wave ode ecomposition using array ensors · except where due reference is made in the text of the...
TRANSCRIPT
Wave Mode Decomposition
using
Array Sensors
POSEARN SEO
This thesis is submitted in the fulfilment of the requirements for the degree of
Master of Engineering Science (Research)
November 2015
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Copyright Notice
© Posearn Seo 2015. Except as provided in the Copyright Act 1968, this thesis may not be
reproduced in any form without the written permission of the author.
I certify that I have made all reasonable efforts to secure copyright permissions for third-party
content included in this thesis and have not knowingly added copyright content to my work without
the owner’s permission.
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Abstract
Most physical structure undergoes wear and tear as a result of usage over time. As means to ensure
the safety of structures, regular inspections are a necessity to evaluate and maintain the integrity of
structures. However, this process is time consuming and requires disassembling of structures which is
economically inviable. Besides that, these conventional inspection techniques are restrained as some
areas that are out of reach has the potential to not be examined. Fortunately, technology
development has a resolution to detect deterioration which is known as Structural Health Monitoring
(SHM). SHM features an important role in the field of Non-Destructive Testing (NDT) and Non-
Destructive Evaluation (NDE).
SHM has introduced the exploitation of piezoelectric transducers that are permanently attached to
structures and Lamb wave propagation as a solution to a lot of issues encountered in this field of study.
This has led to the focus of this research where the combination of ultrasonic Lamb waves with the
continuous monitoring using piezoelectric sensors forms the basis of this study. The thesis is motivated
by the aim of the application of piezoelectric array sensors to locate the point of origin of the wave
and also its wave mode using post processing techniques. This produces a fundamental study of the
application of piezoelectric array sensors and Lamb waves in a SHM system. The research utilises
piezoelectric sensors that are very versatile and reliable as a tool to record information about the
propagating Lamb waves. The approach that was suggested in this thesis is to arrange the piezoelectric
sensors in a square matrix array so that more information can be gathered. Upon acquiring vital
information about the propagating Lamb waves, post processing techniques are then introduced into
the study so that a systematic approach in analysing these information can be carried out.
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Finite Element (FE) studies are used as an instrument to investigate the capabilities of the proposed
method. This thesis aims to propose a strategy or a systematic approach in analysing Lamb waves for
the identification of wave mode and its location of origin using array sensors. The analysis that is
proposed involves resolving two main goals which are locating the point of origin of the wave and also
revealing the modal content of the propagating Lamb wave. The post processing of the results
employs the use of 2D Fast Fourier Transform (2D FFT) coupled with propagation angles to help
achieve the aim of the study.
This thesis compose of validation models, strategy for analysis, FE simulations for the case studies and
signal processing techniques. The fundamental symmetric Lamb mode, S0 is used in the FE simulations
to model the propagation of Lamb waves. The gathered data from the array sensors formed by
piezoelectric transducers are post processed using 2D FFT that is carried out in multiple directions to
produce meaningful information from those acquired results. As such, frequency-wavenumber plots
are obtained and this is compared with the theoretical dispersion curves to reveal or identify the
propagating wave mode. Finally, in order to attain the second goal which is to locate the origin of the
wave source, propagation angles are introduced so that the data collected can be used to predict and
redirect the user back to the origin of the wave source. It was found that the predictions of the location
of the origin matches well with the actual location of the origin of the wave and thus, the accuracy of
those predictions are well justified.
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Declaration
I hereby declare that this thesis contains no material which has been accepted for the award of any
other degree or diploma in any university or equivalent institution and that, to the best of my
knowledge and belief, this thesis contains no material previously published or written by person,
except where due reference is made in the text of the thesis.
Signed: _________________________
POSEARN SEO
Date: _________________________
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Acknowledgements
I wish to thank and express my gratitude to Professor Wing Kong Chiu, for his counsel and guidance
throughout this tenure that provided me the chance to complete my research work on “Wave Mode
Decomposition Using Array Sensors”.
I would also like to thank my fellow peers as this project would not have seen light without their
assistance coupled with constant encouragement and feedback while making my experience more
gratifying.
My sincere appreciation goes to the Department of Mechanical Engineering of Monash University for
their scholarship funding which have granted me with this opportunity. I would also like to thank the
staff members of the Department who have always been helpful and resourceful.
I would like to express my heartfelt gratitude to my family and friends for their unconditional support
throughout these years of hard work. Their support has enabled me to keep persevering all the way
towards the completion of this research and thesis.
Last but not least, much appreciation to those that may have been directly or indirectly involved
towards the completion of this thesis.
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Table of Contents
Copyright Notice .......................................................................................................................... 1
Abstract ....................................................................................................................................... 2
Declaration .................................................................................................................................. 4
Acknowledgements...................................................................................................................... 5
List of Figures ............................................................................................................................... 9
List of Tables .............................................................................................................................. 13
1.0 Introduction ................................................................................................................... 14
2.0 Background & Literature Review ..................................................................................... 17
2.1 Structural Health Monitoring (SHM) ..................................................................................... 17
2.1.1 Introduction & Advantages of SHM .............................................................................. 17
2.1.2 Types and Techniques for SHM ..................................................................................... 19
2.2 Acoustic Emission (AE) .......................................................................................................... 22
2.3 Ultrasonic Testing ................................................................................................................. 22
2.3.1 Pitch Catch Method ...................................................................................................... 23
2.3.3 Pulse-Echo Method ....................................................................................................... 24
2.4 Acousto-Ultrasonic (AU) ....................................................................................................... 24
2.5 Piezoelectric .......................................................................................................................... 25
2.6 Lamb Waves .......................................................................................................................... 29
2.6.1 Mathematical Solution of Lamb Waves ........................................................................ 32
2.6.2 Lamb Waves Characteristics ......................................................................................... 35
2.7 Wave Scattering & Mode Conversion ................................................................................... 38
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2.8 Conclusion ............................................................................................................................. 41
3.0 Computational Modelling ................................................................................................ 43
3.1 Hanning Window................................................................................................................... 44
3.2 2D Fast Fourier Transform (2D FFT) ...................................................................................... 46
3.3 Finite Element (FE) Analysis .................................................................................................. 47
3.3.1 Simulation setup ........................................................................................................... 47
3.4 Validation Model ................................................................................................................... 53
3.4.1 Validation of S0 , A0 and S0A0 ......................................................................................... 54
3.4.2 Problem Identification .................................................................................................. 57
3.5 Conclusion ............................................................................................................................. 60
4.0 Mode Identification & Source Location ............................................................................ 62
4.1 Multiple Array Sensors .......................................................................................................... 62
4.2 Strategy proposed for the identification of wave mode and its location of origin .............. 63
4.2.1 Guide for Source Location Identification ...................................................................... 64
4.3 Conclusion ............................................................................................................................. 73
5.0 Case Studies using S0 Input .............................................................................................. 74
5.1 Source of Excitation from Region A ...................................................................................... 76
5.1.1 S0 Simulation along Column 8 ....................................................................................... 76
5.1.2 S0 Simulation along Column 32 ..................................................................................... 87
5.2 Source of Excitation from Region B ...................................................................................... 96
5.2.1 S0 Simulation along Row 16 .......................................................................................... 96
5.2.2 S0 Simulation along Row 25 ........................................................................................ 105
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5.3 Conclusion ........................................................................................................................... 114
6.0 Conclusion & Future Work ............................................................................................ 116
6.1 Conclusion ........................................................................................................................... 116
6.2 Future Work ........................................................................................................................ 119
7.0 References .................................................................................................................... 121
8.0 Appendices ................................................................................................................... 125
8.1 Example of ABAQUS Script .................................................................................................. 125
8.2 Example of MATLAB Script .................................................................................................. 143
8.2.1 Acquiring voltage-time signal from sensors ................................................................ 143
8.2.2 Extracting wave packet of interest ............................................................................. 146
8.2.3 2D Fast Fourier Transform .......................................................................................... 152
8.2.4 Guide for Source Location & Prediction...................................................................... 156
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List of Figures
Figure 1: Analogy between the human nervous system and a SHM structure [3] .......................................... 18
Figure 2: Types of SHM system (Passive monitoring) ..................................................................................... 20
Figure 3: Types of SHM system (Active monitoring) ....................................................................................... 20
Figure 4: Example of Wave Generation in Acoustic Emission (AE) [10] ........................................................... 22
