INTELLIGENT DIAGNOSIS AND SMART DETECTION OF CRACK IN A STRUCTURE

INTELLIGENT DIAGNOSIS AND SMART DETECTION OF CRACK IN A STRUCTURE
INTELLIGENT DIAGNOSIS AND SMART
DETECTION OF CRACK IN A STRUCTURE
FROM ITS VIBRATION SIGNATURES
Harish Chandra Das
Intelligent Diagnosis and Smart Detection of Crack
in a Structure from its Vibration Signatures
Thesis Submitted to the
Department of Mechanical Engineering
National Institute of Technology, Rourkela
for award of the degree
of
Doctor of Philosophy
by
Harish Chandra Das
under the guidance of
Prof. Dayal R. Parhi
&
Prof. R.C.Kar
Department of Mechanical Engineering
National Institute of Technology Rourkela
Orissa (India)-769008
December 2009
Declaration
I hereby declare that this submission is my own work and that, to the best of my
knowledge and belief, it contains no material previously published or written by another
person nor material which to a substantial extent has been accepted for the award of any
other degree or diploma of the university or other institute of higher learning, except where
due acknowledgement has been made in the text.
(Harish Chandra Das)
Date:
Certificate
This is to certify that the thesis entitled, “Intelligent Diagnosis and Smart Detection of
Crack in a Structure from its Vibration Signatures”, being submitted by Mr. Harish
Chandra Das to the Department of Mechanical Engineering, National Institute of
Technology, Rourkela, for the partial fulfillment of award of the degree Doctor of
Philosophy, is a record of bona fide research work carried out by him under our supervision
and guidance.
This thesis in our opinion, is worthy of consideration for award of the degree of Doctor of
Philosophy in accordance with the regulation of the institute. To the best of our knowledge,
the results embodied in this thesis have not been submitted to any other University or
Institute for the award of any degree or diploma.
Prof. D.R. Parhi
(Supervisor)
Prof. R.C. Kar
(Co-Supervisor)
iv
Acknowledgements
I would like to express my gratitude to all those people who made this research possible and
those who have made my experience in the Institute one that I will always remember fondly.
First and foremost I wish to thank my supervisor, Prof. Dayal R. Parhi, Professor of
Mechanical Engineering, N.I.T, Rourkela, for kindly providing me an opportunity to work
under his supervision and guidance. His patience, stimulating suggestions and
encouragement helped me in all the time of research for and writing of this thesis.
I wish to thank my second supervisor, Dr. R.C. Kar for his priceless suggestions and for
directing the PhD on to the right track. He has been supportive since the days I began
working on this research.
I am thankful to Prof. Sunil Kumar Sarangi, Director of National Institute of Technology, for
giving me an opportunity to work under the supervision of Prof. Dayal R. Parhi. I am
thankful to Prof. R.K. Sahoo, Head of the Department, Department of Mechanical
Engineering, for his moral support and valuable suggestions regarding the research work.
I express my deepest gratitude to Prof. Manojranjan Nayak, President, Siksha O Anusandhan
University, Bhubaneswar, Orissa, who gave me the opportunity of pursuing this research
work .His constant inspiration, encouragement and valuable advice have profoundly
contributed to the completion of the present thesis.
I am thankful to Mr. Amiya kumar Dash, research scholar, who has helped me a lot in
compilation of this thesis.
Finally I would like to thank my wife, Mrs. Lipika Das, for all her support and
encouragement. I thank my daughter, Ms. Barsha Das and son Mr. Adarsh Das, for giving me
so much joy now and so much to look forward to.
v
Synopsis
In recent years, there has been a growing interest in the development of structural health
monitoring for vibrating structures, especially crack detection methodologies and on-line
diagnostic techniques. In the current research, methodologies have been developed for
damage detection of a cracked cantilever beam using analytical, fuzzy logic, neural network
and fuzzy neuro techniques. The presence of a crack in a structural member introduces a
local flexibility that affects its dynamic response. For finding out the deviation in the
vibrating signatures of the cracked cantilever beam the local stiffness matrices are taken into
account. Theoretical analyses have been carried out to calculate the natural frequencies and
mode shapes of the cracked cantilever beam using local stiffness matrices. Strain energy
release rate has been used for calculating the local stiffness of the beam. The fuzzy inference
system has been designed using the first three relative natural frequencies and mode shapes
as input parameters. The output from the fuzzy controller is relative crack location and
relative crack depth. Several fuzzy rules have been developed using the vibration signatures
of the cantilever beam. A Neural Network technique using multi layered back propagation
algorithm has been developed for damage assessment using the first three relative natural
frequencies and mode shapes as input parameters and relative crack location and relative
crack depth as output parameters. Several training patterns are derived for designing the
Neural Network. A hybrid fuzzy-neuro intelligent system has been formulated for fault
identification.
The fuzzy controller is designed with six input parameters and two output parameters. The
input parameters to the fuzzy system are relative deviation of first three natural frequencies
and first three mode shapes. The output parameters of the fuzzy system are initial relative
crack depth and initial relative crack location. The input parameters to the neural controller
are relative deviation of first three natural frequencies and first three mode shapes along with
the interim outputs of fuzzy controller. The output parameters of the fuzzy-neuro system are
final relative crack depth and final relative crack location. A series of fuzzy rules and
training patterns are derived for the fuzzy and neural system respectively to predict the final
crack location and final crack depth.To diagnose the crack in the vibrating structure multiple
vi
adaptive neuro-fuzzy inference system (MANFIS) methodology has been applied. The final
outputs of the MANFIS are relative crack depth and relative crack location. Several hundred
fuzzy rules and neural network training patterns are derived using natural frequencies, mode
shapes, crack depths and crack locations.
The proposed research work aims to broaden the development in the area of fault detection of
dynamically vibrating structures. This research also addresses the accuracy for detection of
crack location and depth with considerably low computational time. The objective of the
research is related to design of an intelligent controller for prediction of damage location and
severity in a uniform cracked cantilever beam using AI techniques (i.e. Fuzzy, neural,
adaptive neuro-fuzzy and Manfis).
vii
Table of Contents
Declaration................................................................................................................. …………iii
Certificate .................................................................................................................................. iv
Acknowledgements .....................................................................................................................v
Synopsis...................................................................................................................................... vi
Contents ................................................................................................................................... viii
List of Tables ............................................................................................................................ xii
List of Figures .......................................................................................................................... xiii
List of Symbols ...................................................................................................................... xxvi
1
1 INTRODUCTION
1.1
Background and Motivation
1
1.2
Aims & Objectives of this Research
3
1.3
Outline of the Research Work
6
8
2 LITERATURE REVIEW
2.1
Introduction
8
2.2
History and Development of Dynamic Analysis of Cracked Structure
9
2.2.1
Structural Vibration and its Analysis
10
2.2.2
Dynamics of Cracked Structures
11
2.3
Effect of Different Parameters on Dynamic Response of Cracked Structures
12
2.4
Dynamic Characteristics of Beam with Transverse Crack
13
2.5
Damage Diagnosis by Artificial Intelligence Technique
26
2.5.1 Fuzzy Logic Technique for Damage Diagnosis
27
2.5.1.1
Fuzzy History
27
2.5.1.2
Application of Fuzzy Logic
28
2.5.1.3
Application of Fuzzy Logic for Fault Diagnosis
28
2.5.2 Neural Network Technique for Damage Diagnosis
2.5.2.1
Neural Network History
viii
30
31
2.5.2.2
Application of Neural Network
31
2.5.2.3
Application of Neural Network for Fault Diagnosis
33
2.5.3 Neuro-Fuzzy Technique for Damage Diagnosis
2.5.3.1
Neuro-Fuzzy Technique History
38
2.5.3.2
Application of Neuro-Fuzzy Technique
39
2.5.3.3
Appilication of Neuro-Fuzzy Technique for Fault Diagnosis
39
2.5.4 Multiple Adaptive Neuro Fuzzy Inference Technique (MANFIS)
for Damage Diagnosis
2.5.4.1 MANFIS History
40
Application of MANFIS
41
2.5.4.3
Application of MANFIS for Fault Diagnosis
41
44
44
Dynamic Characteristics of a Cantilever Beam with a Transverse Crack
45
3.2.1
Theoretical Analysis
45
3.2.1.1
Local Flexibility of a Cracked Cantilever Beam under Bending
and Axial Loading
45
3.2.1.2
Free Vibration Analysis of the Cracked Cantilever Beam
49
3.2.1.3
Forced Vibration analysis of Cracked Cantilever Beam
52
3.2.2
3.3
40
2.5.4.2
3 ANALYSIS OF DYNAMIC CHARACTERISTICS OF BEAM WITH
TRANSVERSE CRACK
3.1 Introduction
3.2
38
Numerical Analysis
53
3.2.2.1
53
Results of Numerical Analysis
Analysis of Experimental Results
129
129
3.3.1
Experimental Results
3.3.2
Comparison among the Results of Numerical and Experimental Analyses 134
3.4
Discussions
135
3.5
Summary
137
4 ANALYSIS OF FUZZY LOGIC TECHNIQUE FOR CRACK DETECTION
138
4.1
Introduction
138
4.2
Fuzzy Inference System
140
4.2.1
Membership Functions
140
4.2.2
Fuzzy Logic Controllers (FLC) and Fuzzy Reasoning Rules
142
ix
4.2.3
4.3
143
Defuzzification
Analysis of the Fuzzy Controller used for Crack Detection
144
4.3.1
Fuzzy Mechanism for Crack Detection
144
4.3.2
Fuzzy Controller for Finding out Crack Depth and Crack Location
145
4.3.3
Results of Fuzzy Controller
146
4.4
Discussions
147
4.5
Summary
157
5 ANALYSIS OF ARTIFICIAL NEURAL NETWORK FOR CRACK DETECTION 158
5.1
Introduction
158
5.2
Neural Network Technique
161
5.2.1
Design of Neural Network
161
5.2.2
Activation Function
163
5.2.2.1
Threshold Activation Function
164
5.2.2.2
Ramping Activation Function
164
5.3
5.2.2.3 Hyperbolic Tangent Activation Function
5.2.3
Modeling of Back Propagation Neural Network
165
Analysis of Neural Network Controller used for Crack Detection
166
5.3.1
Neural Controller Mechanism for Crack Detection
167
5.3.2
Neural Controller for Finding out Crack Depth and Crack Location
171
165
5.4
Results of Neural Controller
175
5.5
Discussions
175
5.6
Summary
176
6 ANALYSIS OF HYBRID FUZZY-NEURO SYSTEM FOR CRACK DETECTION
177
6.1
Introduction
177
6.2
Analysis of Fuzzy-Neuro Controller
178
6.2.1
Analysis of the Fuzzy Segment of Fuzzy-Neuro Controller
179
6.2.2
Analysis of the Neural Segment of Fuzzy-Neuro Controller
183
6.2.3
Results of Fuzzy-Neuro Controller
184
6.3
Discussions
187
6.4
Summary
187
x
7 ANALYSIS OF MANFIS FOR CRACK DETECTION
188
7.1
Introduction
189
7.2
Analysis of Multiple Adaptive Neuro-Fuzzy Inference System for Crack Detection 190
7.3
Results of MANFIS Controller
197
7.4
Discussions
200
7.5
Summary
200
8 ANALYSIS AND DESCRIPTION OF EXPERIMENTAL SETUP
201
8.1
Description of Instruments used in the Experimental Analysis
201
8.2
Experimental Set-up
204
8.3
Experimental Procedure
207
8.4
Experimental Results and Discussions
207
209
9 RESULTS & DISCUSSIONS
9.1
Introduction
209
9.2
Discussions of Results
209
10 CONCLUSIONS AND FURTHER WORK
213
10.1
Contributions
213
10.2
Conclusions
214
10.3
Applications
215
10.4
Scope for Future Work
216
REFERENCES
217
PUBLISHED PAPERS
240
xi
List of Tables
Table 3.3.1
Comparison of results between numerical and experimental analyses… 135
Table 4.3.1
Description of fuzzy linguistic terms…………………………………… 151
Table 4.3.2
Examples of twenty fuzzy rules being used in fuzzy controller………… 152
Table 4.3.3
Comparison of results between triangular, gaussian and trapezoidal
fuzzy controller, numerical analysis and experimental analysis………... 156
Table 5.3.1
Examples of the training patterns for training of the neural network
controller………………………………………………………………... 171
Table 5.3.2
Comparison of results between neural controller, fuzzy controller,
numerical analysis and experimental analysis…………………………... 174
Table 6.2.1
Comparison of the results of the fuzzy-neuro controllers with the results
of numerical and experimental analyses………………………………... 185
Table 6.2.2
Comparison of the results of the fuzzy-neuro controllers with the results
of neural and fuzzy controllers………………………………………….. 186
Table 7.3.1
Comparison of the results of the MANFIS with the results of numerical
and experimental analysis ……………………………………………… 198
Table 7.3.2
Comparison of the results of the MANFIS with the results of fuzzuneuro, neural and fuzzy controller analysis …………………………….. 199
xii
List of Figures
Fig. 3.2.1
Geometry of beam, (a) cantilever beam, (b) cross-sectional view of the
beam. (c) segments taken during integration at the crack section………. 45
Fig. 3.2.2
Relative crack depth (a1/W) vs. dimensionless compliance (ln ( C xy ))…. 48
Fig. 3.2.3
Beam model……………………………………………………………... 49
Fig.3.2.4 (a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W=0.1, L1/L=0.0256…………………………………….. 54
Fig. 3.2.4(a1)
Magnified view of Fig. 3.2.4(a) at the vicinity of the crack location…… 54
Fig.3.2.4 (b)
Relative amplitude vs. relative distance from the fixed end (2nd mode of
vibration), a1/W=0.1, L1/L=0.0256……………………………………... 55
Fig. 3.2.4(b1)
Magnified view of Fig. 3.2.4(b) at the vicinity of the crack location…...
Fig.3.2.4 (c)
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.1, L1/L=0.0256……………………………………... 56
Fig. 3.2.4(c1)
Magnified view of Fig. 3.2.4(c) at the vicinity of the crack location…… 56
Fig.3.2.5 (a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W=0.2, L1/L=0.0256…………………………………….. 57
Fig. 3.2.5(a1)
Magnified view of Fig. 3.2.5(a) at the vicinity of the crack location…… 57
Fig.3.2.5 (b)
Relative amplitude vs. relative distance from the fixed end (2nd mode of
vibration), a1/W=0.2, L1/L=0.0256……………………………………... 58
Fig. 3.2.5 (b1)
Magnified view of Fig. 3.2.5(b) at the vicinity of the crack location.
Fig.3.2.5 (c)
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.2, L1/L=0.0256……………………………………...
55
58
59
Fig. 3.2.5(c1)
Magnified view of Fig. 3.2.5(c) at the vicinity of the crack location…… 59
Fig.3.2.6 (a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W=0.3, L1/L=0.0256……………………………………... 60
Fig. 3.2.6(a1)
Magnified view of Fig. 3.2.6(a) at the vicinity of the crack location…… 60
xiii
Fig.3.2.6 (b)
Relative amplitude vs. relative distance from the fixed end (2nd mode of
vibration), a1/W=0.3, L1/L=0.0256……………………………………... 61
Fig. 3.2.6(b1)
Magnified view of Fig. 3.2.6(b) at the vicinity of the crack location…...
Fig.3.2.6 (c)
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.3, L1/L=0.0256……………………………………... 62
Fig. 3.2.6(c1)
Magnified view of Fig. 3.2.6(c) at the vicinity of the crack location…..
Fig.3.2.7 (a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W=0.4, L1/L=0.0256……………………………………... 63
Fig. 3.2.7(a1)
Magnified view of Fig. 3.2.7(a) at the vicinity of the crack location…… 63
Fig.3.2.7 (b)
Relative amplitude vs. relative distance from the fixed end (2nd mode of
vibration), a1/W=0.4, L1/L=0.0256……………………………………... 64
Fig. 3.2.7(b1)
Magnified view of Fig. 3.2.7(b) at the vicinity of the crack location…...
Fig.3.2.7 (c)
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.4, L1/L=0.0256……………………………………... 65
Fig. 3.2.7(c1)
Magnified view of Fig. 3.2.7(c) at the vicinity of the crack location…… 65
Fig.3.2.8 (a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W=0.1, L1/L=0.0513……………………………………... 66
Fig. 3.2.8(a1)
Magnified view of Fig. 3.2.8 (a) at the vicinity of the crack location…. 66
Fig.3.2.8 (b)
Relative amplitude vs. relative distance from the fixed end (2nd mode of
vibration), a1/W=0.1, L1/L=0.0513……………………………………... 67
Fig. 3.2.8(b1)
Magnified view of Fig. 3.2.8 (b) at the vicinity of the crack location…
Fig. 3.2.8 (c)
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.1, L1/L=0.0513……………………………………... 68
Fig. 3.2.8 (c1)
Magnified view of Fig. 3.2.8 (c) at the vicinity of the crack location…..
Fig.3.2.9 (a)
Relative amplitude vs. relative distance from the fixed end(1st mode of
vibration), a1/W=0.2, L1/L=0.0513……………………………………... 69
Fig. 3.2.9 (a1)
Magnified view of Fig. 3.2.9 (a) at the vicinity of the crack location…. 69
xiv
61
62
64
67
68
Fig.3.2.9 (b)
Relative amplitude vs. relative distance from the fixed end (2nd mode of
vibration), a1/W=0.2, L1/L=0.0513……………………………………... 70
Fig. 3.2.9(b1)
Magnified view of Fig. 3.2.9 (b) at the vicinity of the crack location...
Fig.3.2.9 (c)
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.2, L1/L=0.0513……………………………………... 71
Fig. 3.2.9 (c1)
Magnified view of Fig. 3.2.9 (c) at the vicinity of the crack location…... 71
Fig.3.2.10 (a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W= 0.3, L1/L=0.0513…………………………………….. 72
Fig. 3.2.10 (a1) Magnified view of fig. 3.2.10 (a) at the vicinity of
Fig.3.2.10 (b)
the crack location. 72
Relative amplitude vs. relative distance from the fixed end (2nd mode of
vibration), a1/W=0.3, L1/L=0.0513……………………………………
73
Fig. 3.2.10 (b1) Magnified view of Fig. 3.2.10 (b) at the vicinity of the crack location...
Fig.3.2.10 (c)
70
73
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.3, L1/L=0.0513……………………………………... 74
Fig. 3.2.10 (c1) Magnified view of Fig. 3.2.10 (c) at the vicinity of the crack location…. 74
Fig.3.2.11 (a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W=0.4, L1/L=0.0513……………………………………... 75
Fig. 3.2.11 (a1) Magnified view of Fig. 3.2.11 (a) at the vicinity of the crack location…. 75
Fig. 3.2.11 (b)
Relative amplitude vs. relative distance from the fixed end (2nd mode of
vibration), a1/W=0.4, L1/L=0.0513……………………………………
76
Fig. 3.2.11(b1) Magnified view of Fig. 3.2.11 (b) at the vicinity of the crack location...
Fig.3.2.11 (c)
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.4, L1/L=0.0513……………………………………... 77
Fig. 3.2.11 (c1) Magnified view of Fig. 3.2.11 (c) at the vicinity of the crack location.
Fig.3.2.12 (a)
76
77
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W=0.1, L1/L=0.1795……………………………………... 78
Fig. 3.2.12 (a1) Magnified view of Fig. 3.2.12 (a) at the vicinity of the crack location... 78
xv
Fig. 3.2.12 (b)
Relative amplitude vs. relative distance from the fixed end (2nd mode of
vibration), a1/W=0.1, L1/L=0.1795……………………………………... 79
Fig. 3.2.12 (b1) Magnified view of Fig. 3.2.12 (b) at the vicinity of the crack location..
Fig. 3.2.12 (c)
79
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.1, L1/L=0.1795……………………………………... 80
Fig. 3.2.12 (c1) Magnified view of Fig. 3.2.12 (c) at the vicinity of the crack location... 80
Fig.3.2.13 (a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W=0.2, L1/L=0.1795…………………………………….. 81
Fig. 3.2.13 (a1) Magnified view of Fig. 3.2.13 (a) at the vicinity of the crack location.. 81
Fig. 3.2.13 (b)
Relative amplitude vs. relative distance from the fixed end (2nd mode of
vibration), a1/W=0.2, L1/L=0.1795……………………………………... 82
Fig. 3.2.13(b1) Magnified view of Fig. 3.2.13 (b) at the vicinity of the crack location…
Fig.3.2.13 (c)
82
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.2, L1/L=0.1795……………………………………... 83
Fig. 3.2.13 (c1) Magnified view of Fig. 3.2.13 (c) at the vicinity of the crack location…. 83
Fig.3.2.14 (a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W=0.3, L1/L=0.1795……………………………………... 84
Fig. 3.2.14 (a1) Magnified view of Fig. 3.2.14 (a) at the vicinity of the crack location... 84
Fig.3.2.14 (b)
Relative amplitude vs. relative distance from the fixed end (2nd mode of
vibration), a1/W=0.3, L1/L=0.1795……………………………………... 85
Fig. 3.2.14(b1) Magnified view of Fig. 3.2.14(b) at the vicinity of the crack location….
Fig.3.2.14 (c)
85
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.3, L1/L=0.1795……………………………………... 86
Fig. 3.2.14(c1) Magnified view of Fig. 3.2.14(c) at the vicinity of the crack location….. 86
Fig.3.2.15 (a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W=0.4, L1/L=0.1795……………………………………... 87
Fig. 3.2.15 (a1) Magnified view of Fig. 3.2.15 (a) at the vicinity of the crack location... 87
xvi
Fig.3.2.15 (b)
Relative amplitude vs. relative distance from the fixed end (2nd mode of
vibration), a1/W=0.4, L1/L=0.1795……………………………………... 88
Fig. 3.2.15(b1) Magnified view of Fig. 3.2.15(b) at the vicinity of the crack location….
Fig.3.2.15 (c)
88
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.4, L1/L=0.1795…………………………………….. 89
Fig. 3.2.15(c1) Magnified view of Fig. 3.2.15(c) at the vicinity of the crack location….. 89
Fig.3.2.16 (a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W=0.1, L1/L=0.2564……………………………………... 90
Fig. 3.2.16(a1) Magnified view of Fig. 3.2.16(a) at the vicinity of the crack location….. 90
Fig.3.2.16 (b)
Relative amplitude vs. relative distance from the fixed end (2nd mode of
vibration), a1/W=0.1, L1/L=0.2564……………………………………... 91
Fig. 3.2.16(b1) Magnified view of Fig. 3.2.16(b) at the vicinity of the crack location….
Fig. 3.2.16 (c)
91
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.1, L1/L=0.2564……………………………………... 92
Fig. 3.2.16 (c1) Magnified view of Fig. 3.2.16 (c) at the vicinity of the crack location…. 92
Fig.3.2.17 (a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W=0.2, L1/L=0.2564……………………………………... 93
Fig. 3.2.17 (a1) Magnified view of Fig. 3.2.17 (a) at the vicinity of the crack location…. 93
Fig.3.2.17 (b)
Relative amplitude vs. relative distance from the fixed end (2nd mode of
vibration), a1/W=0.2, L1/L=0.2564……………………………………... 94
Fig. 3.2.17(b1) Magnified view of Fig. 3.2.17(b) at the vicinity of the crack location….
Fig. 3.2.17 (c)
94
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.2, L1/L=0.2564……………………………………... 95
Fig. 3.2.17 (c1) Magnified view of Fig. 3.2.17 (c) at the vicinity of the crack location…. 95
Fig.3.2.18 (a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W=0.3, L1/L=0.2564……………………………………... 96
Fig. 3.2.18 (a1) Magnified view of Fig. 3.2.18 (a) at the vicinity of the crack location…. 96
xvii
Fig.3.2.18 (b)
Relative amplitude vs. relative distance from the fixed end (2nd mode of
vibration), a1/W=0.3, L1/L=0.2564……………………………………... 97
Fig. 3.2.18(b1) Magnified view of Fig. 3.2.18 (b) at the vicinity of the crack location…
Fig. 3.2.18 (c)
97
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.3, L1/L=0.2564……………………………………... 98
Fig. 3.2.18 (c1) Magnified view of Fig. 3.2.18 (c) at the vicinity of the crack location... 98
Fig.3.2.19 (a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W=0.4, L1/L=0.2564……………………………………... 99
Fig. 3.2.19 (a1) Magnified view of Fig. 3.2.19 (a)
Fig.3.2.19 (b)
at the vicinity of the crack location.. 99
Relative amplitude vs. relative distance from the fixed end (2nd mode of
vibration), a1/W=0.4, L1/L=0.2564……………………………………... 100
Fig. 3.2.19(b1) Magnified view of Fig. 3.2.19(b) at the vicinity of the crack location…. 100
Fig.3.2.19 (c)
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.4, L1/L=0.2564…………………………………… 101
Fig. 3.2.19 (c1) Magnified view of Fig. 3.2.19 (c) at the vicinity of the crack location…. 101
Fig.3.2.20 (a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W=0.1, L1/L=0.3846……………………………………... 102
Fig. .3.2.20 (a1) Magnified view of Fig. 3.2.20 (a) at the vicinity of the crack location…. 102
Fig.3.2.20 (b)
Relative amplitude vs. relative distance from the fixed end (2nd mode of
vibration), a1/W=0.1, L1/L=0.3846……………………………………... 103
Fig. 3.2.20(b1) Magnified view of Fig. 3.2.20 (b) at the vicinity of the crack location… 103
Fig.3.2.20 (c)
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.1, L1/L=0.3846……………………………………... 104
Fig. 3.2.20 (c1) Magnified view of Fig. 3.2.20 (c) at the vicinity of the crack location…. 104
Fig.3.2.21 (a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W=0.2, L1/L=0.3846……………………………………... 105
Fig. 3.2.21 (a1) Magnified view of Fig. 3.2.21 (a)
xviii
at the vicinity of the crack location.. 105
Fig.3.2.21 (b)
Relative amplitude vs. relative distance from the fixed end (2nd mode of
vibration), a1/W=0.2, L1/L=0.3846…………………………………….. 106
Fig. 3.2.21 (b1) Magnified view of Fig. 3.2.21 (b)at the vicinity of the crack location…. 106
Fig.3.2.21(c)
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.2, L1/L=0.3846……………………………………... 107
Fig. 3.2.21(c1) Magnified view of Fig. 3.2.21(c) at the vicinity of the crack location... 107
Fig.3.2.22 (a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W=0.3, L1/L=0.3846……………………………………... 108
Fig. 3.2.22 (a1) Magnified view of Fig. 3.2.22 (a) at the vicinity of the crack location…. 108
Fig.3.2.22 (b)
Relative amplitude vs. relative distance from the fixed end (2nd mode of
vibration), a1/W=0.3, L1/L=0.3846……………………………………... 109
Fig. 3.2.22 (b1) Magnified view of Fig. 3.2.22 (b) at the vicinity of the crack location… 109
Fig.3.2.22 (c)
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.3, L1/L=0.3846……………………………………... 110
Fig. 3.2.22 (c1) Magnified view of Fig. 3.2.22 (c) at the vicinity of the crack location…. 110
Fig.3.2.23 (a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W=0.4, L1/L=0.3846……………………………………... 111
Fig. 3.2.23 (a1) Magnified view of Fig. 3.2.23 (a) at the vicinity of the crack location…. 111
Fig.3.2.23 (b)
Relative amplitude vs. relative distance from the fixed end (2nd mode of
vibration), a1/W=0.4, L1/L=0.3846…………………………………….. 112
Fig. 3.2.23(b1) Magnified view of Fig. 3.2.23(b) at the vicinity of the crack location…. 112
Fig.3.2.23 (c)
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.4, L1/L=0.3846……………………………………... 113
Fig. 3.2.23 (c1) Magnified view of Fig. 3.2.23 (c) at the vicinity of the crack location…. 113
Fig.3.2.24 (a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W=0.1, L1/L=0.5128……………………………………... 114
Fig. 3.2.24 (a1) Magnified view of Fig. 3.2.24 (a) at the vicinity of the crack location…. 114
xix
Fig.3.2.24 (b)
Relative amplitude vs. relative distance from the fixed end (2nd mode of
vibration), a1/W=0.1, L1/L=0.5128…………………………………….. 115
Fig. 3.2.24(b1) Magnified view of Fig. 3.2.24 (b) at the vicinity of the crack location… 115
Fig.3.2.24 (c)
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.1, L1/L=0.5128……………………………………... 116
Fig. 3.2.24 (c1) Magnified view of Fig. 3.2.24 (c) at the vicinity of the crack location…. 116
Fig. 3.2.25(a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W=0.2, L1/L=0.5128…………………………………… 117
Fig. 3.2.25(a1) Magnified view of Fig. 3.2.25(a) at the vicinity of the crack location….. 117
Fig. 3.2.25(b)
Relative amplitude vs. relative distance from the fixed end (2nd mode of
vibration), a1/W=0.2, L1/L=0.5128……………………………………... 118
Fig. 3.2.25(b1) Magnified view of Fig.3.2.25 (b) at the vicinity of the crack location…. 118
Fig. 3.2.25(c)
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.2, L1/L=0.5128……………………………………... 119
Fig. 3.2.25(c1) Magnified view of Fig. 3.2.25(c) at the vicinity of the crack location….. 119
Fig.3.2.26(a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W=0.3, L1/L=0.5128……………………………………... 120
Fig. 3.2.26(a1) Magnified view of Fig. 3.2.26(a) at the vicinity of the crack location….. 120
Fig.3.2.26 (b)
Relative amplitude vs. relative distance from the fixed end 2nd mode of
vibration), a1/W=0.3, L1/L=0.5128……………………………………... 121
Fig. 3.2.26(b1) Magnified view of Fig. 3.2.26(b) at the vicinity of the crack location… 121
Fig. 3.2.26(c)
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.3, L1/L=0.5128……………………………………... 122
Fig. 3.2.26(c1) Magnified view of Fig. 3.2.26(c) at the vicinity of the crack location….. 122
Fig. 3.2.27(a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W=0.4, L1/L=0.5128……………………………………... 123
Fig. 3.2.27(a1) Magnified view of Fig. 3.2.27(a1) at the vicinity of the crack location… 123
xx
Fig.3.2.27 (b)
Relative amplitude vs. relative distance from the fixed end (2nd mode of
vibration), a1/W=0.4, L1/L=0.5128…………………………………….. 124
Fig. 3.2.27(b1) Magnified view of Fig. 3.2.27(b) at the vicinity of the crack location…. 124
Fig. 3.2.27(c)
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.4, L1/L=0.5128……………………………………... 125
Fig. 3.2.27(c1) Magnified view of fig. Fig. 3.2.27(c) at the vicinity of the crack
location………………………………………………………………….. 125
Fig. 3.2.28 (a)
Three dimensional cum contour plot for relative first natural frequency.. 126
Fig.3.2.28 (b)
Three dimensional cum contour plot for relative second natural
frequency………………………………………………………………... 126
Fig. 3.2.28 (c)
Three dimensional cum contour plot for relative third natural
frequency……………………………………………………………… 127
Fig. 3.2.29 (a)
Three dimensional cum contour plot for relative 1st mode shape
difference………………………………………………………………... 127
Fig. 3.2.29 (b)
Three dimensional cum contour plot for relative 2nd mode shape
difference……………………………………………………………….. 128
Fig. 3.2.29 (c)
Three dimensional cum contour plot for relative 3rd mode shape
difference………………………………………………………………... 128
Fig. 3.3.1
Schematic block diagram of experimental set-up………………………. 129
Fig.3.3.2 (a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W=0.4, L1/L=0.026………………………………………. 130
Fig.3.3.2 (b)
Relative amplitude vs. relative distance from the fixed end (2nd mode
of vibration), a1/W=0.4, L1/L=0.026……………………………………. 130
Fig.3.3.2 (c)
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W=0.4, L1/L=0.026………………………………………. 131
Fig.3.3.3 (a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W =0.3, L1/L =0.05128…………………………………... 131
Fig.3.3.3 (b)
Relative amplitude vs. relative distance from the fixed end (2nd mode
of vibration), a1/W =0.3, L1/L =0.05128………………………………... 132
xxi
Fig.3.3.3(c)
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W =0.3, L1/L =0.05128…………………………………... 132
Fig.3.3.4 (a)
Relative amplitude vs. relative distance from the fixed end (1st mode of
vibration), a1/W =0.4, L1/L =0.05128…………………………………... 133
Fig.3.3.4 (b)
Relative amplitude vs. relative distance from the fixed end (2nd mode of
vibration), a1/W =0.4, L1/L =0.05128…………………………………... 133
Fig.3.3.4 (c)
Relative amplitude vs. relative distance from the fixed end (3rd mode of
vibration), a1/W =0.4, L1/L =0.05128…………………………………... 134
Fig. 4.2.1(a)
Triangular membership function………………………………………... 141
Fig.4.2.1 (b)
Gaussian membership function………………………………………… 141
Fig.4.2.1(c)
Trapezoidal membership function………………………………………. 141
Fig. 4.2.2
Schematic diagram of the fuzzy logic controller for crack detection…… 143
Fig. 4.3.1(a)
Triangular fuzzy controller……………………………………………… 148
Fig. 4.3.1(b1)
Triangular membership functions for relative natural frequency for first
mode of vibration……………………………………………………….. 148
Fig. 4.3.1(b2)
Triangular membership functions for relative natural frequency for
second mode of vibration……………………………………………… 148
Fig. 4.3.1(b3)
Triangular membership functions for relative natural frequency for
third mode of vibration…………………………………………………. 148
Fig. 4.3.1(b4)
Triangular membership functions for relative mode shape difference for
first mode of vibration…………………………………………………... 148
Fig. 4.3.1(b5)
Triangular membership functions for relative mode shape difference for
second mode of vibration……………………………………………….. 148
Fig. 4.3.1(b6)
Triangular membership functions for relative mode shape difference for
third mode of vibration………………………………………………….. 148
Fig. 4.3.1(b7)
Triangular membership functions for relative crack depth……………... 148
Fig. 4.3.1(b8)
Triangular membership functions for relative crack location…………... 148
Fig. 4.3.2(a)
Gaussian fuzzy controller……………………………………………….. 149
xxii
Fig. 4.3.2(b1)
Gaussian membership functions for relative natural frequency for first
mode of vibration……………………………………………………….. 149
Fig. 4.3.2(b2)
Gaussian membership functions for relative natural frequency for
second mode of vibration……………………………………………….. 149
Fig. 4.3.2(b3)
Gaussian membership functions for relative natural frequency for third
mode of vibration……………………………………………………….. 149
Fig. 4.3.2(b4)
Gaussian membership functions for relative mode shape difference for
first mode of vibration…………………………………………………... 149
Fig. 4.3.2(b5)
Gaussian membership functions for relative mode shape difference for
second mode of vibration……………………………………………….. 149
Fig. 4.3.2(b6)
Gaussian membership functions for relative mode shape difference for
third mode of vibration………………………………………………… 149
Fig. 4.3.2(b7)
Gaussian membership functions for relative crack depth………………. 149
Fig. 4.3.2(b8)
Gaussian membership functions for relative crack location……………. 149
Fig. 4.3.3(a)
Trapezoidal fuzzy controller……………………………………………. 150
Fig. 4.3.3(b1)
Trapezodial membership functions for relative natural frequency for
first mode of vibration…………………………………………………. 150
Fig. 4.3.3(b2)
Trapezodial Membership functions for relative natural frequency for
second mode of vibration……………………………………………….. 150
Fig. 4.3.3(b3)
Trapezodial membership functions for relative natural frequency for
third mode of vibration………………………………………………….. 150
Fig. 4.3.3(b4)
Trapezodial membership functions for relative mode shape difference
for first mode of vibration………………………………………………. 150
Fig. 4.3.3(b5)
Trapezodial membership functions for relative mode shape difference
for second mode of vibration……………………………………………. 150
Fig. 4.3.3(b6)
Trapezodial membership functions for relative mode shape difference
for third mode of vibration……………………………………………… 150
Fig. 4.3.3(b7)
Trapezodial membership functions for relative crack depth……………. 150
Fig. 4.3.3(b8)
Trapezodial membership functions for relative crack location…………. 150
xxiii
Fig.4.3.4
Resultant values of relative crack depth and relative crack location from
triangular fuzzy controller when Rules 1 and 19 of Table 4.3.2 are
activated…………………………………………………………………. 153
Fig. 4.3.5
Resultant values of relative crack depth and relative crack location from
gaussian fuzzy controller when Rules 1 and 19 of Table 4.3.2 are
activated…………………………………………………………………. 154
Fig. 4.3.6
Resultant values of relative crack depth and relative crack location from
trapezoidal fuzzy controller when Rules 1 and 19 of Table 4.3.2 are
activated…………………………………………………………………. 155
Fig. 5.2.1
Model of a neuron………………………………………………………. 163
Fig.5.2.2
Threshold activation function…………………………………………… 164
Fig.5.2.3
Ramping activation function……………………………………………. 164
Fig.5.2.4
Hyperbolic tangent activation function………………………………… 165
Fig. 5.2.5
Architecture of feed forward multilayer neural network trained by backpropagation algorithm…………………………………………………... 166
Fig. 5.3.1
Multi layer neural network controller…………………………………... 172
Fig. 5.3.2
Ten-layer neural network controller for crack detection………………. 173
Fig.6.2.1
Triangular fuzzy-neuro controller for crack detection………………….. 180
Fig.6.2.2
Gaussian fuzzy-neuro controller for crack detection…………………… 181
Fig.6.2.3
Trapezoidal fuzzy-neuro controller for crack detection………………… 182
Fig. 7.2.1
Bell-shaped membership function………………………………………. 192
Fig. 7.2.2
Multiple ANFIS (MANFIS) controller for crack detection…………….. 195
Fig. 7.2.3
Adaptive Neuro Fuzzy Inference System (ANFIS) for crack detection.. 196
Fig. 8.1
View of complete assembly of the experimental set-up………………… 204
Fig.8.2 (a)
Concrete foundation with beam specimen……………………………… 205
Fig.8.2 (b)
Vibration indicator (PULSE labShop software) with lap top…………... 205
Fig.8.2 (c)
Vibration exciter………………………………………………………… 205
xxiv
Fig.8.2 (d)
Vibration pick-up (accelerometer)……………………………………… 206
Fig.8.2 (e)
Vibration analyser………………………………………………………. 206
Fig.8.2 (f)
Function generator………………………………………………………. 206
Fig.8.2 (g)
Power amplifier…………………………………………………………. 207
Fig.8.2
View of the instruments used in the experimental set-up………………. 207
xxv
List of Symbols
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
ρ
Cy
=(
E
= Young’s modulus of elasticity of the beam material
fmd
= Relative first mode difference
fnf
= Relative first natural frequency
Fi i = 1,2
= experimentally determined function
i, j
= variables
J
= strain-energy release rate
K1,i i = 1,2
= Stress intensity factors for Pi loads
Ku
=
Ky
⎛ ωL2 ⎞
⎟
=⎜
⎜ C ⎟
⎝ y ⎠
Kij
= local flexibility matrix elements
L
= length of the beam
EI 1 / 2
)
μ
ωL
Cu
1/ 2
xxvi
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
smd
= Relative second mode difference
snf
= Relative second natural frequency
tmd
= Relative third mode difference
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
μ
= Aρ
ρ
= mass-density of the beam
ξ1
= relative crack depth
L1
L
a1
W
xxvii
V
= Aggregate (union)
Λ
= Minimum (min) operation
∀
= For every
xxviii
Chapter 1
INTRODUCTION
Research has been carried out in recent years for crack detection of dynamically vibrating
structures from its modal parameters. A brief description about the methodologies that have
been adapted for fault diagnosis has been given in the current chapter. At first background
and motivation in the field of vibration analysis of damaged structures has been out lined.
