kohonen network/ the SOM training Algorithm ................... 56
5.2.1. Initialization ................................................................................................... 56
5.2.2. Competitive Process ...................................................................................... 57
5.2.3. Co-operative Process ..................................................................................... 58
5.2.4. Synaptic adaptation Process .......................................................................... 58
5.3. Mechanism ................................................................................................................ 59
5.3.1. Competition Mechanism ............................................................................... 59
5.3.2. Co-operative Mechanism............................................................................... 59
5.3.3. Adaptive Mechanism ..................................................................................... 60
5.4. Flow chart of Kohonen network ............................................................................... 61
5.5. Comparison of Results .............................................................................................. 62
5.5.1. Discussion...................................................................................................... 62
5.5.2. Summary........................................................................................................ 62
Chapter 6 Experimental Setup for identification of crack ............................................... 63
6.1. Introduction ............................................................................................................... 64
6.2. Experimental Setup ................................................................................................... 64
6.2.1. Instruments Used ........................................................................................... 65
iv
6.2.2. Description .................................................................................................... 68
6.3. Discussion.................................................................................................................. 68
Chapter 7 Results and Discussions ..................................................................................... 69
7.1. Introduction ............................................................................................................... 70
7.2. Discussions ................................................................................................................ 70
7.3. Comparison of Results .............................................................................................. 72
7.4. Characteristic Curves................................................................................................. 72
Chapter 8 Conclusions ......................................................................................................... 75
8.1. Conclusions ............................................................................................................... 76
8.2. Applications ............................................................................................................... 77
8.3. Scope for Future Work .............................................................................................. 77
References ............................................................................................................................ 78
Publications .......................................................................................................................... 83
v
LIST OF TABLES
Table: 4.1 Linguistic terms used for fuzzy membership functions ...................................... 46
Table: 4.2 Fuzzy rules for fuzzy inference system ............................................................... 47
Table: 4.3 Comparison of results between theoretical analysis and different fuzzy controller
analysis ................................................................................................................ 52
Table: 5.1 Comparison of results between theoretical analysis and Kohonen Network
technique.............................................................................................................. 62
Table: 6.1 Instruments used in the experimental analysis .................................................... 66
Table: 7.1 Comparison of results between theoretical analysis, experimental analysis, fuzzy
controller analysis and kohonen network technique............................................ 72
vi
LIST OF FIGURES
Figure: 3.1 Geometry of cracked cantilever beam ................................................................. 22
Figure: 3.2 Beam model ......................................................................................................... 25
Figure: 3.3 Representation of a single crack cantilever beam ............................................... 30
Figure: 4.1 Schematic diagram of operation of a fuzzy inference system ........................... 34
Figure: 4.2 Fuzzy controller architecture ............................................................................. 36
Figure: 4.3 Schematic diagram of fuzzy inference system ................................................... 39
Figure: 4.4 (a)Triangular Membership functions for relative natural frequency for first
mode of vibration ................................................................................................ 40
Figure: 4.4 (b) Triangular Membership functions for relative natural frequency for second
mode of vibration ................................................................................................ 40
Figure: 4.4 (c) Triangular Membership functions for relative natural frequency for third
mode of vibration ................................................................................................ 40
Figure: 4.4 (d) Triangular Membership functions for relative crack location ...................... 41
Figure: 4.4 (e) Triangular Membership functions for relative crack depth .......................... 41
Figure: 4.5 (a) Trapezoidal Membership functions for relative natural frequency for first
mode of vibration ................................................................................................ 41
Figure: 4.5 (b) Trapezoidal Membership functions for relative natural frequency for second
mode of vibration ................................................................................................ 42
Figure: 4.5 (c) Trapezoidal Membership functions for relative natural frequency for third
mode of vibration ................................................................................................ 42
Figure: 4.5 (d) Trapezoidal Membership functions for relative crack location.................... 42
Figure: 4.5 (e) Trapezoidal Membership functions for relative crack depth ........................ 42
Figure: 4.6 (a) Gaussian Membership functions for relative natural frequency for first mode
of vibration .......................................................................................................... 43
Figure: 4.6 (b) Gaussian Membership functions for relative natural frequency for second
mode of vibration ................................................................................................ 43
Figure: 4.6 (c) Gaussian Membership functions for relative natural frequency for third
mode of vibration ............................................................................................... 43
Figure: 4.6 (d) Gaussian Membership functions for relative crack location ........................ 43
Figure: 4.6 (e) Gaussian Membership functions for relative crack depth ............................ 44
vii
Figure: 4.7 (a) Hybrid Membership functions for relative natural frequency for first mode
of vibration .......................................................................................................... 44
Figure: 4.7 (b) Hybrid Membership functions for relative natural frequency for second
mode of vibration ................................................................................................ 44
Figure: 4.7 (c) Hybrid Membership functions for relative natural frequency for third mode
of vibration .......................................................................................................... 44
Figure: 4.7 (d) Hybrid Membership functions for relative crack location ........................... 45
Figure: 4.7 (e) Hybrid Membership functions for relative crack depth................................ 45
Figure: 4.8 Resultant values of relative crack depth and relative crack location of triangular
membership function when Rules 3 and 12 of Table: 4.2 are activated ............. 48
Figure: 4.9 Resultant values of relative crack depth and relative crack location of
trapezoidal membership function when Rules 3 and 12 of Table: 4.2 is activated
............................................................................................................................. 49
Figure: 4.10 Resultant values of relative crack depth and relative crack location of Gaussian
membership function when Rules 3 and 12 of Table: 4.2 are activated ............. 50
Figure: 4.11 Resultant values of relative crack depth and relative crack location of hybrid
membership function when Rules 3 and 12 of Table: 4.2 are activated ............. 51
Figure: 5.1 Possible Architectures in Kohonen Network ..................................................... 56
Figure: 5.2 Process of Initialization in Kohonen Network ................................................... 57
Figure: 5.3 Process of Competition in Kohonen Network ................................................... 57
Figure: 5.4 Process of Co-operative in Kohonen Network .................................................. 58
Figure: 5.5 Flow chart showing the processes of Kohonen Network .................................. 61
Figure: 6.1 (a) Pictorial view of complete assembly of Experimental setup1 ...................... 65
Figure: 6.1 (b) Pictorial view of complete assembly of Experimental setup2 ..................... 65
Figure: 7.1 Relative first natural frequencies versus relative crack depth............................ 72
Figure: 7.2 Relative first natural frequencies versus relative crack location........................ 73
Figure: 7.3 Relative second natural frequencies versus relative crack depth ....................... 73
Figure: 7.4 Relative second natural frequencies versus relative crack location ................... 73
Figure: 7.5 Relative third natural frequencies versus relative crack depth .......................... 74
Figure: 7.6 Relative third natural frequencies versus relative crack location ...................... 74
viii
NOMENCLATURE
a1
= Depth of crack
A
= Cross-sectional area of the beam
Ai i = 1to 12 = Unknown coefficients of matrix A
B
= Width of the beam
B1
= Vector of exciting motion
Cu
E
= ( )1 / 2

Cy
=(
E
= Young‟s modulus of elasticity of the beam material
fnf
= Relative first natural frequency
Fi i = 1, 2
= Experimentally determined function
i, j
= Variables
J
= Strain-energy release rate
EI 1 / 2
)

K1,i i = 1, 2 = Stress intensity factors for Pi loads
L
Cu
Ku
=
Ky
 L2
=
 Cy

Kij
= Local flexibility matrix elements
L
= Length of the beam
L1
= Location (length) of the crack from fixed end
Mi i=1, 4
= Compliance constant
Pi i=1, 2
= Axial force (i=1), bending moment (i=2)
Q
= Stiff-ness matrix for free vibration.
Q1
= Stiff-ness matrix for forced vibration
rcd
= Relative crack depth
rcl
= Relative crack location
1/ 2




ix
snf
= Relative second natural frequency
tnf
= Relative third natural frequency
ui i=1, 2
= Normal functions (longitudinal) ui(x)
x
= Co-ordinate of the beam
y
= Co-ordinate of the beam
Y0
= Amplitude of the exciting vibration
yi i=1, 2
= Normal functions (transverse) yi(x)
W
= Depth of the beam
ω
= Natural circular frequency
β
= Relative crack location

= Aρ
ρ
= Mass-density of the beam
ξ1
= Relative crack depth
V
= Aggregate (union)

= Minimum (min) operation

= For every
argj max
= Argument of maximum value
argj min

x

wj
= Argument of minimum value
 
L1
L
a1
W
= Input vector
= Weight Vector

h j,i ( x )
  d 2 j,i
= exp 
 2 2






h j,i ( x )
=Neighborhood Function
dj,i
= Lateral distance between the winning neuron „i‟ and excited neuron j

=Width of Gaussian function
( t )
 t 
 0 exp   
 1 
1
= Time Constant
t
= Number of iterations
x
t 
= Learning Rate
2
= Time Constant
BMU
= Best Matching Unit
xi
INTRODUCTION
1.1. Theme of Thesis
1.2. Motivation of Work
1.3. Thesis Layout
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Chapter 1
Introduction
CHAPTER 1
Introduction
1.1. Theme of Thesis
Damage is one of the important aspects in structural analysis because of safety reason as
well as economic growth of the industries. Generally damage in a structural element may
occur due to normal operations, accidents, deterioration or severe natural events such as
earth quake or storms. To achieve their industrial goal, now a days the plants as well as
industries are running round the clock fully. During operation, all structures are subjected
to degenerative effects that may cause initiation of structural defects such as cracks which,
as time progresses, lead to the catastrophic failure or breakdown of the structure. Thus, the
importance of inspection in the quality assurance of manufactured products is well
understood. To avoid the unexpected or sudden failure earlier crack detection is essential.
Taking this ideology into consideration crack detection is one of the most important
domains for many researchers. This is basically appears in the vibrating structures while
undergoes operations. The most common structural defect is the existence of a crack in
machine member. The presence of a crack could not only cause a local variation in the
stiffness but it could affect the mechanical behavior of the entire structure to a considerable
extent.
It has been observed that for damage/crack detection non-destructive testing is
preferable over destructive testing. Many researchers have carried out different nondestructive methodologies for crack detection but it has been observed that the vibration
based method is fast and inexpensive for crack/damage identification. Vibration- based
methods can be classified into two categories: linear and nonlinear approaches. Linear
approaches detect the presence of cracks in a target object by monitoring changes in the
resonant frequencies in the mode shapes or in the damping factors. Depending on the
assumptions, the type of analysis, the overall beam characteristics and the kind of loading or
excitation, a number of research papers containing a variety of different approaches have
been reported in the relevant literature. Damage can be analyzed through visual inspection
or by the method of measuring frequency, mode shape and structural damping. Damage
detection by visual inspection is a time consuming method and measuring of mode shape as
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Chapter 1
Introduction
well as structural deflection is difficult rather than measuring frequency. The presence of
crack induces local flexibility, which affects the dynamic behavior of the whole structure as
a result the reduction occurs in natural frequency and mode shape. By considering the
changes in those parameters crack can be identified in terms of crack depth and crack
location.
Many researches have been carried out their research works using open crack models,
which means they have considered that a crack remains open during vibration. The
assumption of an open crack leads to a constant shift of natural frequencies of vibration.
Numerous methodologies investigated over last few decades, however, indicate that a real
fatigue crack opens and closes during vibration. It exhibits non-linear behavior due to the
variation of the stiffness which occurs during the response cycle. As a result, a breathing
crack gives rise to natural frequencies falling between those corresponding to the open and
closed states. Therefore, if an always open crack is assumed, the decrease in experimental
natural frequencies will lead to an underestimation of the crack depth.
Beams are one of the most commonly used structural elements in several
engineering applications and experience a wide variety of static and dynamic loads. Cracks
may develop in beam-like structures due to such loads. Considering the crack as a
significant form of such damage, its modeling is an important step in studying the behavior
of damaged structures. Knowing the effect of crack on stiffness, the beam or shaft can be
modeled using either Euler-Bernoulli or Timoshenko beam theories. The beam boundary
conditions are used along with the crack compatibility relations to derive the characteristic
equation relating the natural frequency, the crack depth and location with the other beam
properties.
In the past few years the problem of health monitoring and fault detection of structures has
received considerable deliberation. The changes can be considered as an indication of the
health of the structure. Subsequently, these methods of fault detection are based on the
comparison of the vibrant response of the healthy structure with the active response of the
deserted structure. The evaluation is carried out through some algorithm, which employs the
modal data of the healthy and deserted structure. Therefore, the fault detection problem is in
need of the modal data for the healthy structure, the modal data for the deserted structure
and the algorithm that uses these data and provides information about the state of the
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Chapter 1
Introduction
structure. Each of these items has its own aspects and associated problems that affect the
results of the fault detection.
The most common structural defect is the existence of a crack. Cracks are present in
structures due to various reasons. The presence of a crack could not only cause a local
variation in the stiffness but it could affect the mechanical behavior of the entire structure to
a considerable extent. Cracks present in vibrating/rotating components could lead to
catastrophic failure. They may also occur due to mechanical defects. Another group of
cracks are initiated during the manufacturing processes. Generally they are small in sizes.
Such small cracks are known to propagate due to fluctuating stress conditions. If these
propagating cracks remain undetected and reach their critical size, then a sudden structural
failure may occur. Hence it is possible to use natural frequency measurements to detect
cracks. To help in a continuous safety assessment of a machine or structure it is very
necessary to constantly assess the health of its critical components. This calls for a
continuous assessment of changes in their static and/or dynamic behavior. The development
of a crack does not necessarily make a component instantly useless, but it is a signal that its
behavior has to be monitored more carefully. Such monitoring can play a significant role in
assuring an uninterrupted operation in service by the component.
A direct procedure is difficult for crack identification and unsuitable in some particular
cases, since they require minutely detailed periodic inspections, which are very costly. In
order to avoid these costs, researchers are working on more efficient procedure in crack
detection through vibration analysis.
1.2. Motivation of Work
The main objective of the present research work is to develop an organized structure for
damage detection of a cracked cantilever beam using fuzzy logic technique as well as
kohonen network technique. These techniques are usually called the intelligent techniques
because the techniques can be processed without human intervention.
It is necessary that structures must safely work during its service life but, damages
initiate a breakdown period on the structures which directly affect the industrial growth. It is
a recognized fact that dynamic behavior of structures changes due to presence of crack. It
has been observed that the presence of cracks in structures or in machine members lead to
operational problem as well as premature failure. A number of researchers around the earth
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Chapter 1
Introduction
are working on structural dynamics and particularly on dynamic characteristics of structures
with the presence of crack. The information available on dynamic characteristics of crack is
not so exhaustive for real application. Therefore an effort has been given to formulate some
Intelligent Techniques for localization and identification of crack in cantilever beam
structures. Due to presence of cracks the dynamic characteristics of structure changes. The
change in dynamic behavior has been utilized as one of the criteria of fault diagnosis for
structures. Major characteristics of the structure which undergo change due to presence of
crack are; natural frequencies, the amplitude responses due to vibration and the mode
shapes. The phases of the process plan for the present investigation are as follows:

Theoretical expressions have been developed for free and forced Vibration analysis of
the single cracked cantilever beam for the detection of crack depth and crack location.
In all these theoretical approach transverse crack has been analyzed.