Figure 5: Pitch Catch Method [10] .................................................................................................................. 23
Figure 6: Pulse Echo Method [10] .................................................................................................................. 24
Figure 7: Example of an application of Acousto-Ultrasonic (AU) [16] ............................................................. 25
Figure 8: Types of Piezoelectric Transducers [44, 45] ..................................................................................... 28
Figure 9: Waves in solids: a) Longitudinal (P) wave b) Shear vertical (SV) wave [20, 47] ............................... 29
Figure 10: Longitudinal and shear wave propagation [23] ............................................................................. 30
Figure 11: Waves Propagating in: (a) Antisymmetrical (b) Symmetrical [48] ................................................. 32
Figure 12: Thin plate geometry of thickness 2b used for solution of Lamb wave [9] ...................................... 32
Figure 13: Symmetric and Antisymmetric motion of Lamb waves across the plate [24] ................................. 35
Figure 14: Wave speed dispersion curves for Symmetric (S) and Antisymmetric (A) Lamb waves mode in an
aluminium plate (cs = shear wave speed, d = half thickness of the plate) [24] ...................................... 36
Figure 15: (a) Space-time map illustrating S0 mode propagation in 1mm thickness aluminium plate; Time
signal received (b) close to the excitation (c) 50mm from the excitation (d) 100mm from the excitation
[29] ........................................................................................................................................................ 37
Figure 16: 2D FFT displaying the ability to separate two different modes that are present [8] ...................... 38
Figure 17: Wave mode conversion in the phase velocity dispersion curves [37] ............................................ 41
Figure 18: a) Hanning windowed 5 cycle sine wave with centre frequency of 200 kHz b) FFT revealing the
spectral content of 200kHz .................................................................................................................... 45
Figure 19: a) Hanning windowed 5 cycle sine wave with centre frequency of 50 kHz b) FFT revealing the
spectral content of 50kHz ...................................................................................................................... 45
Figure 20: Example of Preliminary FE Model .................................................................................................. 50
Figure 21: a) Excitation to generate S0 mode b) Excitation to generate A0 mode c) Excitation to generate both
S0 & A0 mode ......................................................................................................................................... 51
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Figure 22: Voltage-Time response recorded by a piezoelectric sensor ........................................................... 52
Figure 23: Example of a dispersion curve ....................................................................................................... 52
Figure 24: Example of comparing 2D FFT results with dispersion curve after Stage 2 of Post Processing ....... 53
Figure 25: Validation FE Model ...................................................................................................................... 54
Figure 26: Validation of S0 wave .................................................................................................................... 55
Figure 27: Validation of A0 wave mode .......................................................................................................... 56
Figure 28: Validation of combination of S0 & A0 wave mode .......................................................................... 56
Figure 29: Two different point of origin ......................................................................................................... 57
Figure 30: Dispersion curve for Location A with the 2D FFT results superimposed ......................................... 58
Figure 31: Dispersion curve for Location B with the 2D FFT results superimposed indicating skewed results 58
Figure 32: Shifting the wavenumber, K values back to the dispersion curve .................................................. 60
Figure 33: Multiple Array Sensors .................................................................................................................. 63
Figure 34: S0 simulation used as guide for source location ............................................................................. 64
Figure 35: 2D FFT measured in the vertical direction ..................................................................................... 65
Figure 36: 2D FFT measured in the horizontal direction ................................................................................. 66
Figure 37: 2D FFT measured in the diagonal direction .................................................................................... 66
Figure 38: Wavenumbers obtained from measuring vertically indicating Region A ........................................ 68
Figure 39: Plot on dispersion curve to verify the mode of the propagating wave ........................................... 69
Figure 40: Representation of the prediction of the point of origin of the wave.............................................. 70
Figure 41: Relationship between the change in wavenumber and the propagation angle ............................. 72
Figure 42: Example of a voltage response recorded by the array sensors ...................................................... 74
Figure 43: Example of an incident wave packet extracted from its original response .................................... 75
Figure 44: Excitation along Column 8 from Region A ...................................................................................... 76
Figure 45: 2D FFT measured vertically with origin of wave located along Column 8 in the Region A .............. 77
Figure 46: 2D FFT measured horizontally with origin of wave located along Column 8 in the Region A ......... 77
Figure 47: 2D FFT measured diagonally with origin of wave located along Column 8 in the Region A ............ 78
Figure 48: Trend observation of wavenumber obtained when 2D FFT measured vertically which indicates
Region A ................................................................................................................................................ 79
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Figure 49: Wavefront reproduced using the voltage-time signal showing propagation along Column 8 from
Region A ................................................................................................................................................ 80
Figure 50: Wavefront reproduced using array sensors from Row 1 and 32 .................................................... 81
Figure 51: 2D FFT revealing the spectral component of the wave .................................................................. 82
Figure 52: Comparison between predicted and actual propagation angles of Column 8 in Region A ............. 84
Figure 53: Comparison between predicted and actual point of origin measured along Column 8 in Region A 86
Figure 54: Excitation along Column 32 from Region A .................................................................................... 87
Figure 55: 2D FFT measured vertically with origin of wave located along Column 32 in the Region A ............ 88
Figure 56: 2D FFT measured horizontally with origin of wave located along Column 32 in the Region A ....... 89
Figure 57: 2D FFT measured diagonally with origin of wave located along Column 32 in the Region A .......... 89
Figure 58: Trend observation of wavenumber obtained when 2D FFT measured vertically which indicates
Region A ................................................................................................................................................ 90
Figure 59: Wavefront reproduced using the voltage-time signal showing propagation along Column 32 from
Region A ................................................................................................................................................ 91
Figure 60: 2D FFT revealing the spectral component of the wave .................................................................. 92
Figure 61: Comparison between predicted and actual propagation angles of Column 32 in Region A............ 93
Figure 62: Magnification of the comparison in Row 16 and 32 ....................................................................... 94
Figure 63: Comparison between predicted and actual point of origin measured along Column 32 in Region A
.............................................................................................................................................................. 95
Figure 64: Excitation along Row 16 from Region B ......................................................................................... 96
Figure 65: 2D FFT measured vertically with origin of wave located along Row 16 in Region B ....................... 97
Figure 66: 2D FFT measured horizontally with origin of wave located along Row 16 in Region B ................... 98
Figure 67: 2D FFT measured diagonally with origin of wave located along Row 16 in Region B ..................... 98
Figure 68: Wavefront reproduced using the voltage-time signal showing propagation along Row 16 from
Region B ................................................................................................................................................ 99
Figure 69: Trend observation of wavenumber obtained when 2D FFT measured horizontally which indicates
Region B .............................................................................................................................................. 100
Figure 70: 2D FFT revealing the spectral component of the wave ................................................................ 101
Figure 71: Comparison between predicted and actual propagation angles of Row 16 in Region B ............... 102
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Figure 72: Comparison between predicted and actual point of origin measured along Row 16 in Region B . 104
Figure 73: Excitation along Row 25 from Region B ....................................................................................... 105
Figure 74: 2D FFT measured vertically with origin of wave located along Row 25 in Region B ..................... 106
Figure 75: 2D FFT measured horizontally with origin of wave located along Row 25 in Region B ................. 106
Figure 76: 2D FFT measured diagonally with origin of wave located along Row 25 in Region B ................... 107
Figure 77: Wavefront reproduced using the voltage-time signal showing propagation along Row 25 from
Region B .............................................................................................................................................. 108
Figure 78: Trend observation of wavenumber obtained when 2D FFT measured horizontally which indicates
Region B .............................................................................................................................................. 109
Figure 79: 2D FFT revealing the spectral component of the wave ................................................................ 110
Figure 80: Comparison between predicted and actual propagation angles of Row 25 in Region B ............... 111
Figure 81: Comparison between predicted and actual point of origin measured along Row 25 in Region B . 113
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List of Tables
Table 1: Estimated time saved by utilising SHM [3] ........................................................................................ 18
Table 2: Summary of SHM technologies [9] ................................................................................................... 21
Table 3: Modes of waves [23] ........................................................................................................................ 30
Table 4: Representation of terms ................................................................................................................... 46
Table 5: Material properties of Aluminium plate ........................................................................................... 48
Table 6: Predicted and actual propagation angles for Column 8 in Region A .................................................. 84
Table 7: Predicted and actual distances measured along Column 8 in Region A ............................................. 86
Table 8: Predicted and actual propagation angles for Column 32 in Region A ................................................ 93
Table 9: Predicted and actual distances measured along Column 32 in Region A ........................................... 95
Table 10: Predicted and actual propagation angles for Row 16 in Region B ................................................. 102
Table 11: Predicted and actual distances measured along Row 16 in Region B ............................................ 104
Table 12: Predicted and actual propagation angles for Row 25 in Region B ................................................. 111
Table 13: Predicted and actual distances measured along Row 25 in Region B ............................................ 113
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1.0 Introduction
Innovation in materials, science and engineering has impacted the growth of technology and the world
significantly. Besides the fact that it stimulates the economy, it also influences the standard of living
of human and the society. Unfortunately, these have led to fear regarding the impact of the
application of these innovative materials and the impact on the environment. As a result, smart
materials have been researched and developed to alleviate these issues while remaining economically
competitive.