The second part of this chapter describes the aims and objective of the research. Finally the
details of each chapter of the thesis for the current investigation have been explained in the
third part of this chapter.
1.1
Background and Motivation
Most of the researches in the area of crack identification are done to avoid catastrophic
failure of structures or engineering systems. Researchers from different domain of
engineering stream have shown their interest to find out a potential tool for damage
detection. On line condition monitoring, prediction and identification of cracks in structural
systems as well as in mechanical components are the areas of application of the crack
detection methodologies. A number of non destructive testing (NDT) techniques with lower
order accuracy have been developed so far. Techniques developed for damage prediction in
the field of structural health monitoring with practical applications have been demonstrated
in very limited research work with very low success rate.
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 throughout the
world are working on structural dynamics and particularly on dynamic characteristics of
structures with crack. Due to presence of crack the dynamic characteristics of structure
changes. These signatures comprise of natural frequencies; the amplitude responses due to
vibration and the mode shapes. Cracks present a serious threat to proper performance of
structures and machines. Most of the failures are due to material fatigue. For this reason
1
methods used for early detection and localization of cracks have been the subject of intensive
research since several years. Since last two decades a number of experiments and theories
have been developed to elucidate the phenomenon and determine the crack effect on dynamic
structures.
Beams are one of the most commonly used structural elements in numerous engineering
applications and experience a wide variety of static as well as dynamic loads. Cracks may
develop in beam-like structures due to such loads. Considering the crack as a significant form
of such damage, its modelling 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. Beam type
structures are commonly used in steel construction and machinery industries. The current
study is based on crack detection for structural health monitoring in regard to change in
natural frequencies and mode shapes of the beam.
Fatigue cracks are a potential source of catastrophic structural failure. To avoid failure
caused by cracks, many researchers have performed extensive investigations to develop
structural integrity monitoring techniques. Most of these techniques are based on vibration
measurements and analysis because vibration based methods can offer an effective and
convenient way to detect fatigue cracks. Generally, 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. In recent years, transport engineering has experienced serious advances
characterized mainly by parameters like higher speeds and weights of vehicles. These
parameters make the transportation problem more complex. With a race towards high speed,
high power and lightweight in rotating machinery design and operation often impose severe
stress and environmental condition upon rotors. As rotating machinery is designed to operate
2
at higher mechanical efficiency; operating speed, power, and load are increased as weight
and dimensional tolerances are decreased. High torsional and radial loads, together with a
complex pattern of rotor motion, can create a severe mechanical stress condition that may
eventually lead to development of a crack in the shaft. The presence of a transverse crack in
shaft/rotor incurs a potential risk of destruction or collapse. This produces high costs of
production and maintenance. Detection of a crack in its early stages may save the rotor/beam
for use after repair. By monitoring the system, depending upon the type and severity of the
crack, it may be possible in some cases to extend the use of a flawed rotor without risking a
catastrophic failure, while arrangements are being made for a replacement rotor. The method
will also improve safety by helping to prevent major rotor failure. For the time being, the
research on cracked rotor is still at the theoretical stage. With a well known fact the dynamic
behavior of a structure changes due to presence of crack. There are two types of problems
related to this topic: the first may be called direct problem and the second called inverse
problem. The direct problem is to determine the effect of damages on the structural dynamic
characteristics, while the inverse problem is to detect, locate and quantify the extent of the
damages. In the past two decades, both the direct and the inverse problems have attracted
many researchers.
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
Aims and Objectives of this Research
It is required that structures must safely work during its service life but, damages initiate a
breakdown period on the structures. Cracks are among the most encountered damage types in
the structures. It is an established fact that dynamic behavior of structures changes due to
presence of crack in them. 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
investigators round the globe are working on structural dynamics and particularly on
dynamic characteristics of structures with crack. Due to presence of cracks the dynamic
3
characteristics of structure changes. The change in dynamic behavior has been utilized by the
researchers 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.
Scientific study on the changes in these characteristics can be widely utilized for the
identification of crack in structures. In general fault detection in structures can be more
specific with the help of these information.
In the current investigation, a number of literatures published till now have been surveyed,
reviewed and analyzed. It is felt that, the results presented by the researchers have not been
utilized so far in a systematic way for engineering applications. Although information on
some aspect are available but it is not exhaustive for real applications. A systematic attempt
has been made in the present study to investigate the dynamic behavior of cracked cantilever
beam structure using theoretical analysis, experimental analysis and artificial intelligence
techniques for damage diagnosis of cracked structure. The dynamic responses of the system
are used for crack prediction.
The phases of the process plan for the present investigation are as follows:
1. At first theoretical, free and forced vibration analyses of the cracked cantilever beam have
been addressed.
2. Experimental analysis is done to obtain the relative values of first, second and third modal
natural frequencies and mode shapes.
3. Training of the developed controllers has been done using artificial intelligence techniques
with series of data obtained from theoretical and experimental analysis. These controllers
predict relative values of crack depth and crack length from the three inputs such as relative
values of first, second, third natural frequencies and mode shapes.
For developing the analytical expressions on dynamic characteristics of structures, a single
transverse crack in the structure has been considered in the theory and the analyses are
presented in detail in subsequent chapters.
4
In all these theories, the presence of a transverse crack in the structure has been considered.
This crack introduces local flexibility at the vicinity of the crack location. Boundary
conditions are derived from the strain energy equation using castigliano’s theorem. Presence
of crack also reduces the stiffness of the structure which has been derived from stiffness
matrix, the details of which have been presented in the respective sections. For dynamic
behavior of beam with a transverse crack, Timoshenko beam theory with modified boundary
conditions due to presence of crack, have been used to find out the theoretical expressions for
natural frequencies and mode shapes for the beam. The first three relative value of first,
second, third natural frequencies and mode shapes obtained from theoretical analysis are
used as input parameters to the controller (fuzzy, neural network, fuzzy-neuro, MANFIS
controller) for crack identification. The outputs from the controller are relative crack location
and relative crack depth.
In the last stage of the investigation the effect of crack depth and crack location on the modal
values of natural frequencies and mode shapes are obtained with a very convincing manner.
Results obtained from the theoretical, fuzzy, fuzzy-neuro, manfis and experimental analysis
are compared and a close agreement has been found. Suitable numerical methods are used in
order to solve the theoretical equations developed. Useful conclusions are drawn from the
numerical results of respective sections. The results from the various methodologies
mentioned above are validated using the developed experimental set up.
From the vibration analysis of a cracked structure the crack characteristics can be detected.
Smart method can be developed for on line condition monitoring of damaged structure with
the help of artificial intelligence techniques. The system can be developed using fuzzy logic
and neural network techniques. Fuzzy system has the advantage of capturing the imprecise
nature of human knowledge and reasoning processes. The neural network has a different
approach for designing of the intelligent system because of its tremendous learning
capability. These two innovative modeling approaches share some common characteristics
such as i) they assume parallel operations, ii) they are well known for their fault tolerance
capabilities . Researchers from different field of engineering applications have integrated the
capabilities of neural network and fuzzy logic techniques to develop a hybrid method, which
has a better capability than the independent methods. This is one of the most important
5
reasons for carrying out the research work to develop a hybrid technique for crack detection.
Multiple Adaptive Neuro-Fuzzy Inference System (MANFIS) has also been devised for
crack diagnosis.
In the current investigation work has been done to develop methodologies with the aid of
artificial intelligence techniques for crack detection in various structural members.
1.3
Outline of the Research Work
The research work as outlined in this thesis is broadly divided into ten chapters. Following
the introduction, Chapter two presents the literature review of previous work on structural
vibration and its analysis, effect of different parameters on dynamic response of cracked
structures, dynamic characteristics of beam with transverse crack, crack detection by
artificial intelligence technique such as fuzzy logic, neural network, fuzzy neuro, multi
adaptive neuro-fuzzy inference system (MANFIS).
Chapter three analyses the dynamics characteristics of beam with a transverse crack using the
expression of strain energy release rate and strain energy density function. The local
flexibilities generated due to the presence of crack have been evaluated. The free and forced
vibration analyses have been performed to compute the vibration characteristics of the
cracked cantilever beam. The results and discussions of the numerical analysis have also
been presented in this chapter. Finally, the results of experimental and numerical analyses
have been compared for validation of theoretical analysis.
Chapter four defines the concept of the fuzzy logic and outlines the methodology used to
design an intelligent fuzzy logic controller for prediction of relative crack location and
relative crack depth using the relative deviations of first three natural frequencies and first
three mode shapes. The results obtained from the developed fuzzy controller have been
validated with the results from experimental analysis.
Chapter five discusses the neural network technique being used for crack detection in
vibrating structures. Comparisons of the results from numerical, fuzzy controller, neural
controller and experimental analyses have been presented.
6
In Chapter six the application of hybrid intelligent system (fuzzy neuro controller) for crack
detection has been discussed. The analysis of fuzzy and neural segment of the fuzzu-neuro
controller has been presented. Comparisons of results of the numerical, fuzzy, neural, fuzzyneuro and experimental analysis have been discussed.
Chapter seven discusses the concept of the multiple adaptive neuro fuzzy inference system
(MANFIS) and outlines the methodology for prediction of relative crack location and relative
crack depth using deviations of vibration signatures of cracked beam. The results of the
numerical, fuzzy, neural, fuzzy-neuro, MANFIS and experimental analysis have been
discussed.
In chapter eight the details of the developed experimental set-up for vibration analysis along
with the specifications of the different equipments used are presented. Finally the
experimental results are discussed. Chapter nine summarizes the findings of all chapters
discussed above.
Finally in Chapter ten contributions, conclusions of this research and future directions for
further investigation have been discussed. The developed methodologies are found to be
suitable for fault diagnosis of vibrating structures.
7
Chapter 2
LITERATURE REVIEW
This chapter reviews the work related to the analyses of crack in dynamic structures and the
development of crack diagnosis tool in damaged structures. The progress made in the last
few decades in the field of crack diagnosis of dynamically vibrating structures has been
described. This chapter also presents a literature review of past and recent developments in
area of crack identification and prediction with the aid of artificial intelligence techniques.
2.1
Introduction
A significant amount of research has been published in many aspects related to crack
detection. This section discusses the contributions that cover structural vibration and its
analysis, dynamics of cracked structures, fault identification methodologies and artificial
intelligence technique that helps to design an intelligent controller for crack identification. A
large number of researchers have used the free and forced vibration analysis for developing
algorithm for crack detection. The ultimate goal of this research is to establish new
methodologies which will predict the crack location and its intensity in a dynamically
vibrating structure by the help of artificial intelligence technique with considerably less
computational time and high precision. This chapter summarizes the past work, mostly in
computational methods for structures, and discusses possible directions for research.
Another challenge in literature review is that even the perception of what constitutes progress
varies widely in the research community. The representations would be difficult to extend to
different structural and mechanical member for crack detection, where the systems work in
various environments (i.e. with noise, chaos, uncertainty). Despite these challenges, the next
section reviews and highlights some of the interesting, important and experimental
milestones. This chapter provides details survey report within important aspects of what the
researchers have worked in the area of vibration analysis and planning for methodologies to
identify crack using fuzzy logic, neural network, neuro-fuzzy and MANFIS technique.
8
2.2
History and Development of Dynamic Analysis of Cracked Structure
The development of techniques for crack identification in real-world environments
constitutes one of the major trends in the current research on fault diagnosis. Some of the
researchers have analyzed the dynamic response of cracked structure during last few decades.
Ayre et al. [1] have developed a method for calculating the natural frequencies of continuous
beams of uniform span length by vibration analysis. Miles [2] has carried out analysis of
beams on many supports using vibration parameters. Bollinger et al. [3] have presented a
method for analysis and prediction of the static and dynamic behavior of machine tool
spindle systems using finite difference technique. Gladwell [4] has analysed a large structural
system using component mode synthesis method, for vibration analysis and shown the
effectiveness of component mode synthesis method with reference to other approaches.
Mercer et al. [5] have developed a transfer matrix method for the prediction of natural
frequencies and normal modes of a row of skin-stringer panels. They have also presented few
examples. Watrasiewicz [6] has applied Wavefront reconstruction interferometry to
mechanical vibration analysis and validated the results with experimental results. Lin et al.
[7] have briefly surveyed the use of transfer matrix method for analyzing the dynamic
behavior of beam structures. Chun [8] has considered the free vibration of a beam hinged at
one end by a rotational spring (with a constant spring constant) and the other end free. Lee
[9] has evaluated the eigenfrequencies for the fundamental mode of a beam hinged at one end
by a rotational spring by vibration analysis. Thomas et al. [10] have used straight beam finite
elements for the analysis of the vibration of curved beams and concluded that the proposed
method gives superior results than a solid 20 node isoparametric element. Venkateswara et
al. [11] have revealed that the Galerkin finite element method is very accurate and even with
a one element idealization of the beam the fundamental frequencies coincide up to five
significant figures.
Broadly the development of dynamic analysis of cracked structure can be divided into two
parts 1) structural vibration and its analysis 2) Dynamics of cracked structures.
9
2.2.1
Structural Vibration and its Analysis
The development of vibration theory, as a subdivision of mechanics, came as a natural result
of the development of the basic sciences i.e. mathematics and mechanics. The sciences are
founded in the middle of the first millennium B.C. by the ancient Greek philosophers. The
term vibration has been used from Vedic times of India, approximately 10,000 B.C..
Pythagoras of Somas (570-497 B.C.) has conducted several vibration experiments with
hammers, springs, pipes, and shells. He established the first vibration research laboratory.
Moreover, he has invented the monochord, a purely scientific instrument to conduct
experimental research in the vibration of taut strings and to set a standard for vibration
measurements.
Extensive experimental results are available for the vibrating strings since Pythagoras times.
Daniel Bernoulli has explained the experimental results using the principle of superposition
of the harmonics and has introduced the idea of expressing the response as a sum of the
simple harmonics. The problem of the vibrating string is solved mathematically first by
Lagrange considering it as sequence of small masses. The wave equation is introduced by
D’Alembert in a memoir to the Berlin Academic. He has used it in his memoir also for
longitudinal vibration of air columns in pipe organs. Experimental results for the same
problem are obtained by Pythagoras. Euler obtained the differential equation for the lateral
vibration of bars and he has determined the functions that we now call normal functions and
the equation that we now call frequency equation for beams with free, clamped or simply
supported ends, while Daniel Bernoulli has supplied him with experimental verification. The
first systematic treatise on vibration has been written by Rayleigh [12]. He has formalized
the idea of normal functions. He has introduced systematically the energy and approximate
methods in vibration analysis, without solving differential equations. This idea has been
further developed by Ritz [13]. Timoshenko theory accounts for rotary inertia and the
correction due to shear deformation of the lateral vibration of beam. Donaldson [14] has
presented the importance of vibration as a flanking path in airborne noise insulation and the
reduction of airborne noise transmitted through panels damped by friction. Henderson et al.
[15] have formulated analytical techniques, based on transfer matrix methods, and presented
for the analysis of the forced vibrations of cylindrically curved multi-span structures with
10
various examples. Tottenham et al. [16] have used the matrix progression method to
determine eigenvalues and eigenvectors for the free vibration problem of thin circular
cylindrical shells. Petyt et al. [17] have calculated the frequencies and modes of vibration of
a five bay curved beam on hinged supports using the finite element displacement method,
which are confirmed experimentally. Gorman [18] has proposed method to calculate the first
five frequencies and modal shapes for the entire family of beams regardless of the location of
the intermediate support following free lateral vibration of double-span uniform beams.
2.2.2
Dynamics of Cracked Structures
The problem on crack is the central problem of science for several decades. The mechanics
of fracture as an independent branch of the mechanics of deformable solids has originated
quite recently. Galileo Galilei is rightly considered the founder of fracture mechanics. He has
stated that the breaking load is independent of the length of a tension bar and is directly
proportional to its cross sectional area. In general the first stage of the investigation on
fracture mechanics, is associated with the names of Galileo Galilei, Robert Hooke, Charles
Augustin de Coulomb, Barre de Saint Venant and Otto mohr. Their investigation is
characterized by extensive studies of deformation properties of solids and by the
development of various failure criteria termed strength theories. These theories state that
fracture occurs at the moment when at a certain point of a body a particular combination of
parameters, such as stress, strain. etc., reaches its critical value. In this approach the process
of fracture propagation through the volume of the body is completely ignored, which is
justified only in cases where the development of defects causing failure takes place in a small
vicinity of the critical region.
Irwin [19] first has studied about cracked beam for finding out local flexibilities of the beam
at crack location. Later Tada [20] has developed theories for strain energy density function
with the help of stress intensity function at the crack section. With the light of above theory
Pafelias [21], Gasch [22] and Henry et al. [23] have analyzed the dynamic behavior of a
simple cracked rotor. Also Mayes et al. [24] have analyzed the vibrational behavior of a
rotating shaft system containing a transverse crack. Freud et al. [25] have analysed the
dynamic fracture of a beam or plate in plane bending. Subsequently Adeli et al. [26] have
11
analyzed the effect of axial force for dynamic fracture of a beam or plate in pure bending.
Dentsoras et al. [27] in their investigation they have taken coupling effect of various type of
vibration (such as bending vibration, torsional vibration and longitudinal vibration) for
analysis of dynamic behavior of cracked beam. Wong et al. [28] have diagnosed the fracture
damage in structures by modal frequency method. Also Nian et al. [29], Quain et al. [30],
Ismail et al. [31] and Sekhar et al. [32] have used the vibrational diagnosis approach for
detection of structural fault. Sorkin et al. [33] have evaluated the performance of different
methodologies for detecting the initiation and propagation of cracks by cyclic loading of the
structure with particular attention to the requirements for high-performance ship structures.
Rao et al. [34] have developed a finite element model to analyse three typical problems
pertaining to the vibrations of initially stressed thin shells of revolution. Wood [35] has
reviewed significant factors leading to the development of damage tolerance criteria and
illustrate the role of fracture mechanics in the analysis and testing aspects necessary to satisfy
the necessary requirements to prevent damage from growing to catastrophic size prior to
detection in aircrafts.
2.3
Effect of Different Parameters on Dynamic Response of Cracked
Structures
The effect of crack parameters on the dynamic response of the cracked structure can be
further established with the review of the following published papers.
Dimarogonas et al. [36] have studied the dynamic response of a cracked cantilever shaft due
to the local flexibility generated at the crack section. The results from the analytical solution
are validated using developed experimental setup. Nonlinear vibration of beams made of
functionally graded materials (FGMs) containing an open edge crack has been studied by
Kitipornchai et al. [37] based on Timoshenko beam theory and von Kármán geometric
nonlinearity. The cracked section is modeled as a mass less elastic rotational spring. It is
found that the intact and cracked FGM beams show different vibration behavior. Rizos et al.
[38] has determined the crack location and its depth in a cantilever beam from the vibration
modes. Analytical results are used to relate the measured vibration modes to the crack
location and depth. It is stated that the crack location can be found and depth can be
estimated with satisfactory accuracy. Ostachowicz et al. [39] have assumed an open and
12
closed crack with triangular disk finite elements. They have analyzed the forced vibrations of
the beam, the effects of the crack locations and sizes on the vibrational behavior and
discussed a basis for crack identification.
2.4
Dynamic Characteristics of Beam with Transverse Crack
The dynamic characteristics of the cracked structures such as natural frequencies and mode
shapes are dependent on the crack depth and its position. Following review discusses on the
effect of crack on vibrating structure.
Shen et al. [40, 41] have proposed an identification procedure to determine the crack
characteristics by measuring the difference between natural frequencies and mode shapes.
They have tested the method for simulated damage in the form of one-side or symmetric
cracks in a simply supported Bernoulli-Euler beam to evaluate the sensitivity of the solution
of damage identification. The crack can be simulated by an equivalent spring, connecting the
two segments of the beam, as stated by Narkis [42]. Analysis of this approximate model
results in algebraic equations which relate the natural frequencies of beam and crack
characteristics. The robustness of the proposed method is confirmed by comparing it with the
results from finite element calculations. Müller et al. [43] have proposed a model to detect
the crack and establish a clear relation between shaft cracks in turbo rotors and induced
phenomena in vibrations. This model is designed to estimate the nonlinear effects. Different
crack identification techniques have been discussed briefly by Dimarogonas [44] and they
have found that crack in a structural member introduce a local flexibility which affect its
vibration response. Tsai et al. [45] have investigated diagnostic method of determining the
position and size of a transverse open crack on a stationary shaft without disengaging it from
the machine system assuming the crack as a joint of a local spring. To obtain the dynamic
characteristics of a stepped shaft and a multidisc shaft the transfer matrix method is
employed by them on the basis of Timoshenko beam theory. Gounaris et al. [46] have
proposed a method for crack identification in beams assuming the crack to be always open
and the method is based on eigenmodes of the structure. In this paper they have co-related
the mode differences with crack depth and location. A cracked Euler-Bernoulli cantilevered
beam with an edge crack has been formulated by Chondros and Dimarogonas [47, 48] for
13
vibration analysis. Yokoyama et al. [49] have studied the vibration characteristics of a
uniform Bernoulli-Euler beam with a single edge crack using a modified line-spring model.
They have determined the natural frequencies and the corresponding mode shapes for
uniform beams having edge cracks of different depths at different positions. Kisa et al. [50]
present a novel numerical technique applicable to analyze the free vibration analysis of
uniform and stepped cracked beams with circular cross section using finite element and
component mode synthesis methods. To reveal the accuracy and effectiveness of the offered
method, a number of numerical examples are demonstrated for free vibration analysis of
beams. Damage detection in vibrating beam systems has been done by Fabrizio et al. [51] by
measuring the natural frequencies to locate and quantify the damage. Xia et al. [52] have
presented a technique for damage identification by selecting a subset of measurement points
and corresponding modes. They have used two factors for measuring the damage, the
sensitivity of a residual vector to the structural damage and the sensitivity of the damage to
the measured noise. A new method of vibration-based damage identification in structures
exhibiting axial and torsional responses has been proposed by Duffey et al. [53]. The method
has been derived to detect and localize linear damage in a structure using the measured
modal vibration parameters. Cracked beam element method for structural analysis has been
used by Viola et al. [54] for detection of crack location. The local flexibility introduced by
cracks changes the dynamic behavior of the structure and by examining this change, crack
position and magnitude can be identified. In order to model the structure a special finite
element for a cracked Timoshenko beam has been developed. Effect of the cracks on the
stiffness matrix and consistent mass matrix are investigated and the cracks in the structure
were identified using the modal data. Theoretical and experimental dynamic behavior of
different multi-beams systems containing a transverse crack has been performed by Saavedra
and Cuitino [55]. Yang et al. [56] have developed an energy-based numerical model to
investigate the influence of cracks on structural dynamic characteristics during the vibration
of a beam with open crack. Upon the determination of strain energy in the cracked beam, the
equivalent bending stiffness over the beam length is computed. Gounaris et al. [57] have used
a method for rotating cracked shafts to identify the depth and the location of a transverse
surface crack. A local compliance matrix of different degrees of freedom is used to model the
transverse crack in the shaft of circular cross section, based on available expressions of the
14
stress intensity factors and the associated expressions for the strain energy release rates.
Fernandez-saez et al. [58] have presented simplified method of evaluating the fundamental
frequency for the bending vibrations of cracked Euler–Bernouilli beams. Their method is
based on the well-known approach of representing the crack in a beam through a hinge and
an elastic spring, by adding polynomial functions to that of the un-cracked beam. This
approach is applied to simply supported beams with a cracked section in any location of the
span. Yang et al. [59] have developed a method to detect the onset and progression of surface
cracks in rotary shafts. They have used a wavelet -based algorithm effective in identifying
the nonlinear dynamical characteristics of a model-based, cracked rotor. Saavedra et al. [60]
have presented a theoretical and experimental dynamic analysis of a rotor-bearing system
with a transversely cracked shaft by modeling a cracked cylindrical shaft using finite
element. They have proposed a simplified opening and closing crack model and the analysis
is being done using Hilbert, Hughes, and Taylor integration method (HHT) implemented in
Matlab platform. Kim et al. [61] have derived a new algorithm to predict locations and
severities of damage in structures using modal characteristics. They have reviewed two
existing algorithms and formulated a new algorithm by eliminating the erratic assumptions
and limitations in those existing algorithm. As described by them this new proposed method
has been applied to a two span continuous beam and the results shown an improved accuracy
in crack location and severity estimation. An analysis has been performed by Patil et al. [62]
for the detection of multiple cracks using frequency measurements. Their method is based on
transverse vibration modeling through transfer matrix method and representation of a crack
by rotational spring. The procedure gives a linear relationship explicitly between the changes
in natural frequencies of the beam and the damage parameters. Darpe et al. [63] have studied
a simple Jeffcott rotor with two transverse surface cracks. The stiffness of such a rotor is
derived based on the concepts of fracture mechanics. Subsequently, the effect of the
interaction of the two cracks on the breathing behavior and on the unbalance response of the
rotor is studied. Zheng et al. [64] have presented a tool for vibrational stability analysis of
cracked hollow beams. According to them each crack is assigned with a local flexibility
coefficient which is a function of depth of crack. They have used least squared method to
device the formulae for shallow cracks and deep cracks. Zou et al. [65] have presented a
slightly modified version of local flexibility of Dimarogonas [2] which is more suitable for
15
theoretical model. According to them there are extensive research on the vibrational behavior
of the cracked rotor and use of response characteristics to detect crack. Owolabi et al. [66]
have done experimental investigations with two sets of aluminum beams with a view to
detect, quantify, and determine the effect of cracks and locations. According to them the
damage detection depends on the measured changes in the first three natural frequencies and
the corresponding amplitudes. Identification of location and severity of damage in structures
using frequency response function (FRF) data have been formulated by Hwang et al. [67]. To
verify the proposed method, examples for simple cantilever and a helicopter rotor blade are
numerically demonstrated. A method for crack identification in double-cracked beams based
on wavelet analysis has been presented by Loutridis et al. [68] using continuous wavelet
transform. The location of the crack is determined by the sudden changes in the spatial
variation of the transformed response. A comprehensive analysis of the stability of a cracked
beam subjected to a follower compressive load is presented by Wang [69].The beam is fixed
at its left end and restrained by a translational spring at its right end. The vibration analysis
on such cracked beam is conducted to identify the critical compression load for flutter or
buckling instability based on the variation of the first two resonant frequencies of the beam.
Kishen et al. [70] have studied the fracture behavior of cracked beams and columns using
finite element analysis. Assuming that failure occurs due to crack propagation when the
mode I stress intensity factor reaches the fracture toughness of the material, the failure load
of cracked columns are determined for different crack depths and slenderness ratios. Hwang
et al. [71] have presented method to identify the locations and severity of damage in
structures using frequency response function data. Dharmaraju et al. [72] have used Euler–
Bernoulli beam element and finite element modeling for crack identification. The transverse
surface crack is considered to remain open. The present identification algorithms have been
illustrated through numerical examples. Khiem et al. [73] have formulated a method to detect
multiple cracks of beams by analyzing natural frequencies in the form of a non-linear
optimization problem, then solving by using the MATLAB functions. They have applied
spring model of crack to establish the frequency equation based on the dynamic stiffness for
the multiple cracked beam. Kyricazoglou et al. [74] have presented method to detect the
damage in composite laminates by measuring and analyzing the slope deflection curve of
composite beams in flexure. They have provided the damage mechanism and location of
16
damage from comparison of dynamic results with the dynamic response from the damaged
laminates. They suggested that slope deflection curve is a promising technique for detection
initial damage in composites. Zheng et al. [75] have used finite element method to find out
the natural frequencies and mode shapes of a cracked beam. They have used overall
additional flexibility matrix instead of local additional flexibility matrix to find out the
stiffness matrix. According to them the overall additional matrix gives more accurate result
to calculate the natural frequencies. Mackerle [76] has reviewed the finite-element methods
along with electrical, magnetic and electromagnetic methods, sonic methods, mechanical
methods, optical methods, condition monitoring methods applied for the non-destructive
evaluation of materials. Wang et al. [77] have investigated the bending and torsional
vibration of a fiber rein- forced composite cantilever with a surface crack. Their analysis
concluded that changes in natural frequencies and the corresponding mode shapes depends
on both crack location and material properties. Cerri et al. [78] have proposed a method to
detect the structural damage affecting a narrow zone of a doubly hinged plane circular arch
by measuring natural frequencies. Nobile et al. [79] have applied S-theory to determine crack
initiation and direction for cracked T-beams and circumfentially cracked pipes. As stated by
them they have used strain energy density factor(s) which is a function of stress intensity
factor. According to them the strain energy density theory has the ability to describe the multi
scale feature of material damage and in dealing with mixed mode crack propagation problem.
A model-based fault diagnosis methodology for nonlinear systems is presented by Luh et al.
[80]. Their simulation results show that it can detect and isolate actuator faults, sensor faults,
and system component faults efficiently. A formulation has been developed by Chondros
[81] for the torsional vibration analysis of a cylindrical shaft with a circumferential crack.
The work is compared with existing methods. Structural damage detection using transfer
matrix method has been performed by Escobar et al. [82] for locating and estimating
structural damage. A robust fault detection method has been formulated by McAdams et al.
[83] for the damage assessment of turbine blades considering impact of crack damage and
manufacturing variation. The changes in the transverse vibration and associated
eigenfrequencies of the beams are considered. They have observed that changes in fault
detection monitoring signals caused by geometric variation are small with those caused by
damage and impending failure. For beams containing multiple cracks and subjected to axial
17
force a new method has been proposed by Binici [84] which used a set of end conditions as
initial parameters for determining the mode shape functions. Analysis for detection of the
location and size of the cracks has been performed by Chang et al. [85]. The proposed
analysis is able to calculate both the positions and depths of multi-cracks from spatial
wavelet based method. First, the mode shapes of free vibration and natural frequencies of the
multiple cracked beams are obtained. Then the mode shapes are analyzed by wavelet
transformation to get the positions of the cracks. The elastic characteristics of a cantilevered
composite panel of large aspect ratio and with an edge crack are investigated by Wang et al.
[86].The fundamental mode shapes of the cracked cantilever are used to study the crack. It is
observed that the analysis may help the development of an online diagnosis tool. Crack
identification in beam using dynamic response has been proposed by Law et al. [87]. The
crack is modeled following the Dirac delta function. The dynamic responses are calculated
based on modal superposition. The proposed identification algorithm is also verified
experimentally from impact hammer tests on a beam with a single crack. Cam et al. [88,90]
have performed a study to obtain information about the location and depth of the cracks in
cracked beam. Their study suggested determining the location and depth of cracks by
analyzing the vibration signals. Sekhar [89] applied the theory of model based identification
in a rotor system with two cracks. They have analysed the detection and monitoring of slant
crack in the rotor system using mechanical impedance. Loutridis et al. [91] have presented a
new method for crack detection in beams based on instantaneous frequency and empirical
mode decomposition. The dynamic behavior of a cantilever beam with a breathing crack
under harmonic excitation is investigated both theoretically and experimentally. Patil et al.
[92] have utilized a method for prediction of location and size of multiple cracks based on
measurement of natural frequencies and verified experimentally for slender cantilever beams
with two and three normal edge cracks. Their analysis is based on energy method and
representation of a crack by a rotational spring. The equation of motion and corresponding
boundary conditions has been developed by Behzad et al. [93] for forced bending vibration
analysis of a beam with an open edge crack. A uniform Euler-Bernoulli beam and Hamilton
principle have been used in this analysis. They have stated that there is an agreement between
the theoretical results and finite element results. Nahvi et al. [94] have proposed method
based on measured frequencies and mode shapes of the beam. In their experimental set up
18
they have used a hammer as an exciter and the responses are obtained in an accelerometer
assuming the crack to be an open crack. To identify the crack location and depth the
intersection of contours of the normalized frequency with the constant modal natural
frequency planes are used. Leontios et al. [95] have presented a new method of crack
detection in beams based on Kurtosis. As stated by them the location of the crack has been
determined by the abrupt changes in spatial varitation of the analyzed response and the size
of the crack is calculated by the estimation of Kurtosis. In this work the proposed method has
been validated by experiments on crack Plexiglas beams. Vibration analysis of an axially
loaded cracked Timoshenko beam have been performed by Mei et al. [96] considering axial
loading, shear deformation and rotary inertia criteria. The transmission and reflection
matrices for various discontinuities of an axially loaded Timoshenko beam are derived.
These matrices are combined to provide a concise and systematic approach for both free and
forced vibration analyses of beams with cracks and sectional changes. Chasalevris and
Papadopoulos [97] have studied the dynamic behavior of a cracked beam with two transverse
surface cracks. Each crack is characterized by its depth, position and relative angle. A local
compliance matrix of two degrees of freedom, bending in the horizontal and the vertical
planes is used to model the rotating transverse crack in the shaft and is calculated based on
the available expressions of the stress intensity factors and the associated expressions for the
strain energy release rates. The natural frequencies have been obtained by Loya et al. [98] for
Timoshenko cracked beams with different boundary conditions by modeling the beam as two
segments connected by two mass less springs. The results show that the method provides
simple expressions for calculating the natural frequencies of cracked beams and it gives good
results for shallow cracks. Humar et al. [99] have studied the vibration based damage
detection method and tried to find number of difficulties present in those damage
identification methods. According to them vibration frequencies, mode shapes and damping
are directly affected by the physical change in the structure including its stiffness. Vibration
based structural damage detection in flexural members using multi criteria approach has been
presented by Shih et al. [100]. In the analysis computer simulation techniques have been
developed and applied for damage assessment in beams and plates. In addition to changes in
natural frequencies, two methods, called the modal flexibility and the modal strain energy
method have been applied which are based on the vibration characteristics of the structure.
19
Dash and chatterjee [101] have studied the fracture toughness of the epoxy composites using
numerical technique. Harsha [102] has presented a model for investigating structural
vibrations in rolling element bearings due to radial internal clearance using the vibration
parameters and the Hertzian elastic contact deformation theory. Panda et al. [103] have
investigated the nonlinear planar vibration of a pipe conveying pulsatile fluid subjected to
principal parametric resonance in the presence of internal resonance. A numerical technique
based on the global-local hybrid spectral element (HSE) method is proposed by Hu et al.