Experimental Analysis has been performed to obtain the relative values of first, second
and third modal natural frequencies.

The data obtained from the theoretical as well as experimental analysis has been trained
to fuzzy controller for designing the rule base for the detection of crack depth and crack
location.

Using the same data the kohonen competitive learning network has been developed for
the localization and identification of crack.

Finally a comparative study has been made between fuzzy logic technique and
Kohonen network technique after experimental verification.
1.3. Thesis Layout
The research work has been delineated in this thesis by dividing eight chapters.
Following the Introduction presented in the Chapter 1, the Chapter 2 followed the literature
survey which contains the previous studies had been made in the analysis of cracked
structure using vibrational techniques, finite element analysis, fuzzy logic techniques, neuro
genetic techniques and the application of kohonen network.
Chapter 3 analyses the theoretical expressions obtained for free and forced vibration for
computing the localization and identification of crack of a single crack cantilever beam. In
this analysis the transverse crack has been taken into consideration.
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Chapter 1
Introduction
Chapter 4 defines the concept of the fuzzy logic and outlines the methodology used to
design an intelligent fuzzy logic controller to train the data obtain from the theoretical as
well as experimental analysis for prediction of relative crack location and relative crack
depth. A comparative study has been made between fuzzy logic technique and Kohonen
network technique after experimental verification.
Chapter 5 defines the concept of kohonen network. The data obtained from theoretical
analysis has been fed to the kohonen competitive learning network. Kohonen network is
nothing but a competitive learning network is used here for the detection of crack depth and
location. It is processed through a vector quantization algorithm.
Chapter 6 outlines the details of the developed experimental set-up for vibration analysis
along with the specifications.
Chapter 7 sketches the results and discussion of the chapters mentioned above.
Chapter 8 highlights the conclusions drawn from the research work and the scope of the
future work. The reference taken for the research work and paper published related to the
research area has been listed in the last section of the chapter.
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LITERTURE SURVEY
2.1. Introduction
2.2. Overview
2.3. Methodologies uses in the area of research on crack detection
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Chapter 2
Literature Survey
CHAPTER 2
Literature Survey
2.1. Introduction
Now a day‟s crack detection is one of the most important areas of the research. Most of
the researchers are doing their research work related to crack detection using various
techniques. This chapter highlights the various methodologies uses by researchers in their
era of research in the last few decades. The area of research basically includes the
theoretical approach, experimental verification and the intelligent techniques.
2.2. Overview
Cracks are a potential source of catastrophic failure in mechanical machines and in
civil structures and in aerospace engineering. To avoid the failure caused by cracks, many
researchers have performed extensive investigations over the years to develop structural
integrity monitoring techniques. Most of the techniques are based on vibration measurement
and analysis because, in most cases, vibration based methods can offer an effective and
convenient way to detect fatigue cracks in structures. It is always require that structures
must safely work during its service life, however damage initiates a breakdown period on
the structures. It is unanimous that cracks are among the most encountered damage types in
structures. Crack in structures may be hazardous due their dynamic loadings. So crack
detection plays an important for structural health monitoring applications.
Many researchers have used the free and forced vibration techniques for developing
procedures for crack detection. The eventual goal of this research is to establish new
methodologies which will predict the crack location and crack depth in a dynamically
vibrating structure with the help of intelligence technique with considerably less
computational time and high precision. This chapter recapitulates the previous works,
mostly in computational methods for structures, and discusses the possible ways for
research.
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Chapter 2
Literature Survey
2.3. Methodologies uses in the area of research on crack detection
Das and Parhi [1] have performed an analytical study on fuzzy inference system for
detection of crack location and crack depth of a cracked cantilever beam structure using six
input parameters to the fuzzy member ship functions. The six input parameters are
percentage deviation of first three natural frequencies and first three mode shapes of the
cantilever beam. The two output parameters of the fuzzy inference system are relative crack
depth and relative crack location. Experimental setup has been developed for verifying the
robustness of the developed fuzzy inference system. The developed fuzzy inference system
can predict the location and depth of the crack in a close proximity to the real results.
Suresh et al. [2] have considered the flexural vibration in a cantilever beam having a
transverse surface crack. He has stated that modal frequency parameters are analytically
computed for various crack locations and depths using a fracture mechanics based crack
model. These computed modal frequencies are used to train a neural network to identify
both the crack location and depth presented in this paper. First, the crack location is
identified with computed modal frequency parameters. Next, the crack depth is identified
with computed modal frequency parameters and the identified crack location.
Pawar et al. [3] have performed a composite matrix cracking model, which is
implemented in a thin-walled hollow circular cantilever beam using an effective stiffness
approach. Using these changes in frequencies due to matrix cracking, a genetic fuzzy system
for crack density and crack location detection is generated. The genetic fuzzy system
combines the uncertainty representation characteristics of fuzzy logic with the learning
ability of genetic algorithm. It is observed that the success rate of the genetic fuzzy system
in the presence of noise is dependent on crack density (level of damage). It is found that the
genetic fuzzy system shows excellent damage detection and isolation performance, and is
robust to presence of noise in data. Das and Parhi [4] have investigated the detection of the
cracks in beam structures using the fuzzy Gaussian inference technique. Fuzzy-logic
controller here used six input parameters and two output parameters. The input parameters
to the fuzzy controller are the relative divergence of the first three natural frequencies and
first three mode shapes in dimensionless forms. The output parameters of the fuzzy
controller are the relative crack depth and relative crack location in dimensionless forms.
The strain-energy release rate has been used for calculating the local stiffnesses of the beam
for a mode-I type of the crack. Several boundary conditions are outlined that take into
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Chapter 2
Literature Survey
account the crack location. The developed fuzzy controller can predict the location and
depth of the crack in close proximity with the real results.
Taghi et al. [5] have proposed a method in which damage in a cracked structure was
analyzed using genetic algorithm technique. For modeling the cracked-beam structure an
analytical model of a cracked cantilever beam was utilized and natural frequencies were
obtained through numerical methods. A genetic algorithm is utilized to monitor the possible
changes in the natural frequencies of the structure. The identification of the crack location
and depth in the cantilever beam was formulated as an optimization problem. Bakhary et al.
[6] applied Artificial Neural Network for damage detection. In his investigation an ANN
model was created by applying Rosenblueth‟s point estimate method verified by Monte
Carlo simulation, the statistics of the stiffness parameters were estimated. The probability of
damage existence (PDE) was then calculated based on the probability density function of
the existence of undamaged and damaged states. The developed approach was applied to
detect simulated damage in a numerical steel portal frame model and also in a laboratory
tested concrete slab. The effects of using different severity levels and noise levels on the
damage detection results are discussed. Maity and Saha [7] have presented a method called
damage assessment in structures from changes in static parameter using neural network. The
basic strategy applied in this study was to train a neural network to recognize the behavior
of the undamaged structure as well as of the structure with various possible damaged states.
When this trained network was subjected to the measured response; it was able to detect any
existing damage. The idea was applied on a simple cantilever beam. Strain and
displacement were used as possible candidates for damage identification by a back
propagation neural network and the superiority of strain over displacement for identification
of damage has been observed.
Structural damage detection using neural network with learning rate improvement
performed by Fang et al. [8] In this study, he has been explore the structural damage
detection using frequency response functions (FRFs) as input data to the back-propagation
neural network (BPNN).Neural network based damage detection generally consists of a
training phase and a recognition phase. Error back-propagation algorithm incorporating
gradient method can be applied to train the neural network, whereas the training efficiency
heavily depends on the learning rate. While various training algorithms, such as the
dynamic steepest descent (DSD) algorithm and the fuzzy steepest descent (FSD) algorithm,
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Literature Survey
have shown promising features (such as improving the learning convergence speed), their
performance is hinged upon the proper selection of certain control parameters and control
strategy. In this paper, a tunable steepest descent (TSD) algorithm using heuristics
approach, which improves the convergence speed significantly without sacrificing the
algorithm simplicity and the computational effort, is investigated. The analysis results on a
cantilevered beam show that, in all considered damage cases (i.e., trained damage cases and
unseen damage cases, single damage cases and multiple-damage cases), the neural network
can assess damage conditions with very good accuracy.
Saridakis [9] applied neural networks, genetic algorithms and fuzzy logic for the
identification of cracks in shafts by using coupled response measurements. In his research
the dynamic behavior of a shaft with two transverse cracks characterized by three measures:
position, depth and relative angle. Both cracks were considered to lie along arbitrary angular
positions with respect to the longitudinal axis of the shaft and at some distance from the
clamped end. A local compliance matrix of two degrees of freedom (bending in both the
horizontal and the vertical planes) was used to model each crack. The calculation of the
compliance matrix was based on established stress intensity factor expressions and was
performed for all rotation angles through a function that incorporated the crack depth and
position as parameters. Towards this goal, five different objective functions were proposed
and validated; two of these were based on fuzzy logic. More computational intelligence was
added through a genetic algorithm, which was used to find the characteristics of the cracks
through artificial neural networks that approximate the analytical model. Both the genetic
algorithm and the neural networks contributed to a remarkable reduction of the
computational time without any significant loss of accuracy. The final results showed that
the proposed methodology may constitute an efficient tool for real-time crack identification.
An approach for crack detection in beam like structures using RBF (Radial Basis Function)
neural network have been performed by Huijian et al. [10] with an experimental validation.
In the particular research a crack damage detection algorithm was presented using a
combination of global and local vibration-based analysis data as input in artificial neural
networks (ANNs) for location and severity prediction of crack damage in beam like
structures. Finite element analysis has been used to obtain the dynamic characteristics of
intact and damaged cantilever steel beams for the first three natural modes. In the
experimental analysis, several steel beams with six distributed surface bonded electrical
strain gauges and an accelerometer mounted at the tip have been used to obtain modal
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parameters such as resonant frequencies and strain mode shapes. Finally, the Radial Basis
Function ANNs have been trained using the data obtained from the numerical damage case
to predicate the severity and localization of the crack damage.
Buezasa et al. [11] has been investigating the crack detection in structural elements
by means of a generic algorithm optimization method. The present study deals with two and
three dimensional models to handle the dynamics of a structural element with a transverse
breathing crack. The methodology is not restricted to beam-like structures since it may be
applied to any arbitrary shaped 3D element. The crack is simulated as a notch or a wedge
with a unilateral Signorini‟s contact model. The contact may be partial or total. All the
simulations are carried out using the partial differential solver of the general purpose, finite
element code FlexPDE. A genetic algorithm (GA) optimization method is successfully
employed for the crack detection. The dynamic responses at some points of the damaged
structures are compared with the solution of the computational (FE) model using least
squares for each proposed crack depth and location. An objective function arises which is
then optimized to obtain an estimate of both parameters. Physical experiments were
performed with a cantilever damaged beam and the resulting data used as input in the
detection algorithm. Panigrahi et al. [12] has firstly formulate of an objective function for
the genetic search optimization procedure along with the residual force method are
presented for the identification of macroscopic structural damage in an uniform strength
beam. Two cases have been investigated here. In the first case the width is varied keeping
the strength of beam uniform throughout and in the second case both width and depth are
varied to represent a special case of uniform strength beam. The developed model requires
experimentally determined data as input and detects the location and extent of the damage
in the beam. Here, experimental data are simulated numerically by using finite element
models of structures with inclusion of random noise on the vibration characteristics. It has
been shown that the damage may be identified for the said problems with a good accuracy.
Chou et al. [13] stated that, the problem is initially formulated as an optimization
problem, which is then solved by using genetic algorithm (GA). Static measurements of
displacements at few degrees of freedom (DOFs) are used to identify the changes of the
characteristic properties of structural members such as Young's modulus and cross-sectional
area, which are indicated by the difference of measured and computed responses. In order to
avoid structural analyses in fitness evaluation, the displacements at unmeasured DOFs are
also determined by GA. The proposed method is able to detect the approximate location of
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the damage, even when practical considerations limit the number of on-site measurements
to only a few. Suh et al. [14] have presented a method to identify the location and depth of a
crack on a structure by using hybrid neuro-genetic technique. Feed-forward multi-layer
neural networks trained by back-propagation are used to learn the input (the location and
depth of a crack)–output (the structural eigen frequencies) relation of the structural system.
With this trained neural network, genetic algorithm is used to identify the crack location and
depth minimizing the difference from the measured frequencies. The problem of crack
identification in a beam when clamp rigidity is unknown performed by Horibe and Asano
[15]. The identification method is based on the genetic algorithm (GA) and the proposed
method is verified by numerical simulation.
Sahoo and Maity [16] stated that artificial neural networks (ANN) have been proved
to be an effective alternative for solving the inverse problems because of the patternmatching capability. But there is no specific recommendation on suitable design of network
for different structures and generally the parameters are selected by trial and error, which
restricts the approach context dependent. A hybrid neuro-genetic algorithm is proposed in
order to automate the design of neural network for different type of structures. The neural
network is trained considering the frequency and strain as input parameter and the location
and amount of damage as output parameter. Damage detection methods of structures based
on changes in their vibration properties have been widely employed during the last two
decades. Existing methods include those based on examination of changes in natural
frequencies, mode shapes or mode shape curvatures. Doebling et al. [17] published a stateof the- art review on vibration based damage identification methods. Messina et al. [18]
used the sensitivity and a statistical-based method to structural damage detection. Kosmatka
and Ricles [19] presented the modal vibration characterization method using the vibratory
residual forces and weighted sensitivity analysis. Ratcliffe [20] performed the frequency
and curvature based experiments. Vestroni and Capecchi [21] presented the method for
concentrated damage detection based on natural frequency measurement. Gawronski and
Sawicki [22] used the method based on modal and sensor norms. Hu et al. [23] presented a
method using quadratic programming. Law et al. [24] presented a method for large-scale
structures using super-elements with the concept of damage detection orientation modelling.
Sahin and Shenoi [25] have presented a damage detection algorithm using a combination of
global and local vibration-based data as input to artificial neural networks (ANNs) for
location and severity prediction of the damage.
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Chapter 2
Literature Survey
Sadettin Orhan [26] has been performed an analysis of free and forced vibration of a
cracked beam in order to identify the crack in a cantilever. Single- and two-edge cracks
were evaluated. The study results suggest that free vibration analysis provides suitable
information for the detection of single and two cracks, whereas forced vibration can detect
only the single crack condition. However, dynamic response of the forced vibration better
describes changes in crack depth and location than the free vibration in which the difference
between natural frequencies corresponding to a change in crack depth and location only is a
minor effect. Mei et al. [27] has been presented his research work on wave vibration