From the development of smart materials, in recent years, the term Structural Health Monitoring
(SHM) has been increasingly popular and become integral in the field of Non-Destructive Testing (NDT)
or Non-Destructive Evaluation (NDE). Especially in the aviation industry, there are a significant amount
of aerospace structures or of similar sort that are being damaged due to cracks or damages that
remain undiscovered before a catastrophic incident occurs. [1] This leads to the need for continuous
monitoring and inspection of the mechanical structures to prevent these costly and undesirable
events from occurring. By implementing condition based monitoring and inspection on structures, the
non-operational time of structures can be reduced which also results in cost savings. SHM is a
technique that is showing great promise in the evaluation of structural integrity assessment as well as
in-situ structural inspection. The rising demand of this technology is allowing less human interaction
when it comes to constant monitoring of the structural integrity. There have been a variety of sensors
that were studied extensively such as fibre optics, accelerometers and piezoelectric elements in which
these sensors are being actively used for the purpose of SHM. Conventional inspection techniques
that are currently available are costly and ineffective, often requiring the structure that is being
monitored to be grounded and these maintenance checks are conducted periodically. SHM on the
other hand allows in-situ monitoring during flight and on ground. A variety of inspection techniques
are capable of enhancing SHM.
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With the evolution of using ultrasonic waves for on-board structural health monitoring (SHM), the
need for piezoelectric sensors are becoming increasingly important due to their low-cost, low-profile
and also the ease of integration within the structure. [2] This expanding field of SHM with piezoelectric
elements has attracted huge interest from both the industrial and scientific communities. The three
most significant sensing techniques using piezoelectric transducers are acoustic emission, acousto-
ultrasonics and electromechanical impedance. [3]
The application of piezoelectric sensors on structural health monitoring (SHM) relies on the magnitude
of the voltage generated by the sensor or the frequency bandwidth of the signal generated by the
sensor. [4] Therefore, the derivation of the voltage response of a piezoelectric sensor attached to a
plate when being subjected to an excitation input is important as those results are used for post
processing. Additionally, the voltage response of piezoelectric sensors being induced by plane wave
fields or circularly crested wave fields is also an area that is of consideration. The strain field generated
in the plate is the foundation in determining the voltage response of a piezoelectric sensor. [2]
There are several damage and health assessment techniques that are founded on ultrasonics. One of
the more sought after techniques is using guided waves, specifically Lamb waves, to examine plate-
like structures. In this research, a guided wave technique using Lamb waves is studied. Since the
propagation features of Lamb waves are dependent on the material properties of the medium, it is
now an effective inspection instrument to monitor changes in the structure due to damages.
Furthermore, Lamb waves are able to propagate for long distances, which increases their appeal in
long range structural monitoring and the inspection of bonded structures without these structures
having to be disassembled. [1] This study will use finite element analysis to simulate the propagation
of Lamb waves in thin plates. The effectiveness of using Lamb waves to monitor defects is well known
and has shown excellent agreement between numerical and experimental results. [5]
The issue that is being encountered at the moment is the identification of individual Lamb wave modes.
This is due to the dispersive and multimodal nature of Lamb waves. In other words, several different
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frequency components of Lamb waves travel at different speeds contribute to the distorted shape of
wave packets and hence, the data analysis is rendered more difficult [6, 7]. Some proposed the
method of frequency tuning as a solution to obtain a single mode. However, despite using a single-
mode excitation, the propagation does not remain as a single mode due to mode conversion as a
result of wave scattering from defects. [8]
A single point measurement is unable to distinguish the modes that are propagating on a plate-like
structure. The wave modes can only be determined with an array of sensors. However, given that the
orientation of the sensor array is fixed, the actual point of origin of the incident wave modes must be
determined. A 2D Fast Fourier Transform (2D FFT) technique carried out in multiple directions with
the sensor array can be used to identify the origin of the wave and its propagating wave mode. This
technique is discussed in detail in this thesis. From the Rayleigh-Lamb equation, it is certain that as a
wave propagates in a plate, it experiences dispersion. The presence of dispersion causes more wave
modes to appear and complicates the analysis of the incoming wave. As a result of this, the location
of the incoming wave is uncertain as each wave mode has different velocities and varies with the plate
thickness. This Rayleigh-Lamb equation is solved to obtain its real roots which are then used to plot
the dispersion curve. The dispersion curve is used as a guide to compensate for the dispersion of the
received waves. By superimposing the 2D FFT results with the dispersion curve, a reasonable
deduction with some level of tolerance can be made in terms of locating the source.
The aims of this research are:
To produce a fundamental study of the application of using piezoelectric array sensors and
Lamb waves in a SHM system.
To propose a strategy on the application of array sensors to locate the point of origin of the
wave and also its wave mode using post processing techniques such as 2D FFT.
This research may provide verification and confidence in the application of array sensors as a method
to identify Lamb wave mode and locate the origin of the wave in SHM systems.
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2.0 Background & Literature Review
2.1 Structural Health Monitoring (SHM)
2.1.1 Introduction & Advantages of SHM
Structural Health Monitoring (SHM) intends to give a diagnosis of the “state” of a structure during its
life cycle. With this diagnosis, SHM is a new and innovative way to perform Non-Destructive Evaluation
(NDE) and Non-Destructive Testing (NDT). SHM is beyond just that as it may even include the
integration of sensors into the structure. This method is frequently described as a “smart solution”
and is more attractive than the classical NDT methods. Classical NDT methods utilize externally applied
equipment to carry out inspections which often requires the structure to be disassembled and also
often taken out of service during the maintenance period. Therefore, on-going inspection using
traditional methods is not economically viable. [3]
Due to this, the economical motivation for structures with SHM systems is a lot more significant as the
benefits of SHM are envisioned to be constant maintenance costs and reliability regardless of the
lifetime of the structure. On the other hand, classical structures without SHM often incur increasing
maintenance cost and decreasing reliability with time. As a result of this, the main motivation for SHM
is that the economic impact it provides in terms of inspection cost. An evaluation was conducted on a
modern fighter aircraft that includes metallic and composite structures. This study reported that an
estimated 40% or more can be saved on inspection time by utilising SHM systems as shown in Table
1. [3]
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Table 1: Estimated time saved by utilising SHM [3]
Inspection Time Current inspection
time (% of total)
Estimated potential
for smart systems
Time saved
(% of total)
Flight line 16 0.40 6.5
Scheduled 31 0.45 14.0
Unscheduled 16 0.10 1.5
Service instructions 37 0.60 22.0
100 44.0
There are many analogies that were used to compare SHM structures with the human body such as in
Figure 1. One of the most common analogies is that the concept of SHM structures is similar to the
nervous system in human where it is being instrumented by sensors (nervous system) and equipped
with a central processor (brain).