[104] to study wave propagation in beams containing damages in the form of transverse and
lateral cracks. This method is employed to investigate the behaviors of wave propagation in
beams containing multiple transverse cracks and lateral cracks. Yang et al. [105] have
discussed on the free and forced vibration of Euler–Bernoulli beams containing open edge
cracks subjected to an axial compressive force and a concentrated transverse load moving
along the longitudinal direction. Analytical solutions of natural frequencies and dynamic
deflections are obtained for cantilever beams. It is found that the natural frequencies
decreases and the dynamic deflection increases due to the presence of the edge crack and the
axial compressive force. Yoona et al. [106] have investigated the influence of two open
cracks on the dynamic behavior of a double cracked simply supported beam both analytically
and experimentally. The equation of motion is derived by using the Hamilton’s principle and
analyzed by numerical method. The simply supported beam is modeled by the EulerBernoulli beam theory. A combined analytical and experimental study has been conducted by
Wang et al. [107] to develop efficient and effective damage detection techniques for beamtype structures. In combination with the uniform load surface (ULS), two new damage
detection algorithms, i.e., the generalized fractal dimension (GFD) and simplified gappedsmoothing (SGS) methods, have been proposed for prediction of damage location and size
successfully. The results from the proposed algorithm are experimentally validated.
Lissenden et al. [108] have focused on the relationship between a crack and load, which
propagated due to bending loads, and the torsional stiffness of the shaft. They have used a 3D finite element model of a shaft section with a crack to predict the effect of a crack on
stiffness. Al-Said [109] has proposed an algorithm based on a mathematical model to identify
crack location and depth in an Euler-Bernoulli beam carrying a rigid disk. As stated by him,
the lateral vibration of the beam has been described in the mathematical model using
20
Lagrange’s equation. He has used mode shapes for two uniform beams connected by mass
less torsional spring as trial function for the proposed method. The presented method utilizes
the natural frequencies to estimate the crack location and depth. Kisa et al. [110] have
presented a numerical technique to analyze the free vibration of uniform and stepped cracked
beam of circular cross section. They have used finite element and component mode synthesis
methods for the analysis and computed the flexibility matrix taking into account inertia
forces and calculated the inverse of the compliance matrix by following appropriate
expression for stress intensity factor and strain energy release rate. Karthikeyan et al. [111]
have followed a method for crack location and size from the free and forced response of the
beam. They have used Finite Element Method for free and forced vibration analysis of the
open transverse surface crack beam. Orhan [112] has studied the free and forced vibration
analysis of a cracked beam to identify the crack in a single- and two-edge cracks cantilever
beam. Darpe [113] has presented a method using both the typical nonlinear breathing
phenomenon of the crack and the coupling of bending-torsional vibrations to detect fatigue
transverse cracks in rotating shafts. Viola et al. [114] have investigated the in-plane linear
dynamic behavior of multi-stepped and multi-damaged circular arches. They have proposed
analytical and numerical solutions for multi-stepped arches in damaged and undamaged
configurations by adapting Euler characteristics exponent procedure for analytical solutions
and focused on generalized differential quadrature method, generalized differential
quadrature element for numerical solutions. According to them the convergence and stability
characteristics of the generalized differential quadrature element techniques are investigated
and the stability of the numerical procedure is found to be very good. Sinha [115] has
proposed the higher order spectra tools for the identification of presence of harmonics in a
signal which is a typical case of a non linear dynamic behavior in case of a mechanical
system. He has found that for a misaligned rotating shaft or a cracked shaft, they are going to
generate higher harmonics exhibiting non linear behavior. He has concluded that the results
from higher order spectra tools are encouraging and suggested the use of the tool for
condition monitoring of rotating machinery. Peng et al. [116,117,137] have used nonlinear
output frequency response functions (NOFRFs), to explain the occurrence of the nonlinear
phenomena when a cracked structure is subjected to sinusoidal excitations. They have also
applied finite element model to analyze the crack induced nonlinear response of a beam by
21
using NOFRF concept. From this research study it has been concluded that NOFRF concept
can be used in fault diagnosis of mechanical structures. Tandon et al. [118] have compared
the condition monitoring techniques available for rolling element bearing namely vibration,
stator current, acoustic emission and shock pulse method. They have concluded that acoustic
emission proved to be the best among them. Friswell [119] has given an overview of the use
of inverse method in the detection of crack location and size by using vibration data. He has
suggested that the uncertain parameters associated with the model have to be identified. A
number of problems with this method have been discussed for health monitoring, including
modeling error, environmental efforts, damage localization and regularization. Yan et al.
[120] have presented a general summary and review of vibration-based structural damage
detection techniques. Various structural damage detection methods based on structural
dynamic characteristic parameters are summarized and evaluated. The principle of intelligent
damage diagnosis and its application prospects in structural damage detection are introduced.
Naniwadekar et al. [121] have presented a technique based on measurement of change in
natural frequencies and modeling of crack by rotational spring to detect a crack with straight
front in different orientations in a section of straight horizontal steel hollow pipe. Variation
of rotational spring stiffness with crack size and orientation has been obtained experimentally
by deflection and vibration methods. The method is found to be very robust. Trendafilova et
al. [122] have dealt with vibration-based fault detection in structures and suggests a viable
methodology based on principal component analysis (PCA) and a simple pattern recognition
(PR) method. The suggested damage detection methodology is based purely on the analysis
of the vibration response of the structure. Courtney et al. [123] have used the bispectrum
signal processing technique to analyze the nonlinear response of a sample to continuous
excitation at two frequencies. The increased nonlinearity due to defects such as fatigue cracks
is detected. Flexural wave propagation characteristics have been used by Park [124] for
identification of damage in beam structures. The results from the proposed method are
validated by the experimental analysis. The locations of damage on the beam structures with
different magnitudes are identified accurately using the developed method. Different
methodologies for crack detection for multi-crack structures have been analyzed by Sekhar
[125] and the respective influences, identification methods in vibrating structures such as
beams, rotors, pipes etc. are discussed. Waveform fractal method has been proposed by Qiao
22
et al. [126] for mode shape-based damage identification of beam-type structures. In their
analysis a mathematical solution using waveform fractal dimension to higher mode shapes
for crack identification has been demonstrated. The applicability and effectiveness of the
applied method is validated by an experimental program on damage identification of a
cracked composite cantilever beam using smart piezoelectric sensors/actuators. To calculate
the natural frequencies and normal mode shapes of uniform isotropic beam element have
been developed by Li et al. [127] using dynamic stiffness matrix based on trigonometric
shear deformation theory. The numerical results obtained are compared to the available
solutions wherever possible and validate the accuracy and efficiency of the present approach.
Ostachowicz [128] has proposed Spectral finite element method for crack detection in
structures using fracture mechanics, elastic wave propagations and applications of Lamb
waves. The results obtained indicated that the current approach is capable of detecting cracks
of very small size, even in the presence of measurement errors. Damage identification based
on Lamb wave measurements have been introduced by Grabowska et al. [129]. The usage of
wavelet transformation with propagating Lamb waves are for distinguishing between
different types of damage. Reddy et al. [130] have presented fractal finite element based
continuum shape sensitivity analysis for a multiple crack system in a homogeneous,
isotropic, and two dimensional linear-elastic body subjected to mixed-mode loading
conditions. The best feature of this method is that the stress intensity factors and their
derivatives for the multiple crack system can be obtained efficiently. Hearndon et al. [131]
have developed a method to study the influence of crack on dynamic properties of a
cantilever beam subjected to bending with the help of Euler-Bernoulli and Timo- shenko
theories. A finite element model based on the response of the cracked beam element under
static load has been proposed by them to compute the influence of crack location and size on
the structural stiffness. In this work they have revealed that the experimental and
computational natural frequencies decreases with increasing crack length. Al-said [132] has
developed a crack identification algorithm to identify crack location and depth in a stepped
cantilever beam carrying concentrated masses. According to him in vibration analysis the
difference between the natural frequencies are used to locate the crack position and depth. As
stated by him the advantage of the algorithm is to identify the crack by monitoring a single
natural frequency system. According to him the advantage of the algorithm is to identify the
23
crack by monitoring a single natural frequency system. The robustness of the algorithm has
been tested from an experimental work and from finite element analysis in this paper. Shin et
al. [133] have presented vibration analysis of circular arches of variable cross section as they
are widely used in modern architectural and structural requirements. In this study generalized
differential quadrature method and differential transformation method have been applied by
them for deriving governing equation of motion and for obtaining natural frequencies for
different boundary conditions and the results are compared with previously published work.
Cerri et al. [134] have presented a comparison of the experimental results of the dynamic
behavior of a circular arch in undamaged and several damaged configuration with those
obtained by analytical methods. In this work they have proposed an identification procedure
by measuring natural frequencies and natural vibration modes and validated the results
experimentally. Ebersbach et al. [135] have suggested an expert system for condition
monitoring of fixed plant, laboratory and industry testing by using vibration analysis which
will allow a great analysis and enable the technician to perform routine analysis. As
described by them the expert system incorporates tri axial and demodulated frequency and
the time domain vibration data analysis algorithms for high accuracy fault detection. Babu et
al. [136] have addressed the problem of multi-crack assessment for rotors and described that
solutions comprising of parameters characterizing the cracks are more complicated. They
have developed a new technique called amplitude deviation curve, or slope deviation curve,
which is a modification of the operational deflection shape. Yaghin et al. [138] have used the
theory of wavelet analysis including continuous and discrete wavelet transform followed by
its application to structural health monitoring. According to them by using the frequency
analysis response of dam with ABAQUS software, crack detection has been done in dam
structure under Wavelet analyzing in MATLAB software. Bayissa et al. [139] have presented
a new method for damage identification based on the statistical moments of the energy
density function of the vibration responses in time-frequency domain. According to them the
major advantage of this method is that the time-frequency analysis conducted using the
wavelet transform provides a tool to characterize deterministic as well as random responses
and can be used to detect slight changes in the response of local vibration. Finally they have
suggested that the proposed method is more sensitive to damage than the other methods. A
solution to the free vibration problem of a stepped column with cracks is presented by Sukla
24
[140]. The open cracks at step changes in the cross-section of the column or at the
intermediate points of the uniform segments are represented by mass less rotational springs.
The proposed approach measures the vibration parameters assuming the column consists of
an arbitrary number of uniform segments. Dilena et al. [141] have shown that the natural
frequency and anti resonant frequency contains certain generalized fourier coefficients of the
stiffness variation due to damage. According to them the results of numerical simulations on
rods with localized or diffused cracks are in good agreement with theory. Mazanoglu et al.
[142] have followed energy based method for vibration identification of non-uniform Euler –
Bernoulli beams having open cracks. They have estimated the distribution of energy
consumed by considering strain change at the cracked beam surface and the stress field due
to angular displacement of beam because of bending. Rayleigh – Ritz approximation method
has been adapted by them for analysis of beam with crack. Lee [143] has presented a method
to identify crack in a beam by modeling the cracks as rotational springs. Newton-Rapson
method has been adapted by him to identify the locations and sizes of the double as well as
triple cracks in a cantilever beam. He has concluded that the detected crack locations and
sizes are in excellent agreement with the actual ones. Faverjon et al. [144] have presented a
damage assessment technique for detection of size of the open crack in beams. They have
used constitutive relation error updating method for identification of crack location and size
of the beam. According to them even if noise has been added to the simulation the algorithm
can identify the crack location and size with satisfactory precision. He et al. [145] have
presented a method to calculate the stress intensity factor and local flexibility matrix for
cracked pipes by dividing the cracked pipe into series of these annuli. They have described
that the calculation of local flexibility matrix for cracked pipes have been calculated
experimentally without calculating the Stress intensity factor. Further the results from their
method have been compared with the experimental results to verify the effectiveness of the
method. Douka et al. [146] have presented the influence of two transverse open cracks on the
anti resonances of a double cracked cantilever beam both analytically and experimentally.
They have shown that the shift in the anti resonances of the cracked beam can be used as
additional information for crack identification in double cracked beam. The results of
experiments performed by them on Plexiglas beams for crack location and severity are in
good agreement with theoretical predictions. Labuschagne et al. [147] have considered three
25
different linear theories: Euler – Bernoulli, Timo Shenko and Two dimensional elasticity for
three models of cantilever beams. Using natural frequencies and modes as the basis, they
concluded that the Timo Shenko theory is close to the two dimensional theory for practical
purpose, but the applicability of Euler – Bernoulli theory is limited. Gan et al. [148] have
developed a clonal selection programming (CSP)-based fault detection system to perform
induction machine fault detection and analysis. The extracted features are inputs of a CSPbased classifier for fault identification and classification. The proposed CSP-based machine
fault diagnostic system has been intensively tested with unbalanced electrical faults and
mechanical faults operating at different rotating speeds. Ribeiro et al. [149,150] have used
transmissibility concept for a structure and observed the response when the structure is
excited at a given set of coordinates for fault detection.
2.5
Damage Diagnosis by Artificial Intelligence Technique
Intelligence is the computational part of the ability to achieve goals in the world. The aim of
artificial intelligence is to develop algorithms that allow machines to perform tasks that
involve cognition when performed by humans. The word “cognition” comes from the latin
word
“cognitio”, which means “knowledge”. Cognitive sciences concern thinking,
perception, reasoning, creation of meaning, and other functions of a human mind. As cracks
pose as a potential cause of failure for mechanical or structural systems, the early detection of
it will save the systems from catastrophic failure and also save a lot amount of finance
involved in it. Algorithms have been developed which can predict the crack location and its
severity of a system before hand using different Artificial Intelligence techniques. Since the
algorithms use Artificial Intelligence techniques they can be used as non destructive testing
methodology for early detection of crack in dynamically vibrating structures. McCarthy
[151] has stated that Artificial intelligence is the science and engineering of making
intelligent machines, especially intelligent computer programs. It is related to the similar task
of using computers to understand human intelligence, but AI does not have to confine itself
to methods that are biologically observable.
26
2.5.1
Fuzzy Logic Technique for Damage Diagnosis
The fuzzy logic technique is one of the AI methods which can be used for damage diagnosis
in the domain of dynamically vibrating structures. It is very much evident that the crack can
be detected from the vibration analysis of the cracked structure but to develop a
methodology/controller for on line condition monitoring of damaged structure artificial
intelligence techniques has to be adapted. To design an intelligent controller fuzzy logic play
vital role. This is due to the fact that fuzzy if-then rules are well suited for capturing the
imprecise nature of human knowledge and reasoning processes. Fuzzy sets are functions that
map a value, which might be a member of a set, to a number between zero and one,
indicating its actual degree of membership .Fuzzy logic is based on the idea that all things
admit of degrees. A degree of zero means that the value is not in the set and a degree of one
means that the value is completely representative of the set. Fuzzy logic modeling is
primarily based on fuzzy sets and fuzzy if-then rules proposed by Zadeh [152] which are
closely related to perception and cognitive science.
2.5.1.1 Fuzzy History
About 300 B.C., the Greek scholar Aristotle developed binary logic. Aristotle thought that
the world was made up of opposites, for example male versus female, hot versus cold, dry
verus wet, active versus passive. Everything has to be A or not-A, it can't be both.
Subsequently with this idea a new technique was developed wich accommodate the
uncertainty of the problem and gives the solution i.e. fuzzy logic. Fuzzy logic is based on the
idea that A can equal not-A. That means that something can contain a part of its opposite.
Negoită [153] has dealt with a new clustering technique using the concept of fuzzy set [152].
A membership function is proposed and a method to select the cluster elements is derived
using the separation theorem of the fuzzy sets. Kandel [154] has discussed two theorems,
which are the basis of a new technique that generates the complete set of fuzzy implicants,
and used for the minimization of fuzzy functions.
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2.5.1.2 Application of Fuzzy Logic
The fuzzy logic method can be used in a wide spectrum of industrial application as well as in
domestic appliances such as shower heads, rice cookers, vacuum cleaners etc. The
application of fuzzy approach is further expressed with the review of following published
papers.
Fox [155] has studied the use of fuzzy logic in medical diagnosis and raised a broad range of
issues in connection to the role of information-processing techniques in the development of
medical computing. Gologlu [156] has presented a set-up planning module as part of a
feature-based process planning system with the aid of artificial intelligence. Zimmermann
[157] has applied fuzzy linear programming approach for solving linear vector maximum
problem. The solutions are obtained by fuzzy linear programming. These are found to be
efficient solutions then the numerous models suggested to solve the vector maximum
problem. Wada et al. [158] have proposed a fuzzy control method with triangular type
membership functions using an image processing unit to control the level of granules inside a
hopper. They have stated that the image processing unit can be used as a detecting element
and with the use of fuzzy reasoning methods good process responses are obtained. Fuzzy
finite element method for static analysis of engineering systems has been done by Rao et al.
[159] using an optimization-based scheme taking fuzzy parameters into consideration. A
fuzzy arithmetical approach has been used by Hanss et al. [160] for the solution of finite
element problems involving uncertain parameters. Boutros et al. [161] have reported a
simple, effective and robust fusion approach based on fuzzy logic and sugeno-style inference
engine. Using this method, four condition-monitoring indicators, developed for detection of
transient and gradual abnormalities, are fused into one single comprehensive fuzzy fused
index (FFI) for reliable machinery health assessment. This approach has been successfully
tested and validated in two different practical applications.
2.5.1.3 Application of Fuzzy Logic for Fault Diagnosis
In this section a number of journal papers related to fault diagnosis using fuzzy technique has
been reviewed and described.
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A fuzzy finite element approach has been proposed by Akpan et al. [162] for modeling smart
structures with imprecise parameters. A simple method for diagnosis of railway wheel
defects using fuzzy-logic has been proposed by Skarlatos et al. [163]. The method is based
on vibration measurements at different train speeds on healthy wheels and wheels with
known defects. To facilitate the implementation of the method fuzzy-logic is adopted. Liu et
al. [164] have described a new method of grinding burn identification with highly sensitive
acoustic emission (AE) techniques. The wavelet packet transform is used to extract features
from AE signals and fuzzy pattern recognition is employed for optimizing features and
identifying the grinding status. Experimental results show that the accuracy of grinding burn
recognition is satisfactory. Parhi et al. [165] have designed a mobile robot navigation system
using fuzzy logic. Fuzzy rules embedded in the controller of a mobile robot enable it to avoid
obstacles in a cluttered environment that includes other mobile robots. Angelov et al. [166]
have presented a systematic classification of the data-driven approaches for design of fuzzy
systems. The condition monitoring of a lab-scale, single stage, gearbox using different nondestructive inspection methodologies and the processing of the acquired waveforms with
advanced signal processing techniques have been presented by Loutas et al. [167]. Acoustic
emission (AE) and vibration measurements with fuzzy method are utilized for this purpose.
The experimental setup has been used for validation of results from the proposed method. As
stated by the author the system can be used for the early diagnosis of natural wear in gear
systems. Saravanan et al. [168] have used decision tree for selecting best statistical features
that will discriminate the fault conditions of the gear box from the signals extracted to
determine the condition of an inaccessible gear in an operating machine. A rule set is formed
from the extracted features and fed to a fuzzy classifier. A fuzzy classifier is built and tested
with representative data. A fuzzy finite element method has been used by Chen et al. [169]
for vibration analysis of imprecisely defined systems by using a search based algorithm. The
fuzzy approach enhances the computational efficiency for identifying the system dynamic
responses. Pawar et al. [170,174] have used a genetic fuzzy system and finite element model
of a cantilever beam to find the location and extent of damage. Using these changes in
frequencies, a fuzzy system is generated and the rule-base and membership functions are
optimized by genetic algorithm. The genetic fuzzy system gives very good results for hinge
less helicopter rotor blade for frequency as well as mode shape-based data. Taha et al. [171]
29
have introduced a method to improve pattern recognition and damage detection by
supplementing Intelligent Structural Health Monitoring (ISHM) with fuzzy sets. Bayesian
updating is used to demarcate levels of damage into fuzzy sets accommodating the
uncertainty associated with the ambiguous damage states. The new techniques are examined
to provide damage identification using data simulated from finite element analysis of a prestressed concrete bridge. Packianather et al. [172] have proposed a method for identification
of defects in wood veneer using neural network. Dwivedy et al. [173] have discussed about
application of AI techniques in various engineering problems. Kim et al. [175] have
presented a computer assisted crack diagnosis system for reinforced concrete structures. The
system presented adapts fuzzy set theory to reflect fuzzy conditions, for crack symptoms and
characteristics which are difficult to treat using crisp sets. The inputs to the system are mostly
linguistic variables of the crack and some numeric data about concrete and environmental
conditions. An attempt has been made by Sasmal et al. [176] to develop a systematic
procedure and formulations for condition evaluation of existing bridges using analytic
hierarchy process in a fuzzy environment. Fuzzy logic approach has been used to take care of
the uncertainties and imprecision in the bridge observations. Chandrashekhar et al. [177]
have shown that geometric and measurement uncertainty cause considerable problem in
damage assessment which can be alleviated by using a fuzzy logic-based approach for
damage detection. Monte Carlo simulation (MCS) is used to study the changes in the damage
indicator due to uncertainty in the geometric properties of the beam. The paper brings
together the disparate areas of probabilistic analysis and fuzzy logic to address uncertainty in
structural damage detection.
2.5.2
Neural Network Technique for Damage Diagnosis
The human brain is very complex, nonlinear and parallel computer. There are billions of
neurons and trillions of connections between them. The interest in neural network stems from
the wish of understanding principles leading in some manner to the comprehension of the
basic human brain functions, and to building the machines that are able to perform complex
tasks. Neural network theory revolves around the idea that certain key properties of
biological neurons can be extracted and applied to simulations, thus creating a simulated
brain. The neural network technique can be used for damage diagnosis in vibrating cracked
30
structures. For on line condition monitoring of damaged structure artificial intelligence
techniques has to be adapted. To design an intelligent controller neural network play vital
role.
2.5.2.1 Neural Network History
The modern era of neural network research is credited with the work done by neurophysiologist, Warren McCulloch and young mathematical prodigy Walter Pitts in 1943[179].
They wrote a paper on how neurons might work, and they designed and built a primitive
artificial neural network using simple electric circuits. They are credited with the McCullochPitts Theory of Formal Neural Networks. The next major development in neural network
technology have arrived in 1949 with Donald Hebb [180]. A major point bought forward
from his research, described how neural pathways are strengthened each time they were used.
As we shall see, this is true of neural networks, specifically in training a network. Years
later, John von Neumann thought of imitating simplistic neuron functions by using telegraph
relays or vacuum tubes. This led to the invention of the von Neumann machine.
2.5.2.2 Application of Neural Network
This section describes the review of literature from the application area of neural network in
various fields.
Little et al. [181] have solved a linearized version of the model and explicitly showed that the
capacity of the memory is related to the number of synapses rather than the number of
neurons. In addition, they have shown that in order to utilize this large capacity, the neural
network must store the major part of the information in memory to generate patterns which
evolve with time. Urban et al. [182] have described the development of enhanced, high speed
data reduction algorithms using artificial neural networks (ANN). The networks are trained
using computed data and subsequently give values of film parameters in the millisecond time
regime. Athanasiu et al. [183] have used an artificial neural network (ANN) with local
connectivity as a track identifier for high energy physics experiments. The performance of
the ANN is evaluated with data of the experiment, with very encouraging results. Sette et al.
[185] have presented a method to simulate a complex production process using a neural
31
network and the optimization by genetic algorithm for quality control of the end product in a
manufacturing environment. They have used the genetic algorithm with a sharing function
and a Pareto optimization to optimize the input parameters for obtaining the best yarns.
According to them the results from this method are considerably better than current manual
machine intervention. Mohanty et al. [186] have proposed a method for classification of
remotely sensed data using an artificial neural network (ANN) approach. Taib et al. [187]
have applied an artificial neural network (ANN) for the analysis of the response of an optical
fibre pH sensor. A three layer feed forward neural network is used and network training is
performed using the recursive prediction error (RPE) algorithm. An application of the
method has been demonstrated. Kermanshahi [188] have applied two artificial neural
networks, a recurrent neural network (RNN) and a three-layer feed-forward back-propagation
(BP) for long-term load forecasting. In this study, total system load forecast reflecting current
and future trends, tempered with good judgement which is the key to all planning and indeed
financial success is carried out for nine utilities in Japan. Ghiassi et al. [190] have presented a
dynamic neural network model for forecasting time series events that uses a different
architecture than traditional models. To assess the effectiveness of this method, they have
forecasted a number of standard benchmarks in time series research from forecasting
literature. Results show that this approach is more accurate and performs significantly better
than the traditional neural network. Hines et al. [191] have focused on the implementation of
methods and the development of methods for next generation plants and space reactors. As
stated by them the advanced techniques are expected to become increasingly important for
current generation nuclear power plants. Song et al. [192] have presented a method for
comparison of logistic regression and artificial neural network for computer-aided diagnosis
on breast sonograms. They have concluded that there is no difference in performance
between logistic regression and the artificial neural network as measured by the area under
the ROC curve. Kadi et al. [193] have attempted to reflect on the work done in the
mechanical modeling of fiber-reinforced composite materials using ANN during the last
decade. Lucon et al. [194] have utilized artificial neural networks in place of a traditional
micromechanical approach to calculate the global elastic properties of composite materials
given the local properties and local geometry. This approach is shown to be more
computationally efficient than conventional numerical micromechanical approaches. Assaad
32
et al. [195] have adapted an ensemble method to the problem of predicting future values of
time series using recurrent neural networks (RNNs) as base learners. The improvement is
made by combining a large number of RNNs, each of which is generated by training on a
different set of examples. Carbonneau et al. [196] have investigated the applicability of
advanced machine learning techniques, including neural networks, recurrent neural networks,
and support vector machines, to forecast distorted demand at the end of a supply chain. Lolas
et al. [197] have presented the first module of an expert system, a neural network architecture
that could predict the reliability performance of a vehicle at later stages of its life by using
only information from a first inspection after the vehicle’s prototype production. A case
study is presented by them to demonstrate the methodology. Ozerdem et al. [198] have
employed an artificial neural network approach to predict the mechanical properties of cast
alloys. In artificial neural network (ANN), multi layer perceptron (MLP) architecture with
back-propagation algorithm is utilized. ANN system is trained using the prepared training set
and has given satisfactory results. Reddy et al. [199] have developed an artificial neural
network (ANN) model for the analysis and simulation of the correlation between the
mechanical properties and composition and heat treatment parameters of low alloy steels.
Panigrahi et al. [200] have used adaptive bacterial foraging algorithm and AI technique for
monitoring power system. Sun et al. [201] have formulated and solve the force distribution
problem for multi arm systems with flexible-link arms, with particular attention to the non
minimum phase character of the system.
2.5.2.3 Application of Neural Network for Fault Diagnosis
The application of neural network for fault diagnosis in different domain of engineering
system can be established with the review of literatures as follows.
Uhrig et al. [202] have described methods that deal with power plants or parts of plants that
can be isolated. Typically, the measured variables from the plants are analog variables that
must be sampled and normalized to expected peak values before they are introduced into
neural networks. Specific applications using neural network described by them include:
transient identification, plant-wide monitoring, analysis of vibrations, and monitoring of
performance and efficiency. Sreedhar et al. [203] have presented an adaptive neural network
33
for fault detection in nonlinear systems. The scheme used by them provides robust fault
detection in the presence of modeling errors and is validated by simulating faults in a section
of a thermal power plant model. Uhrig et al. [204] have done a comprehensive study of the
application of soft computing technologies, particularly neural networks, fuzzy logic, and
genetic algorithms, to the surveillance, diagnostics and operation of nuclear power plants.
The benefits of combining the use of neutral networks, fuzzy systems and genetic algorithms
are illustrated in several applications. Gao et al. [205] have proposed an Elman neural
network-based method for fault detection in motor drive systems. They have stated that the
Elman neural network has the advantageous time series prediction capability because of its
memory nodes, as well as local recurrent connections. The intelligent computational tools of
feed forward neural networks and genetic algorithms are used to develop a real-time
detection and diagnosis system of specific mechanical, sensor failures in a deep-trough
hydroponic system by Ferentinos et al. [206]. Samanta et al. [207] have presented the
comparision of the performance of bearing fault detection using three types of artificial
neural networks (ANNs), namely, multilayer perception (MLP), radial basis function (RBF)
network, and probabilistic neural network (PNN). They have used vibration signals of a
rotating machine with defective bearings as inputs to all three ANN classifiers for normal or
fault recognition. The characteristic parameters along with the selection of input features are
optimized using genetic algorithms (GA). According to them the procedure has used the
experimental vibration data of a rotating machine with and without bearing faults. The results
show the relative effectiveness of three classifiers in detection of the bearing condition. Choi
et al. [209] have developed a method to estimate the size of a tooth transverse crack for a
spur gear in operation. Zhou et al. [211] have described how the recursive algorithm updates
the BRB (Belief Rule Base) system work, so that the updated BRB cannot only be used for
pipeline leak detection but also satisfy the given patterns. They have also demonstrated that
compared with other methods such as fuzzy neural networks (FNNs), the developed system
has a special characteristic of allowing direct intervention of human experts in deciding the
internal structure and the parameters of a BRB expert system. Nishith et al. [212] have
proposed a method with the application of a counter propagation neural network (CPNN) to
detect single faults and their magnitudes in a non isothermal continuous stirred tank reactor
(CSTR). Stavroulakis et al. [213] have used soft computing and in particular neural network
34
techniques for crack detection and identification problems. Elastostatic and elastodynamic
excitations are modelled by the BEM. Results of inverse calculations obtained by the back
propagation neural network model are presented. Suh et al. [214] have established that a
crack has an important effect on the dynamic behavior of a structure. This effect depends
mainly on the location and depth of the crack. A neural network technique is developed for
identifying the damage occurrence in the side shell of a ship’s structure by Zubaydi et al.
[215]. The side shell is modeled as a stiffened plate. The input to the network is the
autocorrelation function of the vibration response of the structure. The response is obtained
using a finite element model of the structure. The results show that the method presented in
this work is successful in identifying the occurrence of damage. Kao et al. [216] have
employed a novel neural network-based approach for detecting structural damage. The first
step, system identification, uses neural system identification networks (NSINs) to identify the
undamaged and damaged states of a structural system. The second step, structural damage
detection, uses the aforementioned trained NSINs to generate free vibration responses with
the same initial condition. Furthermore, numerical and experimental examples demonstrate
the feasibility of applying the proposed method for detecting structural damage. Yam et al.
[217,218] have presented an integrated method for damage detection of composite structures
using their vibration responses, wavelet transform and artificial neural networks (ANN). The
ANN are applied to establish the mapping relationship between structural damage and
damage status (location and severity). The results of show that the method can be applied to
online structural damage detection and health monitoring for various industrial structures.
Sahin et al. [219] have presented a damage detection algorithm using a combination of global
(changes in natural frequencies) and local (curvature mode shapes) vibration-based analysis
data as input in artificial neural networks (ANNs) for location and severity prediction of
damage in beam-like structures. Different damage scenarios have been introduced by
reducing the local thickness of the selected elements at different locations along finite
element model (FEM) of the beam structure. Zacharias et al. [220] have proposed a crack
detection method by an artificial neural network (ANN) trained exclusively with frequency
response spectra from finite-element simulations. The classification fails for some data sets
of intact crates, due to experimental conditions not accounted for in the finite-element
simulation. Suresh et al. [221] have presented a method considering the flexural vibration in
35
a cantilever beam having transverse crack. They have computed modal frequency parameters
analytically for various crack locations and depths and these parameters are used to train the
neural network to identify the damage location and size. Damage assessment in structures
from changes in static parameter using neural network have been performed by Maity et al.
[222]. The basic strategy applied in their study is to train a neural network to recognize the
behaviour of the undamaged structure as well as of the structure with various possible
damaged states. When this trained network is subjected to the measured response, it is able to
detect any existing damage.
Lee et al. [223] have developed a neural networks-based
damage detection method using the modal properties, which can effectively consider the
modelling errors in the baseline finite element model from which the training patterns are to
be generated. Two numerical example analyses on a simple beam and a multi-girder bridge
are presented to demonstrate the effectiveness of the proposed method. Yeung et al. [224]
have proposed a damage detection procedure, using pattern recognition of the vibration
signature and finite element model of a real structure. They have stated that the neural
networks may be adjusted so that a satisfactory rate of damage detection may be achieved
even in the presence of noisy signals. A new approach for crack detection in beam structures
using neural network (Radial Basis Function) have been performed by Li et al. [225].
Damage detection algorithm is presented using a combination of global and local vibrationbased analysis data as input in artificial neural networks (ANNs) for location and severity
prediction of 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
modes. The results from the proposed method have been validated with the results from
experimental analysis. Neural network based damage detection generally consists of a
training phase and a recognition phase. The relative sensitivities of structural dynamic
parameters are analyzed by He-sheng et al. [226] using neural network. The combined
parameters are presented as the input to the neural network, which computed with the change
rates of the several natural frequencies and the change ratios of the frequencies. Some
numerical simulation examples, such as, cantilever and truss with different damage extends
and different damage locations are analyzed. The results indicated that the combined
parameters are more suitable for the input patterns of neural networks than the other
parameters alone. Choubey et al. [227] have studied to analyze the effect of cracks on natural
36
frequencies in two vessel structures. Finite element analysis has been used by them to obtain
the dynamic characteristics of intact and damaged vessels. The natural frequencies for
different modes have been used as input pattern of ANN (artificial neural network) model.
The output of the ANN model is a crack size for a particular location. Li et al. [228] have
described a new and practical method to estimate the size of a crack on a rotating beam in a
laboratory setting. Their paper consists of selecting and validating a sensor and a
measurement variable, devising a signal processing method for crack size estimation and
carrying out experimental validations. The study employed a diagnostic neural network to
map the frequencies to crack size. The experimental results show that the proposed approach
can provide reasonably good estimates of the crack size using the indirectly excited acoustic
signal. Yu et al. [229] have developed a vibration-based damage detection method for a static
laminated composite shell partially filled with fluid and validated by experiment. An
artificial neural network (ANN) is trained by them using numerically simulated structural
damage index to establish the mapping relationship between the structural damage index and
damage status. The damage status is successfully identified using ANN and the method can
be applied to online structural damage detection and health monitoring. Wang et al. [230]
have presented the numerical simulation and the model experiment upon a hypothetical
concrete arch dam for the research of crack detection based on the reduction of natural
frequencies. Numerical analysis and model experiment show that the crack occurring in the
arch dam will reduce natural frequencies and can be detected by using the statistical neural
network based on the information of such reduction. Pawar et al. [231] have proposed spatial
Fourier analysis and Neural technique for damage detection in beam. Their study
investigated the effect of damage on beams with fixed boundary conditions using Fourier
analysis. A finite element model is used to obtain the mode shapes of a damaged beam. It is
found that damage caused considerable change in the Fourier coefficients of the mode
shapes, which are found to be sensitive to both damage size and location. A neural network is
trained to detect the damage location and size using Fourier coefficients as input. Numerical
studies showed that damage detection using Fourier coefficients and neural networks has the
capability to detect the location and damage size accurately. Reddy et al. [232] have
proposed a method for beams with fixed boundary conditions of a damaged fixed beam by
using Fourier analysis, for identification of crack location and depth. They have also used
37
neural network which is trained to detect the damage location and size using fourier
coefficients as input. They have studied that the method for damage detection using Fourier
coefficients and neural network has the capability to detect the location and damage size even
in the presence of noise parameters satisfactorily and accurately. Bakhary et al. [233] have
used Artificial Neural Network for damage detection. In their analysis an ANN model is
created by applying Rosenblueth’s point estimate method verified by Monte Carlo
simulation. The statistics of the stiffness parameters are estimated. The probability of damage
existence (PDE) is then calculated based on the probability density function of the existence
of undamaged and damaged states. The developed approach is applied to detect simulated
damage in a numerical steel portal frame model and also in a laboratory tested concrete slab.
2.5.3 Neuro-Fuzzy Technique for Damage Diagnosis
In the field of artificial intelligence, Neuro-Fuzzy refers to combinations of artificial neural
networks and fuzzy logic which incorporates the capability of both fuzzy logic and neural
network technique. This hybrid method can give better results than the independent
techniques. Fuzzy systems have the ability to make use of knowledge expressed in the form
of linguistic rules, thus they offer the possibility of implementing expert human knowledge
and experience. Usually, tuning parameters of membership functions is a time consuming
task. Neural network learning techniques can automate this process, significantly reducing
development time, and resulting in better performance. Neuro-fuzzy hybridization results in a
hybrid intelligent system that synergizes these two techniques by combining the human-like
reasoning style of fuzzy systems with the learning and connectionist structure of neural
networks. Hence, this methodology can take the vibration signatures as in put parameters and
predict the crack location and depth. Jantunen [234] has used Neuro-Fuzzy System for
condition monitoring and diagnostic management of mechanical system.
2.5.3.1 Neuro-Fuzzy Technique History
A neuro-fuzzy system is based on a fuzzy system which is trained by a learning algorithm
derived from neural network theory. The learning procedure operates on local information,
and causes only local modifications in the underlying fuzzy system. The strength of neuro38
fuzzy systems involves two contradictory requirements in fuzzy modeling: interpretability
versus accuracy. In practice, one of the two properties prevails. The neuro-fuzzy in fuzzy
modeling research field is divided into two areas: linguistic fuzzy modeling that is focused on
interpretability, mainly the Mamdani model; and precise fuzzy modeling that is focused on
accuracy, mainly the Takagi-Sugeno-Kang (TSK) model.
2.5.3.2 Application of Neuro-Fuzzy Technique
Ichihashi et al. [235] have developed a popular and efficient method of making a decision
tree for classification from symbolic data without much computation. Fuzzy reasoning rules
in the form of a decision tree, which can be viewed as a fuzzy partition, are obtained by fuzzy
ID3. Wang et al. [236] have evaluated the performance of recurrent neural networks (RNNs)
and neuro-fuzzy (NF) systems using two benchmark data sets. Through comparison it is
found that if an NF system is properly trained, it performs better than RNNs in both
forecasting accuracy and training efficiency. The performance of the developed prognostic
system is evaluated by using three test cases.