analysis of an axially loaded cracked Timoshenko beam. It includes the effects of axial
loading, shear deformation and rotary inertia. From wave standpoint, vibrations propagate,
reflect and transmit in a structure. The transmission and reflection matrices for various
discontinuities on an axially loaded Timoshenko beam are derived. Such discontinuities
include cracks, boundaries and changes in section. The matrix relations between the injected
waves and externally applied forces and moments are also derived. These matrices are
combined to provide a concise and systematic approach to both free and forced vibration
analyses of complex axially loaded Timoshenko beams with discontinuities such as cracks
and sectional changes. The systematic method is illustrated using several numerical
examples.
Viola et al. [28] have investigated the changes in the magnitude of natural frequencies
and modal response introduced by the presence of a crack on an axially loaded uniform
Timoshenko beam using a particular member theory. A new and convenient procedure
based on the coupling of dynamic stiffness matrix and line-spring element is introduced to
model the cracked beam. The application of the theory is demonstrated by two illustrative
examples of bending–torsion coupled beams with different end conditions, for which the
influence of axial force, shear deformation and rotatory inertia on the natural frequencies is
studied. Moreover, a parametric study to investigate the effect of the crack on the modal
characteristics of the beam is conducted. A theoretical and experimental dynamic behavior
of different multi-beams systems containing a transverse crack has been presented by
Saavedra and Cuitino [29]. The additional flexibility that the crack generates in its vicinity
is evaluated using the strain energy density function given by the linear fracture mechanic
theory. Based on this flexibility, a new cracked finite element stiffness matrix is deduced,
which can be used subsequently in the FEM analysis of crack systems. The proposed
element is used to evaluate the dynamic response of a cracked free–free beam and a U-
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frame when a harmonic force is applied. The resulting parametrically excited system is nonlinear and the equations of motion are solved using the Hilbert, Hughes and Taylor
integration method implemented using a Matlab software platform.
Kisa and Gurel [30] have investigated the, a novel numerical technique applicable to
analyses the free vibration analysis of uniform and stepped cracked beams with circular
cross section. In this approach in which the finite element and component mode synthesis
methods are used together, the beam is detached into parts from the crack section. These
substructures are joined by using the flexibility matrices taking into account the interaction
forces derived by virtue of fracture mechanics theory as the inverse of the compliance
matrix found with the appropriate stress intensity factors and strain energy release rate
expressions. To reveal the accuracy and effectiveness of the offered method, a number of
numerical examples are given for free vibration analysis of beams with transverse nonpropagating open cracks. Numerical results showing good agreement with the results of
other available studies, address the effects of the location and depth of the cracks on the
natural frequencies and mode shapes of the cracked beams.
An identification procedure to determine the crack characteristics (location and size
of the crack) from dynamic measurements has been developed and tested by Shen and
Taylor [31]. This procedure is based on minimization of either the “mean-square” or the
“max” measure of difference between measurement data (natural frequencies and mode
shapes) and the corresponding predictions obtained from the computational model.
Necessary conditions are obtained for both formulations. The method is tested for simulated
damage in the form of one-side or symmetric cracks in a simply supported Bernoulli-Euler
beam. The sensitivity of the solution of damage identification to the values of parameters
that characterize damage is discussed. Crack detection in beam-like structures has been
presented by Rosales et al. [32]. Two approaches are herein presented: The solution of the
inverse problem with a power series technique (PST) and the use of artificial neural
networks (ANNs). Cracks in a cantilever Bernoulli Euler (BE) beam and a rotating beam are
detected by means of an algorithm that solves the governing vibration problem of the beam
with the PST. The ANNs technique does not need a previous model, but a training set of
data is required. It is applied to the crack detection in the cantilever beam with a transverse
crack. The first methodology is very simple and straightforward, though no optimization is
included. It yields relative small errors in both the location and depth detection. When using
one network for the detection of the two parameters, the ANNs behave adequately.
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However better results are found when one ANN is used for each parameter. Finally, a
combination between the two techniques is suggested.
Gounaris and Dimarogonas [33] performed a finite element of a cracked prismatic
beam for structural analysis. A surface crack on a beam section introduces a local flexibility
to the structural member. In order to model the structure for FEM analysis, a finite element
for the cracked prismatic beam is developed. Strain energy concentration arguments lead to
the development of a compliance matrix for the behavior of the beam in the vicinity of the
crack. This matrix is used to develop the stiffness matrix for the cracked beam element and
the consistent mass matrix. The element developed can be used in any appropriate matrix
analysis of structures program. The element was used to evaluate the dynamic response of a
cracked cantilever beam to harmonic point force excitation. Resonant frequencies and
vibration amplitudes are considerably affected by the existence of moderate cracks.
Onard et al. [34] has been presented free-vibration behavior of a cracked cantilever
beam and crack detection .This study is based on cracks that occurred in metal beams
obtained under controlled fatigue-crack propagation. The beams were clamped in a heavy
vise and struck in order to obtain a clean impulse modal response. Spectrograms of the freedecay responses showed a time drift of the frequency and damping: the usual hypothesis of
constant modal parameters is no longer appropriate, since the latter are revealed to be a
function of the amplitude. Signal processing such as the worm transform and phase
spectrogram methods have been developed with enough accuracy to display the behavior of
an uncracked beam where a slight non-linear stiffness is generated by the clamping.
Moreover, extracted worms show that the second mode of a beam with a deep crack is
modulated in frequency by the "first mode. In fact, the dominant mode opens and closes the
crack, thereby modulating the beam stiffness, which affects higher modal frequencies. With
deep cracks, three vibration states are observed: one where the crack is alternately fully
open and fully closed, a second with a crack partially opened, and a third with alternating
force acting on a closed crack. In the latter case, the peak force is smaller than the intrinsic
closure load of the crack. The "first state is difficult for a small crack to reach since highamplitude excitation is required to fully open the crack. For crack detection purposes, the
damping criterion, harmonic distortion criterion and bispectrum appear less sensitive to
small cracks than the phase spectrogram and coherent-modulated power.
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Detection of the Crack in Cantilever Structures Using Fuzzy Gaussian Inference
Technique has been introduced by Das and Parhi [35]. Detection of the cracks in beam
structures using the fuzzy Gaussian inference technique has been investigated by Das and
Parhi. Fuzzy-logic controller here used six input parameters and two output parameters.
The input parameters to the fuzzy controller are the relative divergence of the first three
natural frequencies and first three mode shapes in dimensionless forms. The output
parameters of the fuzzy controller are the relative crack depth and relative crack location in
dimensionless forms. The strain-energy release rate has been used for calculating the local
stiffness of the beam for a mode-I type of the crack. Several boundary conditions are
outlined that take into account the crack location. The developed fuzzy controller can
predict the location and depth of the crack in close proximity with the real results.
Chandrasekhar et al. [36] has stated that geometric and measurement uncertainty cause
considerable problem in damage assessment which can be alleviated by using a fuzzy logicbased approach for damage detection. Curvature damage factor (CDF) of a tapered
cantilever beam is used as damage indicators. Monte Carlo simulation (MCS) is used to
study the changes in the damage indicator due to uncertainty in the geometric properties of
the beam. Variation in these CDF measures due to randomness in structural parameter,
further contaminated with measurement noise, are used for developing and testing a fuzzy
logic system (FLS). Results show that the method correctly identifies both single and
multiple damages in the structure.
Das and Parhi [37] applied hybrid artificial intelligence technique for fault detection
in a cracked cantilever beam. The hybrid technique used here uses a fuzzy-neuro controller.
Here the fuzzy-neuro controller has two parts. The first part is comprised of the fuzzy
controller, and the second part is comprised of the neural controller. The input parameters of
the neural segment of the fuzzy-neuro controller are relative deviation of the first three
natural frequencies and relative values of percentage deviation for the first three mode
shapes, along with the initial outputs of the fuzzy controller. The output parameters of the
fuzzy-neuro controller are final relative crack depth and final relative crack location. The
results of the developed fuzzy-neuro controller and experimental method are in good
agreement. Hasanzadeh et al. [38] have been proposed an aligning method, which is
formalized by a fuzzy recursive least square algorithm as a learning methodology for
electromagnetic alternative current field measurement (ACFM) probe signals of a crack.
This method along with a set of fuzzy linguistic rules, including adequate adaptation of
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different crack shapes for combining knowledge and data whenever the superposition theory
can be utilized, provides a means to compensate for the lack of sufficient samples in
available crack databases. Salahshoor et al. [39] has been described that the issue of fault
detection and diagnosis has gained widespread industrial interest in process condition
monitoring applications. They have presented an innovative data-driven fault detection and
diagnosis methodology on the basis of a distributed configuration of three adaptive neurofuzzy inference system (ANFIS) classifiers for an industrial 440 MW power plant steam
turbine with once-through Benson type boiler. Each ANFIS classifier has been developed
for a dedicated category of four steam turbine faults. They have also conducted
experimental studies to realize such fault categorization scheme.
Kohonen et al. [40] have been proposed that the Self Organizing Map (SOM)
method is a new powerful software tool for the visualization of high dimensional data. They
explain that SOM converts complex, no-linear statistical relationship between high
dimensional data into simple geometric relationship on a two-dimensional display. It may
also be thought to produce some kind of abstraction. These two aspects visualization and
abstraction occur in a member of complex engineering task such as process analysis,
machine perception, control and communication. Cottrel et al. [41] have been performed an
analytical study on Kohonen Network and suggested that the Kohonen algorithm is a
powerful tool for data analysis. In that case they define a specific algorithm which provides
a simultaneous classification of the observation and of the modalities. Vesanto et al. [42]
have been proposed that the Self-Organizing Map (SOM) is a vector quantization method
which places the prototype vectors on a regular low-dimensional grid in an ordered fashion.
This makes the SOM a powerful visualization tool. Also its performance in terms of
computational load is evaluated and compared to a corresponding C program. Kauppinen et
al. [43] have been proposed a non-segmenting defect detection technique combined with a
self-organizing map (SOM) based classifier and user interface. They have tried to avoid the
problems with adaptive detection techniques, and to provide an intuitive user interface for
classification, helping in training material collection and labeling, and with a possibility of
easily adjusting the class boundaries. The approach is illustrated with examples from wood
surface inspection. Many researchers have been used this kohonen network in different area
of research but in this paper we have proposed the essential processes as well as the
mechanism followed in the Kohonen Network for the detection of crack depth and crack
location.
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Hasanzadeh et al. [44] have developed a methodology for sizing of surface cracks
by AC field measurement technique. They have developed an aligning method is formalized
by a fuzzy recursive least square algorithm as a learning methodology for electromagnetic
alternative current field measurement (ACFM) probe signals of a crack (data). This method
along with a set of fuzzy linguistic rules, including adequate adaptation of different crack
shapes (knowledge) for combining knowledge and data whenever the superposition theory
can be utilized, provide a means to compensate for the lack of sufficient samples in
available crack databases. Buezas et al. [45] have presented a technique for damage
detection in structural element using genetic algorithm optimization method with the
consideration of a crack contact model. In this technique the crack is simulated as a notch or
a wedge with a unilateral Signorini contact model. The contact can be partial or total. All
the simulations are carried out using the general purpose partial differential solver FlexPDE,
a finite element (FE) code. A genetic algorithm (GA) optimization method is successfully
employed for the crack detection. The dynamic responses at some points of the damaged
structures are compared with the solution of the computational (FE) model using least
squares for each proposed crack depth and location. An objective function arises which is
then optimized to obtain the parameters. The method is developed for bi- and threedimensional models to handle the dynamics of a structural element with a transverse
breathing crack. Nobahari and Seyedpoor [46] have developed a technique for Structural
damage detection using an efficient correlation-based index and a modified genetic
algorithm; they have developed an efficient optimization procedure to detect multiple
damage in structural systems. Natural frequency changes of a structure are considered as a
criterion for damage presence. In order to evaluate the required natural frequencies, a finite
element analysis (FEA) is utilized. A modified genetic algorithm (MGA) with two new
operators (health and simulator operators) is presented to accurately detect the locations and
extent of the eventual damage. An efficient correlation-based index (ECBI) as the objective
function for the optimization algorithm is also introduced.
A method for Structural damage detection using fuzzy cognitive maps (FCM) and
Hebian learning techniques have been developed by Beena and Ganguli [47]. In their
analysis a Structural damage is modeled using the continuum mechanics approach as a loss
of stiffness at the damaged location. A finite element model of a cantilever beam is used to
calculate the change in the first six beam frequencies because of structural damage. The
measurement deviations due to damage are fuzzified and then mapped to a set of faults
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using FCM. The input concepts for the FCM are the frequency deviations and the output of
the FCM is at five possible damage locations along the beam. The FCM works quite well
for structural damage detection for ideal and noisy data. Further improvement in
performance is obtained when an unsupervised neural network approach based on Hebbian
learning is used to evolve the FCM. Liu et al. [48] have stated a methodology for structural
damage diagnosis using neural network and feature fusion. In their analysis a structure
damage diagnosis method combining the wavelet packet decomposition, multi-sensor
feature fusion theory and neural network pattern classification was presented. Firstly,
vibration signals gathered from sensors were decomposed using orthogonal wavelet.
Secondly, the relative energy of decomposed frequency band was calculated. Thirdly, the
input feature vectors of neural network classifier were built by fusing wavelet packet
relative energy distribution of these sensors. Finally, with the trained classifier, damage
diagnosis and assessment was realized. The result indicates that, a much more precise and
reliable diagnosis information is obtained and the diagnosis accuracy is improved as well.
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THEORETICAL VIBRATION
ANALYSIS FOR
IDENTIFICATION OF CRACK
3.1. Introduction
3.2. Local flexibility of a cracked cantilever beam under axial load and
bending moment
3.3. Analysis of vibration characteristics of a cracked cantilever beam
3.3.1.
Analysis of free vibration
3.3.2.
Analysis of forced vibration
3.4. Finite element formulation
3.4.1.
Theory
3.4.2.
Governing equations
3.4.3.
Process of crack detection
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Chapter 3
Theoretical Vibration Analysis
CHAPTER 3
Theoretical Vibration Analysis
for Identification of Crack
n
3.1. Introduction
In this present research work it has been analyzed that the crack can be detected in the
various structures through visual inspection or by the method of measuring natural
frequency, mode shape and structural damping. As the measurement of natural frequency
and mode shape is quite easy as compared to other parameters, so in this chapter a logical
approach has been adopted to develop the expression to calculate the natural frequency and
the mode shape of the cantilever beam with the presence of a transverse crack and the effect
of natural frequency in the presence of crack. Experimental analysis has been done over
cracked cantilever beam specimen for validation of the theory established.
3.2. Local flexibility of a cracked cantilever beam under axial load and
bending moment
A cantilever beam with a transverse surface crack of depth „a1‟ on beam of width „B‟ and
height „W‟ is considered for the current research. The beam is subjected to axial force (P 1)
and bending moment (P2) (Fig.3.1) which gives coupling with the longitudinal and
transverse motion. The presence of crack introduces a local flexibility, which can be defined
in matrix form, the dimension of which depends on the degrees of freedom. Here a 2x2
matrix is considered.
Y- Axis
Z- Axis
P2
B
a1
da
W
P1
X- Axis
L1
L
Figure: 3.1 Geometry of Cracked Cantilever Beam
The strain energy release rate at the fractured section can be written as (Tada et al. [49])
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Chapter 3
J
=
Theoretical Vibration Analysis
1 1 v 2
1
(for plane strain condition);