Nervous
System
Brain
Central
Processor
Sensors
Figure 1: Analogy between the human nervous system and a SHM structure [3]
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2.1.2 Types and Techniques for SHM
There are various types of sensors that can be used to monitor structural health such as piezoelectric
transducers, electromagnetic sensors and fibre optic sensors to name a few. For each specific sensor,
several different methods exist which may vary based on the way of the sensor is used or the
characteristics of the damage. For instance, monitoring techniques utilises piezoelectric patches.
These monitoring techniques could vary from electromechanical impedance, acoustic emission,
propagation of ultrasonic waves such as Lamb waves and analysis of modal vibrations. [3]
Similar to any Non-Destructive Evaluation (NDE) system, a SHM system can either be passive or active.
An illustration to present the possible situations that may be involved is shown in Figure 2 & Figure 3.
The act of “passive monitoring” indicates that the structure is being monitored by the experimenter
using embedded sensors only. Acoustic emission is an example of a SHM method that utilises passive
monitoring. On the other hand, when a structure is equipped with both sensors and actuators, “active
monitoring” is then being applied. There will be two piezoelectric patches where one acts as an
actuator to generate perturbations or emit ultrasonic waves while the other acts as a detector. [3]
As aforementioned, there are a considerable variety of SHM techniques such as acoustic emission (AE),
acousto-ultrasonic (AU) and electromechanical impedance. A few important characteristics of a good
SHM technology are as below: [9]
High sensitivity to small defects
Minimal false positives and negatives
High probability of detection
Small footprint and weight
Large area of interrogation
The information in Table 2 summarises the advantages and disadvantages of various SHM
technologies.
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Experimenter
Information
Monitored Structure
Interaction Surrounding
Environment
Embedded
sensor
Figure 2: Types of SHM system (Passive monitoring)
Experimenter
Information
Interaction Surrounding
Environment
Monitored Structure
Embedded
actuator Action
Embedded
sensor
Figure 3: Types of SHM system (Active monitoring)
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Table 2: Summary of SHM technologies [9]
Advantages Disadvantages
Acoustic emission
Mature technology (tested in full scale)
Sensitive to crack growth (1mm detectable)
Accurate and simple triangulation
Any time spent offline may cause a missed defect
Does not give direct information on severity of defects
Susceptible to false positives
Impedance methods
Area monitored > footprint
Defect can be quantified
Baseline required
Not capable of defect localization
Comparative Vacuum
Monitoring
No false positives
Direct measurement
Non-electric
Highly sensitive to small cracks (0.2mm detectable)
Detection based on sensor orientation
Area of interrogation is equal to footprint (Highly localized)
Can only detect cracks on surface (Cannot detect cracks on other side)
Large devices used in CVM system
Tomography
Accurate location of defects
As sensitive as Lamb waves
Sensors can be low profile PZTs
Baseline required
Large amount of sensors required
Effectiveness relies on sensor placement and computational method
Lamb waves
Sensors can be low profile PZTs
Number of sensors can be tailored to application (1-2 for detection, 3 or more for triangulation)
Accurate location of defects
Fewer sensors compared to tomography
Damage can be quantified as well as detected
Can inspect beyond sensor’s footprint
Baseline often required
Current implementation is limited to idealised geometry
Complexity requires care in setting up system (Frequency selection, sensor location, noise floor measurement)
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2.2 Acoustic Emission (AE)
AE was discovered by Kaiser in the 1950s. AE is a passive NDE technique used for detection of impact
or damage that relies on physical phenomenon. A structure that is subjected to stress experiences
some deformation. These deformations in the material eventually forms flaw such as fracture, crack
growth, corrosion and creep. All of these are potential sources to generate elastic waves and hence,
acoustic emissions are produced. The most popular type of elastic waves used is Lamb waves which
contribute majorly to the field of SHM. Ultrasonic transducers are attached to the structures to detect
elastic waves and convert those to electrical signals. Data acquisition system are used to obtain these
electrical signals and then analysed further to locate the impact. [9, 10]
2.3 Ultrasonic Testing
Ultrasonic Testing or Inspection is a form of Non Destructive Testing (NDT) method. It is very useful as
it is very sensitive to both surface and subsurface discontinuities. Apart from that, it has a penetration
depth that is much more significant compared to other NDT methods. [11] The other benefits of
ultrasonic testing are that it is cheap, portable, non-hazardous and only one surface of a structure is
needed to be accessible. [10] However, ultrasonic testing still has its limitations. One of the many
would be that rough or irregular shaped materials are difficult to inspect.
AE
Stress
Wave
Applied
Stress Applied
Stress
Source
AE Signal
AE Sensor
Preamplifier
Detection and
Measurement
Electronics
Figure 4: Example of Wave Generation in Acoustic Emission (AE) [10]
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The procedure of ultrasonic testing involves the participation of experienced technicians and
transducers. [10] A common ultrasonic testing system may include tools such as a pulser & receiver,
transducer and some display devices. Ultrasonic testing utilises a transducer which is usually driven
by a pulser to generate high frequency ultrasonic energy. Sound energy is then introduced and
propagates through materials in the form of waves. [12]
In the event where a discontinuity (crack) is present, a portion of the energy is reflected back from the
flaw surface. This reflected signal is transformed into an electrical signal which will then be displayed
on the screen. [12]
Besides detecting crack, ultrasonic testing is commonly used to determine the shape, size and location
of defect as well. There are two different methods within ultrasonic testing which are Pitch Catch
method and Pulse-Echo method. [10]
2.3.1 Pitch Catch Method
As the name suggest, one contact transducer is responsible for pitching out waves while the other
contact transducer receives the signal that could either be transmitted or reflected from defects. The
active transducer is excited with reasonable amplitude and frequency coupled with suitable number
of excitation cycles. [10]
Ultrasonic
Tester
Contact Transducers
Defect
Figure 5: Pitch Catch Method [10]
24 | P a g e
2.3.3 Pulse-Echo Method
Unlike the Pitch Catch method, Pulse-Echo utilises only one contact transducer that is involved in
generating and detecting ultrasonic waves. As with any excitation transducer, appropriate amplitude,
frequency and number of excitation cycle has to be applied. In the event where defects are present,
waves would be reflected back and received by the transducer. [10] This is a very common NDT
method to detect disband, bond-line voids and porosity in adhesive joints. [13]
2.4 Acousto-Ultrasonic (AU)
Acousto-ultrasonic (AU) technique is a hybrid of acoustic emission monitoring and ultrasonic
characterisation. It is a non-destructive method of evaluating the damage state of a component by
analysing the received simulated stress waves to detect and map variations of mechanical material
properties. [14] There will be both the transmitting transducer and receiving transducer. The
transmitting transducer is positioned relative to the receiving transducer such that the excited stress
waves travel along the principal loading direction of the material interacting with deformations along
the path. The signals detected by the receiving transducer is analysed in the frequency domain to
calculate various AU parameters. [15] AU provides benefits such as being able to quickly assess the
combined effects of various sub-critical flaws. An example shown in Figure 7 is of a recent application
of AU used to detect micro-corrosion between riveted plates. [16]
Ultrasonic
Tester
Contact Transducer
Defect
Figure 6: Pulse Echo Method [10]
25 | P a g e
It was shown that Lamb waves are generated in AU measurement and two different modes, namely,
the lowest symmetrical (S0) and asymmetrical (A0) modes are usually excited in AU inspection. [17]
2.5 Piezoelectric
There has been a wide range of types of transducers that are used for SHM such as strain gauges,
accelerometers, piezoceramics and etc. Piezoelectric materials are particularly useful for aircraft SHM
systems as they are slim, unobtrusive and readily-integrated to structures. The development of
piezoelectric transducers as elements of intelligent structures with Lamb waves has been broadly
researched by various researchers. [8, 18, 19] However, not all materials are piezoelectric materials.