2.5.3.3 Appilication of Neuro-Fuzzy Technique for Fault Diagnosis
Singh et al. [237] have reviewed the progress made in electrical drive condition monitoring
and diagnostic research and development in general and induction machine drive condition
monitoring and diagnostic research and development using expert systems, neural network
and fuzzy logic, in particular, since its inception. Xu et al. [238] have proposed neuro-fuzzy
control strategy, in which the neural-network technique is adopted to solve time-delay
problem and the fuzzy controller is used to determine the control current of MR dampers
quickly and accurately. They have observed that the control effect of the neuro-fuzzy control
strategy is better than that of the bi-state control strategy. A novel integrated classifier has
been developed by Wang et al. [239] for real-time machinery health condition monitoring,
specifically for gear systems. The diagnostic classification is performed by a neural fuzzy
scheme. An online hybrid training technique is adopted based on recursive Levenberg–
Marquet and least-squares estimate (LSE) algorithms to improve the classifier convergence
and adaptive capability to accommodate different machinery conditions. The viability of this
39
new monitoring system is verified by experimental tests under different gear conditions. Test
results show that the proposed integrated classifier provides a robust problem solving
framework. Saridakis et al. [240] have proposed a method using fuzzy logic, genetic
algorithm and neural network for considering the dynamic behavior of a shaft with two
transverse cracks characterized by position, depth and relative angle. They have concluded
that genetic algorithm along with the neural network which form the analytical model for
analysis remarkably reduce the computational time without any loss of accuracy. Rafiee et al.
[241] have presented an optimized gear fault identification system using genetic algorithm
(GA) to investigate the type of gear failures of a complex gearbox system using artificial
neural networks (ANNs) with a well-designed structure suited for practical implementations.
2.5.4
Multiple Adaptive Neuro Fuzzy Inference Technique (MANFIS)for Damage
Diagnosis
Adaptive neuro-fuzzy inference systems (ANFIS), fusing the capabilities of artificial neural
networks and fuzzy inference systems, offer a lot of space for solving different kinds of
problems, and are especially efficient in the domain of signal prediction. However, the
ANFIS technique is sometimes notated as being computationally expensive. After
considering the conventional ANFIS architecture, a new idea came up known as multiple
adaptive neuro-fuzzy inference systems (MANFIS) developed with the intention of making
the ANFIS technique more efficient with regard to root mean square error (RMSE) and/or
computing time by Jovanovic et al. [242]. So this technique can be used for crack diagnosis
effectively in cracked structures with multiple ANFIS system.
2.5.4.1 MANFIS History
Fuzzy inference is the process of formulating the mapping from a given input to an output
using fuzzy logic. The mapping then provides a basis from which decisions can be made, or
patterns discerned. The process of fuzzy inference includes membership functions, fuzzy
logic operators, and if–then rules. ANFIS provides a method for the fuzzy modeling
procedure to learn information about a dataset, in order to compute the membership function
parameters that best allow the associated fuzzy inference system to track the given
input/output data. Jang [243] has exhibited the use of ANFIS technique in engineering
40
application. Random and bootstrap sampling method and ANFIS (Adaptive Network based
Fuzzy Inference System)are integrated into En-ANFIS (an ensemble ANFIS) to predict
chaotic and traffic flow time series to achieve both high accuracy and less computational
complexity for time series prediction by Chen et al. [244]. Hinojosa et al. [245] have
presented four modeling methods of microwave devices using multiple neuro-fuzzy inference
systems (MANFIS) based on space-mapping (SM) approach.
2.5.4.2 Application of MANFIS
In this section, applications of MANFIS technique in various fields have been established
with the literature review of published paper.
Jassar et al. [246] have developed an inferential sensor model, based on adaptive neuro-fuzzy
inference system modeling, for estimating the average air temperature in multi-zone space
heating systems. This modeling technique has the advantage of expert knowledge of fuzzy
inference systems (FISs) and learning capability of artificial neural networks (ANNs). The
average air temperature results estimated by using the developed model are strongly in
agreement with the experimental results. Domenech et al. [247] have applied fuzzy logic for
accurate analog circuit macro model sizing is presented. In the proposed method, multiple
adaptive neuro-fuzzy inference systems (MANFIS) are trained to predict the performance
characteristics. Zhang et al. [248] have presented an investigation into the use of the delay
coordinate embedding technique with adaptive-network-based-fuzzy-inference system
(ANFIS) and MANFIS technique to learn and predict the continuation of chaotic signals
ahead in time.
2.5.4.3 Application of MANFIS for Fault Diagnosis
The papers presented, the use of MANFIS technique for fault diagnosis has been reviewed
and discussed in this section.
Nguyen et al. [249] have developed a bearing diagnostics method using fuzzy inference
based on vibration data. Adaptive Network based Fuzzy Inference System (ANFIS) and
Genetic Algorithm (GA) has been proposed to select the fuzzy model input and output
41
parameters. The result is also tested with other set of bearing data to illustrate the reliability
of the chosen model. Elbaset et al. [250] have presented an application of ANFIS approach
for automated fault detection and classification in transmission lines using measured data
from one terminal of the transmission line. The ANFIS design and implementation are aimed
at high-speed processing which can provide selection of real-time detection and classification
of faults. The ANFIS's are trained and tested using various sets of field data. Ye et al. [251]
have presented an online diagnostic algorithm for mechanical faults of electrical machines
with variable speed drive systems using wavelet packet decomposition. A new integrated
diagnostic system for electrical machine mechanical faults is proposed using multiple
adaptive neuro-fuzzy inference Systems (ANFIS). The diagnostic algorithm is validated on a
three-phase induction motor drive system. Yeo et al. [252] have proposed an algorithm for
fault detection and classification for both low impedance faults and high impedance faults
using Adaptive Network-based Fuzzy Inference System (ANFIS). The inputs into ANFIS are
current signals only based on Root-Mean-Square values of three-phase currents and zero
sequence current. The performance of the proposed algorithm is tested and found to be
encouraging. Sadeh et al. [253] have presented an algorithm for locating faults in a combined
overhead transmission line with underground power cable using Adaptive Network-Based
Fuzzy Inference System (ANFIS). Simulation results confirm that the proposed method can
be used as an efficient means for accurate fault location on the combined transmission lines.
Razavi-Far et al. [254] have described a neuro-fuzzy networks based scheme for fault
detection and isolation of a U-tube steam generator in a nuclear power plant. Experimental
results presented in the final part of the paper confirm the effectiveness of this approach.
Tran et al. [255] have presented a fault diagnosis method based on adaptive neuro-fuzzy
inference system (ANFIS) in combination with decision trees. The crisp rules obtained from
the decision tree are then converted to fuzzy if-then rules that are employed to identify the
structure of ANFIS classifier. In order to evaluate the proposed algorithm, the data sets
obtained from vibration signals and current signals of the induction motors are used. The
results indicate that the ANFIS model has potential for fault diagnosis of induction motors.
Lei et al. [256,257] have presented a method for fault diagnosis based on empirical mode
decomposition (EMD), an improved distance evaluation technique and the combination of
multiple adaptive neuro-fuzzy inference systems (ANFISs). Their proposed method is
42
applied to the fault diagnosis of rolling element bearings, and testing results show that the
multiple ANFIS combination can reliably recognize different fault categories and severities.
From the above literature survey, it is found that the vibration signatures of the cracked
structure can be calculated by using strain energy release rate and stress intensity factor.
Different Artificial Intelligence Techniques can be used for fault detection and condition
monitoring of various engineering applications. It is found from the review that the AI
techniques are not used potentially for on line condition monitoring of crack in vibrating
structures.
So in the subsequent section algorithm have been developed for on line condition monitoring
of a cracked cantilever beam using AI techniques such as Fuzzy Logic, Neural Network,
Fuzzy Neuro and MANFIS techniques.
43
Chapter 3
ANALYSIS OF DYNAMIC CHARACTERISTICS OF BEAM
WITH TRANSVERSE CRACK
Cracks in vibrating components can initiate catastrophic failures. Therefore, there is the need
to understand the dynamic characteristics of cracked structures to save the structure
beforehand by detecting the crack location and its intensity. When a structure suffers
damage, its dynamic properties change. Specifically, damage due to the crack can cause a
stiffness reduction, with an inherent reduction in natural frequencies, an increase in modal
damping, and a change in the mode shapes.
3.1
Introduction
Dynamic characteristics of structures with crack have been studied for last four decades
intensively. Natural frequencies and modes shapes undergo variation due to presence of
crack in terms of its location and intensity. Scientists are focusing their thoughts on various
aspects of cracked structures. The current research addresses the investigation of the dynamic
behavior of a cracked beam with a transverse crack. The presence of a crack in a structural
member introduces a local flexibility that affects its dynamic response. For finding out the
deviation in the vibrating signatures of the cracked cantilever beam the local stiffness
matrices are taken into account. Theoretical expressions have been developed to calculate the
natural frequencies and mode shapes of the cracked cantilever beam using local stiffness
matrices. Strain energy release rate has been used for calculating the local stiffnesses of the
beam. The local stiffnesses are dependent on the crack depth. Different boundary conditions
are outlined which take into account the crack location. Comparisons are made between the
numerical results and corresponding experimental results for validation of the established
theory.
44
3.2
Dynamic Characteristics of a Cantilever Beam with a Transverse Crack
3.2.1 Theoretical Analysis
A systematic approach has been adopted in the present investigation to develop theoretical
expressions for calculation of natural frequencies and mode shapes of cracked cantilever
beam with a transverse crack and to notice the effect of crack on natural frequencies and
mode shapes. Experiments have been conducted over cracked cantilever beam specimen for
validation of the theory established. Natural frequencies and the mode shapes of the cracked
cantilever beam specimen are found out both numerically and experimentally for different
relative crack depth and relative crack location from fixed end of the cantilever beam.
Remarkable variations in mode shapes are noticed at the vicinity of crack location.
3.2.1.1 Local Flexibility of a Cracked Cantilever Beam under Bending and Axial Loading
A cantilever beam with a transverse surface crack of depth ‘a1’ on beam of width ‘B’ and
height ‘W’ is considered in the current research. The beam is subjected to axial force (P1) and
bending moment (P2) (Fig.3.2.1) 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
B
W
P1
P2
da
a1
W
(b)
B
X axis
Y axis
da
L1
a1
Z axis
dz
(a)
Fig. 3.2.1
+B/2
-B/2
L
(c)
Geometry of beam, (a) cantilever beam, (b) cross-sectional view of the beam.
(c) segments taken during integration at the crack section
45
The strain energy release rate at the fractured section can be written as [20];
J=
1 1− v 2
1
=
(for plane strain condition);
( K I1 + K I 2 ) 2 , Where
E′
E
E′
1
(for plane stress condition)
E
=
(3.2.1a)
(3.2.1b)
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 [20] are;
K I1 =
P1
6P2
a
πa (F1 ( )), K I 2 =
BW
W
BW 2
πa (F2 (
a
))
W
(3.2.2)
Where expressions for F1 and F2 are as follows
a
2W
πa 0.5 ⎧ 0.752 + 2.02 (a / W ) + 0.37 (1 − sin(πa / 2 W)) 3 ⎫
F1 ( ), = (
tan(
)) ⎨
⎬
cos(πa / 2W )
W
πa
2W
⎩
⎭
F2 (
(3.2.3)
a
πa 0.5 ⎧ 0.923 + 0.199 (1 − sin( πa / 2 W )) 4 ⎫
2W
)) ⎨
tan(
), = (
⎬
πa
W
2W
cos( πa / 2 W )
⎩
⎭
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
(3.2.4)
a1
a
1
∂U t
da = ∫ J da
The strain energy will have the form, U t = ∫
∂a
0
0
Where J=
∂U t
the strain energy density function.
∂a
46
(3.2.5)
From (Eqs. 3.2.4 and 3.2.5), thus we have
a
⎤
∂ ⎡1
ui =
⎢ ∫ J (a ) da ⎥
∂Pi ⎣⎢ 0
⎦⎥
(3.2.6)
The flexibility influence co-efficient Cij will be, by definition
a
∂u i
∂2 1
C ij =
J (a ) da
=
∂Pj ∂Pi ∂Pj ∫0
(3.2.7)
To find out the final flexibility matrix we have to integrate over the breadth ‘B’
∂u
∂2
C ij = i =
∂Pj ∂Pi ∂Pj
+ B / 2 a1
∫ ∫ J(a ) da dz
(3.2.8)
−B / 2 0
Putting the value, strain energy release rate from above, Eq. 3.2.8 modifies as
a
C ij =
B ∂2 1
(K l1 + K l 2 ) 2 da
∫
E ′ ∂Pi ∂Pj 0
Putting ξ = (a/w), dξ =
(3.2.9)
da
,
W
We get da = Wdξ and when a = 0, ξ = 0; a = a1, ξ = a1/W = ξ1
From the above condition Eq. 3.2.9 converts to,
ξ
BW ∂ 2 1
C ij =
(K l1 + K l 2 ) 2 dξ
∫
E ′ ∂Pi ∂Pj 0
(3.2.10)
From the Eq. 3.2.10, calculating C11, C12 (=C21) and C22 we get,
BW
C11 =
E′
ξ1
πa
∫B W
2
2
2(F1 (ξ)) 2 dξ
0
47
ξ
2π 1
=
ξ(F1 (ξ)) 2 dξ
BE ′ ∫0
12π
C12 = C 21 =
E ′BW
C 22 =
72π
E ′BW 2
(3.2.11)
ξ1
∫ ξF (ξ)F (ξ) dξ
1
(3.2.12)
2
0
ξ1
∫ ξF (ξ)F (ξ) dξ
2
(3.2.13)
2
0
Converting the influence co-efficient into dimensionless form
C11 = C11
E ′BW 2
E ′BW
BE ′
C12 = C12
= C 21 ; C 22 = C 22
72π
12π
2π
(3.2.14)
The local stiffness matrix can be obtained by taking the inversion of compliance matrix. i.e.
⎡K
K = ⎢ 11
⎣K 21
K 12 ⎤ ⎡ C11
=
K 22 ⎥⎦ ⎢⎣C 21
C12 ⎤
C 22 ⎥⎦
−1
(3.2.15)
Relative Crack Depth a1/W
.
11 12 21 22 Dimensionless Compliance (ln ( C xy ))
Fig. 3.2.2 Relative crack depth (a1/W) vs. dimensionless compliance (ln ( C xy ))
48
Fig. 3.2.2 shows the variation of dimension-less compliances to that of relative crack depth
From Fig. (3.2.2) it can be observed that as the crack depth increases, the compliances (C11,
C12=C21, C22) also increase.
3.2.1.2 Free Vibration Analysis of the Cracked Cantilever Beam
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. 3.2.1). 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. 3.2.3).
U2
U1
L1
L
Y1
Y2
Fig. 3.2.3 Beam model
The normal function for the system can be defined as
u 1 ( x ) = A1 cos ( K u x ) + A 2 sin(K u x )
(3.2.16a)
u 2 ( x ) = A 3 cos (K u x ) + A 4 sin(K u x )
(3.2.16b)
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 )
(3.2.16c)
y 2 ( x ) = A 9 cosh ( K y x ) + A10 sinh(K y x ) + A11 cos (K y x ) + A12 sin(K y x )
(3.2.16d)
49
Where x =
L
x
u
y
,u= ,y= , β = 1
L
L
L
L
⎛E⎞
ωL
, C u = ⎜⎜ ⎟⎟
Ku =
Cu
⎝ρ⎠
1/ 2
⎛ ωL2
, Ky =⎜
⎜ C
⎝ y
⎞
⎟
⎟
⎠
1/ 2
⎛ EI ⎞
, C y = ⎜⎜ ⎟⎟
⎝μ⎠
1/ 2
, μ = Aρ
Ai, (i=1, 12) Constants are to be determined, from boundary conditions. The boundary
conditions of the cantilever beam in consideration are:
u 1 (0) = 0 ;
3.2.17(a)
y1 (0) = 0 ;
3.2.17(b)
y1′ (0) = 0 ;
3.2.17(c)
u ′2 (1) = 0 ;
3.2.17(d)
y"2 (1) = 0 ;
3.2.17(e)
y′2′′(1) = 0 ;
3.2.17(f)
At the cracked section:
u '1 (β) = u ' 2 (β) ;
3.2.18(a)
y1 (β) = y 2 (β) ;
3.2.18(b)
y1′′(β) = y′2′ (β) ;
3.2.18(c)
y1′′′(β) = y′2′′(β) ;
3.2.18(d)
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 ) ⎞
= K 11 (u 2 (L1 ) − u 1 (L1 )) + K 12 ⎜ 2 1 − 1 1 ⎟
dx
dx ⎠
⎝ dx
Multiplying both sides of the above equation by
(3.2.19)
AE
we get;
LK 11 K 12
M1M 2 u ′(β) = M 2 (u 2 (β) − u1 (β)) + M1 ( y′2 (β) − y1′ (β))
(3.2.20)
Similarly at the crack section (due to the discontinuity of slope to the left and right of the
crack)
50
d 2 y1 ( L1 )
⎛ dy (L ) dy (L ) ⎞
EI
= K 21 (u 2 (L1 ) − u 1 (L1 )) + K 22 ⎜ 2 1 − 1 1 ⎟
2
dx ⎠
dx
⎝ dx
(3.2.21)
EI
we get,
L K 22 K 21
Multiplying both sides of the above equation by
2
M 3 M 4 y1′′(β) = M 3 ( u 2 (β) − u 1 (β)) + M 4 ( y′2 (β) − y1′ (β))
Where, M 1 =
(3.2.22)
AE
AE
EI
EI
, M2 =
, M3 =
, M4 = 2
LK11
K 12
LK 22
L K 21
The normal functions, Eq. {3.2.16} along with the boundary conditions as mentioned above,
yield the characteristic equation of the system as:
Q =0
(3.2.23)
Where Q is a 12x12 matrix and is expressed as
Q=
1
0
1
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
G3
G4
-G7
-G8
0
0
0
0
0
0
0
0
G4
G3
G8
-G7
0
0
0
0
G1
G2
-G5
-G6
-G1
-G2
G5
G6
0
0
0
G2
G1
G6
-G5
-G2
-G1
-G6
G5
0
0
0
0
0
G1
G2
G5
G6
-G1
-G2
-G5
-G6
0
0
0
0
S1
S2
S3
S4
-G2
-G1
G6
-G5
S5
S6
S7
S8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-T8
T7
0
0
0
0
0
0
0
0
-T6
T5
T6
-T5
S9
S10
S11
S12
S13
S14
S15
S16
S17
S18
-T5
-T6
51
(3.2.24)
Where G1= Cosh ( K y α ), G2=Sinh ( K y α ), G3= Cosh ( K y ), G4=Sinh ( K y ), G5=Cos ( K y α ),
G6=Sin ( K y α ), G7=Cosh ( K y ), G8=Sin ( K y ),
T5=Cos ( K u α ), T6=Sin ( K u α ), T7=Cos ( K u ), T8=Sin ( K u )
M12=
M1
M2
, M34=
M3
M4
S1=G2 + M3 K y G1, S2= G1 + M3 K y G2, S3= G6
S6=
M 34
Ky
T6, S7=
-M 34
Ky
T5, S8=
-M 34
Ky
M3 K y G5, S4= G5 M3 K y G6, S5 =
M 34
Ky
,
T6, S9= M12 K y G2
S10=M12 K y G1, S11= M12 K y G6, S12= M12 K y G5
S13= M12 K y G2, S14= M12 K y G1,
S15= M12 K y G6, S16= M12 K y G5, S17 = T5 –M1 K u T6, S18= T6 M1 K u T5
This determinant is a function of natural circular frequency (ωn), the relative location of the
crack (L1/L) and the local stiffness matrix (K) which in turn is a function of the relative crack
depth (a1/W).
3.2.1.3 Forced Vibration Analysis of Cracked Cantilever Beam
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
y
as y 2 (1) = 0 = y 0 . Therefore the boundary conditions for the beam remain same as before
L
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
(3.2.25)
52
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] (3.2.26)
3.2.2 Numerical Analysis
The numerical analysis is carried out for the cracked cantilever beam to find the relative
amplitudes of transverse vibration at different crack location and crack depth. The cracked
cantilever beam of the current research has the following dimensions.
Length of the Beam
= 0.8m
Width of the beam
= 0.05m
Height of the Beam
= 0.006m
Relative crack depth (a1/W)
= Varies from 0.05 to 0.8
Relative crack location (L1/L)
= Varies from 0.125 to 0.95
3.2.2.1 Results of Numerical Analysis
The relative amplitudes of transverse vibration for first three mode shapes of cracked
cantilever beam made of Aluminum are obtained at different crack location and crack depth
by numerical solution of (Eqs.3.2.16c and 3.2.161d) of section 3.2.1.2. The results of
numerical analysis are presented in Fig. 3.2.4 to Fig. 3.2.27. The relative amplitudes of
transverse vibration for first three mode shapes of un-cracked cantilever beam made of
Aluminum are also plotted in the corresponding figures for immediate comparison. The three
dimensional variation of relative natural frequencies and relative mode shape difference with
respect to relative crack location and relative crack depth along with the contour plots are
presented in Fig. 3.2.28 to Fig. 3.2.29.
53
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (1st mode of vibration), a1/W=0.1, L1/L=0.0256
Relative amplitude
Fig.3.2.4 (a)
Relative distance from fixed end
Fig. 3.2.4(a1) Magnified view of Fig. 3.2.4(a) at the vicinity of the
crack location.
54
Relative amplitude
Relative distance from fixed end
Relative amplitude
Fig.3.2.4 (b) Relative amplitude vs. relative distance from the fixed end
(2nd mode of vibration), a1/W=0.1, L1/L=0.0256
Relative distance from fixed end
Fig. 3.2.4(b1) Magnified view of Fig. 3.2.4(b) at the vicinity of the
crack location.
55
Relative amplitude
Relative distance from fixed end
Relative amplitude
Fig.3.2.4 (c) Relative amplitude vs. relative distance from the fixed end
(3rd mode of vibration), a1/W=0.1, L1/L=0.0256
Relative distance from fixed end
Fig. 3.2.4(c1) Magnified view of Fig. 3.2.4(c) at the vicinity of the
crack location.
56
Relative amplitude
Relative distance from fixed end
Relative amplitude
Fig.3.2.5 (a) Relative amplitude vs. relative distance from the fixed end
(1st mode of vibration), a1/W=0.2, L1/L=0.0256
Relative distance from fixed end
Fig. 3.2.5(a1) Magnified view of Fig. 3.2.5(a) at the vicinity of the
crack location.
57
Relative amplitude
Relative distance from fixed end
Relative amplitude
Fig.3.2.5 (b) Relative amplitude vs. relative distance from the fixed end
(2nd mode of vibration), a1/W=0.2, L1/L=0.0256
Relative distance from fixed end
Fig. 3.2.5 (b1) Magnified view of Fig. 3.2.5(b) at the vicinity of the
crack location.
58
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (3rd mode of vibration), a1/W=0.2, L1/L=0.0256
Relative amplitude
Fig.3.2.5 (c)
Relative distance from fixed end
Fig. 3.2.5(c1)
Magnified view of Fig. 3.2.5(c) at the vicinity of
the crack location.
59
Relative amplitud
Relative distance from fixed end
Relative amplitude
Fig.3.2.6 (a) Relative amplitude vs. relative distance from the fixed end
(1st mode of vibration), a1/W=0.3, L1/L=0.0256
Relative distance from fixed end
Fig. 3.2.6(a1) Magnified view of Fig. 3.2.6(a) at the vicinity of the
crack location.
60
Relative amplitude
Relative distance from fixed end
Relative amplitude
Fig.3.2.6 (b) Relative amplitude vs. relative distance from the fixed end
(2nd mode of vibration), a1/W=0.3, L1/L=0.0256
Relative distance from fixed end
Fig. 3.2.6(b1) Magnified view of Fig. 3.2.6(b) at the vicinity of the
crack location.
61
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (3rd mode of vibration), a1/W=0.3, L1/L=0.0256
Relative amplitude
Fig.3.2.6 (c)
Relative distance from fixed end
Fig. 3.2.6(c1)
Magnified view of Fig. 3.2.6(c) at the vicinity of
the crack location.
62
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (1st mode of vibration), a1/W=0.4, L1/L=0.0256
Relative amplitude
Fig.3.2.7 (a)
Relative distance from fixed end
Fig. 3.2.7(a1)
Magnified view of Fig. 3.2.7(a) at the vicinity of
the crack location.
63
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (2nd mode of vibration), a1/W=0.4, L1/L=0.0256
Relative amplitude
Fig.3.2.7 (b)
Relative distance from fixed end
Fig. 3.2.7(b1)
Magnified view of Fig. 3.2.7(b) at the vicinity of
the crack location.
64
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (3rd mode of vibration), a1/W=0.4, L1/L=0.0256
Relative amplitude
Fig.3.2.7 (c)
Relative distance from fixed end
Fig. 3.2.7(c1)
Magnified view of Fig. 3.2.7(c) at the vicinity of the
crack location.
65
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (1st mode of vibration), a1/W=0.1, L1/L=0.0513
Relative amplitude
Fig.3.2.8 (a)
Relative distance from fixed end
Fig. 3.2.8(a1)
Magnified view of Fig. 3.2.8 (a) at the vicinity of
the crack location.
66
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (2nd mode of vibration), a1/W=0.1, L1/L=0.0513
Relative amplitude
Fig.3.2.8 (b)
Relative distance from fixed end
Fig. 3.2.8(b1)
Magnified view of Fig. 3.2.8 (b) at the vicinity of
the crack location.
67
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (3rd mode of vibration), a1/W=0.1, L1/L=0.0513
Relative amplitude
Fig. 3.2.8 (c)
Relative distance from fixed end
Fig. 3.2.8 (c1)
Magnified view of Fig. 3.2.8 (c) at the vicinity of
the crack location.
68
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end(1st mode of vibration), a1/W=0.2, L1/L=0.0513
Relative amplitude
Fig.3.2.9 (a)
Relative distance from fixed end
Fig. 3.2.9 (a1)
Magnified view of Fig. 3.2.9 (a) at the vicinity of
the crack location.
69
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (2nd mode of vibration), a1/W=0.2, L1/L=0.0513
Relative amplitude
Fig.3.2.9 (b)
Relative distance from fixed end
Fig. 3.2.9(b1)
Magnified view of Fig. 3.2.9 (b) at the vicinity of
the crack location.
70
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (3rd mode of vibration), a1/W=0.2, L1/L=0.0513
Relative amplitude
Fig.3.2.9 (c)
Relative distance from fixed end
Fig. 3.2.9 (c1)
Magnified view of Fig. 3.2.9 (c) at the vicinity of
the crack location.
71
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (1st mode of vibration), a1/W= 0.3, L1/L=0.0513
Relative amplitude
Fig.3.2.10 (a)
Relative distance from fixed end
Fig. 3.2.10 (a1) Magnified view of fig. 3.2.10 (a) at the vicinity
of the crack location.
72
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (2nd mode of vibration), a1/W=0.3, L1/L=0.0513
Relative amplitude
Fig.3.2.10 (b)
Relative distance from fixed end
Fig. 3.2.10 (b1) Magnified view of Fig. 3.2.10 (b) at the vicinity of
the crack location.
73
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (3rd mode of vibration), a1/W=0.3, L1/L=0.0513
Relative amplitude
Fig.3.2.10 (c)
Relative distance from fixed end
Fig. 3.2.10 (c1) Magnified view of Fig. 3.2.10 (c) at the vicinity of
the crack location.
74
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (1st mode of vibration), a1/W=0.4, L1/L=0.0513
Relative amplitude
Fig.3.2.11 (a)
Relative distance from fixed end
Fig. 3.2.11 (a1) Magnified view of Fig. 3.2.11 (a) at the vicinity of
the crack location.
75
Relative amplitude
Relative distance from fixed end
Relative amplitude
Fig. 3.2.11 (b) Relative amplitude vs. relative distance from the fixed
end (2nd mode of vibration), a1/W=0.4, L1/L=0.0513
Relative distance from fixed end
Fig. 3.2.11(b1) Magnified view of Fig. 3.2.11 (b) at the vicinity of
the crack location.
76
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (3rd mode of vibration), a1/W=0.4, L1/L=0.0513
Relative amplitude
Fig.3.2.11 (c)
Relative distance from fixed end
Fig. 3.2.11 (c1)
Magnified view of Fig. 3.2.11 (c) at the vicinity
of the crack location.
77
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (1st mode of vibration), a1/W=0.1, L1/L=0.1795
Relative amplitude
Fig.3.2.12 (a)
Relative distance from fixed end
Fig. 3.2.12 (a1) Magnified view of Fig. 3.2.12 (a) at the vicinity
of the crack location.
78
Relative amplitude
Relative distance from fixed end
Relative amplitude
Fig. 3.2.12 (b) Relative amplitude vs. relative distance from the fixed
end (2nd mode of vibration), a1/W=0.1, L1/L=0.1795
Relative distance from fixed end
Fig. 3.2.12 (b1)
Magnified view of Fig. 3.2.12 (b) at the vicinity
of the crack location.
79
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (3rd mode of vibration), a1/W=0.1, L1/L=0.1795
Relative amplitude
Fig. 3.2.12 (c)
Relative distance from fixed end
Fig. 3.2.12 (c1) Magnified view of Fig. 3.2.12 (c) at the vicinity
of the crack location.
80
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (1st mode of vibration), a1/W=0.2, L1/L=0.1795
Relative amplitude
Fig.3.2.13 (a)
Relative distance from fixed end
Fig. 3.2.13 (a1) Magnified view of Fig. 3.2.13 (a) at the vicinity
of the crack location.
81
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (2nd mode of vibration), a1/W=0.2, L1/L=0.1795
Relative amplitude
Fig. 3.2.13 (b)
Relative distance from fixed end
Fig. 3.2.13(b1)
Magnified view of Fig. 3.2.13 (b) at the vicinity
of the crack location.
82
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (3rd mode of vibration), a1/W=0.2, L1/L=0.1795
Relative amplitude
Fig.3.2.13 (c)
Relative distance from fixed end
Fig. 3.2.13 (c1)
Magnified view of Fig. 3.2.13 (c) at the vicinity
of the crack location.
83
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (1st mode of vibration), a1/W=0.3, L1/L=0.1795
Relative amplitude
Fig.3.2.14 (a)
Relative distance from fixed end
Fig. 3.2.14 (a1) Magnified view of Fig. 3.2.14 (a) at the vicinity
of the crack location.
84
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (2nd mode of vibration), a1/W=0.3, L1/L=0.1795
Relative amplitude
Fig.3.2.14 (b)
Relative distance from fixed end
Fig. 3.2.14(b1) Magnified view of Fig. 3.2.14(b) at the vicinity of
the crack location.
85
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (3rd mode of vibration), a1/W=0.3, L1/L=0.1795
Relative amplitude
Fig.3.2.14 (c)
Relative distance from fixed end
Fig. 3.2.14(c1)
Magnified view of Fig. 3.2.14(c) at the vicinity
of the crack location.
86
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (1st mode of vibration), a1/W=0.4, L1/L=0.1795
Relative amplitude
Fig.3.2.15 (a)
Relative distance from fixed end
Fig. 3.2.15 (a1) Magnified view of Fig. 3.2.15 (a) at the vicinity
of the crack location.
87
Relative amplitude
Relative distance from fixed end
Relative amplitude
Fig.3.2.15 (b) Relative amplitude vs. relative distance from the fixed
end (2nd mode of vibration), a1/W=0.4, L1/L=0.1795
Relative distance from fixed end
Fig. 3.2.15(b1)
Magnified view of Fig. 3.2.15(b) at the vicinity of
the crack location.
88
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (3rd mode of vibration), a1/W=0.4, L1/L=0.1795
Relative amplitude
Fig.3.2.15 (c)
Relative distance from fixed end
Fig. 3.2.15(c1) Magnified view of Fig. 3.2.15(c) at the vicinity of
the crack location.
89
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (1st mode of vibration), a1/W=0.1, L1/L=0.2564
Relative amplitude
Fig.3.2.16 (a)
Relative distance from fixed end
Fig. 3.2.16(a1)
Magnified view of Fig. 3.2.16(a) at the vicinity
of the crack location.
90
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (2nd mode of vibration), a1/W=0.1, L1/L=0.2564
Relative amplitude
Fig.3.2.16 (b)
Relative distance from fixed end
Fig. 3.2.16(b1) Magnified view of Fig. 3.2.16(b) at the vicinity of
the crack location.
91
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (3rd mode of vibration), a1/W=0.1, L1/L=0.2564
Relative amplitude
Fig. 3.2.16 (c)
Relative distance from fixed end
Fig. 3.2.16 (c1)
Magnified view of Fig. 3.2.16 (c) at the vicinity
of the crack location.
92
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (1st mode of vibration), a1/W=0.2, L1/L=0.2564
Relative amplitude
Fig.3.2.17 (a)
Relative distance from fixed end
Fig. 3.2.17 (a1) Magnified view of Fig. 3.2.17 (a) at the vicinity of
the crack location.
93
Relative amplitude
Relative distance from fixed end
Relative amplitude
Fig.3.2.17 (b) Relative amplitude vs. relative distance from the fixed
end (2nd mode of vibration), a1/W=0.2, L1/L=0.2564
Relative distance from fixed end
Fig. 3.2.17(b1)
Magnified view of Fig. 3.2.17(b) at the vicinity of
the crack location.
94
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (3rd mode of vibration), a1/W=0.2, L1/L=0.2564
Relative amplitude
Fig. 3.2.17 (c)
Relative distance from fixed end
Fig. 3.2.17 (c1)
Magnified view of Fig. 3.2.17 (c) at the vicinity
of the crack location.
95
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (1st mode of vibration), a1/W=0.3, L1/L=0.2564
Relative amplitude
Fig.3.2.18 (a)
Relative distance from fixed end
Fig. 3.2.18 (a1)
Magnified view of Fig. 3.2.18 (a) at the vicinity
of the crack location.
96
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (2nd mode of vibration), a1/W=0.3, L1/L=0.2564
Relative amplitude
Fig.3.2.18 (b)
Relative distance from fixed end
Fig. 3.2.18(b1)
Magnified view of Fig. 3.2.18 (b) at the vicinity of
the crack location.
97
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (3rd mode of vibration), a1/W=0.3, L1/L=0.2564
Relative amplitude
Fig. 3.2.18 (c)
Relative distance from fixed end
Fig. 3.2.18 (c1)
Magnified view of Fig. 3.2.18 (c) at the vicinity
of the crack location.
98
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (1st mode of vibration), a1/W=0.4, L1/L=0.2564
Relative amplitude
Fig.3.2.19 (a)
Relative distance from fixed end
Fig. 3.2.19 (a1)
Magnified view of Fig. 3.2.19 (a)
of the crack location.
99
at the vicinity
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (2nd mode of vibration), a1/W=0.4, L1/L=0.2564
Relative amplitude
Fig.3.2.19 (b)
Relative distance from fixed end
Fig. 3.2.19(b1)
Magnified view of Fig. 3.2.19(b) at the vicinity of
the crack location.
100
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (3rd mode of vibration), a1/W=0.4, L1/L=0.2564
Relative amplitude
Fig.3.2.19 (c)
Relative distance from fixed end
Fig. 3.2.19 (c1)
Magnified view of Fig. 3.2.19 (c) at the vicinity
of the crack location.
101
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (1st mode of vibration), a1/W=0.1, L1/L=0.3846
Relative amplitude
Fig.3.2.20 (a)
Relative distance from fixed end
Fig. .3.2.20 (a1) Magnified view of Fig. 3.2.20 (a) at the vicinity of
the crack location.
102
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (2nd mode of vibration), a1/W=0.1, L1/L=0.3846
Relative amplitude
Fig.3.2.20 (b)
Relative distance from fixed end
Fig. 3.2.20(b1)
Magnified view of Fig. 3.2.20 (b) at the vicinity of
the crack location.
103
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (3rd mode of vibration), a1/W=0.1, L1/L=0.3846
Relative amplitude
Fig.3.2.20 (c)
Relative distance from fixed end
Fig. 3.2.20 (c1) Magnified view of Fig. 3.2.20 (c) at the vicinity of
the crack location.
104
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (1st mode of vibration), a1/W=0.2, L1/L=0.3846
Relative amplitude
Fig.3.2.21 (a)
Relative distance from fixed end
Fig. 3.2.21 (a1) Magnified view of Fig. 3.2.21 (a)
of the crack location.
105
at the vicinity
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (2nd mode of vibration), a1/W=0.2, L1/L=0.3846
Relative amplitude
Fig.3.2.21 (b)
Relative distance from fixed end
Fig. 3.2.21 (b1)
Magnified view of Fig. 3.2.21 (b)at the vicinity
of the crack location.
106
Relative amplitude
Relative distance from fixed end
Relative amplitude
Fig.3.2.21(c) Relative amplitude vs. relative distance from the fixed end
(3rd mode of vibration), a1/W=0.2, L1/L=0.3846
Relative distance from fixed end
Fig. 3.2.21(c1)
Magnified view of Fig. 3.2.21(c)
of the crack location.