(K I1  K I2 ) 2 , Where
E
E
E
1
(for plane stress condition)
E
(3.1a)
(3.1b)
Kl1, Kl2 are the stress intensity factors of mode I (opening of the crack) for load P1 and P2
respectively. The values of stress intensity factors from earlier studies are;
K I1 
P1
6P2
a
a (F1 ( )), K I 2 
BW
W
BW 2
a (F2 (
a
))
W
(3.2)
Where expressions for F1 and F2 are as follows
F1 (
a
2W
a 0.5  0.752  2.02 (a / W)  0.37 (1  sin(a / 2W)) 3 
),  (
tan(
)) 

cos(a / 2W)
W
a
2W


(3.3)
a
2W
a 0.5  0.923  0.199 (1  sin(a / 2W)) 4 
F2 ( ),  (
tan(
)) 

cos(a / 2W)
W
a
2W


Let Ut be the strain energy due to the crack. Then from Castigliano‟s theorem, the additional
displacement along the force Pi is:
ui 
U t
Pi
(3.4)
1
U t
da   J da
The strain energy will have the form, U t  
a
0
0
a1
Where J=
a
(3.5)
U t
the strain energy density function.
a
From (Equations 3.4 and 3.5), we will get
a

 1
ui 
  J(a ) da 
Pi  0

(3.6)
The flexibility influence co-efficient Cij will be, by definition
u i
2 1

J(a ) da
Pj Pi Pj 0
a
C ij 
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(3.7)
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Chapter 3
Theoretical Vibration Analysis
To find out the final flexibility matrix we have to integrate over the breadth „B‟
u i
 2  B / 2 a1
C ij 

  J(a ) da dz
Pj Pi Pj B / 2 0
(3.8)
Taking the value of strain energy release rate from the Equation (3.1a), the equation (3.8)
can be modifies as
B 2 1
(K l1  K l 2 ) 2 da


E Pi Pj 0
a
C ij 
Putting ξ = (a/w), d 
(3.9)
da
W
We get da = W dξ and when a = 0, ξ = 0; a = a1, ξ = a1/W = ξ1
Putting the above condition in Equation (3.9) we will get

C ij 
BW  2 1
(K l1  K l 2 ) 2 d


E Pi Pj 0
(3.10)
From the Equation (3.10), calculating C11, C12 (=C21) and C22 we get,
BW
C11 
E
1
C12  C 21 
C 22 

a
2 1
2
2
(
F
(

))
d


(F1 ()) 2 d
1
0 B2 W 2

BE 0
12 1
 F1 ()F2 () d
E BW 0
72
EBW 2
(3.11)
(3.12)
1
 F ()F () d
2
2
(3.13)
0
Converting the influence co-efficient into dimensionless form, we will get the final
expression as
C11  C11
E BW 2
BE 
EBW
C12  C12
 C 21 ; C 22  C 22
72
2
12
(3.14)
The local stiffness matrix can be obtained by taking the inversion of compliance matrix. i.e.
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Chapter 3
Theoretical Vibration Analysis
K 12   C11 C12 
K
K   11


K 21 K 22  C 21 C 22 
1
(3.15)
3.3. Analysis of Vibration Characteristics of a Cracked Cantilever Beam
3.3.1. Analysis of Free Vibration
A cantilever beam of length „L‟ width „B‟ and depth „W‟, with a crack of depth „a1‟ at a
distance „L1‟ from the fixed end is considered (Fig. 3.1). Taking u1(x,t) and u2(x,t) as the
amplitudes of longitudinal vibration for the sections before and after the crack and y1(x,t),
y2(x,t) are the amplitudes of bending vibration for the same sections (Fig. 3.2).
U2
U1
L1
L
Y1
Y2
Figure: 3.2 Beam Model
The normal function for the system can be defined as
u1 (x)  A1 cos (K u x)  A 2 sin(K u x)
(3.16a)
u 2 (x)  A 3 cos (K u x)  A 4 sin(K u x)
(3.16b)
y1 (x)  A 5 cosh (K y x)  A 6 sinh(K y x)  A 7 cos (K y x)  A 8 sin(K y x)
(3.16c)
y 2 (x)  A 9 cosh (K y x)  A10 sinh(K y x)  A11 cos (K y x)  A12 sin(K y x)
(3.16d)
Where x 
L
x
u
y
,u , y ,   1
L
L
L
L
E
L
, C u   
Ku 
Cu

1/ 2
 L2
, Ky 
 C
 y




1/ 2
 EI 
, C y   

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,  = A
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Chapter 3
Theoretical Vibration Analysis
Ai, (i=1, 12) Constants are to be determined, from boundary conditions. The boundary
conditions of the cantilever beam in consideration are:
u1 (0)  0 ;
(3.17a)
y1 (0)  0 ;
(3.17b)
y1 (0)  0 ;
(3.17c)
u 2 (1)  0 ;
(3.17d)
y"2 (1)  0 ;
(3.17e)
y2(1)  0 ;
(3.17f)
At the cracked section:
u '1 ()  u ' 2 () ;
(3.18a)
y1 ()  y 2 () ;
(3.18b)
y1()  y2 () ;
(3.18c)
y1()  y2()
(3.18d)
Also at the cracked section (due to the discontinuity of axial deformation to the left and
right of the crack), we have:
AE
du 1 (L1 )
 dy (L ) dy (L ) 
 K11 (u 2 (L1 )  u 1 (L1 ))  K12  2 1  1 1 
dx
dx 
 dx
Multiplying both sides of the above equation by
(3.19)
AE
we will get;
LK 11K 12
M1M 2 u ()  M 2 (u 2 ()  u1 ())  M1 ( y2 ()  y1 ())
(3.20)
Similarly at the crack section (due to the discontinuity of slope to the left and right of the
crack)
EI
d 2 y 1 ( L1 )
 dy (L ) dy (L ) 
 K 21 (u 2 (L1 )  u 1 (L1 ))  K 22  2 1  1 1 
2
dx 
dx
 dx
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(3.21)
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Chapter 3
Theoretical Vibration Analysis
Multiplying both sides of the above equation by
EI
we will get,
L K 22 K 21
2
M 3 M 4 y1()  M 3 (u 2 ()  u1 ())  M 4 ( y2 ()  y1 ())
Where, M1 
(3.22)
AE
AE
EI
EI
, M2 
, M3 
, M4  2
LK 22
LK 11
K 12
L K 21
The normal functions as stated in Equation (3.16) along with the boundary conditions as
mentioned above equations (3.17) and (3.18) yield the characteristic equation of the system
as: Q  0
(3.23)
This determinant is a function of natural circular frequency (ω), the relative location of the
crack (β) and the local stiffness matrix (K) which in turn is a function of the relative crack
depth (a1/W).
3.3.2. Analysis of Forced Analysis
If the cantilever beam with transverse crack is excited at its free end by a harmonic
excitation (Y = Y0 sin(ωt) ), the non-dimensional amplitude at the free end may be
expressed as y 2 (1) 
y0
 y 0 . Therefore the boundary conditions for the beam remain same
L
as before except the boundary condition
y2(1)  0 which modified as y 2 (1)  y 0 . The
constants Ai, i=1, to 12 are then computed from the algebraic condition
Q1D=B1
(3.24)
Q1 is the (12 x 12) matrix obtained from boundary conditions as mentioned above,
D is a column matrix obtained from the constants,
B1 is a column matrix, transpose of which is given by, B1T  0 0 0 y 0 0 0 0 0 0 0 0 0
(3.25)
3.4. Finite Element Formulation
3.4.1. Theory
The beam with a transverse edge crack is clamped at left end, free at right end and has uniform
structure with a constant rectangular cross-section of 800 mm X 38 mm X 6 mm. The EulerBernoulli beam model is assumed for the finite element formulation. The crack in this particular
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Chapter 3
Theoretical Vibration Analysis
case is assumed to be an open crack and the damping is not being considered in this theory. Both
single and double edged crack are considered for the formulation.
3.4.2. Governing Equations
The free bending vibration of an Euler-Bernoulli beam of a constant rectangular cross
section is given by the following differential equation as given in:
d4y
EI
 m2 y  0
i
4
dx
(3.26)
Where „m‟ is the mass of the beam per unit length (kg/m), „ωi‟ is the natural frequency of
the ith mode (rad/sec), E is the modulus of elasticity (N/m2) and I is the moment of inertia
m2
4
i equation is rearranged as a fourth-order differential equation
(m ). By defining  
EI
4
as follows:
d4y
 4 y  0
dx 4
(3.27)
The general solution to equation is
y  A cos i x  B sin i x  C cosh  i x  D sinh i x
(3.28)
Where A, B, C, D are constants and „λi‟ is a frequency parameter. Adopting Hermitian
shape functions, the stiffness matrix of the two-noded beam element without a crack is
obtained using the standard integration based on the variation in flexural rigidity.
The element stiffness matrix of the un cracked beam is given as
Ke    [B(x)]T EI[B(x)]dx
(3.29)
[B(x)]  {H1(x)H 2 (x)H3 (x)H 4 (x)}
(3.30)
Where [H1(x), H 2 (x), H3 (x), H 4 (x)] is the Hermitian shape functions defined as
3x 2 2x 3
2x 2 x 3
, H 2 (x)  x 
H1 ( x )  1 


l
l2
l3
l2
H3 (x) 
3x 2 2x 3
x 2 x3
, H 4 (x)  


l
l2
l3
l2
(3.31)
Assuming the beam rigidity EI is constant and is given by EI0 within the element, and then
the element stiffness is
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Chapter 3
Theoretical Vibration Analysis
6l  12 6l 
 12

6l 4l 2  6l 2l 2 
EI
[K e ]  0 
l 3  12  6l 12  6l

2  6l 4l 2 
 6l 2l

(3.32)
[K ec ]  [K e ]  [K c ]
(3.33)
 
 
Here, K ec  Stiffness matrix of the cracked element, K e  Element stiffness matrix,
K c   Reduction in stiffness matrix due to the crack.
According to (Peng et al. [50]), the matrix [Kc] is
K12
 K11
K
K 22
[K c ]   12
 K11  K12

K 24
 K14
 K11
 K12
K11
 K14
K14 
K 24 
 K14 

K 44 
(3.34)
2
12E(I 0  I c )  2l3c
 2L1  

Where, K11 
 3l c 
 1
2
 L2
L4
L

 

2

2 

3
6
L
12E(I 0  I c )  l c
5L

 
K14 
 lc  2  1  1  

2
L

L3
L2  
L

 

2

2 

6
L
12E(I 0  I c )  l 3c
7
L

 
K12 
 lc  2  1  1  

2
L

L3
L2  
L

 

2

2 

9
L
12E(I0  Ic )  3l3c
9
L

 
K 24 
 2lc  2  1  1  

2
L

L2
L2  
L



2
12E(I 0  I c )  3l3c
 3L1
 

K 22 
 2l c 
 2  ,
L
 L2

 
L3

12E(I 0  I c )  3l3c
 3L


K 44 
 2l c  1  1
 L2
 L

L2
Here, lc=1.5W, L=Total length of the beam, L1=Distance between the left node and crack
BW 3
B( W  a ) 3
I0 
 Moment of inertia of the beam cross section, Ic 
 Moment of
12
12
inertia of the beam with crack.
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Chapter 3
Theoretical Vibration Analysis
It is supposed that the crack does not affect the mass distribution of the beam. Therefore, the
consistent mass matrix of the beam element can be formulated directly as
Me   1 A[H(x)]T H(x)dx
(3.35)
0
22l
 156
 22l
4l 2
Al 
e
M 
13l
420  54