The piezoelectric effect is displayed by a specific group of crystalline solid materials whose unit cells
do not possess a center of symmetry. [19] For instance, materials with unaligned dipoles are not
piezoelectric. Poling can be achieved by applying a strong electric field to align the dipoles in the
material at an elevated temperature below the Curie temperature. [4]
Figure 7: Example of an application of Acousto-Ultrasonic (AU) [16]
26 | P a g e
Piezoelectric principle operates on the basis of generating electrical energy from mechanical energy
and vice versa. The direct piezoelectric effect is defined as the amount of electric field produced by a
given mechanical stress. This direct effect is utilised in piezoelectric sensors. Alternatively, the
converse piezoelectric effect is defined as the amount of mechanical strain generated by a given
electric field. This effect is useful in piezoelectric actuators. [4]
Piezoelectric are govern by these two constitutive equations where Equation 1 signifies Direct
Piezoelectric Effect while the Equation 2 is for Converse Piezoelectric Effect. [4] These general
constitutive equations of linear piezoelectric materials as supplied by the ANSI/IEE Standard 176-1987
explain the tonsorial relationship between mechanical and electric variables. [18]
Equation 1
𝐷 = 𝑒𝜎𝐸 + 𝑑𝜎
Equation 2
𝜀 = 𝑠𝐸𝜎 + 𝑑′𝐸
D = Electric Displacement Field (𝐶ℎ𝑎𝑟𝑔𝑒
𝐴𝑟𝑒𝑎)
eσ = Permittivity @ constant stress
E = Electric Field (𝑉𝑜𝑙𝑡𝑎𝑔𝑒
𝐿𝑒𝑛𝑔𝑡ℎ)
d = Strain Constant
σ = Stress
ɛ = Strain
SE = Elastic Compliance @ constant electric field
27 | P a g e
A piezoelectric sheet can be treated as a parallel capacitor. Ultimately, as the piezoelectric material is
being induced by stress or strain, it produces a voltage. The dimension of the piezoelectric material is
directly related to the capacitance of the material. The capacitance of a piezoelectric material can be
described as;
𝐶 =𝑒33
𝜎𝐴𝑐
𝑡𝑐
𝐶 =𝑒33
𝜎𝑙𝑐𝑏𝑐
𝑡𝑐
Where e33σ is the permittivity while lc, bc and tc are the length, width and thickness of the sensor
respectively.
The capacitance can be then related to the Voltage generated by Q = CV.
𝑄 = 𝑑31𝑌𝑐𝑏𝑐𝑙𝑐𝜀
𝑄 = 𝑑31𝑌𝑐 ∬(𝜀11 + 𝜀22
𝑙𝑐,𝑏𝑐
) 𝑑𝑥 𝑑𝑦
By combining the equations,
𝑑31𝑌𝑐 ∬(𝜀11 + 𝜀22
𝑙𝑐,𝑏𝑐
) 𝑑𝑥 𝑑𝑦 =𝑒33
𝜎𝑙𝑐𝑏𝑐
𝑡𝑐 𝑉𝑐
𝑉𝑐 = (𝑑31𝑌𝑐𝑡𝑐
𝑒33𝜎𝑙𝑐𝑏𝑐
) ∬(𝜀11 + 𝜀22
𝑙𝑐,𝑏𝑐
) 𝑑𝑥 𝑑𝑦
Equation 3
𝑉𝑐 =𝑑31. 𝑡𝑐 . 𝑌𝑐
𝑒33𝜎
.𝑙𝑒𝑓𝑓
𝑙𝑐.𝑏𝑒𝑓𝑓
𝑏𝑐. (𝜀11 + 𝜀22)
28 | P a g e
Therefore, the relationship between the strain experienced by a piezoelectric sensor and the voltage
output of the piezoelectric sensor can be governed by Equation 3.
There are several different types of PZT which varies in terms of shape, properties and its application.
The two common ones are the tube and strip type of PZT as shown in Figure 8.
A standard piezoelectric transducer has an allowable field strength ranging from 1 to2kV/mm in the
poling direction and up to 300V/mm inverse to the poling direction. All these are dependent on the
ceramic properties and the insulating materials. If the maximum voltage is exceeded, the PZT may
experience dielectric breakdown and irreversible damage. For instance, in a high stress environment,
the performance of PZT changes according to its properties where the PZT may suffer changes in
permittivity and piezoelectric constant.
Figure 8: Types of Piezoelectric Transducers [44, 45]
29 | P a g e
2.6 Lamb Waves
Lamb waves are made up of two wave components namely longitudinal (P) waves and shear vertical
(SV) waves. For the longitudinal (P) waves, the particle motion is in the direction of propagation while
the shear vertical (SV) waves moves in sort of a direction perpendicular to the direction of propagation.
[20] The particle motion of both waves is depicted in Figure 9.
There have been various interests and studies in using Lamb waves on NDT plates. It has been
demonstrated that Lamb waves technique may be used for long range NDT application such as large
plate-like aircraft structures. [21, 22]Depending on the way the particles oscillate in solids, several
different types of wave propagation may exist. [23] The modes of ultrasonic waves that can possibly
be present and propagate in solids are summarised in Table 3.
Figure 9: Waves in solids: a) Longitudinal (P) wave b) Shear vertical (SV) wave [20, 47]
30 | P a g e
Table 3: Modes of waves [23]
Type of Modes Direction of Propagation
Compression/Longitudinal Parallel to the direction of propagation
Transverse/Shear/Distortional Wave Perpendicular to the direction
Surface - Rayleigh Wave Elliptical orbit – Symmetrical mode
Plate - Lamb Wave Perpendicular to surface
Plate - Love Wave Parallel to plane layer, perpendicular to wave
direction
Stoneley (Leaky Rayleigh Waves) Wave guided along interface
Fundamental modes such as longitudinal and shear waves propagation are most commonly used in
ultrasonic testing. As can be seen from Table 3, the longitudinal waves oscillate parallel to the
direction of wave propagation while transverse waves oscillate at a right angle to the direction of
propagation. Transverse waves are relatively weaker compared to longitudinal waves as they require
an acoustically solid material for effective propagation. Rayleigh and Lamb waves are also useful for
ultrasonic inspection. Rayleigh waves are able to penetrate a reasonably thick solid material up to a
depth of one wavelength as it travels along the surface. [23]
Direction of
wave propagation
Shear wave
Particles at rest
position
Longitudinal wave Direction of particle motion
Direction of
wave propagation
λ
λ
λ
Direction of particle motion
Figure 10: Longitudinal and shear wave propagation [23]
31 | P a g e
Lamb waves, also known as plate waves are an extension of Rayleigh waves. It is a complicated
vibrational wave that is able to propagate through the entire thickness of a material. [23] Lamb waves
are a type of ultrasonic waves that remains guided between two parallel free surfaces. [24] As its name
suggest, the plate acts as a guide that leads them along the structure. Lamb waves have displacements
occurring in the direction of the wave propagation and out of plane. The propagation mechanism is
fairly complicated and includes elliptical particle motion. Lamb waves are dispersive and minor
disruptions can cause complicated outputs. Lamb waves are one of the most widely used guided waves
for damage detection. [9] The benefits of using Lamb waves are abundance. A few key benefits of
Lamb waves are such as its capabilities in damage detection and the other advantages or
disadvantages of it are summarised in Table 2.