107
at the vicinity
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (1st mode of vibration), a1/W=0.3, L1/L=0.3846
Relative amplitude
Fig.3.2.22 (a)
Relative distance from fixed end
Fig. 3.2.22 (a1)
Magnified view of Fig. 3.2.22 (a) at the vicinity
of the crack location.
108
Relative amplitude
Relative distance from fixed end
Relative amplitude
Fig.3.2.22 (b) Relative amplitude vs. relative distance from the fixed
end (2nd mode of vibration), a1/W=0.3, L1/L=0.3846
Relative distance from fixed end
Fig. 3.2.22 (b1)
Magnified view of Fig. 3.2.22 (b) at the vicinity
of the crack location.
109
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (3rd mode of vibration), a1/W=0.3, L1/L=0.3846
Relative amplitude
Fig.3.2.22 (c)
Relative distance from fixed end
Fig. 3.2.22 (c1)
Magnified view of Fig. 3.2.22 (c) at the vicinity
of the crack location.
110
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (1st mode of vibration), a1/W=0.4, L1/L=0.3846
Relative amplitude
Fig.3.2.23 (a)
Relative distance from fixed end
Fig. 3.2.23 (a1)
Magnified view of Fig. 3.2.23 (a) at the vicinity
of the crack location.
111
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (2nd mode of vibration), a1/W=0.4, L1/L=0.3846
Relative amplitude
Fig.3.2.23 (b)
Relative distance from fixed end
Fig. 3.2.23(b1) Magnified view of Fig. 3.2.23(b) at the vicinity of
the crack location.
112
Relative amplitude
Relative distance from fixed end
Relative amplitude
Fig.3.2.23 (c) Relative amplitude vs. relative distance from the fixed
end (3rd mode of vibration), a1/W=0.4, L1/L=0.3846
Relative distance from fixed end
Fig. 3.2.23 (c1)
Magnified view of Fig. 3.2.23 (c) at the vicinity
of the crack location.
113
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (1st mode of vibration), a1/W=0.1, L1/L=0.5128
Relative amplitude
Fig.3.2.24 (a)
Relative distance from fixed end
Fig. 3.2.24 (a1) Magnified view of Fig. 3.2.24 (a) at the vicinity of
the crack location.
114
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (2nd mode of vibration), a1/W=0.1, L1/L=0.5128
Relative amplitude
Fig.3.2.24 (b)
Relative distance from fixed end
Fig. 3.2.24(b1)
Magnified view of Fig. 3.2.24 (b) at the vicinity of
the crack location.
115
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (3rd mode of vibration), a1/W=0.1, L1/L=0.5128
Relative amplitude
Fig.3.2.24 (c)
Relative distance from fixed end
Fig. 3.2.24 (c1)
Magnified view of Fig. 3.2.24 (c) at the vicinity
of the crack location.
116
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (1st mode of vibration), a1/W=0.2, L1/L=0.5128
Relative amplitude
Fig. 3.2.25(a)
Relative distance from fixed end
Fig. 3.2.25(a1) Magnified view of Fig. 3.2.25(a) at the vicinity of
the crack location.
117
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (2nd mode of vibration), a1/W=0.2, L1/L=0.5128
Relative amplitude
Fig. 3.2.25(b)
Relative distance from fixed end
Fig. 3.2.25(b1)
Magnified view of Fig.3.2.25 (b) at the vicinity of
the crack location.
118
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (3rd mode of vibration), a1/W=0.2, L1/L=0.5128
Relative amplitude
Fig. 3.2.25(c)
Relative distance from fixed end
Fig. 3.2.25(c1)
Magnified view of Fig. 3.2.25(c) at the vicinity of
the crack location.
119
Relative amplitude
Relative distance from fixed end
Relative amplitude
Fig.3.2.26(a) Relative amplitude vs. relative distance from the fixed end
(1st mode of vibration), a1/W=0.3, L1/L=0.5128
Relative distance from fixed end
Fig. 3.2.26(a1)
Magnified view of Fig. 3.2.26(a) at the vicinity
of the crack location.
120
Relative amplitude
Relative distance from fixed end
Fig.3.2.26 (b) Relative amplitude vs. relative distance from the fixed
end 2nd mode of vibration), a1/W=0.3, L1/L=0.5128
Relative amplitude
(
Relative distance from fixed end
Fig. 3.2.26(b1)
Magnified view of Fig. 3.2.26(b) at the vicinity
of the crack location.
121
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (3rd mode of vibration), a1/W=0.3, L1/L=0.5128
Relative amplitude
Fig. 3.2.26(c)
Relative distance from fixed end
Fig. 3.2.26(c1) Magnified view of Fig. 3.2.26(c) at the vicinity of
the crack location.
122
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (1st mode of vibration), a1/W=0.4, L1/L=0.5128
Relative amplitude
Fig. 3.2.27(a)
Relative distance from fixed end
Fig. 3.2.27(a1) Magnified view of Fig. 3.2.27(a1) at the vicinity of
the crack location.
123
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (2nd mode of vibration), a1/W=0.4, L1/L=0.5128
Relative amplitude
Fig.3.2.27 (b)
Relative distance from fixed end
Fig. 3.2.27(b1)
Magnified view of Fig. 3.2.27(b) at the vicinity of
the crack location.
124
Relative amplitude
Relative distance from fixed end
Relative amplitude vs. relative distance from the fixed
end (3rd mode of vibration), a1/W=0.4, L1/L=0.5128
Relative amplitude
Fig. 3.2.27(c)
Relative distance from fixed end
Fig. 3.2.27(c1) Magnified view of fig. Fig. 3.2.27(c) at the vicinity
of the crack location.
125
Relative first natural frequency
99999
R
el
at
iv
e
cr
ac
k
po
si
tio
n
th
ack dep
r
c
e
v
i
t
R e la
Relative second natural frequency
Fig. 3.2.28 (a) Three dimensional cum contour plot for relative first natural frequency
R e la ti v e c
ra c k p o s it
io n
Re
r
ec
v
ti
la
ac
k
Fig.3.2.28 (b) Three dimensional cum contour plot for relative second natural frequency
126
de
pt
h
la t
iv e
c ra
ck
de
p th
Relative third natural frequency
c k p o sitio n
Re
R e la tiv e c ra
la t
iv e
c ra
ck
de
p th
Relative 1st mode shape difference
Fig. 3.2.28 (c) Three dimensional cum contour plot for relative third natural frequency
Re
R el at iv e cr ac k po
si tio n
Fig. 3.2.29 (a) Three dimensional cum contour plot for relative 1st mode shape difference
127
Relative 2nd mode shape difference
R e la t
iv e c r
ack
p o s iti
iv
lat
e
R
on
c
ra
c
e
e
kd
h
pt
R ela tiv e cra ck p o siti o n
Re
lati
ve
cra
ck
dep
th
Relative 3rd mode shape difference
Fig. 3.2.29 (b) Three dimensional cum contour plot for relative 2nd mode shape difference
Fig. 3.2.29 (c) Three dimensional cum contour plot for relative 3rd mode shape difference
128
3.3
Analysis of Experimental Results
3
2
1
8
9
6
4
5
78
7
1. Vibration Pick-up
(Accelerometer)
4. Distribution box
7. Power amplifier
2. Vibration analyser
(PULSE Lite Type 3560L)
5. Power supply
8. Vibration exciter
3. Vibration indicator
with software
(PULSE labshop software)
6. Function generator
9. Cantilever beam
specimen
Fig. 3.3.1 Schematic block diagram of experimental set-up
An experimental set-up is used for carrying out the experiment as shown in the schematic
diagram (Fig. 3.3.1). A number of tests are conducted on aluminum beam specimen (800 x
50 x 6 mm) with a transverse crack as shown in Fig. 3.3.1 for determining the amplitude of
vibration, natural frequencies and mode shapes. The beam is allowed to vibrate under 1st, 2nd
and 3rd modes of vibration.
3.3.1
Experimental Results
The experimental results for relative amplitude at different relative crack location (0.026,
0.05128) and relative crack depths (0.3, 0.4) are depicted in Fig.3.3.2 to Fig. 3.3.4.
Corresponding numerical results are also plotted for cracked and un-cracked beam in the
same graphs for immediate comparison.
129
Relative amplitude Relative distance from fixed end Fig.3.3.2 (a) Relative amplitude vs. relative distance from the fixed end
Relative amplitude (1st mode of vibration), a1/W=0.4, L1/L=0.026
Relative distance from fixed end Fig.3.3.2 (b) Relative amplitude vs. relative distance from the fixed end
(2nd mode of vibration), a1/W=0.4, L1/L=0.026
130
Relative amplitude
Relative distance from fixed end
Fig.3.3.2 (c) Relative amplitude vs. relative distance from the fixed end
Relative amplitude
(3rd mode of vibration), a1/W=0.4, L1/L=0.026
Relative distance from fixed end
Fig.3.3.3 (a) Relative amplitude vs. relative distance from the fixed end
(1st mode of vibration), a1/W =0.3, L1/L =0.05128
131
Relative amplitude
Relative distance from fixed end
Relative amplitude
Fig.3.3.3 (b) Relative amplitude vs. relative distance from the fixed end
(2nd mode of vibration), a1/W =0.3, L1/L =0.05128
Relative distance from fixed end
Fig.3.3.3(c) Relative amplitude vs. relative distance from the fixed end
(3rd mode of vibration), a1/W =0.3, L1/L =0.05128
132
Relative amplitude
Relative distance from fixed end
Relative amplitude
Fig.3.3.4 (a) Relative amplitude vs. relative distance from the fixed end
(1st mode of vibration), a1/W =0.4, L1/L =0.05128
Relative distance from fixed end
Fig.3.3.4 (b) Relative amplitude vs. relative distance from the fixed end
(2nd mode of vibration), a1/W =0.4, L1/L =0.05128
133
Relative amplitude
Relative distance from fixed end
Fig.3.3.4 (c) Relative amplitude vs. relative distance from the fixed end
(3rd mode of vibration), a1/W =0.4, L1/L =0.05128
3.3.2. Comparison among the Results of Numerical and Experimental Analyses
It is evident from the comparison of experimental and numerical results depicted in the
Fig.3.3.2 to Fig. 3.3.4. and Table 3.3.1 that they are in good agreement. Table 3.3.1
delineates the comparison of results between numerical and experimental results. This table
depicts ten sets of data out of hundreds of data set recorded to show the comparison between
the numerical and experimental results. In Table 3.3.1 column “one” for relative first natural
frequency, column “two” for relative second natural frequency, column “three” for relative
third natural frequency, column “four” for relative first mode shape difference, column “five”
for relative second mode shape difference, column “six” for relative third mode shape
difference, column “seven” for relative crack depth and relative crack location of numerical
134
analysis, column “eight” for relative crack depth and relative crack location of experimental
analysis.
Various numerical and experimental results are found out from the above theoretical and
experimental analysis. The numerical, experimental results and comparison between the
numerical and experimental results are put forward in the graphical and tabular format. The
relative natural frequency and relative mode shape difference used in the above analysis can
be defined as follows.
(Natural frequency of cracked beam)
(Natural frequency of uncracked beam)
Relative natural frequency =
(Uncrack mode – Crack mode)
Uncrack mode
Relative mode shape difference =
Relative
first
natural
frequency
Relative
second
natural
frequency
Relative
third
natural
frequency
“tnf”
Relative
first
mode
shape
difference
“fmd”
Relative
second
mode
shape
difference
“smd”
“fnf”
“snf”
0.9848
0.9958
0.9975
0.2709
0.2372
0.3158
0.202
0.06888
0.205
0.0725
0.9673
0.9874
0.9943
0.3969
0.3247
0.3923
0.427
0.079
0.43
0.08388
0.9623
0.9948
0.9983
0.1814
0.0279
0.0774
0.537
0.15988
0.568
0.1575
0.9756
0.9976
0.9972
0.1383
-0.0823
0.1898
0.394
0.18675
0.391
0.18775
0.9852
0.9984
0.9967
0.01
-0.8678
0.2572
0.231
0.23625
0.23
0.24
0.9723
0.9961
0.9818
0.1947
0.0672
0.4105
0.556
0.2825
0.545
0.28625
0.9823
0.9872
0.9919
0.0726
0.2567
0.3994
0.451
0.40388
0.447
0.40513
0.981
0.9809
0.9931
0.0898
0.3154
0.392
0.497
0.42388
0.495
0.4235
0.986
0.9842
0.9988
-0.032
0.322
0.3965
0.426
0.50125
0.425
0.50375
0.9834
0.9685
0.9974
0.038
0.4558
0.3507
0.542
0.535
0.535
0.53313
Table 3.3.1
3.4
Relative Numerical results
third
mode
(relative crack
shape
depth “rcd” and
difference
location“rcl”)
“tmd”
rcd
rcl
Experimental
results
(relative crack
depth “rcd” and
location“rcl”)
rcd
rcl
Comparison of results between numerical and experimental analyses
Discussions
From above theoretical, numerical and experimental analysis the following discussions are
made. Fig.3.2.1 (a) represents the cracked cantilever beam, Fig.3.2.1 (b) shows the cross
sectional view and Fig.3.2.1(c) shows the segment at crack section. Fig. 3.2.2 shows the
135
variation of dimension-less compliances to that of relative crack depth. It is observed that as
the crack depth increases the compliance value increases. Fig.3.2.4 to Fig.3.2.27 show the
plots of relative beam distance from fixed end verses relative amplitudes for first, second,
third modes of vibration, for different relative crack locations and different relative crack
depths. Fig.3.2.28 depicts the variation of relative natural frequencies with respect to relative
crack locations and relative crack depth in three dimensional forms, along with the contour
plots. Fig.3.2.29 exhibits the variation of relative mode shapes with respect to relative crack
locations and relative crack depth in three dimensional forms along with the contour plot.
The observations made from the above results are depicted below.
(1) From Fig.3.2.2 it is observed that as the crack depth increases, the compliances (C11,
C12=C21, C22) also increase. This is due to decrease in local stiffness at the crack
section.
(2) It is observed from Fig. 3.2.4 to Fig. 3.2.27 that there are remarkable changes in the
mode shapes of the beam due to the presence of crack as compared to un-cracked
beam.
(3) It is evident from Fig.3.2.4 that up to the relative crack depth of 0.1 and relative crack
location 0.0256, there is no appreciable change in the mode shapes as compared with
similar un-cracked beam. However with the magnification of ordinates at the vicinity
of crack location as shown in Fig.3.2.4 to Fig.3.2.27, significant variation is noted in
mode shapes.
(4) Similarly for relative crack depth of 0.3 the mode shape variations are more
prominent (Fig.3.2.6 to Fig.3.2.27).
(5) Again with the increase in the relative crack depth up to 0.50, keeping the relative
crack location same, it is observed that there is appreciable variation in the 1st and 2nd
mode shapes as depicted in Fig.3.26 to Fig.3.2.27. With the magnification of
ordinates at the vicinity of crack location (Fig.3.24 to 3.2.27) abrupt changes in mode
shape are observed.
(6) Fig.3.3.2 to Fig.3.3.4 shows the comparison between the experimental and numerical
results for the cracked and un-cracked beam.
136
3.5
Summary
From the above analyses and discussions, the conclusions drawn are depicted as follows.
Crack depth and crack location have got effect on mode shapes and natural frequencies of the
vibrating structures. At the crack location significant changes in mode shapes are observed in
magnified views. These changes will help in depicting the location and intensity of the crack.
The mode shapes for the beam with crack obtained theoretically are compared with the
experimental results for cross verification. This methodology has been applied for collection
of rule and training data set for inverse problem in the subsequent section. The results
obtained from various analyses mentioned above shows a very good agreement. The
methodology can be utilized for condition monitoring of vibrating structures. Artificial
intelligence embedded technique can be developed for smart detection of fault in structures.
In the next sections fuzzy inference technique, Neural technique and hybrid technique are
applied for predicting crack location and crack depth as an inverse problem for condition
monitoring.
Publications
•
Das H.C. and Parhi D.R., Modal analysis of vibrating structures impregnated with crack,
International Journal of Applied Mechanics & Engineering, 13(3), 2008, 639-652.
137
Chapter 4
ANALYSIS OF FUZZY LOGIC TECHNIQUE FOR CRACK
DETECTION
Much research effort has been spent on various structural health monitoring techniques in
order to develop a reliable, efficient and economical approach to increase the safety and
reduce the maintenance cost of elastic structures. Although improved design methodologies
have significantly enhanced the reliability and safety of structures in recent years, it is still
not possible to build structures that have zero percent probability of failure. There is an
increasing interest in the development of smart structures with built-in fault detection
systems that would provide failure warnings. This current research presents methodologies
for structural damage detection and assessment using fuzzy logic. The approach for damage
detection is based on monitoring various system responses to determine the condition
monitoring of a vibrating structure. In this chapter an intelligent controller has been proposed
for crack detection algorithm employing fuzzy theory.
4.1
Introduction
It is observed that the human beings do not need precise, numerical information input to
make a decision, but they are able to perform highly adaptive control. Humans have a
remarkable capability to perform a wide variety of physical and mental tasks without any
explicit measurements or computations. Examples of everyday tasks are parking a car,
driving in city traffic, playing golf, and summarizing a story. In performing such familiar
tasks, humans use perceptions of time, distance, speed, shape, and other attributes of physical
and mental objects [152]. Fuzzy logic is a problem-solving control system methodology that
lends itself for implementation in systems ranging from simple, small, embedded microcontrollers to large, networked, workstation-based data acquisition and control systems. The
theory of fuzzy logic systems is inspired by the remarkable human capability to operate on
and reason with perception-based information. The rule-based fuzzy logic provides a
scientific formalism for reasoning and decision making with uncertain and imprecise
138
information. This methodology can be implemented in hardware, software, or a combination
of both. Fuzzy logic approach to control problems mimics how a person would make
decisions. Fuzzy systems allow for easier understanding as they are expressed in terms of
linguistic variables [178]. Damage detection is one of the key aspects in structural
engineering both for safety reasons and because of economic benefits that can result. Many
non-destructive testing methods for health monitoring have been proposed and investigated.
These methods include modal analysis, strain analysis, photo- elastic techniques, ultrasound
and acoustic emissions [170]. A fuzzy logic methodology can be presented for structural fault
detection based on eigen value, and dynamic responses of vibrating structure.
This chapter proposes an on-line crack detection methodology embedded with a new
intelligent fuzzy inference system. In this approach, the fuzzy logic controller is designed and
is used to detect the relative crack location and relative crack depth. The designed fuzzy
controller has six inputs and two outputs. The inputs to the designed fuzzy controller are
relative deviation of first three natural frequencies and relative deviation of first three mode
shapes and the out puts are relative crack location and relative crack depth. The fuzzy logic
system learns the full dynamics of the cracked beam. The inputs have ten membership
functions each and the outputs have forty seven membership functions for relative crack
location and nineteen membership functions relative crack depth. Each membership function
consists of triangular, trapezoidal and Gaussian membership functions. In this methodology
six hundred and ninety two rules have been used to design the fuzzy controllers. This
research focuses a fuzzy logic framework to be implemented for on-line crack detection. The
results of the proposed fuzzy controller have been compared with the numerical method
which shows the effectiveness of the developed method. It is also concluded that the current
method can be successfully employed for crack detection. This fuzzy controller designed for
crack detection has been authenticated by experimental results.
This chapter organized into five sections following the introduction, the analysis of fuzzy
inference system is described in section 4.2. The fuzzy controller design for crack detection
and corresponding results are discussed in section 4.3. In section 4.4, the results of the fuzzy
controller are compared with experimental and numerical results to demonstrate the
superiority of the proposed methodology and finally summary is given in section 4.5.
139
4.2
Fuzzy Inference System
Fuzzy inference is the process of formulating the mapping from a given input to an out put
using fuzzy logic. The mapping then provides a basis from which decisions can be made, or
patterns discerned. The process of fuzzy inference involves: membership functions, fuzzy
logic operators, and if-then rules. Fuzzy inference systems have been successfully applied in
fields such as automatic control, fault diagnosis, data classification, decision analysis, expert
systems, and computer vision.
In general, there are five parts of the fuzzy inference process.
(i) Input fuzzification: The step is to take the inputs and determine the degree to which they
belong to each of the appropriate fuzzy sets via membership functions.
(ii) Antecedent matching: Once the inputs have been fuzzified, the degree to which each
part of the antecedent has been satisfied for each rule is known. If the antecedent of a given
rule has more than one part, the fuzzy operator is applied to obtain one number that
represents the result of the antecedent for that rule. This number will then be applied to the
output function.
(iii) Rule fulfillment: A consequent of a rule is a fuzzy set represented by a membership
function. In this step, the consequent is reshaped using a function associated with the
antecedent.
(iv) Consequent aggregation: Since decisions are based on all the rules in a fuzzy inference
system, the rules must be combined in some manner in order to make a decision.
Aggregation is the process by which the fuzzy sets that represent the outputs of each rule are
combined into a single fuzzy set.
(v) Output defuzzification: Taking fuzzy sets as input, defuzzification outputs a crisp value,
which is suitable for analysis and control.
4.2.1
Membership Functions
The membership function of a fuzzy set is a generalization of the indicator function in
classical sets. In fuzzy logic, it represents the degree of truth as an extension of valuation.For
any set X, a membership function on X is any function from X to the real unit interval [0, 1].
Membership functions on X represent fuzzy subsets of X. The membership function which
140
represents a fuzzy set A is usually denoted by μA. For an element x of X, the value μA(x) is
called the membership degree of x in the fuzzy set A. The membership degree μA(x)
quantifies the grade of membership of the element x to the fuzzy set A. The value 0 means
that x is not a member of the fuzzy set; the value 1 means that x is fully a member of the
fuzzy set. The values between 0 and 1 characterize fuzzy members, which belong to the
fuzzy set only partially.
The Triangular membership function with straight lines is defined as
f (u, α, β, γ) =0 u<α
= (u-α)/ (β-α) α<=u<=β
= (α - u)/ (β-α) β<=u<=γ
=0 u>γ
One typical plot of the triangular membership function is given in Fig. 4.2.1(a).
A Gaussian membership function is defined by
f (u: m, σ) =exp [-{(u-m)/√2σ} 2]
Where the parameters m and σ control the center and width of the membership function. A
plot of the Gaussian membership function is presented if Fig. 4.2.1(b).
Trapezoidal membership function is defined as
f (u, a, b, c, d) = 0 when u < a and u > d
= (u - a) / (b - a) when a <= u <= b
= 1 when b <= u <= c
= (d - u) / (d - c) when c <= u <= d
A plot of the Trapezoidal membership function furnished in Fig. 4.2.1(c).
1.0
1.0
1.0
α
Fig. 4.2.1(a)
u
-σ
σ
m
Triangular
Fig.4.2.1 (b)
membership
function Gaussian
membership
function
141
u
a
b
Fig.4.2.1(c)
c
d
Trapezoidal
membership
function
u
4.2.2
Fuzzy Logic Controllers (FLC) and Fuzzy Reasoning Rules
In a fuzzy logic controller (FLC), the dynamic behavior of a fuzzy system is characterized by
a set of linguistic description rules based on expert knowledge. The expert knowledge is
usually of the form IF (a set of conditions are satisfied) THEN (a set of consequences can be
inferred). Since the antecedents and the consequents of these IF-THEN rules are associated
with fuzzy concepts (linguistic terms), they are often called fuzzy conditional statements. In
our terminology, a fuzzy control rule is a fuzzy conditional statement in which the antecedent
is a condition in its application domain and the consequent is a control action for the system
under control. Basically, fuzzy control rules provide a convenient way for expressing control
policy and domain knowledge.
Furthermore, several linguistic variables might be involved in the antecedents and the
conclusions of these rules. When this is the case, the system will be referred to as a multiinput- multi-output (MIMO) fuzzy system. For example, in the case of two-input-singleoutput (MISO) fuzzy systems, fuzzy control rules have the form
Rule-1: if x is A1 and y is B1 then z is C1
also
Rule-2: if x is A2 and y is B2 then z is C2
also
also
Rule-n: if x is An and y is Bn then z is Cn
where x and y are the process state variables, z is the control variable, Ai, Bi, and Ci are
linguistic values of the linguistic variables x, y and z.
The inputs to the fuzzy logic controller for crack detection comprises of
Relative first natural frequency = “fnf”; Relative second natural frequency = “snf”;
Relative third natural frequency = “tnf”; Relative first mode shape difference = “fmd”;
Relative second mode shape difference = “smd”;
Relative third mode shape difference = “tmd”.
142
The linguistic term used for the outputs are as follows;
Relative crack location = “rcl” and Relative crack depth = “rcd”
Set of Inputs
fnf
snf
tnf
Knowledge
Base
system
Fuzzy
Interface
Engine
Fuzzy
Rule Base
Defuzzyfication
Output Module
Defuzzyfication
Rcl
Set of Outputs
4.2.3
tmd
smd
Fuzzy Input Module
Fuzzyfication
Fig. 4.2.2
fmd
Rcd
Schematic diagram of the fuzzy logic controller for crack detection
Defuzzification
The output of the inference process so far is a fuzzy set, specifying a possibility distribution
of control action. In the on-line control, a nonfuzzy (crisp) control action is usually required.
Consequently, one must defuzzify the fuzzy control action (output) inferred from the fuzzy
control algorithm, namely:
z0 = defuzzifier(C);
where z0 is the nonfuzzy control output and defuzzifier is the defuzzification operator.
Defuzzification is a process to select a representative element from the fuzzy output C
inferred from the fuzzy control algorithm. The widely used defuzzification operators are:
(a) Centroid of area method
(b) Mean of maxima method
143
4.3
Analysis of the Fuzzy Controller used for Crack Detection
fuzzy controllers (Fig. 4.3.1(a), 4.3.2(a), 4.3.3(a)) developed for crack detection has got six
input parameters and two output parameters.
The linguistic term used for the inputs are as follows;
Relative first natural frequency = “fnf”; Relative second natural frequency = “snf”;
Relative third natural frequency = “tnf”; Relative first mode shape difference = “fmd”;
Relative second mode shape difference = “smd”;
Relative third mode shape difference = “tmd”.
The linguistic term used for the outputs are as follows;
Relative crack location = “rcl” and Relative crack depth = “rcd”
In the current section the fuzzy controllers are developed with triangular, Gaussian and
trapezoidal membership functions. The linguistic terms used in the fuzzy inference system
for the membership functions are described in the Table 4.3.1. For each input parameter ten
membership functions are taken. For the output parameter “relative crack location (rcl)” forty
six membership functions are taken and for the output parameter “relative crack depth (rcd)”
nineteen membership functions are taken.
4.3.1
Fuzzy Mechanism for Crack Detection
For the fuzzy subsets, the fuzzy control rules are defined in a general form as follows:
If (fnf is fnfi and snf is snfj and tnf is tnfk and fmd is fmdl and smd is smdm
and tmd is tmdn) then rcl is rclijklmn and rcd is rcdijklmn
(4.3.1)
where i=1 to 10, j=1 to 10, k = 1 to 10, l= 1 to 10, m= 1 to 10, n= 1 to 10
Because “fnf”, “snf”, “tnf”, “fmd”, “smd”, “tmd” have ten membership functions each.
From expression (4.3.1), two set of rules can be written
If (fnf is fnfi and snf is snfj and tnf is tnfk and fmd is fmdl and smd is smdm
and tmd is tmdn) then rcd is rcdijklmn
(4.3.2)
If (fnf is fnfi and snf is snfj and tnf is tnfk and fmd is fmdl and smd is smdm
and tmd is tmdn) then rcl is rclijklmn
144
According to the usual fuzzy logic control method [165], a factor is defined for the rules as
follows:
Wijklmn = μ fnf i (freq i ) Λ μ snf j (freq j ) Λ μ tnf k (freq k ) Λ μ fmd l (moddif l ) Λ μ smd m ( moddif m ) Λ μ tmd n ( moddif n )
Where freqi , freqj and freqk are the first , second and third relative natural frequencies of the
cantilever beam with crack respectively ; moddifl, moddifm and moddifn are the first, second
and third mode relative differences of the cantilever beam with crack respectively. By
applying the composition rule of inference [165] the membership values of the relative crack
location and relative crack depth, (location)rcl and (depth)rcd can be computed as;
μ rclijklmn (location) = Wijklmn Λ μ rclijklmn (location)
∀
∈ rcl
length
∀
∈ rcd
depth
μ rcdijklmn (depth ) = Wijklmn Λ μ rcdijklmn (depth )
(4.3.3)
The overall conclusion by combining the outputs of all the fuzzy rules can be written as
follows:
μ rcl (location) = μ rcl111111 (location) ∨ .... ∨ μ rclijklmn (location) ∨ ..... ∨ μ rcl10 10 10 10 10 10 (location)
μ rcd (depth) = μ rcd111111 (depth) ∨ .......... ∨ μ rcdijklmn (depth) ∨ .......... ∨ μ rcd10 10 10 10 10 10 (depth)
(4.3.4)
The crisp values of relative crack location and relative crack depth are computed using the
centroid of area method [165] as:
relative crack location = rcl =
relativecrackdepth = rcd =
4.3.2
∫ (location ⋅ μ rcl (location) ⋅ d(location)
∫ μ rcl (location) ⋅ d(location)
(4.3.5)
∫ (depth) ⋅ μ rcd (depth) ⋅ d(depth)
∫ μ rcd (depth) ⋅ d(depth)
Fuzzy Controller for Finding out Crack Depth and Crack Location
The inputs to the fuzzy controller are relative deviation of first natural frequency; relative
deviation of second natural frequency; relative deviation of third natural frequency; relative
145
first mode shape difference; relative second mode shape difference and relative third mode
shape difference. 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
numbers of the fuzzy rules out of several hundred fuzzy rules are being listed in Table 4.3.2.
4.3.3
Results of Fuzzy Controller
In the current research the fuzzy controller is designed with three types of membership
functions i.e triangular (Fig. 4.3.1), gaussian (Fig. 4.3.2) and trapezoidal (Fig. 4.3.3).
Fig.4.3.4 shows the defuzzyfication of triangular membership function fuzzy controller
results when the rule-1 and rule-19 are activated from Table 4.3.2. Fig.4.3.5 shows the
defuzzyfication of Gaussian membership function fuzzy controller results when the rule-1
and rule-19 are activated from Table 4.3.2. Fig. 4.3.6 shows the defuzzyfication of
trapezoidal membership function fuzzy controller results when the rule-1 and rule-19 are
activated from Table 4.3.2. Table 4.3.3 presents the comparison of results between
triangular, Gaussian and trapezoidal fuzzy controller, numerical analysis and experimental
analysis. In this table ten sets of inputs out of several hundred sets are taken. Corresponding
ten set of outputs through the fuzzy controller, numerical analysis and experimental analysis
are depicted in the same table. In the Table 4.3.3 the first column represents the relative first
natural frequency (fnf), the second column represents the relative of second natural
frequency (snf), the third column represents the relative of third natural frequency (tnf), the
fourth column represents the relative first mode shape difference (fmd), the fifth column
represents the relative second mode shape difference (smd), the sixth column represents the
relative third mode shape difference (tmd) as inputs ,the seventh column presents the outputs
relative crack location(rcl), relative crack depth(rcd) from the triangular membership
function fuzzy controller, the eighth column presents the outputs, relative crack location(rcl),
relative crack depth(rcd) from the gaussian membership function fuzzy controller ,the ninth
column presents the outputs, relative crack location(rcl), relative crack depth(rcd) from the
trapezoidal membership function fuzzy controller, the tenth column presents the outputs,
relative crack location(rcl), relative crack depth(rcd) from the numerical analysis and the
146
eleventh column presents the outputs, relative crack location(rcl), relative crack depth(rcd)
from the experimental analysis.
4.4
Discussions
In this chapter fuzzy controller has been addressed for prediction of crack location and crack
depth. Three types of membership functions such as triangular function (Fig. 4.2.1(a)),
gaussian function (Fig. 4.2.1(b)) and trapezoidal function (Fig. 4.2.1(c)) have been used for
designing the fuzzy controllers. The working principle for the fuzzy controller has been
depicted in Fig. 4.2.2. The complete architecture of triangular, Gaussian and trapezoidal
fuzzy controller are presented in Fig. 4.3.1, Fig. 4.3.2 and Fig. 4.3.3 respectively. The
linguistic terms of the fuzzy membership function have been given in Table 4.3.1. Table
4.3.2 gives twenty sets of the fuzzy rules sets being used for the fuzzy controller. Fig. 4.3.4
to Fig. 4.3.6 exhibits the fuzzy results after defuzzification when rule 1 and 19 of the Table
4.3.2 are activated for triangular, gaussian, trapezoidal membership functions respectively.
Table 4.3.3 gives the comparison of the results obtained from numerical, experimental,
fuzzy controller with triangular membership function, fuzzy controller with gaussian
membership function and fuzzy controller with trapezoidal membership function. During
comparison a good agreement is seen between the results. It is evident from the Table 4.3.3
that the average percentage deviation of the results of the triangular membership function
fuzzy controller is 2.9%, for gaussian membership function fuzzy controller is 0.9% and for
trapezoidal membership function fuzzy controller is 1.5%.