2
 13l  3l
 
 13l 
13l  3l 2 
156  22l 

 22l 4l 2 
54
(3.36)
The natural frequency then can be calculated from the relation.
[2 M  K]{q}  0
(3.37)
Where,
q=displacement vector of the beam
3.4.3. Process of Crack Detection
Detection of crack in a cantilever beam has been performed in two methods. First the finite
element model of the cracked cantilever beam is developed and the beam is discretized into
a number of elements, and the crack position is assumed to be in each of the elements. Next,
for each position of the crack in each element, depth of the crack is varied. Modal analysis
for each position and depth is then performed to determine the natural frequencies of the
beam.
Y- Axis
Z- Axis
P2
B
W
P1
X- Axis
L1
L
Figure: 3.3 Representation of a Single Crack Cantilever Beam
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ANALYSIS OF FUZZY
INFERENCE SYSTEM FOR
IDENTIFICATION OF CRACK
4.1. Introduction
4.2. Fuzzy sets and membership functions
4.3. Fuzzy inference system
4.3.1.
Fuzzy linguistic variables
4.3.2.
Fuzzy controller/ Fuzzy If-then rule
4.3.3.
Creating Fuzzy rules
4.3.4.
Fuzzification
4.3.5.
Defuzzification of output distribution
4.4. Fuzzy mechanism used for localization and identification of crack
4.5. Function of fuzzy controller for localization and identification of
crack
4.5.1.
Discussion
4.5.2.
Comparison of Results
4.5.3.
Summary
4.6. Why fuzzy logic is used
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Chapter 4
CHAPTER 4
Fuzzy Inference System
Analysis of Fuzzy Inference
System for Identification of Crack
4.1. Introduction
In order to develop a reliable, efficient and economical approach to increase the safety and
reduce the maintenance cost of elastic structures, many researchers have been developed
various structural health monitoring techniques. Although improved design methodologies
have significantly enhanced the reliability and safety of structures, it is still not possible to
build a structure that has zero percent probability of failure. In the recent years, researchers
have motivated to words the development of intelligent techniques for the fault detection.
This current research presents methodologies for structural damage detection and
assessment using fuzzy interface system. In this chapter an intelligent controller has been
projected for localization and identification of crack.
Fuzzy sets originated in the year 1965 and this concept was proposed by Lofti A. Zadeh.
Since then it has grown and is found in several application areas. According to Zadeh, The
notion of a fuzzy set provides a convenient point of departure for the construction of a
conceptual framework which parallels in many respects of the framework used in the case
of ordinary sets, but is more general than the latter and potentially, may prove to have a
much wider scope of applicability, specifically in the fields of pattern classification and
information processing.” Fuzzy logics are multi-valued logics that form a suitable basis for
logical systems reasoning under uncertainty or vagueness that allows intermediate values to
be defined between conventional evaluations like true/false, yes/no, high/low, etc. These
evaluations can be formulated mathematically and processed by computers, in order to
apply a more human-like way of thinking in the programming of computers. Fuzzy logic
provides an inference morphology that enables approximate human reasoning capabilities to
be applied to knowledge-based systems. The theory of fuzzy logic provides a mathematical
strength to capture the uncertainties associated with human cognitive processes, such as
thinking and reasoning. Fuzzy systems are suitable for uncertain or approximate reasoning,
especially for the system with a mathematical model that is difficult to derive. Fuzzy logic
allows decision making with estimated values under incomplete or uncertain information.
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Chapter 4
Fuzzy Inference System
4.2. Fuzzy Sets and Membership Functions
The concept of a fuzzy set is an extension of the concept of a crisp set. Similar to a
crisp set a universe set U is defined by its membership function from U to [0, 1]. Consider
U to be a non-empty set also known as the universal set or universe of discourse or domain.
A fuzzy set on U is defined as  A (x) : U  [0, 1] . Here  A is known as the membership
function, and  A ( x ) is known as the membership grade of x. Membership function is the
degree of truth or degree of compatibility. The membership function is the crucial
component of a fuzzy set. Therefore all the operations on fuzzy sets are defined based on
their membership functions.
The membership function is a graphical representation of the magnitude of participation of
each input. It associates a weighting with each of the inputs that are processed, define
functional overlap between inputs, and ultimately determines an output response. The rules
use the input membership values as weighting factors to determine their influence on the
fuzzy output sets of the final output conclusion. Once the functions are inferred, scaled, and
combined, they are defuzzified into a crisp output which drives the system. There are
different membership functions associated with each input and output response. Reasonable
functions are often piecewise linear function, such as triangular or trapezoidal functions.
The value for the membership function can be taken in the interval [0, 1]. When the
functions are nonlinear the Gaussian membership function will be taken for the smooth
operation.
4.3. Fuzzy Inference System
A fuzzy inference system is the process of formulating the mapping from a given input to an
output using fuzzy logic. To compute the output of this FIS for the given the inputs, we
must go through six steps:
 Determining a set of fuzzy rules.
 Fuzzifying the inputs using the input membership functions.
 Combining the fuzzified inputs according to the fuzzy rules to establish rule strength.
 Finding the consequence of the rule by combining the rule strength and the output
membership function.
 Combining the consequences to get an output distribution, and
 Defuzzifying the output distribution.
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Chapter 4
Fuzzy Inference System
Fuzzy inference systems have been successfully applied in fields such as automatic control,
fault diagnosis, data classification, decision analysis, expert systems, and computer vision.
If
Then
Rule Strength
And
And
Input Distributions
x0
Output Distributions
y0
Figure: 4.1 Schematic Diagram of Operation of a Fuzzy Interface System
4.3.1.Fuzzy Linguistic Variables
Just like an algebraic variable takes numbers as values, a linguistic variable takes words or
sentences as values. The set of values that it can take is called its term set. Each value in the
term set is a fuzzy variable defined over a base variable. The base variable defines the
universe of discourse for all the fuzzy variables in short. In short the hierarchy is as follows:
Linguistic variable → Fuzzy variable → Base variable.
In 1973, Professor Lotfi Zadeh [51] proposed the concept of linguistic or "fuzzy"
variables. Think of them as linguistic objects or words, rather than numbers. Suppose that X
= "age." Then we can define fuzzy sets "young," "middle aged,"
and "old" that are
characterized by MFs  young( x) ,  middleaged (x) , and  old ( x ) , respectively. Just as a
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Chapter 4
Fuzzy Inference System
variable can assume various values, a linguistic variable "Age” can assume different
linguistic values, such as "young," "middle aged “and” old" in this case. If "age" assumes
the value of "young," then we have the expression "age is young," and so forth for the other
values.
4.3.2. Fuzzy Controller/ Fuzzy If-Then Rule
Fuzzy logic controllers are based on the combination of Fuzzy set theory and fuzzy logic.
Systems are controlled by fuzzy logic controllers based on rules instead of equations. This
collection of rules is known as the rule base usually in the form of IF-THEN-ELSE
statements. Here the IF part is known as the antecedent and the THEN part is the
consequent. The antecedents are connected with simple Boolean functions like AND, OR,
NOT etc., Figure 4.2 outlines a simple architecture for a fuzzy logic controller [52]. The
outputs from a system are converted into a suitable form by the fuzzification block. Once all
the rules have been defined based on the application, the control process starts with the
computation of the rule consequences. The computation of the rule consequences takes
place within the computational unit. Finally, the fuzzy set is defuzzified into one crisp
control action using the defuzzification module.
A fuzzy if-then rule (also known as fuzzy rule, fuzzy implication or fuzzy
conditional statement) assumes the form “if x is A then y is B”. Where A and B are
linguistic values defined by fuzzy sets on universes of discourse x and y respectively. Often
“x is A” is called the antecedent or premise, while “y is B” is called the consequence or
conclusion. (Some of the linguistic terms used are shown in Table: 4.1).
4.3.3. Creating Fuzzy Rules
Fuzzy rules are a collection of linguistic statements that describe how the FIS should make a
decision regarding classifying an input or controlling an output. Fuzzy rules are always
written in the following form:
if (input1 is membership function1) and/or (input2 is membership function2) and/or ….
then (outputn is output membership functionn).
For example: one could make up a rule that says: if temperature is high and humidity is high
then room is hot.
There would have to be membership functions that define what we mean by high
temperature (input1), high humidity (input2) and a hot room (output1). This process of
taking an input such as temperature and processing it through a membership function to
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Chapter 4
Fuzzy Inference System
determine what we mean by "high" temperature is called fuzzification. Also, we must define
what we mean by "and" / "or" in the fuzzy rule. This is called fuzzy combination.
Rule Base
Input
Output
Computational
Unit
Fuzzification
Defuzzification
Figure: 4.2 Fuzzy Controller Architecture
4.3.4. Fuzzification
The purpose of fuzzification is to map the inputs from a set of devices (for example sensors
or features of those sensors such as amplitude or spectrum) to values from 0 to 1 using a set
of input membership functions. In the schematic diagram shown in Figure 4.1, there are two
inputs, x0 and y0 shown at the lower left corner. These inputs are mapped into fuzzy numbers
by drawing a line up from the inputs to the input membership functions above and marking
the intersection point.
These input membership functions, can represent fuzzy concepts such as "large" or "small",
"old" or "young", "hot" or "cold", etc. The membership functions could then represent
"large" amounts of tension coming from a muscle or "small" amounts of tension. When
choosing the input membership functions, the definition of what we mean by "large" and
"small" may be different for each input.
4.3.5. Defuzzification of Output Distribution
In many situations, for a system whose output is fuzzy, it is easier to take a crisp decision if
the output is represented as a single scalar quantity. This conversion of a fuzzy set to single
crisp value is called defuzzification and is the reverse process of fuzzification.
There are two common methods for defuzzification generally followed:
Centroid Method: Also known as the centre of gravity or the centre of area method, it
obtains the centre of area (x*) occupied by the fuzzy set. It is given by the expression;
x* 
 ( x ) x dx
(For a continuous membership function)
 ( x ) dx
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(4.1)
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Chapter 4
Fuzzy Inference System
n
 x i ( x i )
x *  i 1
(For a discrete membership function)
n
 ( x i )
i 1
(4.2)
Here, „n‟ represents the numbers of element in the sample, xi‟s are the elements, and ( x i )
is its membership function.
Mean of maxima (MOM) defuzzification method: One simple way of defuzzifying the
output is to take the crisp value with the highest degree of membership. In case with more
than one element having the maximum value, the mean value of the maxima is taken. The
equation of the defuzzified value x* is given by;
 xi
x M
x*  i
M
(4.3)