Lamb waves are also known as guided elastic waves that are able to propagate in the plane of a plate
over long distances. [25] This advantageous characteristic makes them suitable for the purpose of NDE
& NDT especially in hard to inspect regions. However, due to its highly dispersive nature and the ability
to generate multiple modes, Lamb waves are described as challenging. Difficulties when dealing with
Lamb waves arise in ensuring the appropriate waves are generated as they are highly dependent on
the material properties and frequency. Wave propagation in a material is complex too as the waves
interact in a different manner with the diverse type of interferences within the material. [23]
Lamb waves can be generated in a plate with free boundaries for both the symmetric and anti-
symmetric mode within the plate. The symmetric modes can also be referred to as longitudinal modes
because the average displacement over the thickness of the plate is in the longitudinal direction. On
the other hand, the anti-symmetric modes are called flexural modes due to the average displacement
in the transverse direction. An example on the movement of the symmetrical and antisymmetrical
waves is shown in Figure 11.
32 | P a g e
2.6.1 Mathematical Solution of Lamb Waves
Lamb waves were initially discovered by an English mathematician, Sir Horace Lamb. Ever since then,
many others have been investigating Lamb waves and even produce a thorough analysis on it. [26, 27]
The complete Lamb wave theory is fully documented in a number of textbooks. [27] The beginning of
the whole analysis is from the wave equations [24]
Equation 4
𝜕2∅
𝜕𝑥2+
𝜕2∅
𝜕𝑦2+
𝜔2
𝑐𝑃2 ∅ = 0
𝜕2𝛹
𝜕𝑥2+
𝜕2𝛹
𝜕𝑦2+
𝜔2
𝑐𝑆2 𝛹 = 0
Where φ and Ψ are two potential functions, 𝑐𝑃2 = (𝜆 + 2𝜇)/𝜌 and 𝑐𝑆
2 = 𝜇/𝜌 are the pressure
(longitudinal) and shear (transverse) wavespeeds, λ and µ are the Lame constants and ρ is the mass
density.
(a)
(b)
Figure 11: Waves Propagating in: (a) Antisymmetrical (b) Symmetrical [48]
Figure 12: Thin plate geometry of thickness 2b used for solution of Lamb wave [9]
2b x
y
33 | P a g e
The boundary conditions for Lamb waves are defined as
𝜏𝑦𝑦 = 𝜏𝑧𝑦 = 𝜏𝑥𝑦 = 0 when 𝑦 = ±𝑏
Where b is half of the thickness of the plate in Figure 12 and τyy, τzy, τxy are the stresses in its respective
directions.
The time dependence is assumed to be harmonic in nature and is in the form of e-iωt. This yields the
solution for the potentials from Equation 4 to be in the form of
Equation 5
∅ = (𝐴1 sin 𝑝𝑦 + 𝐴2 cos 𝑝𝑦)𝑒𝑖(𝜉𝑥−𝜔𝑡)
𝜓 = (𝐵1 sin 𝑞𝑦 + 𝐵2 cos 𝑞𝑦)𝑒𝑖(𝜉𝑥−𝜔𝑡)
Where 𝜉 = 𝜔 𝑐⁄ is the wave number and
Equation 6
𝑝2 =𝜔2
𝑐𝐿2 − 𝜉2
𝑞2 =𝜔2
𝑐𝐿2 − 𝜉2
The integration constants, A1, A2, B1 & B2 are obtained from the boundary conditions. The potential
functions have a relationship between the displacements, stresses and strains.
𝑢𝑥 =𝜕∅
𝜕𝑥+
𝜕𝜓
𝜕𝑦
𝑢𝑦 =𝜕∅
𝜕𝑦−
𝜕𝜓
𝜕𝑥
𝜀𝑥 =𝜕𝑢𝑥
𝜕𝑥
𝜏𝑦𝑥 = 𝜇(𝜕2∅
𝜕𝑥𝜕𝑦−
𝜕2𝜓
𝜕𝑥2+
𝜕2𝜓
𝜕𝑦2)
𝜏𝑦𝑦 = 𝜆 (𝜕2∅
𝜕𝑥2+
𝜕2∅
𝜕𝑦2) + 2𝜇(𝜕2∅
𝜕𝑥2−
𝜕2𝜓
𝜕𝑥𝜕𝑦)
34 | P a g e
With those relations, we will obtain
Equation 7
𝑢𝑥 = [(𝐴2𝑖𝜉 cos 𝑝𝑦 + 𝐵1𝑞 cos 𝑞𝑦) + (𝐴1𝑖𝜉 sin 𝑝𝑦 − 𝐵2𝑞 sin 𝑞𝑦)𝑒𝑖(𝜉𝑥−𝜔𝑡)
𝑢𝑦 = [−(𝐴2𝑝 sin 𝑝𝑦 + 𝐵1𝑖𝜉 sin 𝑞𝑦) + (𝐴1𝑝 cos 𝑝𝑦 − 𝐵2𝑖𝜉 cos 𝑞𝑦)𝑒𝑖(𝜉𝑥−𝜔𝑡)
The terms in Equation 7 have been divided into two parts where the first part represents the
symmetric motion and the second corresponds to the antisymmetric motion.
In a free wave motion situation, the homogenous solution is derived by applying the stress-free
boundary conditions at both the upper and lower surfaces as defined earlier. By solving it, the
characteristic equations for both the symmetric and antisymmetric motion can be obtained. The
characteristic equations obtained are [24]
Equation 8
𝐷𝑆 = (𝜉2 − 𝑞2)2 cos 𝑝𝑑 sin 𝑞𝑑 + 4𝜉2𝑝𝑞 sin 𝑝𝑑 cos 𝑞𝑑 = 0
𝐷𝐴 = (𝜉2 − 𝑞2)2 sin 𝑝𝑑 cos 𝑞𝑑 + 4𝜉2𝑝𝑞 cos 𝑝𝑑 sin 𝑞𝑑 = 0
Both the symmetric and antisymmetric equations in Equation 8 can be rewritten in a more compact
form which is commonly known as the Rayleigh-Lamb equation as in Equation 9. [24]
Equation 9
tan 𝑝𝑑
tan 𝑞𝑑= − [
4𝜉2𝑝𝑞
(𝜉2 − 𝑞2)2]
±1
Where +1 corresponds to symmetric motion and -1 to antisymmetric motion (Figure 13).
35 | P a g e
2.6.2 Lamb Waves Characteristics
The complexity of Lamb wave propagation in monitored structures has caused difficulty in analysing
and interpreting damage detection results. Dispersion characteristics of Lamb waves of a material can
be represented by solving the frequency relations equation iteratively. Apart from that, the phase and
group velocities of Lamb waves are also commonly plotted against the frequency thickness product.
An example of that is illustrated in Figure 14. This dispersion characteristic of Lamb waves is able to
produce an infinite number of symmetric (Si) and anti-symmetrical (Ai) Lamb wave modes where i = 0,
1, 2,…. Among all the possible modes, the fundamental modes namely; S0 and A0 modes are the most
widely used mode for damage detection. This is due to the fact that they are easier to be generated
and it produces the largest amplitudes in most situations. [9]
The phase velocity is a fundamental characteristic of Lamb waves. From a known phase velocity, the
wave number, stresses and displacements of the Lamb wave at any location in the plate can be
determined. In addition to that, there is another kind of velocity known as the group velocity. It is the
speed of the guided wave packet at a particular frequency. Whereas, phase velocity is the speed at
which the individual wave peaks within the travelling wave packet. The group velocity and the phase
velocity can be expressed as below. [28]
𝑉𝑔𝑟𝑜𝑢𝑝 = 𝑉𝑝ℎ𝑎𝑠𝑒 + 𝑘𝜕𝑉𝑝ℎ𝑎𝑠𝑒
𝜕𝑘
Uy
Uy ux
Symmetric motion Anti-symmetric motion
x
y
ux
Uy
Uy ux
ux
Figure 13: Symmetric and Antisymmetric motion of Lamb waves across the plate [24]
36 | P a g e
Wave dispersion is produced as the wave speed changes with frequency. The speed of Lamb waves in
a material is dependent on the product between the frequency, f and the plate thickness, d. Each
frequency thickness product, fd for each solution of the Rayleigh-Lamb equation corresponds to a
Lamb wave speed and wave mode. [24] The plot of the Lamb wave speeds against the fd product yields
the wave speed dispersion curves as in Figure 14.