147
Inputs
Outputs
fnf
Fuzzy Controller
rcd
snf
tnf
fmd
rcl
smd
tmd
Fig. 4.3.1(a) Triangular fuzzy controller
L1F4
L1F3
L1F2
L1F1
M1F1
M1F2
H1F1
H1F2
H1F3
L2F4
H1F4
0.0, 0.912 0.92
0.928
0.936
0.944
0.952
0.96
0.968
0.976
0.984
0.992
1.0
Fig. 4.3.1(b1) Triangular membership functions for
relative natural frequency for first mode of vibration
L3F4
L3F3
L3F2
L3F1
M3F1 M3F2
H3F1
H3F2
H3F3
L2F3
L2F2
L2F1
M2F1
M2F2
H2F1 H2F2
H2F3
H2F4
0.0, 0.934 0.940 0.946
0.952
0.958
0.964
0.970
0.976
0.988
0.994
1.0
1.0
0.982
1.0
Fig. 4.3.1(b2) Triangular membership functions for
relative natural frequency for second mode of vibration
H3F4
S1M4
1.0
S1M3
S1M2
S1M1 M1M1 M1M2 H1M1 H1M2 H1M3
H1M4
1.0
0.0,0.934 0.940
0.946
0.952
0.958
0.964
0.970
0.976
0.982
0.988
0.994
1.0
0.0,-1.0 - 0.81818 -0.63636 -0.45454 -0.27272 -0.0909 0.09092 0.27272 0.45454 0.63636 0.81818
Fig. 4.3.1(b3) Triangular membership functions for
relative natural frequency for third mode of vibration
S2M4
S2M3 S2M2
S2M1 M2M1 M2M2 H2M1 H2M2
1.0
Fig. 4.3.1(b4) Triangular membership functions
for relative mode shape difference for first mode of
vibration
H2M3 H2M4
S3M4
1.0
S3M3
S3M2
S3M1 M3M1 M3M2 H3M1 H3M2 H3M3 H3M4
1.0
0.0,-1.0 -0.81818 -0.63636 -0.45454 -0.27272 -0.0909 0.09092 0.27272 0.45454 0.63636 0.81818
1.0
0.0,-1.0 -0.81818 -0.63636 -0.45454 -0.27272 -0.0909 0.09092 0.27272 0.45454 0.63636 0.81818
Fig. 4.3.1(b5) Triangular membership functions for
relative mode shape difference for second mode of
vibration
SD9
SD8
SD7
SD6
SD5
0.0,0.01 0.0545
0.099
0.1435
0.188
0.2325
SD4
SD3
SD2
0.277
0.3215
0.366
1.0
Fig. 4.3.1(b6) Triangular membership functions for
relative mode shape difference for third mode of
vibration
SD1
MD
LD1
LD2
0.4105
0.455
0.4995
0.5440
LD3
LD4
LD5
LD6
LD7
0.5885
0.633
0.6775
0.722
0.7665
LD8
LD9
0.8110
0.8555
1.0
0.9
Fig. 4.3.1(b7) Triangular membership functions for relative crack depth
SL22
SL20
SL21
SL18
SL16
SL14
SL12
SL10
SL8
SL6
SL4
SL2
SL19
SL17
SL15
SL13
SL11
SL9
SL7
SL5
SL3
SL1
ML1
BL1
BL3
BL5
BL7
BL9
BL10
BL12
BL14
BL16
BL18
BL20
ML2
BL2
BL4
BL6
BL8
BL11
BL13
BL15
BL17
BL19
BL21 BL22
1.0
0.0,.01
.052
.0943
.1364 .1785
.2206
.2628
.3049
.3470 .3891
.4312
.4734
.5155
.5576
.5997
.6418
.6840
.7261 .7682
.8103
.8524
.8946
.9367
.9789
.0311 .0732
.1153
.1575
.1996
.2417
.2838
.3259
.3681 .4102
.4523
.4944
.5365
.5787
.6208
.6629 .7050
.7471
.7893
.8314
.8735
.9156
.9578
1.0
Fig. 4.3.1(b8) Triangular membership functions for relative crack location
148
Inputs
Outputs
fnf
Fuzzy Controller
rcd
snf
tnf
fmd
rcl
smd
tmd
Fig. 4.3.2(a) Gaussian fuzzy controller
1.0
L1F4
L1F3
L1F2
L1F1
M1F1
M1F2 H1F1 H1F2
H1F3
L2F4
H1F4
0.0, 0.912 0.92
0.928
0.936 0.944
0.952
0.96
0.968
0.976
0.984
0.992
1.0
0.0,0.934 0.940 0.946
Fig. 4.3.2(b1) Gaussian membership functions
for relative natural frequency for first mode of
vibration
1.0
L2F3
L2F2
L2F1
M2F1 M2F2
H2F1
H2F2
H2F3
H2F4
1.0
L3F4
L3F3
L3F2
L3F1 M3F1
M3F2 H3F1
H3F2
H3F3
0.952
0.958
0.964
0.970
0.976
0.982
0.988
0.994
1.0
Fig. 4.3.2(b2) Gaussian membership functions for
relative natural frequency for second mode of
vibration
H3F4
S1M4 S1M3 S1M2 S1M1 M1M1 M1M2 H1M1
H1M2 H1M3
H1M4
1.0
0.0,0.934 0.940 0.946
0.952
0.958 0.964
0.970
0.976 0.982
0.988 0.994
1.0
Fig. 4.3.2(b3) Gaussian membership functions for
relative natural frequency for third mode of
vibration
1.0
S2M4 S2M3 S2M2
S2M1
M2M1 M2M2 H2M1 H2M2 H2M3
0.0,-1.0 -0.81818 -0.63636 -0.45454 -0.27272 -0.0909 0.09092 0.27272 0.45454 0.63636 0.81818
1.0
Fig. 4.3.2(b4) Gaussian membership functions for
relative mode shape difference for first mode of
vibration
S3M4
H2M4
S3M3
S3M2 S3M1 M3M1 M3M2 H3M1 H3M2 H3M3
H3M4
1.0
0.0,-1.0 -0.81818 -0.63636 -0.45454 -0.27272
-0.0909
0.09092 0.27272 0.45454 0.63636
0.81818
0.0,-1.0 -0.81818 -0.63636 -0.45454 -0.27272 -0.0909 0.09092 0.27272 0.45454 0.63636
1.0
Fig. 4.3.2(b5) Gaussian membership functions
for relative mode shape difference for second
mode of vibration
SD9
SD8
SD7
SD6
SD5
SD4
SD3
0.277
0.3215
SD2
0.81818
1.0
Fig. 4.3.2(b6). Gaussian membership functions for
relative mode shape difference for third mode of
vibration
SD1
MD
LD1
LD2
LD3
0.4105
0.455
0.4995
0.5440
0.5885
LD4
LD5
LD6
LD7
LD8
0.6775
0.722
0.7665
0.8110
LD9
1.0
0.0,0.01
0.0545
0.099
0.1435
0.188
0.2325
0.366
0.633
0.8555
0.9
Fig. 4.3.2(b7) Gaussian membership functions for relative crack depth
SL22
SL20
SL18
SL16
SL14
SL12
SL10
SL8
SL6
SL4
SL2
ML1
BL1
BL3
BL5
BL7
BL9
BL10
BL12
BL14
BL16 BL18 BL20
SL21
SL19
SL17
SL15
SL13
SL11
SL9
SL7
SL5
SL3
SL1
ML2
BL2
BL4
BL6
BL8
BL11
BL13
BL15
BL17
BL19 BL21
BL22
1.0
0.0,.01 .0522 .0943 .1364 .1785 .2206 .2628 .3049 .3470 3891 .4312 .4734 .5155 .5576 .5997 .6418 .6840 .7261 .7682 .8103 .8524 .8946 .9367 .9789
.0311 .0732 .1153 .1575 .1996 .2417 .2838 .3259 .3681 .4102 .4523 .4944 .5365 .5787 .6208 .6629 .7050 .7471 .7893 .8314 .8735 .9156 .9578 1.0
Fig. 4.3.2(b8.) Gaussian membership functions for relative crack location
149
Inputs
fnf
snf
tnf
fmd
smd
tmd
Outputs
Fuzzy Controller
rcd
rcl
Fig. 4.3.3(a) Trapezoidal fuzzy controller.
L1F4
L1F3
L1F2
L1F1
M1F1
M1F2
H1F1
H1F2
H1F3
H1F4
L2F4
1.0
L2F3
L2F2
L2F1
M2F1
M2F2
H2F1
H2F2
H2F3
H2F4
1.0
0.0,0.912 0.92
0.928
0.936
0.944
0.952
0.96
0.968
0.976
0.984
0.992
1.0
0.0, 0.934 0.940 0.946
Fig. 4.3.3(b1) Trapezodial membership functions for
relative natural frequency for first mode of vibration.
L3F4
L3F3
L3F2
L3F1
M3F1
M3F2
H3F1
H3F2
H3F3
0.952
0.958
0.964
0.970
0.976
0.982
0.988
0.994
1.0
Fig. 4.3.3 (b2) Trapezodial Membership functions for
relative natural frequency for second mode of
vibration.
H3F4
S1M4
H1M3
H1M4
0.0, -1.0 -0.81818 -0.63636 -0.45454 -0.27272 -0.0909 0.09092 0.27272 0.45454 0.63636
0.81818
1.0
S1M3
S1M2
S1M1
M1M1
M1M2
H1M1
H1M2
1.0
0.0, 0.934 0.940
0.946
0.952
0.958
0.964
0.970
0.976
0.982
0.988
0.994
1.0
Fig. 4.3.3 (b3) Trapezodial membership functions for
relative natural frequency for third mode of vibration.
S2M4
S2M3
S2M2
S2M1
M2M1
M2M2
H2M1
H2M2
H2M3
1.0
Fig. 4.3.3 (b4) Trapezodial membership functions
for relative mode shape difference for first mode of
vibration.
H2M4
S3M4
1.0
S3M3
S3M2
S3M1
M3M1
M3M2
H3M1
H3M2
H3M3
0.45454
0.63636
H3M4
1.0
0.0, -1.0 -0.81818 -0.63636 -0.45454 -0.27272 -0.0909
0.09092
0.27272
0.45454
0.63636
0.81818
0.0, -1.0 -0.81818 -0.63636 -0.45454 -0.27272 -0.0909 0.09092
1.0
Fig. 4.3.3 (b5) Trapezodial membership functions for
relative mode shape difference for second mode of
vibration.
SD9
SD8
SD7
SD6
SD5
SD4
SD3
SD2
SD1
MD
0.27272
0.81818
LD1
LD2
LD3
LD4
LD5
LD6
0.633
0.6775
0.722
LD7
LD8
LD9
0.8110
0.8555
1.0
0.0, 0.01
0.0545
0.099
0.1435
0.188
0.2325
0.277
0.3215
0.366
0.4105
0.455
1.0
Fig. 4.3.3 (b6) Trapezodial membership functions for
relative mode shape difference for third mode of
vibration.
0.4995
0.5440
0.5885
0.7665
0.9
Fig. 4.3.3 (b7) Trapezodial membership functions for relative crack depth.
SL22
SL20
SL18
SL16
SL14
SL12
SL10
SL8
SL6
SL4
SL2
ML1
BL1
BL3
BL5
BL7
BL9
BL10
BL12
BL14
BL16
BL18
BL20
SL21
SL19
SL17
SL15
SL13
SL11
SL9
SL7
SL5
SL3
SL1
ML2
BL2
BL4
BL6
BL8
BL11
BL13
BL15
BL17
BL19
BL21
BL22
1.0
0.0, .01
.0522
.0943
.1364
.1785
.2206
.2628
.3049
.3470
.3891
.4312
.4734
.5155
.5576
.5997
.6418
.6840
.7261
.7682
.8103
.8524
.8946
.9367
.9789
.0311
.0732
.1153
.1575
.1996
.2417
.2838
.3259
.3681
.4102
.4523
.4944
.5365
.5787
.6208
.6629
.7050
.7471
.7893
.8314
.8735
.9156
.9578
1.0
Fig. 4.3.3 (b8) Trapezodial membership functions for relative crack location.
150
Memberships Function
Name
L1F1,L1F2,L1F3,L1F4
Linguistic
Terms
fnf 1 to 4
M1F1,M1F2
fnf 5,6
H1F1,H1F2,H1F3,H1F4
fnf 7 to 10
L2F1,L2F2,L2F3,L2F4
snf 1 to 4
M2F1,M2F2
snf 5,6
H2F1,H2F2,H2F3,H2F4
snf 7 to 10
L3F1,L3F2,L3F3,L3F4
tnf 1 to 4
M3F1,M3F2
tnf 5,6
H3F1,H3F2,H3F3,H3F4
tnf 7 to 10
S1M1,S1M2,S1M3,S1M4
fmd 1 to 4
M1M1,M1M2
fmd 5,6
H1M1,H1M2,H1M3,H1M4
fmd 7 to 10
S2M1,S2M2,S2M3,S2M4
smd 1 to 4
M2M1,M2M2
smd 5,6
H2M1,H2M2,H2M3,H2M4
smd 7 to10
S3M1,S3M2,S3M3,S3M4
tmd 1 to 4
M3M1,M3M2
tmd 5,6
H3M1,H3M2,H3M3,H3M4
tmd 7 to 10
SL1,SL2……SL22
rcl 1 to 22
ML1,ML2
rcl 23,24
BL1,BL2…….BL22
rcl 25 to 46
SD1,SD2……SD9
rcd 1 to 9
MD
LD1,LD2……LD9
rcd 10
rcd 11 to 19
Description and range of the Linguistic terms
Low ranges of relative natural frequency for first mode
of vibration in descending 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 descending order respectively
Medium ranges of relative natural frequency for second
mode of vibration in ascending order respectively
Higher ranges of relative natural frequencies for second
mode of vibration in ascending order respectively
Low ranges of relative natural frequencies for third
mode of vibration in descending order respectively
Medium ranges of relative natural frequencies for third
mode of vibration in ascending order respectively
Higher ranges of relative natural frequencies for third
mode of vibration in ascending order respectively
Small ranges of first relative mode shape difference in
descending order respectively
medium ranges of first relative mode shape difference
in ascending order respectively
Higher ranges of first relative mode shape difference in
ascending order respectively
Small ranges of second relative mode shape difference
in descending order respectively
medium ranges of
second relative mode shape
difference in ascending order respectively
Higher ranges of second relative mode shape difference
in ascending order respectively
Small ranges of third relative mode shape difference in
descending order respectively
medium ranges of third relative mode shape difference
in ascending order respectively
Higher ranges of third relative mode shape difference in
ascending order respectively
Small ranges of relative crack location in descending
order respectively
Medium ranges of relative crack location in ascending
order respectively
Bigger ranges of relative crack location in ascending
order respectively
Small ranges of relative crack depth in descending order
respectively
Medium relative crack depth
Larger ranges of relative crack depth in ascending order
respectively
Table 4.3.1 Description of fuzzy linguistic terms
151
Sl.No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Examples of some rules used in the fuzzy controller
If fnf is H1F3,snf is H2F4,tnf is H3F1,fmd is H1M3,smd is H2M3,tmd is
H3M3, then rcd is SD8,and rcl is SL22
If fnf is H1F3,snf is H2F3,tnf is H3F1,fmd is H1M3,smd is H2M3,tmd is
H3M3, then rcd is SD8,and rcl is SL22
If fnf is H1F1,snf is M2F1,tnf is L3F2,fmd is H1M4,smd is H2M4,tmd is
H3M4, then rcd is MD,and rcl is SL22
If fnf is M1F1,snf is L2F2,tnf is L3F2,fmd is H1M4,smd is H2M4,tmd is
H3M4, then rcd is LD5,and rcl is SL22
If fnf is H1F1,snf is M2F1,tnf is L3F1,fmd is H1M3,smd is H2M3,tmd is
H3M3, then rcd is LD1,and rcl is SL21
If fnf is M1F1,snf is L2F1,tnf is L3F2,fmd is H1M4,smd is H2M3,tmd is
H3M4, then rcd is LD6,and rcl is SL21
If fnf is H1F2,snf is M2F2,tnf is M3F2,fmd is H1M3,smd is H2M3,tmd is
H3M9, then rcd is SD1,and rcl is SL20
If fnf is H1F3,snf is H2F4,tnf is H3F2,fmd is H1M2,smd is H2M1,tmd is
H3M2, then rcd is SD7,and rcl is SL19
If fnf is H1F1,snf is M2F2,tnf is H3F2,fmd is H1M2,smd is H2M2,tmd is
H3M2, then rcd is LD2,and rcl is SL18
If fnf is H1F3,snf is H2F4,tnf is H3F3,fmd is S1M1,smd is S2M1,tmd is S3M1,
then rcd is SD3,and rcl is SL17
If fnf is H1F1,snf is H2F3,tnf is H3F2,fmd is H1M2,smd is H2M1,tmd is
H3M1, then rcd is LD2,and rcl is SL16
If fnf is H1F2,snf is H2F4,tnf is H3F2,fmd is H1M2,smd is M2M2,tmd is
H3M2, then rcd is LD1,and rcl is SL15
If fnf is H1F2,snf is H2F4,tnf is H3F1,fmd is H1M2,smd is M2M2,tmd is
H3M2, then rcd is MD,and rcl is SL14
If fnf is H1F1,snf is H2F4,tnf is M3F1,fmd is H1M2,smd is M2M2,tmd is
H3M3, then rcd is LD4,and rcl is SL13
If fnf is H1F2,snf is H2F4,tnf is L3F1,fmd is H1M2,smd is M2M1,tmd is
H3M3, then rcd is LD3,and rcl is SL12
If fnf is H1F3,snf is H2F4,tnf is M3F1,fmd is H1M2,smd is H2M1,tmd is
H3M3, then rcd is SD2,and rcl is SL10
If fnf is M1F1,snf is M2F2,tnf is L3F3,fmd is H1M2,smd is H2M2,tmd is
H3M3, then rcd is LD9,and rcl is SL9
If fnf is M1F2,snf is M2F1,tnf is L3F3,fmd is H1M2,smd is H2M2,tmd is
H3M3, then rcd is LD9,and rcl is SL7
If fnf is H1F4,snf is H2F4,tnf is H3F1,fmd is H1M3,smd is H2M3,tmd is
H3M3, then rcd is SD9,and rcl is SL21
If fnf is H1F4,snf is H2F3,tnf is M3F2,fmd is M1M1,smd is H2M2,tmd is
H3M3, then rcd is SD4,and rcl is BL7
Table 4.3.2 Examples of twenty fuzzy rules being used in fuzzy controller
152
Inputs
Rule no 19 of Table 4.3.2 is activated
Rule no 1 of Table 4.3.2 is activated
Outputs
0.0983
0.0466
Relative crack depth
Relative crack location
Fig.4.3.4 Resultant values of relative crack depth and relative crack location from triangular
fuzzy controller when Rules 1 and 19 of Table 4.3.2 are activated
153
Inputs
Rule no 19 of Table 4.3.2 is activated
Rule no 1 of Table 4.3.2 is activated
Outputs
0.07457
0.04403
Fig. 4.3.5
Relative crack depth
Relative crack location
Resultant values of relative crack depth and relative crack location from gaussian
fuzzy controller when Rules 1 and 19 of Table 4.3.2 are activated
154
Inputs
Rule no 1 of Table 4.3.2 is activated
0.08392
0.04523
Rule no 19 of Table 4.3.2 is activated
Relative crack depth
Relative crack location
Fig. 4.3.6 Resultant values of relative crack depth and relative crack location from trapezoidal fuzzy
controller when Rules 1 and 19 of Table 4.3.2 are activated.
155
156
0.9809
0.9842
0.9685
0.981
0.986
0.9834
0.9974
0.9988
0.9931
0.9919
0.9818
0.9967
0.9972
0.9983
0.9943
0.9975
“tnf”
Relative
third
natural
frequency
0.038
-0.032
0.0898
0.0726
0.1947
0.01
0.1383
0.1814
0.3969
0.2709
“fmd”
Relative
first
mode
shape
difference
0.4558
0.322
0.3154
0.2567
0.0672
-0.8678
-0.0823
0.0279
0.3247
0.2372
“smd”
Relative
second
mode
shape
difference
0.3507
0.3965
0.392
0.3994
0.4105
0.2572
0.1898
0.0774
0.3923
0.3158
“tmd”
Relative
third
mode
shape
difference
0.534
0.428
0.498
0.446
0.545
0.229
0.382
0.557
0.441
0.202
rcd
0.53538
0.50363
0.42263
0.40338
0.27625
0.23613
0.18475
0.15825
0.08238
0.07163
rcl
Triangular Fuzzy
Controller
(relative crack depth
“rcd” and
location“rcl”)
0.537
0.425
0.495
0.449
0.552
0.227
0.389
0.548
0.431
0.203
rcd
0.53313
0.50188
0.42363
0.40438
0.28375
0.23688
0.18725
0.15913
0.08138
0.07138
rcl
Gaussian Fuzzy
Controller
(relative crack depth
“rcd” and
location“rcl”)
0.534
0.428
0.498
0.446
0.545
0.229
0.382
0.557
0.441
0.202
rcd
0.53538
0.50363
0.42263
0.40338
0.27625
0.23613
0.18475
0.15825
0.08238
0.07163
rcl
Trapezoidal Fuzzy
Controller
(relative crack depth
“rcd” and
location“rcl”)
0.542
0.426
0.497
0.451
0.556
0.231
0.394
0.537
0.427
0.202
rcd
0.53500
0.50125
0.42388
0.40388
0.28250
0.23625
0.18675
0.15988
0.07900
0.06888
rcl
Numerical
(relative crack depth
“rcd” and
location“rcl”)
0.535
0.425
0.495
0.447
0.545
0.23
0.391
0.568
0.43
0.205
rcd
0.53313
0.50375
0.42350
0.40513
0.28625
0.24000
0.18775
0.15750
0.08388
0.07250
rcl
Experimental
(relative crack
depth “rcd” and
location“rcl”)
Table 4.3.3 Comparison of results between triangular, gaussian and trapezoidal fuzzy controller, numerical analysis
and experimental analysis
0.9872
0.9976
0.9756
0.9823
0.9948
0.9623
0.9961
0.9874
0.9673
0.9723
0.9958
0.9848
0.9984
“snf”
“fnf”
0.9852
Relative
second
natural
frequency
Relative
first
natural
frequency
4.5
Summary
From the above analyses and discussions, the conclusions drawn are depicted as follows.
Crack depth and crack location have got effect on mode shapes and natural frequencies of the
vibrating structures. The fuzzy controllers developed here take natural frequencies and mode
shape differences for prediction of crack location and crack depth. The predicted results from
fuzzy controllers for crack location and crack depth are compared with the theoretical and
experimental results for cross verification. They show a very good agreement. The result of
gaussian membership function fuzzy controller is more accurate in comparison to other two
controllers. The developed fuzzy controller along with the methodology can be used as a
robust tool for fault detection in cracked structures.
Publications
•
Das H.C. and Parhi D.R., Damage Analysis of Cracked Structure Using Fuzzy Control
Technique, International Journal of Acoustics and Vibration, 13(2) , 2008 , 3-14.
•
Parhi D.R. and Das H.C., Structural damage detection by fuzzy- gaussian technique,
International Journal of Applied Mathematics and Mechanics, 4(2), 2008, 39-59.
•
Das H.C. and Parhi D.R., Identification of Crack Location and Intensity in a Cracked
Beam by Fuzzy Reasoning, International Journal of Intelligent Systems Technologies
and Applications (In Press).
•
Das H.C. and Parhi D.R., Online fuzzy logic crack detection of a cantilever beam,
International Journal of Knowledge-Based and Intelligent Engineering Systems, 12(2),
2008, 157-171.
•
Parhi D.R. and Das H.C. Smart crack detection of a beam using fuzzy logic controller,
International Journal of Computational Intelligence: Theory and Practice, 3(1), 2008, 9-
21.
•
Das H.C. and Parhi D.R., Detection of crack in cantilever structures using fuzzy-gaussian
interface technique, American Institute of Aeronautics and astronautics journal, 47(1),
2009, 105-115.
157
Chapter 5
ANALYSIS OF ARTIFICIAL NEURAL NETWORK FOR
CRACK DETECTION
Engineering structures used in wide range of civil, mechanical and aeronautical fields are
prone to damage such as crack formation and deterioration during their service period. An
effective and reliable structural crack detection methodology can be a useful tool for timely
determination of damage and deterioration in structural members. Crack detection
methodologies attempt to determine whether structural damage has occurred, as well as the
location and extent of any such damage. The information obtained by crack detection
methodology can play a vital role in the development of economical repair and retrofit
programmes. Crack can be detected and quantified by on-line crack detection techniques
using vibration-based analysis data in the service life of a structure. The effects of crack on a
structure changes the natural frequencies and mode shapes. Hence, crack can be detected
using artificial intelligence technique from dynamic analysis using natural frequencies and
mode shapes. This chapter presents on-line diagnostic technology for crack detection in
terms of crack location and its intensity in elastic structures using artificial neural network
technique. An intelligent controller has been proposed in this chapter using feed forward
multilayer neural network trained by back-propagation algorithm for crack diagnosis. The
results of the developed neural controller are compared with experimental results, which are
satisfactory and show a very good agreement.
5.1
Introduction
Artificial Neural Networks (ANN) [184,189] are relatively crude electronic models based on
the neural structure of the brain. The field goes by many names, such as connectionism,
parallel distributed processing, neurocomputing, natural intelligent systems, machine
learning algorithms, and artificial neuralnetworks. According to [189] “A neural network is a
massively parallel distributed processor made up of simple processing units, called neurons,
which have a naturaltendency for storing experiential knowledge and making it available for
158
use”. This knowledge is acquired from the environment through a learning process and is
stored in connections between the neurons, know as synaptic weights.
Generalization ability: Generalization [189] refers to the neural network producing
reasonable outputs for inputs not encountered during training (learning). This informationprocessing capability makes it possible for neural networks to solve complex (large-scale)
problems that are currently intractable.
Non-linearity: Artificial neural networks can be used even for non-linear problems
[175,180] as the interconnected neurons can be either linear or non-linear. This feature is
extremely important in the field of structural health monitoring as the signals from complex
structures under variable loading may be non-linear.
Input-Output mapping: This is the most powerful feature of the neural network which
involves supervised learning. The network tries to correlate a unique input signal with a
desired response. It modifies the synaptic weights by a learning process in order to achieve
the desired response. Training the network involves feeding the network with a set of input
signals and the corresponding desired response.
Adaptivity: Neural networks adapt easily to changes in the environment, by adjusting the
synaptic weight accordingly on retraining the system. Real-time networks which are capable
of changing its synaptic weights automatically can be designed for non-stationary
environmental conditions.
Evidential response: For pattern classification purposes, networks can be designed to
provide information about the pattern to be selected, as well as the confidence of the decision
made. This helps in rejecting indistinct patterns, thereby improving the classification
performance of the network.
Contextual information: Related information is dealt naturally by the network as the
knowledge is represented by the very structure and activation state of the network.
159
Fault tolerance: Neural networks are inherently fault tolerant. If the neural networks are
implemented in hardware form, and if a neuron or a connecting link is damaged, then only
the output quality deteriorates [189] rather than the system failing completely.
Over the past few years researchers around the globe have been focusing their attention
towards the development of real-time structural health monitoring systems. This field is of
paramount importance especially with the structures that are prone to in-service defects. This
necessitates the need for an online structural health monitoring system, which is capable of
determining the presence of the damage such as crack (in the incipient stage itself),
determining the location of the damage and the size of the damage. Various non-destructive
techniques which are capable of achieving the goal have been discovered. For complex
situations (variable loading, variable damage level, and variable type of damage) using
complex structures the response signal obtained from the sensors (damage signature vector)
will be complicated. This makes it difficult to decode the signal to determine the damage
location and damage. Moreover, it is also difficult or impossible to create accurate
mathematical models for complex structures due to both geometric and material property
non-linearity. This is where the role of Artificial Neural Networks is of utmost importance.
The following features of neural networks make it an effective tool in structural health
monitoring.Scores of researchers have documented the use of artificial neural networks in
tandem with existing non-destructive testing techniques for the purpose of structural health
monitoring. Fang et al. [208] in their research have explored the structural damage detection
using frequency response functions (FRFs) as input data to the back-propagation neural
network (BPNN). Their analysis results on a cantilevered beam show that, in damage cases
the neural network can assess damage conditions with very good accuracy. Rajakarunakaran
et al. [210] in their paper have presented the development of artificial neural network-based
model for the fault detection of centrifugal pumping system. The fault detection model is
developed by using two different artificial neural network approaches, namely feed forward
network with back propagation algorithm and binary adaptive resonance network (ART1).
They have tested the performance of the developed back propagation and ART1 model for a
total of seven categories of faults in the centrifugal pumping system.
160
In this chapter an on-line crack detection methodology has been proposed using feed forward
multilayer neural network trained by back-propagation algorithm. The artificial neural
network controller is designed and is used to detect the relative crack location and relative
crack depth. The developed neural network controller has six inputs and two outputs. The
inputs to the designed neural network controller are relative first three natural frequencies
and relative first three mode shape differences and the out puts are relative crack location and
relative crack depth. This chapter describes the application of neural network technique for
on-line crack detection. The results of the developed neural network controller have been
compared with the results of fuzzy controller and numerical method and found to be most
accurate. Hence it is concluded that the developed method can be more accurately applied for
crack detection. The developed neural network controller has been authenticated by
experiments.
This chapter is outlined into six sections following the introduction; the neural network
technology and its importance in the field of structural health monitoring along with multilayer perceptron architecture trained by back-propagation algorithm is described in section
5.2. The chapter 5.3 presents the analysis of neural network controller design for crack
detection and corresponding results are discussed. In section 5.4, the results of the neural
network controller are compared with the results of fuzzy controller, experimental and
numerical analysis to demonstrate the superiority of the proposed methodology and finally
the discussions and summary are described in section 5.5 and 5.6.
5.2
Neural Network Technique
The field of neural network technique can be thought of as being related to artificial
intelligence, machine learning, fault detection, parallel processing, statistics, and other fields.
The attraction of neural networks is that they are best suited to solve the problems that are the
most difficult to solve by traditional computational methods.
5.2.1
Design of Neural Network
A neural network is a massively parallel distributed processor that has a natural propensity
for storing experimental knowledge and making it available for further use. It resembles the
brain in two respects:
161
1.
Knowledge is acquired by the network through a learning process.
2.
Interneuron connection strengths known as synaptic weights and are used to store the
knowledge.
The procedure used to perform the learning process is called a learning algorithm, the
function of which is to modify the synaptic weights of the network in an orderly fashion so as
to attain a desired design objective. In neural network a neuron is an information processing
unit. Fig.5.2.1 shows the model of a neuron. A neuron model can be identified by three basic
elements, which is described below.
i)
A set of synapses or connecting links, each of which is characterized by a
weight or strength of its own. Specifically, a signal xj at the input of synapse j
connected to neuron k is multiplied by the synaptic weight wkj. In the synaptic
weight wkj the subscript refers to the neuron in question and the second
subscript refers to the input end of the synapse to which the weight refers.
ii).
An adder for summing the input signals, weighted by the respective synapses
of the neuron.
iii).
An activation function for limiting the amplitude of the output of a neuron.
Generally the normalised amplitude range of the output of a neuron is given as
the closed unit interval [0,1] or alternatively [-1,1].
In mathematical terms, we can describe a neuron k by writing the following pair of
equations:
p
u k = ∑ w kj x j
(5.2.1)
y k = f (u k )
(5.2.2)
j=1
Where x1, x2,…..,xp are the input signals; wk1, wk2,…..,wkp are the synaptic weights of neuron
k; uk is the linear combiner output; f (⋅) is the activation function; and yk is the output signal
of the neuron.
162
x1
wk1
x2
wk2
Activation
Input
∑
uk
f (⋅)
Output
Summing
xp
wkp
Synaptic
Fig. 5.2.1
Model of a neuron
The use of neural networks in various fields offers the following useful properties due to
which it can be applied to many fields.
•
In a neural network modification of the synaptic weights can be done by a set of training
samples (i.e. supervised learning). For this, a training set will be presented to the neural
network for training and the synaptic weights of the network are modified so as to
minimise the difference between the desired response and the actual response.
•
Neural networks have a built-in capability to adapt their synaptic weights in the
surrounding environment. In particular, a neural network trained to operate in a specific
environment can deal with minor changes in the operating environmental conditions.
•
A neural network implemented has the potential to be inherently fault tolerant in the
sense that its performance is degraded gracefully under adverse operating conditions.
5.2.2
Activation Function
The activation function, denoted by f(.), defines the output of a neuron in terms of the
activity level at its input. Generally three types of activation functions are used (e.g.
163
Threshold Activation Function, Ramping Activation Function, Hyperbolic tangent Activation
function).
5.2.2.1 Threshold Activation Function
f(x)
(0, 1)
(0, 0)
x
Fig.5.2.2
Threshold activation function
The Threshold Activation Function is shown in Fig.5.2.2. This function limits the output of
the neuron to either 0, if the net input argument x is less than zero; or 1, if x is greater than or
equal to zero.
Mathematically the Threshold Activation Function can be described as:
f(x) =
1
if x 0
0
if x 0
(5.2.3)
5.2.2.2 Ramping Activation Function
f (x)
(0, -0.05) (0,0)
Fig.5.2.3
(0, 0.05)
x
Ramping activation function
164
The Ramping Activation Function is shown in Fig.5.2.3.
Mathematically the Ramping Activation Function can be described as:
1
2
1
-1
if > x >
2
2
−1
if x ≤
2
5.2.2.3 Hyperbolic Tangent Activation Function
⎧
⎪1
⎪
f x = ⎨x
⎪0
⎪⎩
if x ≥
(5.2.4)
f (x)
(0, 1.0)
(0, 0)
x
(0, -1.0)
Fig.5.2.4 Hyperbolic tangent activation function
The hyperbolic tangent activation function is shown in Fig.5.2.4.
Mathematically the Ramping Activation Function can be described as:
f(x) =
5.2.3
ex - e-x
ex + e-x
(5.2.5)
Modeling of Back Propagation Neural Network
The back propagation paradigm trains a neural network using a gradient descent algorithm in
which the mean square error between the network's output and the desired output is
minimized. This creates a global cost function that is minimized iteratively by 'back
propagating’ the error from the output nodes to the input nodes. Once the network's error has
decreased to less than or equal to the specified threshold, the network has converged and is
165
considered to be trained. A simpler version of back propagation, the simple delta rule or the
perceptron convergence procedure, which are applicable only in networks with only one
modifiable connection layer can be proven to find a solution for all input-output mappings
which are realizable in that simpler architecture. The error surface in such networks has only
one minimum, and the system moves on this error surface towards this minimum (and stays
there after it has reached it). This is not true for back propagation in a network with more
than one modifiable connection layers. That is, although in practice one can almost always
find a network architecture (even with only two layers of modifiable connections) that can
realize a particular input-output mapping, this is not guaranteed. This is because in a network
with hidden layer(s) of nodes, i.e. nodes that are neither in the input nor in the output layer –
the error surface has, in addition to the global, ``lowest'', minimum also local minima, and the
system can get stuck in such local error minima. The Fig. 5.2.5 shows a typical architecture
for networks with back propagation as the learning rule.
OUTPUT LAYER
HIDDEN LAYER
b1
Activation
Function
b2
S1
R
Fig. 5.2.5
5.3
W2
Activation
Function
S2
OUTPUT
INPUT
W1
S2
Architecture of feed forward multilayer neural network trained by
back- propagation algorithm
Analysis of Neural Network Controller used for Crack Detection
A feed forward multilayer neural network controller trained by back-propagation algorithm
has been developed for detection of the relative crack location and relative crack depth
(Fig.5.3.1) for the cracked cantilever beam. The neural network controller has got six input
parameters and two output parameters.
166
The inputs to the neural network controller are as follows;
Relative first natural frequency = “fnf”; Relative second natural frequency = “snf”;
Relative third natural frequency = “tnf”; Relative first mode shape difference = “fmd”;
Relative second mode shape difference = “smd” and
Relative third mode shape difference = “tmd”.
The outputs from the neural network are as follows;
Relative crack location = “rcl” and Relative crack depth = “rcd”
The back propagation neural network has got ten layers (i.e. input layer, output layer and
eight hidden layers). The neurons associated with the input and output layers are six and two
respectively. The neurons associated in the eight hidden layers are twelve, thirty-six, fifty,
one hundred fifty, three hundred, one hundred fifty, fifty and eight respectively. The input
layer neurons represent relative deviation of first three natural frequencies and first three
relative mode shape difference. The output layer neurons represent relative crack location
and relative crack depth. The neurons are taken in order to give the neural network a
diamond shape (Fig.5.3.2).
5.3.1 Neural Controller Mechanism for Crack Detection
The neural network used is a ten-layer perceptron [189]. The chosen number of layers was
found empirically to facilitate training. The input layer has six neurons, three for first three
relative natural frequencies and other three for first three relative mode shape difference. The
output layer has two neurons, which represent relative crack location and relative crack
depth. The first hidden layer has 12 neurons, the second hidden layer has 36 neurons, the
third hidden layer has 50 neurons, the fourth hidden layer has 150 neurons, the fifth hidden
layer has 300 neurons, the sixth hidden layer has 150 neurons, the seventh hidden layer has
50 neurons and the eighth hidden layer has 8 neurons. These numbers of hidden neurons are
also found empirically. Fig. 5.3.2 depicts the neural network with its input and output signals.
The neural network is trained with 800 patterns representing typical scenarios, some of which
are depicted in Table5.3.1. For example, from Table5.3.1, when the first three relative natural
frequencies and first three mode shape differences are 0.9839, 0.9903, 0.9938, 0.0127,
167
0.8437, and 0.2639 respectively then the relative crack location and relative crack depth are
0.225 and 0.25 respectively. The neural network is trained to give outputs such as relative
crack depth and relative crack location.
During training and during normal operation, the input patterns fed to the neural network
comprise the following components:
y1{1} = relative deviation of first natural frequency
(5.3.1(a))
y {21} = relative deviation of second natural frequency
(5.3.1(b))
y {31} = relative deviation of third natural frequency
(5.3.1(c))
y {41} = relative deviation of first mode shape
(5.3.1(d))
y {51} = relative deviation of second mode shape
(5.3.1(e))
y {61} = relative deviation of third mode shape
(5.3.1(f))
These input values are distributed to the hidden neurons which generate outputs given by
[189]:
(
y {jlay} = f Vj{lay}
)
(5.3.2)
where
Vj{lay} = ∑ W ji{lay} .y {i lay−1}
(5.3.3)
i
lay = layer number
j = label for jth neuron in hidden layer ‘lay’
168
i = label for ith neuron in hidden layer ‘lay-1’
W ji{lay} = weight of the connection from neuron i in layer ‘lay-1’
to neuron j in layer ‘lay’
f (.) = activation function, chosen in this work as the hyperbolic tangent function:
f (x ) =
ex − e−x
e x + e −x
(5.3.4)
During training, the network output θactual, n may differ from the desired output θdesired,n as
specified in the training pattern presented to the network. A measure of the performance of
the network is the instantaneous sum-squared difference between θdesired, n and θactual, n for the
set of presented training patterns:
Err =
(
)
1
θ desired, n − θ actual, n 2
∑
2 all training
(5.3.5)
patterns
Where θactual, n (n=1) represents Relative crack location (“rcl”)
θactual, n (n=2) represents Relative crack depth (“rcd”)
The error back propagation method is employed to train the network [189]. This method
requires the computation of local error gradients in order to determine appropriate weight
corrections to reduce Err. For the output layer, the error gradient δ {10} is:
(
)
δ {10} = f ′ V1{10} (θ desired, n − θ actual, n )
(5.3.6)
The local gradient for neurons in hidden layer {lay} is given by:
169
⎛
⎞
δ {jlay} = f ′(Vj{lay} )⎜ ∑ δ {klay+1} Wkj{lay+1} ⎟
⎝ k
⎠
(5.3.7)
The synaptic weights are updated according to the following expressions:
W ji (t + 1) = W ji (t ) + ΔW ji (t + 1)
(5.3.8)
and ΔW ji (t + 1) = αΔW ji (t ) + ηδ {jlay} y {i lay−1}
(5.3.9)
where
α = momentum coefficient (chosen empirically as 0.2 in this work)
η = learning rate (chosen empirically as 0.35 in this work)
t = iteration number, each iteration consisting of the presentation of a training
pattern and correction of the weights.
The final output from the neural network is:
(
θ actual, n = f Vn{10}
)
(5.3.10)
where
{10}y{9}
Vn{10} = ∑ Wni
i
(5.3.11)
i
170
Input to the Neural Network Controller
Sl.
no
.