Where M= x i (x i ) is equal to the height of fuzzyset
M is the cardinality of the set M. In the continuous case, M could be defined as
M= x i [-c, c] (x i ) is equal to the height of the fuzzyset
In such a case, the mean of maxima is the arithmetic average of mean values of all intervals
contained in M including zero length intervals.
The height of a fuzzy set A, i.e. h(A) is the largest membership grade obtained by
any element in that set.
4.4. Fuzzy Mechanism used for localization and identification of crack
The fuzzy controller has been developed (as shown in Figure: 4.3) where there are 3 inputs
and 2 outputs parameter. The natural linguistic representations for the input are as follows
Relative first natural frequency = “FNF”
Relative second natural frequency = “SNF”
Relative third natural frequency = “TNF”
The natural linguistic term used for the outputs are
Relative crack depth = “RCD”
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Chapter 4
Fuzzy Inference System
Relative crack length= “RCL”
Based on the above fuzzy subset the fuzzy rules are defined in a general form as follows:
If (FNF is FNFi and SNF is SNFj and TNF is TNFk) then (CD is CDijk and CL is CLijk)
Where i= 1to 9, j=1 to 9, k=1 to 9
(4.4)
Because of “FNF”, “SNF”, “TNF” have 9 membership functions each.
From the above expression (4.4), two set of rules can be written
If (FNF is FNFi and SNF is SNFj and TNF is TNFk) then CD is CDijk
(4.5a)
If (FNF is FNFi and SNF is SNFj and TNF is TNFk) then CL is CLijk
(4.5b)
According to the usual Fuzzy logic control method (Das and Parhi [1]), a factor Wijk is
defined for the rules as follows:
Wijk=μfnfi (freqi) Λ μsnfj (freqj) Λ μtnfi (freqk)
Where freqi, freqj and freqk are the first, second and third natural frequency of the
cantilever beam with crack respectively ; by Appling composition rule of interference (Das
and Parhi [1]) the membership values of the relative crack location and relative crack depth
(location)CL.
μrclijk (location) = Wijk Λ μrclijk (location) length CL
As;
μrclijk (depth) = Wijk Λ μrclijk (depth) depth CD
The overall conclusion by combining the output of all the fuzzy can be written as follows:
μrclijk (location) = μrcl111 (location) V.….V μrclijk (location)
V.V μrcl9 9 9 (location)
(4.6a)
μrclijk (location) = μrcl111 (depth) V.…..V μrclijk (depth)
V….V μrcl9 9 9 (depth)
(4.6b)
The crisp values of relative crack location and relative crack depth are computed using the
center of gravity method (Das and Parhi [1]) as:
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Chapter 4
Fuzzy Inference System
Relative crack location=rcl=
 location. rcl (location ).d(location )
  rcl (location ).d(location )
(4.7a)
Relative crack depth=rcd=
 depth. rcd (depth).d(depth)
  rcd (depth).d(depth)
(4.8b)
Membership
Functions
Fuzzy (if-then)
rules, Linguistic
variables
Crisp Values
Crisp Values
fnf
snf
I
N
P
U
T
Fuzzifier
Fuzzy Controller
Defuzzifier
O
U
T
P
U
T
rcl
rcd
tnf
Figure: 4.3 Schematic diagram of Fuzzy Inference System
4.5. Function of Fuzzy Controller for Localization and Identification of
Crack
The inputs to the fuzzy controller are relative first natural frequency; relative second natural
frequency; relative third natural frequency. The outputs from the fuzzy controller are
relative crack depth and relative crack location. Several hundred fuzzy rules are outlined to
train the fuzzy controller. Twenty four numbers of the fuzzy rules out of several hundred
fuzzy rules are being listed in Table: 4.2. The output data has been generated from the input
data and the rule base.
Relative Natural Frequency =
Natural frequency of uncracked beam
Natural frequency of cracked beam
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Chapter 4
Fuzzy Inference System
L1F1 L1F2 L1F3
L1F4
M1F1 M1F2 M1F3
H1F1
H1F2
H1F3
H1F4
0.96
0.98
1
0.5
0.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
1
Figure: 4.4(a) Triangular Membership functions for relative natural frequency for 1st mode of vibration
L2F1 L2F2 L2F3
L2F4
M2F1 M2F2 M2F3
H2F1
H2F2
H2F3
H2F4
0.98
0.99
1
0.5
0.9
0.91
0.92
0.93
0.94
0.95
0.96
0.97
1
Figure: 4.4(b) Triangular Membership functions for relative natural frequency for 2nd mode of vibration
L3F1 L3F2 L3F3
L3F4
M3F1 M3F2 M3F3
H3F1
H3F2
H3F3
H3F4
0.98
0.99
1
0.5
0.9
0.91
0.92
0.93
0.94
0.95
0.96
0.97
1
Figure: 4.4(c) Triangular Membership functions for relative natural frequency for 3rd mode of vibration
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Chapter 4
Fuzzy Inference System
SL1 SL2
SL3
SL4
ML1
ML2
ML3
BL1
BL2
BL3
BL4
1
0.5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Figure: 4.4(d) Triangular Membership functions for relative crack location
SD1 SD2
SD3
SD4
MD1
0.15
0.2
MD2
MD3
LD1
LD2
LD3
LD4
1
0.5
0
0.05
0.1
0.25
0.3
0.35
0.4
0.45
0.5
Figure: 4.4(e) Triangular Membership functions for relative crack depth
L1F1 L1F2
L1F3
L1F4
M1F1
M1F2
M1F3
0.88
0.9
0.92
H1F1
H1F2
H1F3
H1F4
1
0.5
0.8
0.82
0.84
0.86
0.94
0.96
0.9 8
1
Figure: 4.5(a) Trapezoidal Membership functions for relative natural frequency for 1st mode of vibration
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Chapter 4
Fuzzy Inference System
L2F1 L2F2
L2F3
L2F4
M2F1
M2F2
M2F3
H2F1
H2F2
H2F3
H2F4
1
0.5
0.9
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.9 9
1
Figure: 4.5(b) Trapezoidal Membership functions for relative natural frequency for 2nd mode of vibration
L3F1 L3F2
L3F3
L3F4
M3F1 M3F2 M3F3 H3F1 H3F2 H3F3
H3F4
1
0.5
0.9
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
Figure: 4.5(c) Trapezoidal Membership functions for relative natural frequency for 3rd mode of vibration
SL1
SL2
SL3
SL4
ML1
ML2
ML3
0.2
0.25
0.3
BL1
BL2
BL3
BL4
1
0.5
0
0.05
0.1
0.15
0.35
0.4
0.45
0.5
Figure: 4.5(d) Trapezoidal Membership functions for relative crack location
SD1
SD2
SD3
SD4
MD1
MD2
MD3
LD1
LD2
LD3
LD4
1
0.5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Figure: 4.5(e) Trapezoidal Membership functions for relative crack depth
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Chapter 4
Fuzzy Inference System
L1F1 L1F2 L1F3
L1F4
M1F1 M1F2 M1F3 H1F1 H1F2 H1F3 H1F4
1
0.5
0.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
Figure: 4.6(a) Gaussian Membership functions for relative natural frequency for 1st mode of vibration
L2F1 L2F2 L2F3
L2F4
M2F1 M2F2 M2F3 H2F1 H2F2 H2F3 H2F4
1
0.5
0.9
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
Figure: 4.6(b) Gaussian Membership functions for relative natural frequency for 2nd mode of vibration
L3F1 L3F2 L3F3
L3F4
M3F1 M3F2 M3F3 H3F1 H3F2 H3F3 H3F4
1
0.5
0.9
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
Figure: 4.6(c) Gaussian Membership functions for relative natural frequency for 3rd mode of vibration
SL1
SL2
0.05
0.1
SL3
SL4
ML1
ML2
ML3
BL1
BL2
BL3
BL4
1
0.5
0
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Figure: 4.6(d) Gaussian Membership functions for relative crack location
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Chapter 4
Fuzzy Inference System
SD1
SD2
0.05
0.1
SD3
SD4
MD1
MD2
MD3
LD1
LD2
LD3
LD4
1
0.5
0
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Figure: 4.6(e) Gaussian Membership functions for relative crack depth
L1F1 L1F2 L1F3
L1F4 M1F1 M1F2 M1F3 H1F1 H1F2 H1F3 H1F4
1
0.5
0.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
Figure: 4.7(a) Hybrid Membership functions for relative natural frequency for 1st mode of vibration
L2F1 L2F2 L2F3
L2F4 M2F1 M2F2 M2F3 H2F1 H2F2 H2F3 H2F4
1
0.5
0.9
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
Figure: 4.7(b) Hybrid Membership functions for relative natural frequency for 2nd mode of vibration
L3F1 L3F2 L3F3
L3F4 M3F1 M3F2 M3F3 H3F1 H3F2 H3F3 H3F4
1
0.5
0.9
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
Figure: 4.7(c) Hybrid Membership functions for relative natural frequency for 3rd mode of vibration
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Chapter 4
Fuzzy Inference System
SL1
SL2
SL3
SL4
ML1
ML2
ML3
BL1
BL2
BL3
BL4
1
0.5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Figure: 4.7(d) Hybrid Membership functions for relative crack location
SD1
SD2
SD3
SD4
MD1
MD2
MD3
LD1
0.3
0.35
LD2
LD3 LD4
1
0.5
0
0.05
0.1
0.15
0.2
0.25
0.4
0.45
0.5
Figure: 4.7(e) Hybrid Membership functions for relative crack depth
4.5.1. Discussion
In this current research the fuzzy controller has been designed using four types of
membership functions, i.e. triangular (Figure: 4.4), trapezoidal (Figure: 4.5), Gaussian
(Figure: 4.6) and hybrid membership function (Figure: 4.7) which combines trapezoidal,
Gaussian as well as triangular membership functions. By taking the consideration of
triangular membership functions the Figure: 4.4(a) shows the various linguistic terms used
for the Relative 1st mode natural frequency. It has total 11 number of membership
functions. Similarly Figure: 4.4(b) and 4.4(c) show the membership functions and the
respective linguistic terms used for 2nd and 3rd mode relative natural frequencies. And both
are having 11 membership functions each. Figure: 4.4(d) and 4.4(e) represents the
membership functions and respective linguistic terms used for output of fuzzy controller,
i.e. the linguistic terms used for relative crack location and relative crack depth. The similar
linguistic terms are also used for other membership functions are as shown in the figures
(Figure: 4.5 to 4.7). The working principle for the fuzzy inference system has been depicted
in Figure: 4.3. The linguistic terms used in the fuzzy membership function has been
specified in Table: 4.1. The fuzzy rules being used for the fuzzy inference system are
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Chapter 4
Fuzzy Inference System
specified in the Table: 4.2. Out of several hundreds of fuzzy rules only twenty four fuzzy
rules has been indicated in the table. Figure: 4.8 to Figure: 4.11 shows the operation of
fuzzy inference system to exhibits the fuzzy results after defuzzification when rule 3 and 12
of the Table: 4.2 are activated for triangular, trapezoidal, Gaussian and hybrid membership
functions respectively. The comparison of the results obtained from theoretical and the
fuzzy controller with triangular membership function, fuzzy controller with trapezoidal
membership function, fuzzy controller with Gaussian membership function and fuzzy
controller with hybrid membership functions are presented in Table: 4.3 at the end of this
Chapter and also a comparison of result shown between different fuzzy controllers and the
kohonen network technique in Chapter 7.
Table: 4.1 Linguistic Terms used for Fuzzy Membership Functions
Name of the
Membership functions
L1F1,L1F2,L1F3,L1F4
Linguistic
terms
fnf1to4
M1F1,M1F2,M1F3
fnf5to7
H1F1,H1F2,H1F3,H1F4
fnf8to11
L2F1,L2F2,L2F3,L2F4
snf1to4
M2F1,M2F2,M2F3
snf5to7
H2F1,H2F2,H2F3,H2F4
snf8to11
L3F1,L3F2,L3F3,L3F4
tnf1to4
M3F1,M3F2,M3F3
tnf5to7
H3F1,H3F2,H3F3,H3F4
tnf8to11
SD1,SD2,SD3,SD4
rcd1to4
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Description and range of the linguistic terms
Low ranges of relative natural frequency for
first mode of vibration in ascending order
respectively.
Medium ranges of relative natural frequency
for first mode of vibration in ascending order
respectively.
Higher ranges of relative natural frequency for
first mode of vibration in ascending order
respectively
Low ranges of relative natural frequency for
second mode of vibration in ascending order
respectively.
Medium ranges of relative natural frequency
for second mode of vibration in ascending
order respectively.
Higher ranges of relative natural frequency for
first mode of vibration in ascending order
respectively
Low ranges of relative natural frequency for
second mode of vibration in ascending order
respectively
Medium ranges of relative natural frequency
for second mode of vibration in ascending
order respectively
Higher ranges of relative natural frequency for
first mode of vibration in ascending order
respectively
Small ranges of relative crack depth in
ascending order respectively.
46
Chapter 4
Fuzzy Inference System
MD1,MD2,MD3
rcd5to7
LD1,LD2,LD3,LD4
rcd8to11
SL1,SL2,SL3,SL4
rcl1to4
ML1,ML2,ML3
rcl5to7
BL1,BL2,BL3,BL4
rcl8to11
Medium ranges of relative crack depth
ascending order respectively
Larger ranges of relative crack depth
ascending order respectively.
Small ranges of relative crack location
ascending order respectively.
Medium ranges of relative crack location
ascending order respectively.
Bigger ranges of relative crack location
ascending order.
in
in
in
in
in
Table: 4.2 Fuzzy Rules for Fuzzy Inference System
Sl. No
Some Examples of Fuzzy rule used in the Fuzzy Controller
1
If fnf is L1F1, snf is L2F1, tnf is L3F1 then rcd is SD1 and rcl is SL1
2
If fnf is L1F1, snf is L2F2, tnf is L3F2 then rcd is SD2 and rcl is SL2
3
If fnf is L1F1, snf is L2F2, tnf is L3F3 then rcd is SD1 and rcl is SL2
4
If fnf is M1F1, snf is M2F1, tnf is M3F1 then rcd is MD1 and rcl is ML1
5
If fnf is M1F1, snf is M2F2, tnf is M3F2 then rcd is MD2 and rcl is ML2
6
If fnf is M1F1, snf is M2F2, tnf is M3F3 then rcd is MD1 and rcl is ML2
7
If fnf is M1F2, snf is M2F1, tnf is M3F1 then rcd is MD2 and rcl is ML1
8
If fnf is M1F2, snf is M2F2, tnf is M3F2 then rcd is MD2 and rcl is ML3
9
If fnf is M1F3, snf is M2F1, tnf is M3F2 then rcd is MD3 and rcl is ML1
10
If fnf is M1F2, snf is M2F3, tnf is M3F2 then rcd is MD1 and rcl is ML3
11
If fnf is L1F2, snf is L2F1, tnf is L3F1 then rcd is SD2 and rcl is SL1
12
If fnf is L1F2, snf is L2F3, tnf is L3F3 then rcd is SD2 and rcl is SL3
13
If fnf is L1F3, snf is L2F1, tnf is L3F2 then rcd is SD3 and rcl is SL1
14
If fnf is L1F2, snf is L2F3, tnf is L3F2 then rcd is SD1 and rcl is SL3
15
If fnf is L1F3, snf is L2F3, tnf is L3F3 then rcd is SD3 and rcl is SL3
16
If fnf is M1F3, snf is M2F3, tnf is M3F3 then rcd is MD3 and rcl is ML3
17
If fnf is H1F1, snf is H2F1, tnf is H3F1 then rcd is LD1 and rcl is BL1
18
If fnf is H1F1, snf is H2F2, tnf is H3F2 then rcd is LD2 and rcl is BL2
19
If fnf is H1F1, snf is H2F3, tnf is H3F3 then rcd is LD1 and rcl is BL2
20
If fnf is H1F2, snf is H2F1, tnf is H3F1 then rcd is LD2 and rcl is BL1
21
If fnf is H1F2, snf is H2F2, tnf is H3F2 then rcd is LD2 and rcl is BL3
22
If fnf is H1F3, snf is H2F1, tnf is H3F2 then rcd is LD3 and rcl is BL1
23
If fnf is H1F2, snf is H2F3, tnf is H3F2 then rcd is LD1 and rcl is BL3
24
If fnf is H1F3, snf is H2F3, tnf is H3F3 then rcd is LD3 and rcl is BL3
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Chapter 4
Fuzzy Inference System
Inputs for triangular membership function
Rule No. 3 of Table: 4.2 is activated
Rule No.12 of Table: 4.2 is activated
Outputs obtain from triangular membership function
0.169
Relative Crack Depth
0.255
Relative Crack Location
Figure: 4.8 Resultant values of relative crack depth and relative crack location of triangular membership
function when Rules 3 and 12 of Table: 4.2 are activated
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Chapter 4
Fuzzy Inference System
Inputs for trapezoidal membership function
Rule No. 3 of Table: 4.2 is activated
Rule No.12 of Table: 4.2 is activated
Outputs obtain from trapezoidal membership function
0.170
Relative Crack Depth
0.260
Relative Crack Location
Figure: 4.9 Resultant values of relative crack depth and relative crack location of trapezoidal membership
function when Rules 3 and 12 of Table: 4.2 are activated
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Chapter 4
Fuzzy Inference System
Inputs for Gaussian membership function
Rule No. 3 of Table: 4.2 is activated
Rule No.12 of Table: 4.2 is activated
Outputs obtain from Gaussian membership function
0.165
Relative Crack Depth
0.245
Relative Crack Location
Figure: 4.10 Resultant values of relative crack depth and relative crack location of Gaussian membership
function when Rules 3 and 12 of Table: 4.2 are activated
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Chapter 4
Fuzzy Inference System
Inputs for Hybrid membership function
Rule No. 3 of Table: 4.2 is activated
Rule No.12 of Table: 4.2 is activated
Outputs obtain from Hybrid membership function
0.161
Relative Crack Depth
0.262
Relative Crack Location
Figure: 4.11 Resultant values of relative crack depth and relative crack location of hybrid membership
function when Rules 3 and 12 of Table: 4.2 are activated
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Chapter 4
Fuzzy Inference System
4.5.2. Comparison of Results
Table: 4.3 Comparison of Results between Theoretical Analysis and different Fuzzy Controller Analysis
Relative
First
natural
frequency
fnf
Relative
Second
natural
frequency
snf
Relative
Third
natural
frequency
tnf
0.8142
0.9537
0.8635
Theoretical
Triangular
Fuzzy
Controller
rcd
rcl
Trapezoidal
Fuzzy
Controller
rcd
rcl
Gaussian
Fuzzy
Controller
rcd
rcl
Hybrid
Fuzzy
Controller
rcd
rcl
Relative
crack
depth
rcd
Relative
crack
location
rcl
0.9266
0.3167
0.125
0.314
0.124
0.315
0.126
0.316
0.125
0.316
0.128
0.9737
0.9335
0.3
0.1875
0.296
0.189
0.297
0.190
0.298
0.190
0.305
0.186
0.9013
0.9813
0.9470
0.2834
0.25
0.280
0.255
0.285
0.260
0.281
0.245
0.287
0.262
0.9315
0.9867
0.9523
0.2667
0.3125
0.270
0.320
0.272
0.323
0.268
0.313
0.261
0.312
0.9544
0.9888
0.9664
0.25
0.375
0.252
0.379
0.252
0.382
0.245
0.372
0.245
0.379
0.9692
0.9905
0.9757
0.2334
0.4375
0.240
0.442
0.239
0.445
0.233
0.440
0.224
0.425
0.9839
0.9917
0.9845
0.2167
0.5
0.224
0.505
0.227
0.495
0.214
0.512
0.215
0.498
0.9908
0.9946
0.9855
0.2
0.5625
0.213
0.571
0.212
0.565
0.21
0.561
0.23
0.567
0.9964
0.9967
0.9993
0.1834
0.625
0.185
0.635
0.179
0.637
0.182
0.629
0.182
0.632
0.9986
0.9980
0.9994
0.1667
0.6875
0.169
0.69
0.170
0.72
0.165
0.687
0.161
0.681
4.5.3. Summary
A fuzzy controller has been designed, which uses three natural frequencies as inputs where
as the crack depth and crack location as output. It has been observed that the natural
frequency of the beam is changing with Crack depth and crack location. The predicted
results from fuzzy controllers for crack location and crack depth are compared with the
theoretical results. It is observed from the Table: 4.3 that the results obtained from Gaussian
membership function fuzzy controller predict more accurate result in comparison to other
three controllers.
4.6. Why Fuzzy Logic is Used
 Provides an easy to use interface for applying modern fuzzy logic techniques.
 Does not require mathematical formulation.
 Powerful tool for dealing with imprecision, uncertainty.
 Precision is traded for tractability, robustness and low cost solution.
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Chapter 4
Fuzzy Inference System
 Provides the ability to use fuzzy logic when appropriate with other control techniques.
 It may be used for real world model like behavior of a human being.
 Supplies a fuzzy inference engine that can execute the fuzzy system as a stand-alone
application.
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ANALYSIS OF KOHONEN
NETWORK FOR
IDENTIFICATION OF CRACK
5.1. Introduction
5.2. Essential processes of kohonen network/ the SOM training Algorithm
5.2.1.
Initialization
5.2.2.
Competitive Process
5.2.3.
Co-operative Process
5.2.4.
Synaptic adaptation Process
5.3. Mechanism
5.3.1.
Competition Mechanism
5.3.2.
Co-operative Mechanism
5.3.3.
Adaptive Mechanism
5.4. Flow chart of Kohonen network
5.5. Comparison of Results
5.5.1.
Discussion
5.5.2.
Summary
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Chapter 5
CHAPTER 5
Kohonen Network
Analysis of Kohonen Network
for Identification of Crack
5.1. Introduction
The Self-Organizing Map (SOM), commonly also known as Kohonen network (Kohonen
1982, Kohonen 2001) is a computational method for the visualization and analysis of highdimensional data, especially experimentally acquired information [53]. The self-organizing
map (SOM) network was originally designed for solving problems that involve tasks such
as clustering, visualization, and abstraction. While Kohonen‟s SOM networks have been
successfully applied as a classification tool to various problem domains, their potential as a
robust substitute for clustering and visualization analysis remains relatively un researched.
The self-organizing map (SOM) network is a special type of neural network that can learn
from complex, multi-dimensional data and transform them into visually decipherable
clusters. The Kohonen network (Kohonen, 1982, 1984) can be seen as an extension to the
competitive learning network, although this is chronologically incorrect. Also, the Kohonen
network has a different set of applications. In the Kohonen network, the output units are
ordered in some fashion, often in a two dimensional grid or array, although this is
application-dependent. The ordering, which is chosen by the user1, determines which output
neurons are neighbors. The main function of SOM networks is to map the input data from
an n-dimensional space to a lower dimensional (usually one or two-dimensional) plot while
maintaining the original topological relations. The physical location of points on the map
shows the relative similarity between the points in the multi-dimensional space.
Competitive learning (Kohonen, 1982) is a special case of Self organizing Map.
In Self-Organizing Map, it Transform as input signal pattern of arbitrary dimension into one
or two dimensional discrete map, Perform this transformation adaptively in a topological
ordered fashion. Winner takes all neuron. Two possible architectures are existing in
Kohonen Network, which are shown in Figure: 5.1.
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Chapter 5
Kohonen Network
One Dimensional Output
Two Dimensional Outputs
Figure: 5.1 Possible Architectures in Kohonen Network
5.2. Essential Processes of Kohonen Network/ the SOM Training
Algorithm
The SOM network typically has two layers of nodes, the input layer and the Kohonen layer.
The input layer is fully connected to a two-dimensional Kohonen layer. During the training
process, input data are fed to the network through the processing elements (nodes) in the
input layer. An input pattern xm (m=1, 2, 3…, m) is denoted by a vector of order m as: x m
=(x1, x2… xm), where xm is the mth input signal in the pattern and m is the number of input
signals in each pattern. An input pattern is simultaneously incident on the nodes of a twodimensional Kohonen layer. Associated with the N nodes in the m × l (N =m × l) Kohonen
layer, is a weight vector, also of order l; denoted by: wj = (wj1, wj2… wjl), where wjl is the
weight value associated with node j corresponding to the lth signal of an input vector. As
the training process proceeds, the nodes adjust their weight values according to the
topological relations in the input data. The node with the minimum distance is the winner
and adjusts its weights to be closer to the value of the input pattern. In this study, Euclidean
distance is the most common way of measuring distance between vectors, is used. The
procedure with details being as follows.
5.2.1. Initialization
Each nodes weight is initialized. A vector is chosen at random from the set of training data
and presented to the lattice. When the input vector is presented to the map, its distance to
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Chapter 5
Kohonen Network
the weight vector of each node is computed. The map returns the closest node which is
called the Best Matching Unit (BMU).
BMU
Figure: 5.2 Process of Initialization in Kohonen Network
5.2.2. Competitive Process
It is also called long range inhibition*. For each input pattern, the neurons in the output
layer will determine the value of a function. That function will be calling as the discriminant
function. Each neuron computes a discriminant function. The neuron with largest
discriminant function is the winner.
“A continuous input space of activation pattern is mapped onto a discrete output space of
neurons by a process of competition among the neurons in the network”.
 