Dispersion curves of the different modes are required for the study of Lamb waves. Extensive research
has been carried out on the dispersive behaviour of Lamb waves. The dispersion characteristic of Lamb
wave packets increases the complexity of using Lamb waves. It was evident that the effects of
dispersion are more prominent as the distance between the point of origin of the wave and sensor
increases. The dispersion of waves causes wave packets to spread out both in time as well as in space
while propagating through a structure. [29] This phenomenon is as depicted in Figure 15 where it is
obvious that the amplitude of the signals decreases further from the origin of the wave due to the
attenuation of the energy of the signal. On the other hand, the width of the wave packet increases
which indicates that the response signal is dispersed.
Figure 14: Wave speed dispersion curves for Symmetric (S) and Antisymmetric (A) Lamb waves mode in an aluminium plate
(cs = shear wave speed, d = half thickness of the plate) [24]
37 | P a g e
Frequency-Wavenumber representation is another technique to illustrate dispersion curves. This
method offers an alternative way to measure dispersion curves of Lamb waves as it enhances the issue
of multi modes and dispersion by converting the amplitude-time frequency response to amplitude-
wavenumber at specific frequencies. The result of this is that it resolves every single Lamb wave mode
with their amplitudes. [1]
It is evident that as the frequency of excitation increases, there will be a significant increase in the
possible modes that can be excited. Therefore, it is common for engineering applications to operate
at lower frequency regions so that only the fundamental S0 and Ao modes can be excited. However,
despite those efforts, both the fundamental modes can eventually cause multiple reflections and even
mode conversions in engineering structures that are complex. [9]
Figure 15: (a) Space-time map illustrating S0 mode propagation in 1mm thickness aluminium plate;
Time signal received (b) close to the excitation (c) 50mm from the excitation (d) 100mm from the excitation [29]
38 | P a g e
As stated previously, one of the issues with Lamb waves is the existence of multiple modes at any
given frequency which makes it difficult to differentiate each mode for analysis. These modes will be
present on different spatial wavelengths but they may all exist on the same frequency. A standard
Fourier transform will be unable to separate the amplitudes of the modes clearly. Hence, Cawley et al
developed a method of utilising a two-dimensional Fourier transform (2D FFT) to measure propagating
multimode signals. This technique has shown that it is a viable technique for measuring amplitudes of
Lamb wave modes. [8] Despite having the same temporal frequency, with the presence of the spatial
frequency axis, the 2D FFT method is able to distinguish the modes as depicted in Figure 16.
2.7 Wave Scattering & Mode Conversion
The phenomenon often known as scattering or diffraction arise when incident waves that are
propagating through an infinite medium encounter defects, cavities, inclusions which could be due to
a flaw in the material or cracks which are commonly represented by a slit or wedge. These waves are
then propagated back in the plate. The interaction of elastic waves with discontinuities or boundaries
of more complex shape are interesting in the field of SHM. [27]
Figure 16: 2D FFT displaying the ability to separate two different modes that are present [8]
39 | P a g e
Wave scattering are dependent on the different locations and severity of the damages as these
produces unique scattering phenomena. Wave scattering problems in plate like structures have been
getting a lot of attention and studies have been carried out on it. Some of the studies that involves
wave scattering includes observing the scattering field in plates with a single damage or multiple
damages. There has also been studies which looked at scattering problems with different kinds of
defects such as scattering problem on an incident plane Lamb wave in plates with a circular partly
through-thickness hole or guide wave scattering from non-symmetric blind holes in isotropic plate.
[30, 31]
Apart from wave scattering, the interaction of Lamb waves with structural damage can significantly
influence the propagation properties such as mode conversion. In addition, with the presence of
multiple damages, secondary scattering of guided waves will take place. [30] These several
complications cause the theoretical analysis of the received signals difficult to interpret and analyse.
Most SHM applications utilise only the S0 and A0 modes to ensure that the wave propagation patterns
are maintained and the data analysis is simple. These fundamental modes are excited by choosing
frequencies below a critical value to ascertain that only the S0 and A0 modes exist. Despite having a
single mode for inspection, several other modes may still be produced due to mode conversion when
there is interaction with structural features namely, boundaries, notches, stiffeners and thickness
changes. [32]
These issues occur with the presence of single damage (crack, hole, delamination and etc), multiple
damages, which are all visible from surface. However, there are other issues which are caused by
corrosion or even initial defect in plate structures that are invisible from the surface and these provide
a concerning interest in the interaction of guided Lamb waves with these damages. [30]
40 | P a g e
In addition, wave scattering problems can be more complicated in the presence of a varying plate
thickness compared to a single plate with uniform thickness. Wave scattering occurs on both the
incident mode as well as all the other propagating modes that may exist at a particular frequency
through mutual interference. [33]
When a certain form of wave energy travels in a solid material, it can be transformed into another
form. For instance, if a longitudinal wave strikes an interface at an angle, due to the energy from the
wave causing particle movement in the transverse direction, shear (transverse) waves may be present.
In other words, mode conversion may occur when a wave encounters an interface between materials
of different acoustic impedances and the incident angle is not normal to the interface. Ultrasonic
waves can be confusing and difficult to interpret at times, as a result of mode conversion occurring
every time a wave interacts with an interface at an angle. [11]
In Lamb wave scattering, mode conversion redistributes the energy among multi modes of Lamb
waves. This causes the distortion of the received modes from the incident modes. Mode conversion
is often observed in the form of frequency and phase velocity shift. By analysing the characteristics of
Lamb wave mode conversion, there is a possibility to size the defects for NDT of plates. [34]
Besides Lamb waves, there is a possibility of other additional waves being mode converted when the
incident waves encounters a hole. These Rayleigh-like waves were introduced and called
“circumferential creeping waves”. [35] These waves re-radiate some of the energy back into the
structure and the re-radiated field can be regarded as a weak scattering perturbation. Furthermore,
the waves are able to be locate and size cracks. The technique that is used for crack sizing is by utilising
the amplitude of the circumferential waves. As for the location of the crack, it is done by calculating
the time delays between a perfect hole and its backscattered duplicate. The time delay indicates the
distance between the defect and the receiver, provided that the size of the hole is known. [36] These
waves have been researched and studied in relation to incident shear waves; however, this thesis will
not be addressing these waves.
41 | P a g e
Wave scattering has different mode conversion phenomena with respect to frequency, thickness and
reflector shape. As a result, the shape variation of the plate could cause unique scattering at a certain
frequency. In the event where the plate thickness is fixed, the mode conversion occurs only in the
vertical direction at its specific fd value. On the other hand, when the thickness of the plate varies,
this affects both the vertical and horizontal mode conversion on the dispersion curve. A simple
representation of these conversions is depicted in Figure 17. [37]
2.8 Conclusion
In Chapter 2, a comprehensive literature review of the work related to this research was presented.
The broader view of the research which is Structural Health monitoring (SHM) was first discussed. The
various techniques, advantages and disadvantages of the wide range of SHM technologies were also
looked into. Some of the techniques that are more applicable in this research were examined further
and soon the focus was shifted towards using ultrasonic waves specifically Lamb waves.
Figure 17: Wave mode conversion in the phase velocity dispersion curves [37]
42 | P a g e
The theory of Lamb waves were presented in detail and a few key areas of Lamb waves were discussed.
Those areas include the mathematical solution of Lamb waves and the characteristics of Lamb waves.
Additionally, some common properties or phenomenon of Lamb waves such as wave scattering and
mode conversion were investigated and all the information about it were included in this chapter.