Relative
first
natural
frequency
Relative
second
natural
frequency
Relative
third
natural
frequency
“fnf”
“snf”
“tnf”
Desired output
from the Neural
Network
Controller
Relative
first
mode
shape
difference
“fmd”
Relative
second
mode shape
difference
Relative
third
mode shape
difference
Relative
crack
Location
Relative
crack
depth
“smd”
“tmd”
“rcl”
“rcd”
1
0.9592
0.9616
0.9801
0. 4013
0. 8437
0. 4071
0.15
0.525
2
0.9632
0.9886
0.9927
0. 2852
0. 4466
0. 3642
0.1
0.425
3
0.9715
0.9903
0.9931
0. 2016
0. 3248
0.4127
0.175
0.4
4
0.9728
0.9905
0.9938
0. 1917
0. 3186
0.4103
0.075
0.2
5
0.9789
0.9931
0.9939
0. 1418
0. 2983
0.3937
0.275
0.55
6
0.9831
0.9936
0.9947
0. 092
0. 2611
0.3872
0.225
0.25
7
0.9839
0.9954
0.9968
0. 0702
0. 2439
0.3207
0.4
0.525
8
0.9863
0.9961
0.9969
0. 0364
0. 0917
0.2639
0.425
0.5
9
0.9902
0.9968
0.9982
0. 0294
0. 0598
0.1823
0.525
0.55
10
0.9941
0.9990
0.9991
0. 0127
0. 0263
0.0698
0.5
0.45
Table 5.3.1 Examples of the training patterns for training of the neural network controller
5.3.2
Neural Controller for finding out Crack Depth and Crack Location
The inputs to the neural controller are relative first natural frequency; relative second natural
frequency; relative third natural frequency; relative first mode shape difference; relative
second mode shape difference and relative third mode shape difference. The outputs from the
fuzzy controller are relative crack depth and relative crack location. The neural network
controller is trained with 800 patterns representing typical scenarios, out of which ten rules
are depicted in Table5.3.1.
171
INPUTS
OUTPUTS
fnf
snf
rcl
tnf
fmd
rcd
smd
tmd
Input layer
Hidden layer
Output layer
Fig. 5.3.1 Multi layer neural network controller
172
173
tmd
smd
fmd
tnf
snf
fnf
Fourth Hidden
Layer
(150 neurons)
Sixth Hidden
Layer
(150 neurons)
rcd
rcl
Output Layer
(2 neurons)
Eight Hidden
Layer
(8 neurons)
Seventh Hidden
Layer
(50 neurons)
Ten-layer neural network controller for crack detection
Second Hidden
Layer
(36 neurons)
Fig. 5.3.2
Input Layer
(6 neurons)
First Hidden
Layer
(12 neurons)
Third Hidden
Layer
(50 neurons)
Fifth Hidden
Layer
(300 neurons)
174
0.9809
0.9842
0.9685
0.981
0.986
0.9834
Table 5.3.2
0.9872
0.9976
0.9756
0.9823
0.9948
0.9623
0.9961
0.9874
0.9673
0.9723
0.9958
0.9848
0.9984
“snf”
“fnf”
0.9852
Relative
second
natural
frequency
Relative
first
natural
frequency
0.0279 0.0774 0.548
0.3247 0.3923 0.431
0.2372 0.3158 0.203
0.038
-0.032
0.0898
0.0726
0.1947
0.01
0.4558 0.3507 0.537
0.322 0.3965 0.425
0.3154 0.392 0.495
0.2567 0.3994 0.449
0.0672 0.4105 0.552
-0.8678 0.2572 0.227
0.1383 -0.0823 0.1898 0.389
0.1814
0.3969
0.2709
0.534
0.503
0.424
0.405
0.286
0.238
0.187
0.159
0.081
0.072
rcl
Relative Neural Network
third
Controller (relative
mode
crack depth “rcd”
shape
and location“rcl”)
difference
“fmd” “smd” “tmd” rcd
Relative first Relative
mode shape second
difference mode
shape
difference
rcl
rcd
0.534 0.53538 0.537
0.428 0.50363 0.425
0.498 0.42263 0.495
0.446 0.40338 0.449
0.545 0.27625 0.552
0.229 0.23613 0.227
0.382 0.18475 0.389
0.557 0.15825 0.548
0.441 0.08238 0.431
0.53313
0.50188
0.42363
0.40438
0.28375
0.23688
0.18725
0.15913
0.08138
0.07138
rcl
Gaussian Fuzzy
Controller (relative
crack depth “rcd” and
location“rcl”)
0.202 0.07163 0.203
rcd
Triangular Fuzzy
Controller (relative
crack depth “rcd” and
location“rcl”)
rcl
rcd
rcl
Numerical
(relative crack depth
“rcd” and
location“rcl”)
rcd
rcl
Experimental
(relative crack depth
“rcd” and
location“rcl”)
0.43
0.08388
0.23
0.24000
0.534 0.53538 0.542 0.53500 0.535 0.53313
0.428 0.50363 0.426 0.50125 0.425 0.50375
0.498 0.42263 0.497 0.42388 0.495 0.42350
0.446 0.40338 0.451 0.40388 0.447 0.40513
0.545 0.27625 0.556 0.28250 0.545 0.28625
0.229 0.23613 0.231 0.23625
0.382 0.18475 0.394 0.18675 0.391 0.18775
0.557 0.15825 0.537 0.15988 0.568 0.15750
0.441 0.08238 0.427 0.07900
0.202 0.07163 0.202 0.06888 0.205 0.07250
rcd
Trapezoidal Fuzzy
Controller (relative
crack depth “rcd” and
location“rcl”)
Comparison of results between neural controller, fuzzy controller, numerical analysis and experimental analysis.
0.9974
0.9988
0.9931
0.9919
0.9818
0.9967
0.9972
0.9983
0.9943
0.9975
“tnf”
Relative
third
natural
frequency
5.4
Results of Neural Controller
The feed forward neural network controller developed in the current research is a ten-layer
perceptron. The artificial neural network controller is designed and is used to detect the
relative crack location and relative crack depth. The results of the developed neural controller
are compared (Table 5.3.2) with the fuzzy controller results of chapter-4, experimental and
numerical results of chapter-3. In the Table 5.3.2, ten sets of inputs out of several hundred
sets are taken. The inputs to different analyses made above are relative first three natural
frequencies and relative first three mode shape differences and the out puts are relative crack
location and relative crack depth. Corresponding ten set of outputs from the developed neural
network controller, fuzzy controller, numerical analysis and experimental analysis are
depicted in the Table 5.3.2. In the Table 5.3.2, the first column represents the relative 1st
natural frequency (fnf), the second column represents the relative of 2nd natural frequency
(snf), the third column represents the relative of 3rd natural frequency (tnf), the fourth column
represents the relative 1st mode shape difference (fmd), the fifth column represents the
relative 2nd mode shape difference (smd), the sixth column represents the relative 3rd mode
shape difference (tmd) as inputs and the rest coloumns represents the outputs as relative
crack location and relative crack depth obtained from different analyses.
5.5
Discussions
This section describes the application of artificial neural network controller for prediction of
crack size and severity. The working principles of neural network technique (Fig.5.2.1) and
activation function (Fig.5.2.2) have been depicted in section 5.2. The feed forward multi
layer neural network trained by back propagation algorithm (Fig.5.2.5) has been used for
designing the neural network controller. The ten layer feed forward controller and a
schematic diagram of multi layer neural network controller for crack diagnosis are depicted
in Fig.5.3.2 and Fig.5.3.1 respectively. These two figures express the complete architecture
of the neural controller for crack detection. Few of the examples of training patterns out of
several hundreds training patterns for neural network controller are given in the Table 5.3.1.
175
The Fig.5.3.2 represents a multi layer controller with ten set of rules for training with 1st
three relative natural frequencies and 1st three relative mode shape differences as inputs and
relative crack location and relative crack depth as outputs. The comparison of the results
from neural controller, fuzzy controllers, numerical analysis and experimental analysis are
expressed in Table 5.3.2.
It is evident from the Table 5.3.2 that the average percentage
deviation of the results of neural network controller is 1%.
5.6
Summary
From the analysis mentioned above and discussions, the summaries drawn are depicted
below. The neural network controller trained with eight hundred training patterns consist of
different crack location and crack depth. The neural network gives out puts such as relative
crack depth and relative crack location, very close to the experimental results. The ten layer
perceptron neural network has different number of neurons in the ten layers for processing
the input data like relative natural frequencies and mode shapes. It is observed that the error
in the output of the controller is considerably reduced from the desired output by employing
back propagation method. The developed controller predicts the crack location and its
intensity very closely to the actual results. The result from the controller is compared with the
output from numerical, fuzzy and experimental analysis for checking the robustness of the
developed system. The data collected from the controller is used for training the hybrid
technique such as fuzzy- neuro and MANFIS methods in next chapters for on line condition
monitoring of dynamically vibrating structures with higher accuracy and less computational
time.
Publications
•
Das H.C. and Parhi D.R., Application of Neural network for fault diagnosis of cracked
cantilever beam, IEEE International Symposium on Biologically Inspired Computing and
Applications (BICA-2009), Bhubaneswar, India, December 21-22, 2009, 353-358.
176
Chapter 6
ANALYSIS OF HYBRID FUZZY-NEURO SYSTEM
FOR CRACK DETECTION
Research on hybrid systems is one of the key issues of developing intelligent systems. It can
be applied by hybridising artificial neural networks, fuzzy logic, knowledge-based systems,
genetic algorithms and evolutionary computation. Neural network technology and fuzzy
inference system are becoming well recognized tools of designing an identifier/controller
capable of perceiving the operating environment and imitating a human operator with high
performance. The motivation behind the use of fuzzy neuro approaches is based on the
complexity of real life systems. In this respect, fuzzy neuro design approaches combine
architectural and philosophical aspects of an expert resulting in an artificial brain which can
be used as a controller. It is known that the fuzzy inference systems and neural networks are
universal approximators. In the following section fuzzy inference technique and neural
network hybridized together to produce fuzzy-neuro controller for fault diagnosis.
6.1
Introduction
Hybrid intelligent systems being the product of fuzzy logic and neural networks are
computational machines with unique capabilities for dealing with both numerical data and
linguistic knowledge (fuzzy) information. As the hybrid system refers to combinations of
artificial neural networks and fuzzy logic it incorporates the capability of both fuzzy logic
and neural network technique. This hybrid method can give better results than the
independent techniques. Fuzzy systems make use of knowledge expressed in the form of
linguistic rules, thus they offer the possibility of implementing expert human knowledge and
experience. Neural network learning techniques automate this process, significantly reducing
development time, and resulting in better performance. Fuzzy neuro hybridization results in
a hybrid intelligent system that synergizes these two techniques by combining the human-like
reasoning style of fuzzy systems with the learning and connectionist structure of neural
networks. Hence, this methodology can be effectively utilized for prediction of crack location
and crack depth in engineering structures with the vibration signatures as in put parameters.
177
This current research addresses the fault detection of a cracked cantilever beam using hybrid
fuzzy neuro intelligence technique. The fuzzy-neuro controller has two parts. The first part
comprises of fuzzy controller and the second part addresses the neural controller. The input
parameters to the fuzzy controller are first three relative natural frequencies and first three
mode shape differences. The output parameters of the fuzzy controller are initial relative
crack depth and initial relative crack location. The input parameters to the neural segment of
fuzzy-neuro controller are first three relative natural frequencies and first three mode shape
differences along with the interim outputs of fuzzy controller. The output parameters of the
fuzzy-neuro controller are final relative crack depth and final relative crack location. For
deriving the fuzzy rules and training patterns of natural frequencies, mode shapes, crack
depths and crack locations, theoretical expressions have been developed. Several fuzzy rules
and training patterns for the fuzzy segment and neural segment of fuzzy-neuro controller are
derived respectively. Experiments have been conducted for verifying the robustness of the
developed fuzzy-neuro controller. The results of the developed fuzzy-neuro controller and
experimental method are in very good agreement.
This chapter is divided into six sections. The section 6.1 briefs the hybrid intelligent system
and its importance in advance computing. The analysis of the fuzzy-neuro controller, the
mechanism of fuzzy controller and neural controller for crack detection are depicted in
section 6.2. In section 6.3 the results of the hybrid intelligent controller are compared with
the results of neural controller (chapter 5), fuzzy controller (chapter 4), experimental and
numerical analysis (chapter 3) to demonstrate the effectiveness of the proposed methodology.
Finally the discussions and summary are described in section 6.4 and 6.5 respectively.
6.2
Analysis of Fuzzy-Neuro Controller
In the current investigation damage analysis of dynamic structures has been addressed using
inverse approach i.e. hybrid computational fuzzy neuro technique. This hybrid technique
comprises of two parts; i.e. fuzzy controller and neural controller. The fuzzy controller has
six input parameters and two output parameters. The input parameters to the fuzzy controller
are first three relative natural frequencies and first three mode shape differences. The output
parameters of the fuzzy controller are initial relative crack depth and relative crack location.
178
The input parameters to the neural segment of fuzzy-neuro controller are first three relative
natural frequencies and first three mode shape differences. The output parameters of the
fuzzy-neuro controller are final relative crack depth and relative crack location. Three types
of membership functions i.e Triangular, Gaussian and Trapezoidal are used in the fuzzyneuro controller and accordingly three fuzzy-neuro controllers such as triangular membership
function fuzzy-neuro controller (Fig.6.2.1), gaussian membership function fuzzy-neuro
controller (Fig.6.2.2) and trapezoidal membership function fuzzy-neuro controller (Fig.6.2.3)
are designed for prediction of crack location and crack depth.
6.2.1
Analysis of the Fuzzy Segment of Fuzzy-Neuro Controller
The fuzzy part of the developed fuzzy-neuro controller has got six input parameters and two
output parameters. The linguistic terms used for the inputs in the fuzzy system of the fuzzyneuro controller are as follows;
Relative first natural frequency = “fnf”
Relative second natural frequency = “snf”;
Relative third natural frequency = “tnf”
Relative first mode shape difference = “fmd”
Relative second mode shape difference = “smd”
Relative third mode shape difference = “tmd”.
The linguistic term used for the outputs are as follows;
Initial relative crack location = “rclinitial” and Initial relative crack depth = “rcdinitial”
The membership functions used in the fuzzy segment of fuzzy-neuro controller are shown
pictorially in Fig. 4.3.1, Fig. 4.3.2 and Fig. 4.3.3. The linguistic terms of membership
functions used in the fuzzy segment of fuzzy-neuro controller are described in the Table
4.3.1. The fuzzy controller mechanism for crack detection has been given in section 4.3.1 and
4.3.2 of chapter 4.
179
180
tmd
smd
fmd
tnf
snf
fnf
Triangular Fuzzy
Controller
Fuzzy Controller
rcdinitial
tmd
smd
fmd
tnf
snf
fnf
Fourth Hidden
Layer
(150 neurons)
Second Hidden
Layer
(36 neurons)
Sixth Hidden
Layer
(150 neurons)
rcd
rcl
Eight Hidden
Layer
(8 neurons)
Output Layer
(2 neurons)
Fig. 6.2.1 Fuzzy-Neural
fordetection
fault detection
Fig.6.2.1 Triangular
fuzzy-neuro
controllersystem
for crack
Input
Layer
rclinitial
(Interim Outputs)
First Hidden
Layer
(12 neurons)
Fifth Hidden
Layer
Third Hidden (300 neurons) Seventh Hidden
Layer
Layer
(50 neurons)
(50 neurons)
181
tmd
smd
fmd
tnf
snf
fnf
Gaussian Fuzzy
Controller
Fuzzy
Controller
Second Hidden
Layer
(36 neurons)
Fourth Hidden
Layer
(150 neurons)
Sixth Hidden
Layer
(150 neurons)
rcd
rcl
Eight Hidden
Layer
(8 neurons)
Output Layer
(2 neurons)
Fig. 6.2.2. Fuzzy-Neural system for fault detection
Fig.6.2.2 Gaussian
fuzzy-neuro
controller for crack detection
rcdinitial Input
Layer
tmd
smd
fmd
tnf
snf
fnf
rclinitial
(Interim Outputs)
First Hidden
Layer
(12 neurons)
Third Hidden
Layer
(50 neurons)
Fifth Hidden
Layer
(300 neurons) Seventh Hidden
Layer
(50 neurons)
182
tmd
smd
fmd
tnf
snf
fnf
Trapezoidal Fuzzy
Fuzzy
Controller
Controller
Fourth Hidden
Layer
(150 neurons)
Sixth Hidden
Layer
(150 neurons)
rcd
rcl
Output Layer
(2 neurons)
Eight Hidden
Layer
(8 neurons)
Fig. 6.2.3 fuzzy-neuro
Fuzzy-Neural
system
for fault
detection
Fig.6.2.3 Trapezoidal
controller
for crack
detection
Second Hidden
Layer
(36 neurons)
rcdinitial Input Layer
(6 neurons)
tmd
smd
fmd
tnf
snf
fnf
rclinitial
(Interim Outputs)
First Hidden
Layer
(12 neurons)
Fifth Hidden
Layer
(300
neurons) Seventh Hidden
Third Hidden
Layer
Layer
(50 neurons)
(50 neurons)
6.2.2
Analysis of the Neural Segment of Fuzzy-Neuro Controller
In the fuzzy-neuro controller, the fuzzy segment will be inherited from chapter four, section
4.2.2. The fuzzy segment in the fuzzy-neuro controller will give the intermittent result for
relative crack depth and relative crack location. The neural segment of the fuzzy-neuro
controller has eight inputs such as intermittent relative crack depth and relative crack location
obtained from the fuzzy segment along with first three relative natural frequencies and first
three relative mode shape difference. The output from the fuzzy-neuro controller is the
refined result for relative crack depth and relative crack location. The analysis of the neural
network used in the fuzzy-neuro controller is given below.
The neural segment of the fuzzy-neuro controller is a ten layer feed forward neural network
trained by back propagation algorithm. The fuzzy-neuro controller has been developed for
detection of the relative crack location and relative crack depth. The neural network has got
eight input parameters and two output parameters.
The inputs to the neural segment of the fuzzy-neuro controller are as follows;
Relative first natural frequency = “fnf”; Relative second natural frequency = “snf”;
Relative third natural frequency = “tnf”; Relative first mode shape difference = “fmd”;
Relative second mode shape difference = “smd” and
Relative third mode shape difference = “tmd”.
Initial relative crack depth(output of the fuzzy segment)= “rcdinitial”
Initial relative crack location (output of the fuzzy segment)= “rclinitial”
The final outputs from the fuzzy-neuro controller are;
Final relative crack location = “rclfinal” and Final Relative crack depth = “rcdfinal”
The back propagation neural network used in fuzzy-neuro controller has got ten layers (i.e.
input layer, output layer and eight hidden layers). The neurons associated with the input and
output layers are eight and two respectively. The neurons associated in the eight hidden
layers are twelve, thirty-six, fifty, one hundred fifty, three hundred, one hundred fifty, fifty
and eight respectively. The input layer neurons represent first three relative natural
frequencies and first three relative mode shape difference along with the two interim outputs
from the the fuzzy segment. The output layer neurons represent final relative crack location
183
and final relative crack depth. The neurons are taken in order to give the neural network a
diamond shape. The neural controller mechanism for crack detection may be referred from
section 5.3.1 and 5.3.2 of chapter 5.
6.2.3
Results of Fuzzy-Neuro Controller
The results obtained after analyzing the fuzzy segment and neural segment of fuzzy-neuro
controller are given in Tables 6.2.1 and 6.2.2. A comparison of results between the triangular
membership fuzzy-neuro controller, gaussian membership fuzzy-neuro controller,
trapezoidal membership fuzzy-neuro controller, numerical analysis and experimental
analysis is depicted in Table 6.2.1. Again the comparison of results between the three fuzzyneuro controllers, neural network controller, triangular, gaussian and trapezoidal fuzzy
controller is presented in Table 6.2.2. Ten sets of random inputs out of several hundred sets
are taken in all the above tables for comparison of accuracy of the results. The inputs to
different analyses made above are first three relative natural frequencies and first three
relative mode shape differences and the out puts are relative crack location and relative
crack depth. Corresponding ten set of outputs from the developed fuzzy-neuro controllers,
neural network controller, fuzzy controllers, numerical analysis and experimental analysis
are presented in the Table 6.2.1 and Table 6.2.2. In the Tables 6.2.1 and 6.2.2 the first
column represents the relative 1st natural frequency (fnf), the second column represents the
relative 2nd natural frequency (snf), the third column represents the relative 3rd natural
frequency (tnf), the fourth column represents the relative 1st mode shape difference (fmd),
the fifth column represents the relative 2nd mode shape difference (smd), the sixth column
represents the relative 3rd mode shape difference (tmd) as inputs and the rest coloumns
represents the outputs (i.e. relative crack location and relative crack depth). It is observed
from the Tables 6.2.1 and 6.2.2 that the average percentage deviation of the results of
gaussian membership fuzzy-neuro controller is 0.55%. For the triangular membership fuzzyneuro controller and trapezoidal membership fuzzy-neuro controller the average percentage
deviation of the results are 0.95% and 0.85% respectively.
184
185
0.9809
0.9842
0.9685
0.981
0.986
0.9834
Table 6.2.1
0.9872
0.9976
0.9756
0.9823
0.9948
0.9623
0.9961
0.9874
0.9673
0.9723
0.9958
0.9848
0.9984
“snf”
“fnf”
0.9852
Relative
second
natural
frequency
Relative
first
natural
frequency
0.038
-0.032
0.0898
0.0726
0.1947
0.01
0.1383
0.1814
0.3969
0.2709
“fmd”
Relative
first
mode
shape
difference
0.4558
0.322
0.3154
0.2567
0.0672
-0.8678
-0.0823
0.0279
0.3247
0.2372
“smd”
Relative
second
mode
shape
difference
0.3507
0.3965
0.392
0.3994
0.4105
0.2572
0.1898
0.0774
0.3923
0.3158
“tmd”
Relative
third
mode
shape
difference
0.538
0.425
0.496
0.449
0.551
0.229
0.393
0.552
0.429
0.204
0.534
0.503
0.424
0.404
0.285
0.239
0.188
0.158
0.082
0.071
rcl
0.537
0.425
0.495
0.449
0.552
0.227
0.389
0.548
0.431
0.203
rcd
0.53313
0.50188
0.424
0.404
0.283
0.2364
0.18725
0.16
0.079
0.069
rcl
0.534
0.426
0.497
0.446
0.545
0.23
0.385
0.542
0.431
0.202
rcd
0.53538
0.50163
0.42363
0.40338
0.28325
0.23613
0.18475
0.15825
0.08038
0.07163
rcl
Fuzzy_Neuro
Controller
(relative crack
depth “rcd” and
location“rcl”)
Fuzzy_Neuro
Controller
(relative crack
depth “rcd” and
location“rcl”)
Fuzzy_Neuro
Controller
(relative crack
depth “rcd” and
location“rcl”)
rcd
Trapezoidal
Gaussian
Triangular
0.542
0.426
0.497
0.451
0.556
0.231
0.394
0.537
0.427
0.202
rcd
0.535
0.501
0.424
0.404
0.283
0.236
0.187
0.160
0.079
0.069
rcl
Numerical
(relative crack
depth “rcd” and
location“rcl”)
0.535
0.425
0.495
0.447
0.545
0.23
0.391
0.568
0.43
0.205
rcd
0.533
0.504
0.424
0.405
0.286
0.240
0.188
0.158
0.084
0.073
rcl
Experimental
(relative crack
depth “rcd” and
location“rcl”)
Comparison of the results of the fuzzy-neuro controllers with the results of numerical and experimental analyses
0.9974
0.9988
0.9931
0.9919
0.9818
0.9967
0.9972
0.9983
0.9943
0.9975
“tnf”
Relative
third
natural
frequency
186
0.9961
0.9872
0.9809
0.9842
0.9685
0.9723
0.9823
0.981
0.986
0.9834
0.9974
0.9988
0.9931
0.9919
0.9818
0.9967
0.9972
Table 6.2.2
0.9984
0.9852
0.9976
0.9756
0.9983
0.0774
0.4558
0.322
0.3154
0.2567
0.0672
0.3507
0.3965
0.392
0.3994
0.4105
-0.8678 0.2572
-0.0823 0.1898
0.0279
0.3923
0.16
0.079
rcl
rcd
0.542 0.15825 0.548
0.431 0.08038 0.431
0.202 0.07163 0.203
rcd
0.424
0.404
0.283
0.2364
0.497 0.42363 0.495
0.446 0.40338 0.449
0.545 0.28325 0.552
0.23 0.23613 0.227
0.538 0.534 0.537 0.53313 0.534 0.53538 0.537
0.425 0.503 0.425 0.50188 0.426 0.50163 0.425
0.496 0.424 0.495
0.449 0.404 0.449
0.551 0.285 0.552
0.229 0.239 0.227
rcd
rcl
Triangular Fuzzy
Controller
(relative crack
depth “rcd” and
location“rcl”)
rcd
rcl
Gaussian
Fuzzy
Controller
(relative crack
depth “rcd” and
location“rcl”)
rcd
rcl
Trapezoidal Fuzzy
Controller
(relative crack
depth “rcd” and
location“rcl”)
0.534 0.534 0.53538 0.537 0.53313 0.534 0.53538
0.503 0.428 0.50363 0.425 0.50188 0.428 0.50363
0.424 0.498 0.42263 0.495 0.42363 0.498 0.42263
0.405 0.446 0.40338 0.449 0.40438 0.446 0.40338
0.286 0.545 0.27625 0.552 0.28375 0.545 0.27625
0.238 0.229 0.23613 0.227 0.23688 0.229 0.23613
0.187 0.382 0.18475 0.389 0.18725 0.382 0.18475
0.159 0.557 0.15825 0.548 0.15913 0.557 0.15825
0.081 0.441 0.08238 0.431 0.08138 0.441 0.08238
0.072 0.202 0.07163 0.203 0.07138 0.202 0.07163
rcl
Neural Network
Controller
(relative crack
depth “rcd” and
location“rcl”)
0.393 0.188 0.389 0.18725 0.385 0.18475 0.389
0.552 0.158 0.548
0.429 0.082 0.431
0.069
rcl
Trapezoidal
Fuzzy_Neuro
Controller
(relative crack
depth “rcd” and
location“rcl”)
Comparison of the results of the fuzzy-neuro controllers with the results of neural and fuzzy controllers
0.038
-0.032
0.0898
0.0726
0.1947
0.01
0.1383
0.1814
0.3247
rcd
Gaussian
Fuzzy_Neuro
Controller
(relative crack
depth “rcd” and
location“rcl”)
0.204 0.071 0.203
0.9948
0.3969
0.3158
0.9623
0.9943
0.2372
0.9874
0.2709
rcl
Triangular
Fuzzy_Neuro
Controller
(relative crack
depth “rcd” and
location“rcl”)
0.9673
Relative
third
mode
shape
difference
0.9975
Relative
second
mode
shape
difference
0.9958
Relative
first mode
shape
difference
0.9848
Relative
third
natural
frequency
“tnf” “fmd” “smd” “tmd” rcd
Relative
second
natural
frequency
“fnf” “snf”
Relative
first
natural
frequency
6.3
Discussion
Results obtained from the analysis of the developed fuzzy-neuro controller and their
comparison with neural controller, fuzzy controllers, numerical and experimental analyses,
following discussions are made.
Fig. 6.2.1, Fig. 6.2.2 and Fig. 6.2.3 represent the architecture of developed fuzzy-neural
controllers with triangular, gaussian and trapezoidal membership functions respectively.
These three controllers are used for prediction of crack location and crack depth. Table 6.2.1
and Table 6.2.2 show the comparison of the results of triangular, Gaussian and trapezoidal
fuzzy-neuro controllers with the numerical, experimental, neural controller and triangular,
gaussian and trapezoidal fuzzy controller results.
6.4
Summary
The results and discussions made above, show that the crack location and its size can be
predicted by the help of a fuzzy-neuro controller developed. The fuzzy-neuro controller is
based on the natural frequencies and mode shape differences of the structures with crack. The
predicted values of crack location and its size are compared with the numerical,
experimental, neural and fuzzy controllers results and are found to be in well agreement. This
fuzzy-neuro controller can be used as an effective tool for fault diagnosis of the vibrating
structures.
Publications
•
Das H.C. and Parhi D.R., Fuzzy-Neuro Controller for Smart Fault Detection of A Beam,
International Journal of Acoustics and Vibration, 13(2), 2009, 55-66.
187
Chapter 7
ANALYSIS OF MANFIS FOR CRACK DETECTION
It has been established that dynamic behavior of a structure changes due to the presence of
crack in the structure. The effect of crack on the vibration signatures of the structure depends
mainly on the location and depth of the crack. To identify the location and depth of a crack in
a structure, a new method is presented in this chapter which uses multiple adaptive neurofuzzy-evolutionary technique (MANFIS). With this MANFIS, it is possible to formulate the
inverse problem. MANFIS is used to obtain the outputs (the relative crack location and
relative crack depth) from the inputs (the first three natural frequencies and first three mode
shapes). This new method has been applied to diagnose fault on a cracked cantilever beam
and the results are promising.
In the MANFIS controller after the input layer there are five layers out of which three layers
are fixed layers and two layers are adaptive layers. The adaptive neuro-fuzzy hybrid system
combines the advantages of fuzzy logic system, which deal with explicit knowledge that can
be explained and understood, and neural networks, which deal with implicit knowledge,
which can be acquired by learning. The merger of neural networks and fuzzy logic led to the
creation of neuro-fuzzy controllers which are currently one of the most popular research
fields. The inputs to MANFIS are relative deviation of first three natural frequencies and
relative values of percentage deviation for first three mode shapes and outputs are relative
crack depth and relative crack location. A learning algorithm based on neural network
technique has been developed to tune the parameters of fuzzy membership functions. The
experimental results agree well with the MANFIS results, proves the authenticity of the
theory developed.
188
7.1
Introduction
Recently sophisticated vibration monitoring techniques have been available for monitoring
and diagnosis of faulty vibrating structures. Among them, the artificial intelligence
techniques such as neural networks, fuzzy logic, expert systems and so on are at the priority.
One of the hybrid artificial intelligence techniques i.e. multiple adaptive neuro-fuzzyevolutionary technique have woken up a lot among the researchers in the recent years. The
MANFIS approach is becoming one of the major areas of interest because it gets the benefits
of neural networks as well as of fuzzy logic systems and it removes the individual
disadvantages by combining them on the common features.
In this chapter for diagnosis of the crack in the structure multiple adaptive neuro-fuzzy
inference system methodology has been applied. The adaptive neuro-fuzzy controller has got
input layer, hidden layers and out put layer. The input layer is the fuzzy layer. The other
layers are neural layers. The inputs to the fuzzy layer are relative deviation of first three
natural frequencies and relative values of percentage deviation for first three mode shapes.
The final outputs of the MANFIS controller are relative crack depth and relative crack
location. Several hundreds fuzzy rules and neural network training patterns are derived using
natural frequencies, mode shapes, crack depths and crack locations. Real results have been
obtained using the experimental setup. Comparison between the simulation and experimental
results, exhibits a good agreement between them. This methodology can be effectively used
for condition monitoring of dynamic structures.
This chapter is organised into five sections following the introduction; the entire analysis
of MANFIS architecture has been discussed in section 7.2. The analysis of MANFIS used for
crack detection has been discussed in section 7.3. The results of MANFIS controller are
presented in section 7.4 and comparisons of results of MANFIS with other methods
discussed previously are analyzed in section 7.5. Finally, the discussions and summary are
given in section 7.6 and 7.7 respectively.
189
7.2
Analysis of Multiple Adaptive Neuro-Fuzzy Inference System for
Crack Detection
The multiple adaptive neuro fuzzy inference system is an integrated system of artificial
neural network (ANN) and fuzzy inference system (FIS). The ANFIS controller used is a
first order Takagi Sugeno Fuzzy Model [258]. In the current analysis, there are six inputs and
two outputs. They are as follows:
Relative first natural frequency (x1) =“fnf”; Relative second natural frequency (x2) = “snf”;
Relative third natural frequency (x3)=“tnf” ;Relative first mode shape difference(x4)=“fmd”;
Relative second mode shape difference (x5)=“smd”
Relative third mode shape difference (x6)=“tmd”.
The outputs are as follows;
Relative crack location = “rcl” and Relative crack depth = “rcd”
As in the current investigation there are two out puts, multiple ANFIS (MANFIS)
architecture has been used (Fig. 7.2.1).
The “if then” rules for the MANFIS architecture is defined as follows;
IF x1 is Aj , x2 is Bk , x3 is Cm, x4 is Dn , x5 is Eo , x6 is Fp
THEN
(7.2.1)
fe,i = pe,i x1 + re,i x2 + se,i x3 + te,i x4 + ue,i x5 + ve,i x6 + ze,i
Where;
f1,i = rcli = p1,i x1 + r1,i x2 + s1,i x3 + t1,i x4 + u1,i x5 + v1,i x6 + z1,i
for relative crack length.
;
(7.2.2)
f2,i = rcdi = p2,i x1 + r2,i x2 + s2,i x3 + t2,i x4 + u2,i x5 + v2,i x6 + z2,i ;
for relative crack depth.
e = 1,2 ; j = 1 to q1 ; k = 1 to q2 ; m = 1 to q3 ; n = 1 to q4 ; o = 1 to q5 and p = 1 to q6 and
i = 1 to q1.q2.q3.q4.q5.q6
190
A, B, C, D, E and F are the fuzzy membership sets defined for the input variables x1 (fnf),
x2(snf), x3(tnf), x4(fmd), x5(smd) and x6(tmd). q1, q2, q3, q4, q5 and q6 are the number of
member ship functions for the fuzzy systems of the inputs x1, x2, x3, x4, x5 and x6
respectively.
“rcl” and “rcd” are the linear consequent functions defined in terms of the inputs (x1, x2, x3,
x4, x5 and x6) . p1,i , r1,i, s1,i ,t1,i ,u1,i ,v1,i , z1,i,p2,i , r2,i, s2,i ,t2,i ,u2,i , v2,i and z2,i are the
consequent parameters of the ANFIS fuzzy model. In the ANFIS model nodes of the same
layer have similar functions. The output signals from the nodes of the previous layer are the
input signals for the current layer. The output obtained with the help of the node function will
be the input signals for the subsequent layer.
Layer 1: Every node in this layer is an adaptive node (square node) with a particular fuzzy
membership function (node function) specifying the degrees to which the inputs satisfy the
quantifier. For six inputs the outputs from nodes are given as follows;
O1, g,e = μAg (x) for g = 1, ……, q1
(for input x1)
O1, g,e = μBg (x)
for g = q1+1, ……, q1+q2
(for input x2)
O1, g,e = μCg (x)
for g = q1+q2+1, ……, q1+q2+q3
(for input x3)
(7.2.3)
O1, g,e = μDg (x) for g = q1+q2+q3+1, …, q1+q2+q3+q4
(for input x4)
O1, g,e = μEg (x) for g = q1+q2+q3+q4+1, …, q1+q2+q3+q4+q5
(for input x5)
O1, g,e = μFg (x) for g = q1+q2+q3+q4+q5+1, …, q1+q2+q3+q4+q5+q6
(for input x6)
Here the membership functions for A, B, C, D, E and F considered are the bell shaped
function.
191
The membership function for A,B,C,D,E and F considered in “layer 1” are the bell shaped
function (Fig. 7.2.1) and are defined as follows;
MF
0.1
Slope=-b/2ag
0.5
cg-ag
0.0
cg
cg+ag
X
2ag
Fig. 7.2.1 Bell-shaped membership function
μAg(x)=
μBg(x)=
μCg(x)=
μDg(x)=
1
bg
;
g = 1, ……, q1
bg
;
g = q1+1, ……, q1+q2
(7.2.4 (ii))
bg
;
g = q1+q2+1, ……, q1+q2+q3
(7.2.4 (iii))
bg
;
g = q1+q2+q3+1, …, q1+q2+q3+q4
(7.2.4 (iv))
⎧⎛ x − c ⎞ 2 ⎫
⎪
g
⎟ ⎪
1 + ⎨⎜
⎜ a ⎟ ⎬
⎪⎩⎝ g ⎠ ⎪⎭
1
⎧⎛ x − c ⎞ 2 ⎫
⎪
g
⎟ ⎪⎬
1 + ⎨⎜
⎜
⎟
⎪⎩⎝ a g ⎠ ⎪⎭
1
⎧⎛ x − c ⎞
⎪
g
⎟
1 + ⎨⎜
⎜ a ⎟
⎪⎩⎝ g ⎠
2
⎫
⎪
⎬
⎪⎭
1
⎧⎛ x − c ⎞
⎪
g
⎟
1 + ⎨⎜
⎜ a ⎟
⎪⎩⎝ g ⎠
2
⎫
⎪
⎬
⎪⎭
192
(7.2.4 (i))
μEg(x)=
μFg(x)=
1
⎧⎛ x − c ⎞
⎪
g
⎟
1 + ⎨⎜
⎜ a ⎟
⎪⎩⎝ g ⎠
2
bg
;
g = q1+q2+q3+q4+1, ……, q1+q2+q3+q4+q5
(7.2.4 (v))
bg
;
g = q1+q2+q3+q4+q5+1, ., q1+q2+q3+q4+q5+q6
(7.2.4 (vi))
⎫
⎪
⎬
⎪⎭
1
⎧⎛ x − c ⎞ 2 ⎫
⎪
g
⎟ ⎪
1 + ⎨⎜
⎜ a ⎟ ⎬
⎪⎩⎝ g ⎠ ⎪⎭
Where ag,bg and cg are the parameters for the fuzzy membership function. The ball-shaped
function changes its pattern as per the change of the parameters. This change will give the
various contour of bell shaped function as needed in accord with the data set for the problem
considered.