 
Winner neuron= arg j max( w jT x )  arg i min( x  w j )
(5.1)
fnf
snf
tnf
Winner
Figure: 5.3 Process of Competition in Kohonen Network
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Chapter 5
Kohonen Network
5.2.3. Co-operative Process
It is also called short range excitation*. The winning neuron locates the Centre of a
topological neighborhood of co-operating neurons. A neuron that is firing tends to excite the
neurons in its immediate neighborhood more than those farther away from it. Excitation is a
co-operation. It strengthens the neuron which are closer to winner, where as the process of
competition the neurons which are far apart, they are eliminated.
  d 2 j,i 

h ji  exp 
 2 2 


(5.2)
*Neurons which are there at the output, they are act in a competitive manner, in the sense
that they inhibit the responses of each other. The neurons which are close to the winning
neuron, they tends to have an excitatory response; that means to say around the winning
neuron an excitatory response is generally created where as an inhibitory response is created
for the neurons which are there far distant apart. This short of networks will exhibit long
range inhibition and short range excitation.
Figure: 5.4 Process of Co-operative in Kohonen Network
5.2.4. Synaptic Adaptation Process
It enables the excitation/excited neurons to increase their individual values of discriminant
function in relation to the input pattern. The discriminant function values for the nonexcited will be kept unchanged. Response of winning neuron is increased.
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Chapter 5
Kohonen Network
The output of the map is usually sent to another learning machine which will finish the
process of pattern recognition.
5.3. Mechanism
The training of the Kohonen Network is done by a specific algorithm. The goal is to obtain
a map where two points, which are nearby in the input space are also closed in the map. The
algorithm of Kohonen network is processed through various mechanisms as explained
below.
5.3.1. Competition Mechanism


[ x]  x1, x 2 ,..., x m T Where, x is the input vector, m= Dimensional Input.



[ w j ]  w j1, w j2 ,..., w j T , j=1, 2, 3...l, l= Number of output neurons.
(5.3)
(5.4)
w j = Weight Vector
Every output is connected to all input. There will be m  l number of arrays. We have to

 

determine the best match between x and w j . Compute w jT x for j=1, 2 …l.
 
Winning neuron= arg maximize ( w jT x ) , we have to minimize the Euclidian distance
 
( xwj )
 
 
 Winner neuron= arg j max( w jT x )  arg i min( x  w j )
(5.5)
5.3.2. Co-operative Mechanism
In addition to winning neuron all the neighborhood neuron should adjust their weights.
Winning neuron: i
Topological neighborhood:
h j,i : Topological neighborhood centered on i; encompassing neuron j.
d j,i : Lateral distance between the winning neuron „i‟ and excited neuron j.
Satisfying two properties:
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Chapter 5
Kohonen Network
Symmetric about dj,i→0 and considering monotonically decaying function, we can get
  d 2 j,i 

,
h j,i ( x )  exp 
 2 2 


(5.6)
 =width of Gaussian function,  is not constant with time/iteration. As the iteration
processes,  is going to be decrease. The topological neighborhood hj,i shrinks and narrow
down with time.
 t 
( t )   0 exp    ,
 1 
(5.7)
t = number of iteration.  0 = Initial  (at t=0), 1 = time constant
 is a function of iteration number. When t  1 , ( t ) decreases to 0.37 of its maximum
value. t= 0, 1, 2 . . .
  d 2 j,i 


h j,i ( x )  exp 
 2 2 t  


(5.8)

h j,i t  is called the neighborhood function (the more a node is far from the BMU the
smaller value is returned by this function).
5.3.3. Adaptive Mechanism
In this step we have to update the weights in relation to inputs.