Layer 2: Every node in this layer is a fixed node (circular) labeled as “Π”. The out put
denoted by O2,i,e. The output is the product of all incoming signal.
O2,i,e = wi ,e = μAg(x) μBg(x) μCg(x) μDg(x) μEg(x) μFg(x) ;
(7.2.5)
for i = 1,…., q1.q2.q3.q4.q5.q6 and g = 1 ,….., q1+q2+q3+q4+q5+q6
The output of each node of the second layer represents the firing strength ( degree of
fulfillment) of the associated rule. The T-nom operator algebraic product { Tap(a,b) = ab},
has been used to obtain the firing strength (wi,e).
Layer 3: Every node in this layer is a fixed node (circular) labeled as “N”. The output of the
i th. node is calculated by taking the ratio of firing strength of i th. rule (wi,e) to the sum of all
rules’ firing strength.
193
w i, e
O3,i,e = w i , e = r = q1.q2.q3.q4.q5.q6
w r, e
∑
(7.2.6)
r -1
This output gives a normalized firing strength.
Layer 4: Every node in this layer is an adaptive node (square node) with a node function.
O4,i, e = w i ,e fe,i = w i ,e (pe,i x1 + re,i x2 + se,i x3 + te,i x4 + ue,i x5 + ve,i x6 + ze,i ) (7.2.7)
Where w i ,e is a normalized firing strength form (output) from layer 3 and {pe,i , re,i , se,i , te,i,
ue,i , ve,i , ze,i}is the parameter set for relative crack location(e=1) and relative crack depth
(e=2). Parameters in this layer are referred to as consequent parameters.
Layer 5: The single node in this layer is a fixed node (circular) labeled as “Σ”, which
computes the overall output as the summation of all incoming signals.
i = q1.q2.q3.q4.q5.q6
O5,1,e =
i = q1.q2.q3.q4.q5.q6
∑
i -1
w i, e f e,i =
∑
w i, e f e,i
i -1
i = q1.q2.q3.q4.q5.q6
∑
i -1
(7.2.8)
w i, e
In the current developed ANFIS structure there are six dimensional space partition and has
“q1 x q2 x q3 x q4 x q5 x q6” regions. Each region is governed by a fuzzy if then rule. The first
layer (consists of premise or antecedent parameters) of the ANFIS is dedicated to fuzzy sub
space. The parameters of the fourth layer are referred as consequent parameters and are used
to optimize the network. During the forward pass of the hybrid learning algorithm node
outputs go forward until layer four and the consequent parameters are identified by least
square method. In the backward pass, error signals propagate backwards and the premise
parameters are updated by a gradient descent method.
194
X1
X2
O5, 1, 1
X3
ANFIS (1)
Σ
O5, 1, 2
X4
ANFIS (2)
Σ
Output Layer
X5
ANFIS Layer
X6
Input Layer
Fig. 7.2.2 Multiple ANFIS (MANFIS) controller for crack detection
195
X1, X2, X3, X4, X5, X6 fe, i A1 П N Aq1 П N fe, i B1 П N fe, i Bq2 П N fe, i C1 П N fe, i Cq3 П N fe, i D1 П N fe, i Dq4 П N fe, i Fifth Layer E1 П N fe, i (Summation Evaluation output) Eq5 П N fe, i F1 П N fe, i Fq6 П N fe, i Second Layer Third Layer X1 X2 X3 O5,1,e
ΣΣ X4 X5 X6 Input Layer First Layer (Fuzzification) Fourth Layer (Rule (Normallization (Consequent Inference) evaluation Inference) Inference) Fig. 7.2.3Fig. 7.2.3 Adaptive‐Neuro‐Fuzzy‐Inference System (ANFIS) for crack detection Adaptive Neuro Fuzzy Inference System (ANFIS) for crack detection
196
7.3
Results of MANFIS Controller
The multiple adaptive neuro-fuzzy inference system is designed and developed to predict the
relative crack location and relative crack depth. A comparison of results between the
developed MANFIS, numerical analysis and experimental analysis is presented in Table
7.3.1. Finally the comparisons of results between the developed MANFIS, triangular fuzzyneuro controller, gaussian fuzzy-neuro controller, trapezoidal fuzzy-neuro controller of
chapter-6, neural controller of chapter-5 and triangular fuzzy controller, gaussian fuzzy
controller and trapezoidal fuzzy controller of chapter-4 are presented in Table 7.3.2. In the
Tables 7.3.1 and 7.3.2 ten sets of random inputs out of several hundred sets of inputs are
taken. The inputs to different analyses described above, are relative first three natural
frequencies and relative first three mode shape differences and the out puts are relative crack
location and relative crack depth. Corresponding ten set of output results from the developed
MANFIS controller, triangular fuzzy-neuro controller, gaussian fuzzy-neuro controller,
trapezoidal fuzzy-neuro controller, neural controller, triangular fuzzy controller, gaussian
fuzzy controller , trapezoidal fuzzy controller, numerical analysis and experimental analysis
are given in the Table 7.3.1 and Table 7.3.2. In the Tables 7.3.1 and 7.3.2 the first column
represents the relative 1st natural frequency (fnf), the second column represents the relative
2nd natural frequency (snf), the third column represents the relative 3rd natural frequency
(tnf), the fourth column represents the relative 1st mode shape difference (fmd), the fifth
column represents the relative 2nd mode shape difference (smd), the sixth column represents
the relative 3rd mode shape difference (tmd) as inputs and the rest coloumns represents the
outputs of relative crack location and relative crack depth.
197
198
0.9958
0.9874
0.9948
0.9976
0.9984
0.9961
0.9872
0.9809
0.9842
0.9685
0.9848
0.9673
0.9623
0.9756
0.9852
0.9723
0.9823
0.981
0.986
0.9834
0.9974
0.9988
0.9931
0.9919
0.9818
0.9967
0.9972
0.9983
0.9943
0.9975
“tnf”
Relative
third
natural
frequency
0.038
-0.032
0.0898
0.0726
0.1947
0.01
0.1383
0.1814
0.3969
0.2709
“fmd”
Relative
first mode
shape
difference
0.4558
0.322
0.3154
0.2567
0.0672
-0.8678
-0.0823
0.0279
0.3247
0.2372
“smd”
0.3507
0.3965
0.392
0.3994
0.4105
0.2572
0.1898
0.0774
0.3923
“tmd”
0.3158
Relative
Relative third
second mode mode shape
shape
difference
difference
0.538
0.425
0.497
0.449
0.555
0.231
0.392
0.543
0.43
0.202
rcd
0.5349
0.503
0.424
0.404
0.2842
0.2362
0.1865
0.159
0.08
0.0712
rcl
MANFIS Controller
(relative crack depth
“rcd” and
location“rcl”)
0.542
0.426
0.497
0.451
0.556
0.231
0.394
0.537
0.427
0.202
rcd
0.535
0.50125
0.42388
0.40388
0.2825
0.23625
0.18675
0.15988
0.079
0.06888
rcl
Numerical
(relative crack
depth “rcd” and
location“rcl”)
Comparison of the results of the MANFIS with the results of numerical and experimental analysis
“snf”
“fnf”
Table 7.3.1
Relative
second
natural
frequency
Relative
first
natural
frequency
0.535
0.425
0.495
0.447
0.545
0.23
0.391
0.568
0.43
0.205
rcd
0.53313
0.50375
0.4235
0.40513
0.28625
0.24
0.18775
0.1575
0.08388
0.0725
rcl
Experimental
(relative crack depth
“rcd” and
location“rcl”)
199
0.9872
0.9809
0.9842
0.9685
0.9823
0.981
0.986
0.9834
Table 7.3.2
0.9961
0.9976
0.9756
0.9723
0.9948
0.9623
0.9984
0.9874
0.9673
0.9852
0.9958
0.9848
0.038
-0.032
0.0898
0.0726
0.1947
0.01
0.1383
0.1814
0.3969
0.2709
0.4558
0.322
0.3154
0.2567
0.0672
-0.8678
-0.0823
0.0279
0.3247
0.2372
0.3507
0.3965
0.392
0.3994
0.4105
0.2572
0.1898
0.0774
0.3923
0.3158
rcd
0.429 0.082 0.431
0.159 0.552 0.158 0.548
0.08
0.16
0.079
rcl
rcd
0.542 0.15825 0.548
0.431 0.08038 0.431
0.424
0.404
0.497 0.42363 0.495
0.446 0.40338 0.449
0.545 0.28325 0.552
0.503 0.425 0.503 0.425 0.5018 0.426 0.50163 0.425
0.424 0.496 0.424 0.495
0.404 0.449 0.404 0.449
0.283
0.23 0.23613 0.227
0.538 0.5349 0.538 0.534 0.537 0.5331 0.534 0.53538 0.537
0.425
0.497
0.449
0.555 0.2842 0.551 0.285 0.552
0.231 0.2362 0.229 0.239 0.227 0.2364
rcd
rcl
Triangular Fuzzy
Controller
(relative crack
depth “rcd” and
location“rcl”)
rcd
rcl
Gaussian
Fuzzy
Controller
(relative crack
depth “rcd” and
location“rcl”)
rcd
rcl
Trapezoidal
Fuzzy Controller
(relative crack
depth “rcd” and
location“rcl”)
0.534 0.534 0.53538 0.537 0.53313 0.534 0.53538
0.503 0.428 0.50363 0.425 0.50188 0.428 0.50363
0.424 0.498 0.42263 0.495 0.42363 0.498 0.42263
0.405 0.446 0.40338 0.449 0.40438 0.446 0.40338
0.286 0.545 0.27625 0.552 0.28375 0.545 0.27625
0.238 0.229 0.23613 0.227 0.23688 0.229 0.23613
0.187 0.382 0.18475 0.389 0.18725 0.382 0.18475
0.159 0.557 0.15825 0.548 0.15913 0.557 0.15825
0.081 0.441 0.08238 0.431 0.08138 0.441 0.08238
0.072 0.202 0.07163 0.203 0.07138 0.202 0.07163
rcl
Neural Network
Controller
(relative crack
depth “rcd” and
location“rcl”)
0.202 0.07163 0.203
rcd
Trapezoidal
Fuzzy_Neuro
Controller
(relative crack
depth “rcd” and
location“rcl”)
0.392 0.1865 0.393 0.188 0.389 0.1872 0.385 0.18475 0.389
0.543
0.43
0.069
rcl
Gaussian
Fuzzy_Neuro
Controller
(relative crack
depth “rcd” and
location“rcl”)
0.202 0.0712 0.204 0.071 0.203
rcd rcl
Triangular
Fuzzy_Neuro
Controller
(relative crack
depth “rcd”
and
location“rcl”)
Comparison of the results of the MANFIS with the results of fuzzu-neuro, neural and fuzzy controller analysis
0.9974
0.9988
0.9931
0.9919
0.9818
0.9967
0.9972
0.9983
0.9943
0.9975
rcl
MANFIS
Controller
(relative crack
depth “rcd” and
location“rcl”)
“fnf” “snf” “tnf” “fmd” “smd” “tmd” rcd
Relative Relative Relative Relative Relative Relative
first
second
third
first
second
third
natural natural natural mode
mode
mode
frequency frequency frequency shape
shape
shape
difference difference difference
7.4
Discussions
Results obtained for the fault diagnosis from the developed MANFIS, triangular fuzzy-neuro
controller, gaussian fuzzy-neuro controller, trapezoidal fuzzy-neuro controller, neural
controller , triangular fuzzy controller, gaussian fuzzy controller , trapezoidal fuzzy
controller, numerical and experimental analysis, following discussions are drawn.
Fig. 7.2.1 shows the
Bell shaped membership function used as membership functions in
layer-1 of ANFIS controller. Fig. 7.2.2 represents Multiple ANFIS (MANFIS) architecture
for crack detection. The architecture of ANFIS for crack detection is shown in Fig. 7.2.3.
Table 7.3.1 shows the comparison of results between the developed MANFIS, numerical
analysis and experimental analysis. Table 7.3.2 represents comparison of results between the
developed MANFIS, triangular fuzzy-neuro controller, gaussian fuzzy-neuro controller,
trapezoidal fuzzy-neuro controller, neural controller and triangular fuzzy controller, gaussian
fuzzy controller and trapezoidal fuzzy controller analysis. It is evident from the Tables 7.3.1
and 7.3.2 that the average percentage deviation of the results of MANFIS is 0.5%.
7.5
Summary
Following conclusions are drawn on the basis of analysis and results obtained from multiple
adaptive neuro-fuzzy inference system;
Using the above analysis and MANFIS methodology condition monitoring of dynamic
structures can be addressed effectively. The crack size and its location have got significant
effect on the natural frequencies and mode shapes of the vibrating structures. MANFIS can
predict the crack location and its size with the help of the natural frequencies and mode shape
differences of the dynamic structures. The developed controller predicted the results are in
close proximity with theoretical and experimental results.
Publications
•
Parhi D.R. and Das H.C., Diagnosis of fault and condition monitoring of dynamic
structures using MANFIS technique, Journal of Aerospace Engineering Proceedings of
the Institution of Mechanical Engineers, Part G, Vol. 223, In Press.
200
Chapter 8
ANALYSIS AND DESCRIPTION OF EXPERIMENTAL
SETUP
In order to justify the validation of the theoretical analysis and numerical analysis discussed
in chapter-3 and different artificial intelligence methodologies proposed for prediction of
crack location and crack depth discussed in chapter-3 to chapter-7, experimental
investigations have been carried out. For the experimental investigations an experimental setup has been developed in order to measure the dynamic response of the cantilever beam with
a transverse crack. The details of the instruments used in the experimental set-up, test
specimens and experimental procedure are presented in the subsequent sections.
8.1
Description of Instruments used in the Experimental Analysis
A (800 x 50 x 6mm) aluminum beam specimen is selected for the experimental
investigations. The schematic block diagram of the whole experimental setup is shown in
Fig. 3.3.1. It can be noted from the Fig. 3.3.1 that the electro-dynamic exciter is driven by a
function generator connected to a signal amplifier. Vibration indicator connected to a
vibration analyzer shows the vibration responses of cracked cantilever beam through the
signal which comes from the accelerometer. The detailed specifications of the instruments
used in this investigation are given below.
1. Vibration pick-up
(Accelerometer)
-
Delta Tron Accelerometer
Type
:
4513-001
Make
:
Bruel & kjaer
Sensitivity
:
10mv/g-500mv/g
Frequency
Range
:
1Hz-10KHz
Supply voltage:
24volts
Operating
temperature
Range
-500C to +1000c
201
:
2. Vibration Analyzer
-
Type
:
Product Name :
Pocket front end
Make
:
Bruel & kjaer
Frequency
Range
:
7 Hz to 20 Khz
ADC Bits
:
16
Simultaneous
Channels
:
Input Type
3. Vibration indicator
4. Function Generator
-
-
3560L
:
2 Inputs,
2 Tachometer
Direct/CCLD
PULSE LabShop Software Version 12
Make
:
Bruel & kjaer
Model
:
FG200K
Frequency
Range
:
0.2Hz to 200 KHz
VCG IN connector for Sweep Generation
Sine, Triangle, Square, TTL outputs
Output Attenuation up to 60dB
Output Level :
5. Power Amplifier
-
Rise/Fall Time :
15Vp-p into 600 ohms
Square Wave
<300nSec
Make
:
Aplab
Type
:
2719
Power
Amplifier
:
180VA
Make
:
Bruel & kjaer
202
6.
Vibration Exciter
-
Type
:
4808
Permanent Magnetic Vibration Exciter
Force rating 112N (25 lbf) sine peak
(187 N (42 lbf) with cooling)
Frequency
Range
:
5Hz to 10 kHz
First axial
resonance
:
10 kHz
Maximum bare table
Acceleration :
700 m/s2 (71 g)
Continuous 12.7 mm (0.5 in)
peak-to-peak displacement
with over travel stops
Two high-quality, 4-pin
Neutrik® Speakon® connectors
Make: Bruel & kjaer
7.
Specimen
-
cantilever type cracked aluminum
beam specimen of dimension
(800 x 50 x 6mm)
203
8.2
A
Experimental Set-up
cracked cantilever beam has been rigidly clamped to the concrete foundation base as
shown in the Fig.8.1. The free end of the 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 amplitude of vibration of
the uncracked and cracked cantilever beam is taken by the accelerometer and is fed to the
vibration indicator (PULSE LabShop Software Version 12) for vibration analysis. The
vibration signatures are analyzed graphically by PULSE LabShop Software loaded in the
laptop. The views of the instruments used in the experimental set-up are shown in Figs.
8.2(a) - 8.2(g) .
4
5
3
2
6
7
1
9
8
Fig. 8.1 View of complete assembly of the experimental set-up
204
Fig.8.2 (a) Concrete foundation with beam specimen
Fig.8.2 (b) Vibration indicator (PULSE labShop
software) with lap top
Fig.8.2 (c) Vibration exciter
205
Fig.8.2 (d) Vibration pick-up (accelerometer)
Fig.8.2 (e) Vibration analyser
Fig.8.2 (f) Function generator
206
Fig.8.2 (g) Power amplifier
Fig.8.2 View of the instruments used in the experimental set-up
8.3
Experimental Procedure
Several tests are conducted using the experimental setup (Fig. 8.1) on Aluminum beam
specimens (800 x 50 x 6mm) with a transverse crack for determining the natural frequencies
and mode shapes for different crack locations and crack depths. These specimens are set to
vibrate under 1st, 2nd and 3rd mode of vibrations and the corresponding amplitudes are
recorded in the vibration indicator.
Experimental results for amplitude of transverse
vibration at various locations along the length of the beam are recorded by positioning the
vibration pick-up and tuning the vibration generator at the corresponding resonant
frequencies.
207
8.4
Experimental Results and Discussions
The experimental results of relative amplitude for different relative crack locations (0.026,
0.05128) and different crack locations (0.3, 0.4) for 1st,2nd and 3rd modes of vibration are
presented graphically in Fig.3.3.2, Fig.3.3.3 and Fig.3.3.4 in section 3.3.1 of chapter 3.
Corresponding results of numerical analysis are also presented in the same graph for cracked
and uncracked beam for immediate comparison. The experimental results for relative crack
location and relative crack depth are compared with the corresponding results of the
triangular, gaussian and trapezoidal fuzzy controller in Table 4.3.6 of chapter-4 and are found
to be in good agreement. The experimental results for relative crack location and relative
crack depth are compared with the corresponding results of the back propagation neural
controller in Table 5.3.4 of chapter-5 and are found to be in agreement. The experimental
results for relative crack location and relative crack depth are also compared with the
corresponding results of the triangular, gaussian and trapezoidal fuzzy-neuro controllers in
Table 6.2.1 of chapter-6 and are found to be in good agreement. The experimental results for
relative crack location and relative crack depth are also compared with the corresponding
results of the multiple adaptive neuro-fuzzy controller in Table 7.2.1 of chapter-7 and are
found to be in good agreement.
208
Chapter 9
RESULTS AND DISCUSSIONS
9.1
Introduction
In this section the results obtained from different analyses performed on the cracked
cantilever beam have been analyzed and discussed. The effects of crack parameters on the
dynamic response of the structure have been elaborated.
9.2
Discussions of Results
The current research has been carried out in seven stages. The stages comprised of are 1)
Literature survey. 2) Analysis of dynamic characteristics of beam with a transverse crack. 3)
Analysis of fuzzy logic technique for crack detection. 4) Analysis of artificial neural network
for crack detection. 5) Analysis of hybrid fuzzy neuro system for crack detection. 6) Analysis
of MANFIS for crack detection 7) Analysis and description of experimental setup. These
stages are presented in chapter forms from chapter two to chapter eight. The outcomes of
results of different chapters during analysis are presented below systematically.
Chapter two depicts the various methodologies adapted since last five decades for crack
detection in damaged structures. Many condition monitoring tools used by authors for fault
detection in different domain of engineering applications with the help of artificial
intelligence techniques have been discussed.
In chapter three theoretical analysis has been carried out on the cracked cantilever beam (Fig.
3.2.1). During analysis it is observed that the crack position and crack depth have a
considerable effect on the dynamic response of the beam. The variation of the mode shapes
for the 1st three modes of vibration and the significant changes in the mode shapes at the
crack location can be seen with magnifying views in Fig. 3.2.4 to Fig. 3.2.27. From the
analysis it is found that with the increase in relative crack depth there is an increase in
209
dimensionless compliances (Fig. 3.2.2), which clearly shows the relationship between
relative crack depth and vibration parameters of the structure. The results from the numerical
analysis have been validated with the results obtained from the developed experimental set
up (Fig. 3.3.1). The comparisons of results from theoretical and experimental analysis for the
cracked and uncracked beam are presented in Fig.3.3.2 to Fig.3.3.4, which shows a close
agreement. The variation of relative natural frequencies, relative mode shapes with respect to
relative crack locations and relative crack depth in three dimensional forms, along with the
contour plots are depicted in Fig.3.2.28 and Fig.3.2.29 respectively.
Fuzzy controllers have been designed for prediction of crack location and its severity using
three different types of membership functions such as triangular function (Fig. 4.2.1(a)),
Gaussian function (Fig. 4.2.1(b)) and trapezoidal function (Fig. 4.2.1(c)) and are presented in
chapter four. The fuzzy controllers developed here take the 1st three natural frequencies and
mode shape differences as input parameters and relative crack depth and relative crack
location as output parameters as shown in Fig. 4.2.2. Several fuzzy rules and fuzzy linguistic
terms have been developed to design the fuzzy controller, some of them are described in
Table 4.3.1 and Table 4.3.2. The complete architecture of triangular, Gaussian, trapezoidal
fuzzy controller are presented in Fig. 4.3.1, Fig. 4.3.2 and Fig. 4.3.3 respectively. Fig. 4.3.4
to Fig. 4.3.6 exhibits the fuzzy results after defuzzification when rule 1 and 19 of the Table
4.3.2 are activated for triangular, Gaussian and trapezoidal membership function
respectively. Table 4.3.3 gives the comparison among the results obtained from numerical,
experimental, fuzzy controller with triangular membership function, fuzzy controller with
gaussian membership function and fuzzy controller with trapezoidal membership functions.
Results obtained from gaussian membership function fuzzy controller is more accurate in
comparison to other two controllers and the computational time for crack prediction in
considerably lower as compared to numerical analysis. The predicted results from fuzzy
controllers for crack location and crack depth are compared with the theoretical and
experimental results for cross verification. A close agreement between the results is found.
Chapter five describes the design of neural network controller with back propagation method
(Fig.5.2.5) for prediction of crack location and its size for the cracked cantilever beam. The
details of neural network technique (Fig.5.2.1), reasons for using neural network, activation
210
function (Fig.5.2.2), modeling of multi layer perceptron have been depicted in section 5.2.
Neural network approach has been used to develop the controller so as to take the advantage
of neuron for high accuracy results and faster computation. The neural network controller has
been trained with eight hundred patterns of data of different crack location and crack depth
gives out put such as relative crack depth and relative crack location. Few of the examples of
training patterns out of several hundreds to train the controller are given in the Table 5.3.1.
The working principle of the ten layer feed forward controller and a schematic diagram of
multi layer neural network controller is depicted in Fig.5.3.2 and Fig.5.3.1 respectively. The
controller is made of ten layers with one input layer, eight numbers of hidden layer and one
output layer. The input layer takes the relative natural frequencies and relative mode shapes
as input parameters where as relative crack location and relative crack depth are the results
from the output layer. The controller is designed in a diamond shape for convergence of
results. It is observed that the error in the output of the controller is considerably reduced
from the desired output by employing error back propagation method. The result from the
controller is compared with the outputs from numerical, fuzzy and experimental analysis for
checking the robustness of the developed system. The comparison of the results from neural
controller, fuzzy controllers, numerical analysis and experimental analysis are expressed in
Table 5.3.2. The prediction of crack location and its intensity from the neural network
controller is very close to the actual results.
Fuzzy neuro controllers have been developed in chapter six by integrating the capabilities of
fuzzy logic and neural network for prediction of crack in damaged structures. Fig. 6.2.1, Fig.
6.2.2 and Fig. 6.2.3 represent the developed fuzzy-neural controllers with triangular, gaussian
and trapezoidal membership functions respectively. These three controllers are used for
prediction of crack location and crack depth. The fuzzy-neuro controller is based on the
natural frequencies and mode shape differences of the structures with crack. Table 6.2.1 and
Table 6.2.2 show the comparison of the results of triangular, gaussian and trapezoidal fuzzyneuro controllers with the results from numerical, experimental, neural controller and
triangular, gaussian and trapezoidal fuzzy controllers. The predicted values from the
designed hybrid controller of crack location and its size are compared with the numerical,
experimental, neural and fuzzy controllers results and are found to be well in agreement. This
211
fuzzy-neuro controller can be used as an effective tool for fault diagnosis of vibrating
structures.
Multiple adaptive neuro fuzzy inference system (MANFIS) has been developed for effective
condition monitoring in dynamically vibrating structures and is discussed in chapter seven.
Bell shaped membership function has been adapted for designing the MANFIS controller.
Fig. 7.2.1 shows the Bell shaped membership function used in layer-1 of ANFIS controller.
Relative natural frequencies and relative mode shapes are the input parameters for the
controller and relative crack depth and relative crack location are the output parameters from
the controller. Fig. 7.2.2 represents multiple ANFIS (MANFIS) controller for crack
detection. The architecture of Adaptive neuro-fuzzy inference system (ANFIS) for crack
detection is shown in Fig. 7.2.3. Table 7.3.1 shows the comparison of results between the
developed MANFIS controller, numerical analysis and experimental analysis. From the
comparison it is evident that the results from the MANFIS controller have higher accuracy
with respect to numerical analysis. Table 7.3.2 represents comparison of results between the
developed MANFIS controller, triangular fuzzy-neuro controller, gaussian fuzzy-neuro
controller, trapezoidal fuzzy-neuro controller, neural controller and triangular fuzzy
controller, gaussian fuzzy controller and trapezoidal fuzzy controller analysis. From the
analysis of the Table 7.3.2 it is observed that the results obtained from the MANFIS
technique yields better results with least amount of error as compared to the other methods
cited in the Table 7.3.2.
Chapter eight discusses on various instruments used in the experimental setup for carrying
out the experimental analysis. The instruments used are 1. Concrete foundation with beam
specimen 2. Vibration indicator (PULSE labShop software) with lap top 3. Vibration exciter
4. Vibration pick-up (accelerometer) 5. Vibration analyzer 6. Function generator 7. Power
amplifier and are given in Fig. 8.2(a) to Fig. 8.2(g) respectively. This chapter also discusses
the experimental procedure in section 8.3. During experimental analysis care has been taken
to reduce error and noise signal.
The contributions, conclusions and scope for future work of the above analysis have been
given in the next chapter.
212
Chapter 10
CONCLUSIONS AND FURTHER WORK
The aim of this research has been to develop an efficient methodology for diagnosis of crack
in a vibrating structure. To achieve the above objective a comprehensive investigation has
been carried out to study the effect of crack on the vibration signatures of a dynamically
vibrating uniform cracked cantilever beam. The vibration analysis has been carried out in
several stages, such as theoretical, numerical, and experimental analysis. The influence of
cracks on the dynamic behavior of the beam is found to be very sensitive in regards to crack
location, crack depth and mode number. A number of inverse methods have been developed
comprising of artificial intelligence techniques such as fuzzy logic, neural network, fuzzy
neuro and MANFIS techniques for predicting the crack location and its severity based on
changes in the vibration signatures (natural frequencies, mode shapes).
On the basis of analyses and discussions done in previous chapters, the following
contributions and conclusions of the research are drawn.
10.1 Contributions
Theoretical analysis of the cracked beam on the basis of strain energy release rate has been
carried out to find out the effect of crack depth and crack location on vibration signatures of
the beam.
Numerical analysis has been carried out on the basis of above theoretical and experimental
analysis for studying the influence of crack depth and crack location on the dynamic response
of the cracked beam. Four inverse methods comprising of artificial intelligence techniques
such as fuzzy logic, neural network, fuzzy neuro and MANFIS have been developed for
diagnosis of crack depth and crack location.
213
10.2 Conclusions
•
Small crack depth ratios have little effects on the natural frequencies of the cracked
cantilever beam. Deviations in mode shapes are noticeable due to presence of crack.
Analysis of change in natural frequencies and mode shape in combination is effective
for prediction of crack in beam structure containing small crack.
•
A clearcut deviation in the mode shapes and natural frequencies at the vicinity of
crack location has been observed from the comparison of the results of the un cracked
and cracked beam during the vibration analysis.
•
Unique changes have been observed in the natural frequencies and mode shapes with
the change of crack depth and crack location. The changes in the vibration signatures
become more prominent as the crack grows bigger. It is observed that the results from
theoretical, numerical and experimental analysis are in good agreement.
•
Three types of fuzzy controllers have been designed with triangular, gaussian and
trapezoidal membership functions to predict the crack location and its size with the
help of natural frequencies and mode shape differences. During this design several
rules are formed and are used with various membership functions.
•
It has been observed that the developed fuzzy controllers can predict the relative
crack location, relative crack depth of the beam with a considerably less amount of
computational time.
•
Comparisons of fuzzy controller results with the experimental results show the
effectiveness of the proposed methods towards the identification of location and
extent of damage in vibrating structures. Fuzzy controller with gaussian membership
function is found to be more suitable.
•
The neural network controller has been developed using back propagation algorithm
to predict the crack location and size by using relative deviation of first three natural
frequencies and first three mode shapes as inputs. The neural network controller
predicted results are reasonably acceptable and in agreement with the experimental
214
data. The successful detection of crack and it’s intensity in cantilever beam
demonstrates that the new technique developed can be used as a smart fault detecting
tool for different types of vibrating structures.
•
From the comparison of fuzzy and neural network controller results, the neural
controller is found to deliver closer result with respect to actual result.
•
The fuzzy neuro hybrid intelligent systems have been designed with relative deviation
of first three natural frequencies and first three mode shapes as input parameters and
relative crack depth and relative crack location as output parameters. The predicted
results are found to be of higher accuracy than the results obtained from independent
fuzzy and neural controller.
•
The multiple adaptive neuro fuzzy inference system (MANFIS) has been designed
and is used as an effective tool for diagnosis of crack in vibrating structures. The
results obtained from MANFIS controller are found to be of higher accuracy than
fuzzy and neural controllers results. The developed MANFIS controller predicted
results are in close proximity with theoretical and experimental results.
•
After analysis the errors obtained from various methodologies developed using
artificial intelligence techniques, it can be stated that the fuzzy-neuro hybrid
intelligence controller based on Gaussian membership function and the MANFIS
controller are two most efficient controllers for fault diagnosis.
10.3 Applications
•
The developed controllers can be used as effective tools for online condition
monitoring of engineering systems.
•
The present study can be utilized for inverse engineering application/problems, and
can also be used in biomedical engineering system for fault detection.
215
•
The methodologies formulated using artificial intelligence techniques can be used for
prediction of fatigue crack of offshore structure, flow lines, turbo machinery, nuclear
plants, ship structures etc.
10.4 Scope for Future Work
•
In this research fault diagnosis and structural health monitoring systems have been
derived using vibration signatures. These developed techniques can be extended to
predict the health of complex structures with multiple cracks.
•
Genetic algorithm can be hybridized with the current developed controller for design
of more robust fault diagnosis system.
•
Systems can be developed for fault diagnosis of structures subjected to moving load.
216
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239
Published Papers
•
Das H.C. and Parhi D.R., Modal analysis of vibrating structures impregnated with crack,
International Journal of Applied Mechanics & Engineering, 13(3), 2008, 639-652.
•
Parhi D.R. and Das H.C., Diagnosis of fault and condition monitoring of dynamic
structures using MANFIS technique, Journal of Aerospace Engineering Proceedings of
the Institution of Mechanical Engineers, Part G (Accepted for publication).
•
Das H.C. and Parhi D.R., Damage Analysis of Cracked Structure Using Fuzzy Control
Technique, International Journal of Acoustics and Vibration, 13(2) , 2008 , 3-14.
•
Parhi D.R. and Das H.C., Structural damage detection by fuzzy- gaussian technique,
International Journal of Applied Mathematics and Mechanics, 4(2), 2008, 39-59.
•
Das H.C. and Parhi D.R., Identification of Crack Location and Intensity in a Cracked
Beam by Fuzzy Reasoning, International Journal of Intelligent Systems Technologies
and Applications (In Press).
•
Das H.C. and Parhi D.R., Online fuzzy logic crack detection of a cantilever beam,
International Journal of Knowledge-Based and Intelligent Engineering Systems, 12(2),
2008, 157-171.
•
Parhi D.R. and Das H.C. Smart crack detection of a beam using fuzzy logic controller,
International Journal of Computational Intelligence: Theory and Practice, 3(1), 2008, 9-
21.
•
Das H.C. and Parhi D.R., Detection of crack in cantilever structures using fuzzy-gaussian
interface technique, American Institute of Aeronautics and astronautics journal, 47(1),
2009, 105-115.
•
Das H.C. and Parhi D.R., Fuzzy-Neuro Controller for Smart Fault Detection of A Beam,
International Journal of Acoustics and Vibration, 13(2), 2009, 55-66.
•
Das H.C. and Parhi D.R., Application of Neural network for fault diagnosis of cracked
cantilever beam, IEEE International Symposium on Biologically Inspired Computing and
Applications (BICA-2009), Bhubaneswar, India, December 21-22, 2009, 353-358.
240
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242
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10 PAPERS IN PRESS:
Identification of Crack Location and Intensity in a Cracked Beam
by Fuzzy Reasoning
by Harish Das, Dayal Parhi
Abstract: In this paper a soft-computing-fuzzy-logic approach for
crack identification in cantilever beam has been considered. The fuzzy
controller consists of six input parameters and two output parameters
The input parameters are first three relative natural frequencies and
first three relative mode shape differences in dimensionless forms. Th
output parameters are relative crack location and relative crack depth
Theoretical analyses have been done including the effects of crack
depths and crack locations on natural frequencies and mode shapes.
Several fuzzy rules are outlined for the fuzzy controller. Gaussian
member ship functions are used for the fuzzy controller. The local
stiffnesses at crack location of the beam have been calculated using
strain energy release rate. The fuzzy rules are used to identify the
location and depth of the crack. Finally the effectiveness of the
developed fuzzy controller has been verified by results obtained from
the developed experimental setup.
Keywords: beam, vibration, crack, natural frequency, strain energy,
mode shape, fuzzy gaussian controller
The Design of H8 control methodology for Nonlinear Systems to
Guarantee the Tracking Behavior in the Sense of Input-Output
Spheres
by Shun-Min Wang
Abstract: A general nonlinear system usually contains some
245
Journal Article
Online fuzzy logic crack detection of a cantilever beam 6
Journal
International Journal of Knowledge-Based and Intelligent
Engineering Systems
Publisher
IOS Press
ISSN
1327-2314 (Print) 1875-8827 (Online)
Issue
Volume 12, Number 2 / 2008
Pages
157-171
Subject
Group
Computer & Communication Sciences
Pay-Per-View Copyright Statement
1.3.1.1.1
Authors
Harish Ch. Das1, Dayal R. Parhi2
1
Department of Mechanical Engineering, I.T.E.R, Bhubaneswar, Orissa, 751010, India
Department of Mechanical Engineering, N.I.T. Rourkela, Orissa, 769008, India
2
1.3.1.1.2
Abstract
Premature failure of beam structure is observed due to presence of crack. In the current analysis a
fuzzy inference system has been developed for detection of crack location and crack depth of a
cracked cantilever beam structure. The six input parameters to the fuzzy member ship functions 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. Strain energy release rate at the crack section of the beam has been used for
calculating its local stiffnesses. Different boundary conditions for the cracked beam structure are
outlined during theoretical analysis for deriving the vibration signatures (mode shapes and natural
frequencies). These signatures are subsequently used for deriving the fuzzy rules. Several fuzzy
rules are derived and the Fuzzy inference system has been designed accordingly. 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.
246
International Journal of Computational Intelligence: Theory and Practice, Vol 3, No
1 (2008)
Smart Crack Detection of a Beam Using Fuzzy Logic
Controller
Dayal R. Parhi, Harish Ch. Das
Abstract
Smart detection method to find out the fault in a cracked beam is addressed here. In
the present investigation a fuzzy logic approach is used. In the fuzzy inference
system six input parameters and two output parameters are used. The input
parameters to the fuzzy member ship functions are percentage deviation of first
three natural frequencies and first three mode shapes. The output parameters of the
fuzzy inference system are relative crack depth and relative crack location. For
deriving the fuzzy rules for natural frequencies, mode shapes, crack depths and
crack locations theoretical expressions have been developed. Strain energy release
rate has been used for calculating the local stiffnesses of the beam. The local
stiffnesses are dependent on the crack depth. Different boundary conditions are
outlined which take into account the crack location. Several fuzzy rules are derived
and the Fuzzy controller has been designed accordingly. An experimental setup has
been developed for verifying the effectiveness of the developed fuzzy controller. The
location and intensity of the crack can be predicted by the developed fuzzy inference
system in a close proximity to the actual results.
247
248
249
250
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