 
w j t  1  w j t   t h j,i t  x  w j t 
(5.9)
Where t  = Learning Rate and
 t 

t   0 exp  


 2
(5.10)
Where,  2 is another time constant
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Chapter 5
Kohonen Network
5.4. Flow Chart of Kohonen Network
Start
Initialization
Draw an Input Sample
Winner neuron=
Update
End
Figure: 5.5 Flow Chart showing the processes of Kohonen Network
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Chapter 5
Kohonen Network
5.5. Comparison of Results
Table: 5.1. Comparison of Results between Theoretical Analysis and Kohonen Network Technique
Relative
First natural
frequency
fnf
Relative
Second
natural
frequency
snf
Relative
Third
natural
frequency
tnf
0.8142
0.9537
0.8635
Theoretical
Kohonen Network
Technique
Relative
crack
depth
rcd
Relative
crack
location
rcl
Relative
Crack
depth
rcd
Relative
crack
location
rcl
0.9266
0.3167
0.125
0.317
0.122
0.9737
0.9335
0.3
0.1875
0.302
0.187
0.9013
0.9813
0.9470
0.2834
0.25
0.291
0.258
0.9315
0.9867
0.9523
0.2667
0.3125
0.267
0.321
0.9544
0.9888
0.9664
0.25
0.375
0.253
0.374
0.9692
0.9905
0.9757
0.2334
0.4375
0.242
0.438
0.9839
0.9917
0.9845
0.2167
0.5
0.217
0.505
0.9908
0.9946
0.9855
0.2
0.5625
0.198
0.563
0.9964
0.9967
0.9993
0.1834
0.625
0.186
0.615
0.9986
0.9980
0.9994
0.1667
0.6875
0.167
0.689
5.5.1. Discussion
Kohonen network technique has been developed for the prediction of crack depth and crack
location. Kohonen network can be viewed as a clustering method so that similar data
samples tend to be mapped to nearby neurons. The complete architecture of the Kohonen
network, the essential processes and the mechanism of the competitive learning algorithm
has been discussed in the different sections. Finally the steps involved for Kohonen network
technique is presented through a flow chart.
5.5.2. Summary
Kohonen network technique is nothing but a competitive learning algorithm is developed
for crack detection, which uses three natural frequencies as inputs where as the crack depth
and crack location as output. The predicted results from Kohonen network technique for
crack location and crack depth are compared with the theoretical results and it is observed
that process of Kohonen network predict the depth and location accurately as close to
theoretical technique.
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EXPERIMENTAL SETUP
FOR
IDENTIFICATION OF CRACK
6.1. Introduction
6.2. Experimental Setup
6.2.1.
Instruments Used
6.2.2.
Description
6.3. Discussion
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Chapter 6
CHAPTER 6
Experimental Analysis
Experimental Setup for
Identification of Crack
6.1. Introduction
Experimental Analysis plays a vital role in the research work. Experimental Analysis is
being carried out to justify the validation of theoretical analysis and different intelligent
techniques projected in the chapter 3 to 5 for localization and identification of crack. For the
analysis, the experimental setup is made to measure the natural frequency and to observe the
response of cantilever beam with the presence of transverse crack. The experimental setup
is discussed in detail in the subsequent sections of this chapter.
An aluminum beam specimen of dimension (800 x 38 x 6mm) is selected for the
experimental analysis. The schematic diagram of the complete experimental setup is shown
in Figure: 6.1. It is shown in the figure that the vibration exciter is driven by a function
generator connected to a power amplifier. An oscilloscope is connected to observe the
vibrational response of cracked cantilever beam after getting the signal from the vibration
pick up. The detailed specifications of the instruments used in this analysis are given below.
6.2. Experimental Setup
The experiment has been conducted in two ways. The pictorial view of experimental setup1
and setup2 are shown in the Figures: 6.1. In the Figure: 6.1(a) a cracked cantilever beam is
rigidly clamped to the concrete foundation base. The free end of the cantilever beam is
excited with a vibration exciter. The vibration exciter is excited by the signal from the
function generator. The signal is amplified by a power amplifier before being fed to the
vibration exciter. The natural frequency is measured from the function generator at the point
of resonance under the excitation.
In the Figure: 6.1(b) the same cantilever beam is taken into consideration. The free end of
the cantilever beam is excited freely with the help of thumb and allowed to vibrate freely.
The amplitude of vibration of un cracked and cracked cantilever beam is taken by the
vibration pick up and is fed to the digital storage oscilloscope. The vibration signatures are
analyzed graphically in the oscilloscope and the natural frequency of the beam is calculated.
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Chapter 6
Experimental Analysis
2
4
5
3
Figure: 6.1(a) Pictorial View of complete assembly of Experimental Setup1
6
5
1
Figure: 6.1(b) Pictorial View of complete assembly of Experimental Setup2
6.2.1. Instruments Used
The various instruments used in the experimental analysis, the specification as well as the
views of the instruments are shown in the Table: 6.1
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Chapter 6
Experimental Analysis
Table: 6.1 Instruments used in the experimental Analysis
Instruments
Specifications
View of the Instrument
1. Vibration Pick up
(Accelerometer)
electromechanical Delta Tron Accelerometer
An
transducer
capable
of
converting mechanical vibration
into
electrical
voltages.
Depending upon their sensing
element
and
Type
:
Make :
4513-001
Bruel & kjaer
Frequency Range
:
1Hz-10KHz
Supply voltage
output
:
24volts
characteristics, such pickups are Operating temperature Range
referred to as accelerometer,
: -500C to +1000c
velocity
pickups
or
displacement pickups.
2. Function Generator
It is a moving coil device with a Model :FG200K
frequency in excess of 0.2Hz to Frequency Range
: 0.2Hz to 200 KHz
200 KHz. In our analysis the
natural frequency is calculated
from the function generator at
the point of resonance.
VCG IN connector for Sweep
Generation Sine, Triangle,
Square, TTL outputs.
Output Attenuation up to
60dB.
Output Level: 15Vp-p into
600 ohms
Rise/Fall Time:
<300nSec
Make :Aplab
3. Power Amplifier
An RF Power Amplifier is a Type
:
2719
type of electronic amplifier used Power Amplifier:
to convert a low power radio
frequency signal into a larger Make :
180VA
Bruel & kjaer
signal of significant power.
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Chapter 6
Experimental Analysis
4. Vibration Exciter
A vibration exciter (shaker) is Type : 4808
an electro-mechanical device Permanent
which transforms electrical a. c.
signals
into
mechanical
vibrations and is used to excite
vibrations
in
bodies
or
Magnetic
Vibration Exciter
Force rating 112N (25 lbf)
sine peak (187 N (42 lbf) with
cooling)
Frequency Range:
structures for testing purposes.
5
Hz to 10Hz
During the past decade a wide First Axial Resonance:
variety of vibration exciters
Hz
have
been
developed,
their Maximum
Bare
10
Table
fields of application ranging Acceleration : 700 m/s2 (71
from
fatigue
testing
of g)
automobile, missile and aircraft Continuous 12.7 mm
(0.5
in) peak-to-peak displacement
components, to the calibration with over travel stops
Two high-quality, 4-pin
of vibration pick-ups.
Neutrik®
Speakon®
connectors
Make :
5. Specimen
with
Bruel & kjaer
the
Concrete Foundation
A
cantilever
type
cracked Dimension
of
the
Aluminum beam specimen is specimen: (800 × 38 × 6
used for the analysis.
mm)
6. Oscilloscope
An
oscilloscope
is
commonly Band Width: DC ~ 60
known as CRO (Cathode Ray MHz
Oscilloscope). It is an electronic Sample Rate : 100 Sa/S
test
instrument
that
allows
observation of constantly varying
signal voltages, usually in a twodimensional graph of one or more
electrical
potential
differences
using the vertical or 'Y' axis,
Channels: 2
Storage Memory: 16 K/Ch
Vertical Sensitivity: 2 mV
~ 5V
Rise Time: 3.5ns
plotted as a function of time using Power Supply: 100V ~
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Chapter 6
the
horizontal
Experimental Analysis
or
'x'
axis. 240V/ 50W max
Oscilloscopes are commonly used Make: Metravi OS 5060
to observe the amplitude of the
signal.
6.2.2. Description
An Aluminum beam specimens of dimension (800 × 38 × 6mm) with a transverse crack is
taken for the experimental analysis for determining the natural frequencies at different crack
locations and crack depths. These specimens are allowed to vibrate under 1st, 2nd and 3rd
mode of vibrations. The experimental results of corresponding amplitudes are recorded in
the digital storage oscilloscope at various locations along the length of the beam and also
observed through function generator at the point of resonance.
6.3. Discussion
The experimental results of relative natural frequencies at different relative crack locations
and relative crack depths for 1st, 2nd and 3rd vibration are presented in Chapter 7. By feeding
the relative natural frequency to the fuzzy controller as well as Kohonen network the
relative crack depth and relative crack location is obtained. The detail mythologies have
been explained in Chapter 4 and 5. A comparison made between the results obtained from
the experiment, fuzzy logic and Kohonen network. The results are shown in a tabular form
in the Chapter 7. A graph has been plotted to show the comparison among all the results.
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RESULTS AND DISCUSSIONS
7.1.
Introduction
7.2.
Discussions
7.3.
Comparison of Results
7.4.
Characteristic Curves
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Chapter 7
Results and Discussions
CHAPTER 7
Results and Discussions
7.1. Introduction
An Aluminum beam (cantilever beam) specimen with transverse crack is used to obtain the
natural frequencies. Further the natural frequencies have been used as the training data for
the fuzzy inference system and Kohonen competitive learning network. The results obtained
from both the techniques have been discussed and analyzed in this chapter. At the end of
this chapter a comparative result has been shown between the theoretical analysis, fuzzy
logic and kohonen network.
7.2. Discussions
Including the introductory part and literature survey the present research work has been
processed in six stages. The other stages are theoretical vibration analysis for identification
of crack, analysis of fuzzy inference system for identification of crack, analysis of kohonen
network for identification of crack and the experimental setup for identification of crack.
The various methodologies used for identification of crack since last few decades has been
stated in the literature survey. In the first phase of theoretical vibration analysis, various
expressions are developed to obtain the natural frequencies under the consideration of free
and forced vibration analysis of the single crack cantilever beam and in the second phase,
the expressions has been developed using finite element analysis. The data obtained from
the theoretical analysis has been used as the training data for the fuzzy inference system and
kohonen network. The detail analysis has been explained in Chapter 3.
In the Chapter 4 a fuzzy inference system has been developed for the identification of crack
(crack depth and crack location) using different types of membership functions, such as
triangular, trapezoidal, Gaussian and hybrid function. The developed fuzzy inference system
uses three natural frequencies as inputs and the crack depth and crack location as output, the
schematic diagram of the fuzzy inference system is stated in Figure: 4.3. Several linguistic
terms and the fuzzy rules have been developed for the design of fuzzy inference system.
Some of the linguistic terms and the fuzzy rules are stated in the Table: 4.1 & 4.2. The
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Chapter 7
Results and Discussions
complete structure of the membership function has been presented in the Figures: 4.4 to 4.7.
The defuzzification results have been obtained by activating the rule no. 3 and 12 of Table:
4.2, which are demonstrated in the Figures: 4.8 to 4.11. It is observed from the Table: 4.3
that the results obtained from Gaussian membership function fuzzy controller predict more
accurate result in comparison to other three controllers and the computational time for crack
detection using fuzzy inference system is considerably lower as compared to theoretical
analysis.
In the Chapter 5 a Kohonen network technique has been developed for the prediction of
crack depth and crack location. Kohonen network can be viewed as a clustering method so
that similar data samples tend to be mapped to nearby neurons. The architecture of the
kohonen network is presented in the Figure: 5.1. The essential processes of kohonen
network technique are presented in the section 5.2 and also explained through the Figures:
5.2 to 5.4. The kohonen network technique is an extension to competitive learning network
and the training of the Kohonen Network is done by a specific algorithm. The processes of
the kohonen network have been presented with the help of a flow chart, which is depicted in
the Figure: 5.5. The data obtained from the theoretical analysis has been trained to the
kohonen network and the predicted results of crack location and crack depth is shown in the
Table: 5.1. It has been observed that the prediction of crack location and crack depth from
the kohonen network technique is very close to the actual results.
Chapter 6 describes the complete architecture of the experimental setup. The complete view
of various instruments used with the descriptions and the specifications are presented in the
Table: 6.1. The experiment has been conducted in two ways and the pictorial view of
complete assembly of setup-1 and setup-2 are shown in Figure: 6.1. Experimental Analysis
is being carried out to justify the validation of theoretical analysis and different intelligent
techniques used in this research work (Fuzzy logic and Kohonen Network).
In the last section of this chapter a comparative result shown between theoretical,
experimental, fuzzy logic and kohonen network.
The conclusions and scope for future work of the above analysis have been discussed in the
next chapter.
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Chapter 7
Results and Discussions
7.3. Comparisons of the Results
Table: 7.1. Comparison of Results between Theoretical Analysis, Experimental Analysis, Fuzzy Controller
Analysis and Kohonen Network Technique
Relative
First
natural
frequency
fnf
Relative
Second
natural
frequency
snf
Relative
Third
natural
frequency
tnf
0.8142
0.8635
0.9013
0.9315
0.9544
0.9692
0.9537
0.9737
0.9813
0.9867
0.9888
0.9905
0.9839
0.9908
0.9964
0.9986
0.9917
0.9946
0.9967
0.9980
Theoretical
Experimental
Fuzzy
Controller
Kohonen
Network
Technique
rcd
rcl
Relative
crack
location
rcl
0.125
0.1875
0.25
0.3125
0.375
0.4375
Rcd
rcl
rcd
rcl
0.9266
0.9335
0.9470
0.9523
0.9664
0.9757
Relative
crack
depth
rcd
0.3167
0.3
0.2834
0.2667
0.25
0.2334
0.315
0.299
0.282
0.267
0.248
0.234
0.124
0.185
0.260
0.315
0.377
0.438
0.316
0.298
0.281
0.268
0.245
0.233
0.125
0.190
0.245
0.313
0.372
0.440
0.317
0.302
0.291
0.267
0.253
0.242
0.122
0.187
0.258
0.321
0.374
0.438
0.9845
0.9855
0.9993
0.9994
0.2167
0.2
0.1834
0.1667
0.5
0.5625
0.625
0.6875
0.219
0.203
0.182
0.165
0.505
0.569
0.623
0.679
0.214
0.21
0.182
0.165
0.512
0.561
0.629
0.687
0.217
0.198
0.186
0.167
0.505
0.563
0.615
0.689
7.4. Characteristic Curves
Relative Crack Depth
0.35
0.3
0.25
0.2
Theoretical
Experimental
0.15
Fuzzy
0.1
Kohonen
0.05
0
0.8 0.820.840.860.88 0.9 0.920.940.960.98 1
Relative First Natural Frequency
Figure: 7.1 Relative First Natural Frequencies versus Relative Crack Depth
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Chapter 7
Results and Discussions
0.8
Relative Crack Location
0.7
0.6
0.5
Theoretical
0.4
Experimental
0.3
Fuzzy
0.2
Kohonen
0.1
0
0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1
Relative First Natural Frequency
Figure: 7.2 Relative First Natural Frequencies versus Relative Crack Location
0.35
Relative Crack Depth
0.3
0.25
0.2
Theoretical
Experimental
0.15
Fuzzy
0.1
Kohonen
0.05
0
0.95
0.96
0.97
0.98
0.99
Relative Second Natural Frequency
1
Figure: 7.3 Relative Second Natural Frequencies versus Relative Crack Depth
Relative Crack Location
0.8
0.7
0.6
0.5
Theoretical
0.4
Experimental
0.3
Fuzzy
0.2
Kohonen
0.1
0
0.95
0.96
0.97
0.98
0.99
1
Relative Second Natural Frequency
Figure: 7.4 Relative Second Natural Frequencies versus Relative Crack Location
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Chapter 7
Results and Discussions
0.35
Relative Crack Depth
0.3
0.25
0.2
Theoretical
Experimental
0.15
Fuzzy
0.1
Kohonen
0.05
0
0.92
0.94
0.96
0.98
Relative Third Natural Frequency
1
Figure: 7.5 Relative Third Natural Frequencies versus Relative Crack Depth
Relative Crack Location
0.8
0.7
0.6
0.5
Theoretical
0.4
Experimental
0.3
Fuzzy
0.2
Kohonen
0.1
0
0.92
0.94
0.96
0.98
Relative Third Natural Frequency
1
Figure: 7.6 Relative Third Natural Frequencies versus Relative Crack Location
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CONCLUSIONS
8.1. Conclusions
8.2. Applications
8.3. Scope for Future Work
References
Publications
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Chapter 8
Conclusion
CHAPTER 8
Conclusions
8.1. Conclusions
The effects of transverse cracks on the vibrating uniform cracked cantilever beam have been
presented in this thesis. The main purpose of this research work has been to develop a
proficient technique for diagnosis of crack in a vibrating structure in short span of time. The
vibration analysis has been done using theoretical, experimental analysis and also it has
been carried through using some intelligent techniques like fuzzy logic and kohonen
network. In this analysis natural frequency plays an important role for the identification of
crack. Crack has been identified in terms of crack depth and crack location.
Based on the results of various analyses performed on the cracked cantilever beam
structure, the following conclusions are drawn:
The analysis has been done on the presence of a transverse crack and it is observed that
the presence of crack affects the natural frequency, as a result the natural frequency
decreases with the increase in crack depth and it increases with the increase in crack
location. So it is concluded that the analysis of change of natural frequencies is effective
for prediction of crack in beam like structures.
The results of the crack depth and crack location have been obtained from the
comparison of the results of the uncracked and cracked beam during the vibration
analysis.
A fuzzy inference system has been developed using different membership functions for
the analysis of crack detection and it is observed that the fuzzy controller predict the
results of crack depth and crack location as close to the theoretical and experimental
analysis. The important factor of the fuzzy inference system is that it is predicting the
results with less computational time.
The natural frequencies obtained from the theoretical analysis are used as the training
data for fuzzy inference system. It shows a good agreement between theoretical,
experimental and fuzzy analysis.
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Chapter 8
Conclusion
The kohonen network technique is developed using a competitive learning algorithm to
predict the crack depth and location by using relative values of three natural frequencies.
The predicted results of kohonen network technique are reasonably adequate and in
agreement with the experimental result. With the successful detection of crack in a
cantilever beam, it has been observed that the new technique developed can be used as an
intelligent fault detecting tool for different types of vibrating structures.
The experimental analysis shows the effectiveness of the proposed methods towards the
identification of location and extent of damage in vibrating structures and it is observed
that the changes in the vibration signatures become more prominent as the crack grows
bigger.
8.2. Applications
As the techniques used for crack detection are non-destructive in nature, so these
techniques can be effectively used for online condition monitoring of engineering
systems.
The techniques developed for crack detection can be used for prediction of crack in flow
lines, turbo machinery, nuclear plants and ship structures, biomedical engineering system
etc.
8.3. Scope for Future Work
Analysis of Fuzzy Inference system and Kohonen Network system can be extended for
localization and identification of crack in complex beam structures with multiple cracks.
The complete analysis of the current research work is carried out based on the Euler
beam like structure and it can be extended for Timoshenko beam like structure.
Kohonen network technique and Fuzzy Logic technique can be hybridized to propose
another technique for identification of crack in beam like structure.
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Publications
Publications
[1] Dayal R. Parhi and Sasanka Choudhury, “Intelligent Fault Detection of a Cracked
Cantilever beam using Fuzzy Logic Technology with Hybrid Membership Functions”,
International Journal of Artificial Intelligence and Computational Research, 3(1), 2011,
Pages:13-20.
[2] Dayal R. Parhi and Sasanka Choudhury, “Analysis of Smart Crack Detection
Methodologies in various structures”, Journal of Engineering and Technology Research
(Academic Journals). Accepted for Publication.
[3] Dayal R. Parhi and Sasanka Choudhury, “Smart crack detection of a Cracked Cantilever
beam Using Fuzzy Logic Technology with hybrid membership functions”, Journal of
Engineering and Technology Research (Academic Journals), Accepted for Publication.
[4] Sasanka Choudhury and Dayal R. Parhi, “Analysis of fault detection methodologies in
various structures”, International Journal of Structural Engineering (Inderscience
Publishers), Accepted for Publication.
[5] Sasanka Choudhury, Sasmita Sahu, Dayal R. Parhi and Satish Choudhury, “ Intelligent
Fault Detection in Beam like Structures using Fuzzy Logic Technology”, International
Conference on Process automation Control and Computing, IEEE,2011, Accepted for
Publication.
[6] Sasmita Sahu, Sasanka Choudhury and Dayal R. Parhi, “Vibration Analysis of Cracked
Beam using Genetic Controller”, International Conference on Process automation
Control and Computing, IEEE, 2011, Accepted for Publication.
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