One of the important element in this research is the usage of piezoelectric transducers and its
usefulness in the field of SHM. Therefore, the derivation of the formula that governs the piezoelectric
effect which enables them to be utilised in this research was also showed in this chapter. This is vital
as this key information is the bridge that ties everything together which enables Lamb waves to be
used as a tool for NDT. The combination of using Lamb waves and piezoelectric transducers has
broaden the development of NDT and NDE in the field of SHM.
The next chapter will look into applying the knowledge and theory from Chapter 2 with the help of
computational modelling to simulate Lamb wave propagation on a plate-like structure. By learning
more about the literature of this research it enables a logical or systematic approach to be set up in
computational modelling for the application of Lamb wave decomposition in mode identification and
source location using array of sensors. In accordance to the aim listed in Chapter 1 coupled with the
knowledge acquired in Chapter 2, it provides the fundamental basis for this research. The work will
focus on how one can use an array of sensors to locate the point of origin of the wave and the
identification of its wave mode can be achieved. The focus of the research is to propose a suitable
methodology to synthesize the signals using a stationary array of sensors to identify the location of
the origin of the incident wave and the type of wave mode.
43 | P a g e
3.0 Computational Modelling
This chapter discusses the numerical method and computational modelling used in this thesis for the
application of Lamb waves in decomposing wave modes. Prior to this, there have been studies on
simulating ultrasonic waves in solids for NDT purposes and wave propagation studies using analytical
methods. [38] These processes are important as it enables the prediction of propagation of Lamb
waves prior to manufacturing and fabrication. It is vital to be able to predict the outcomes of
propagation of Lamb waves for developmental purposes of Structural Health Monitoring. Previously,
in Chapter 2 on literature review, it was discussed that the implementation of guided Lamb waves for
in-situ SHM utilizes piezoelectric transducers. The advantages of using piezoelectric elements as
sensors for SHM are that they require a small power input and the size can be really small which allows
it to be distributed across a structure. Fourier analysis is carried out on the data acquired from the
Lamb waves to predict the modal content of the waves. In the initial stages of this research, there
were several tasks and aims that were identified. A simulation model was first set up to conduct some
Finite Element (FE) Modelling. The FE Modelling is also used for validation purposes. The validation
model consists of different Lamb wave modes being excited separately. This is to ensure that each of
the Lamb wave modes is able to be excited. Subsequently, a linear array made up of 32 sensors is lined
up in the middle of the specimen which is the Aluminium plate. Two different excitation points were
chosen and the response from all the 32 piezoelectric sensors were analysed. This provides a good
understanding on the propagation of Lamb waves in plate-like structures. In addition to that,
Aluminium plate with multiple sensor arrays was also being modelled. This enables more information
to be gathered from the various sensor arrays which results in the proposition of a strategy to
synthesise Lamb wave propagation in the medium. The final arrangement of the sensors was decided
to be a square matrix array comprises of 32 sensors by 32 sensors.
44 | P a g e
3.1 Hanning Window
Hanning window was named after Julius von Hann. It is an apodization function or a tapering function
which is being used to smooth the edges of the sampled region down to zero. [39] Usually, the signal
within the temporal and spatial sampling window will not be periodic and hence, leakages will occur.
The Hann function is able to reduce the leakages or aliasing by creating a “window” for Fourier
transforms filtering. Apart from that, zeros are also padded at the end of the signal so that the
frequency and wavenumber of the maximum amplitude can be determined more accurately. [8]
Hanning window is governed by Equation 10 expressed below.
Equation 10
𝜔(𝑛) = 0.5 [1 − cos (2𝜋𝑛
𝑁 − 1)]
Hanning window is used in the modelling of the amplitude of the excitation signal. Hanning window is
applied onto the sinusoidal wave that was used as the excitation amplitude for the Finite Element (FE)
analysis.
A Hanning windowed 5 cycle sine wave was selected for the excitation of Lamb waves. The smooth
Hanning window function is ideal as the absence of sudden changes in wave packet will allow the the
targeted frequency to be identified without issues such as leakages. Due to the need to satisfy the
stability criterion and ensure accurate FE simulation, two different centre frequencies are used for
different type of excitation. Examples of the Hanning windowed excitation that is used in this
reasearch are depicted in Figure 18 and Figure 19.
45 | P a g e
Figure 19: a) Hanning windowed 5 cycle sine wave with centre frequency of 50 kHz
b) FFT revealing the spectral content of 50kHz
a)
-1
-0.5
0
0.5
1
0 0.00005 0.0001 0.00015 0.0002
0
0.05
0.1
0.15
0.2
0.25
0.3
0 50 100 150 200 250
Am
plit
ud
e
Frequency (kHz)
b)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 200 400 600 800 1000
Am
plit
ud
e
Frequency (kHz)-1
-0.5
0
0.5
1
0 0.00001 0.00002 0.00003 0.00004
Am
plit
ud
e
Time (s)
a) b)
Figure 18: a) Hanning windowed 5 cycle sine wave with centre frequency of 200 kHz
b) FFT revealing the spectral content of 200kHz
46 | P a g e
3.2 2D Fast Fourier Transform (2D FFT)
In the event where a Lamb wave propagates through the plate along the x-direction, the displacement
on the surface of the plate can be represented by Equation 11 below.
Equation 11
𝑢(𝑥, 𝑡) = 𝐴(𝜔)𝑒𝑗(𝜔𝑡−𝑘𝑥−𝜃)
The Fourier analysis is a commonly used technique for engineering analysis in the frequency domain.
This method is widely used in this entire thesis when analysing Lamb waves in the frequency domain.
The Fourier transform of the spatial coordinate, x transforms the spatial domain to wavenumber while
the Fourier transform of the time domain, t converts the time domain to frequency domain. For the
Fourier transform of the spatial coordinate, x, it is wise to obtain a sample series of equally spaced
points along the direction of the propagating Lamb wave. A larger amount of points sampled will yield
a better spatial resolution. By combining both of the individual Fourier transformation, a two
dimensional Fourier Transform of the formula above can be generated. [40] It is represented by
Equation 12 below.
Equation 12
𝐻(𝑘, 𝑓) = ∬ 𝑢(𝑥, 𝑡)𝑒−𝑗(2𝜋𝑓𝑡+𝑘𝑥)𝑑𝑥𝑑𝑡
Each of the representation of the terms is listed in Table 4.
Table 4: Representation of terms
A(ω) Amplitude constant associated with the frequency
k Wavenumber or spatial frequency
x Coordinate on the direction along the wave propagation
ω = 2πf Angular frequency or temporal frequency
θ Initial phase
47 | P a g e
Therefore, Two Dimensional Fast Fourier Transform was implemented on propagating Lamb waves to
analyse both the temporal and spatial frequency. [8] Every Lamb wave mode has its own wavenumber,
k and frequency, f that corresponds to a different H(k,f) where it is a three-dimensional graph. By
projecting the contours of H(k,f) onto the frequency-wavenumber plane, these resulting contours can
be used to compare with the analytical curves to distinguish the different wave modes. This
comparison can be observed in the later sections of this thesis. This is possible because different Lamb
wave has different frequency-wavenumber curve. By revealing the modal content in Lamb waves, the
information obtained can be used for amplitude comparison to detect damages, determining
excitation frequency or method and also verifying the desired modes that are excited. This
characteristic enables 2D FFT to be a vital tool for qualifying Lamb waves.
3.3 Finite Element (FE) Analysis
Finite element analysis is very useful in allowing novel ideas to be examined and tested without
actually manufacturing the product. It proves to be a very cost-saving method and therefore, FE
analysis makes up the basis of this thesis. As discussed earlier, Lamb waves are dispersive and more
than one mode may be present. In addition, the through thickness deflected shape is frequency
thickness dependent. Therefore, it is not easy to simulate Lamb wave propagation and a few
considerations had to be made when constructing the simulation.
3.3.1 Simulation setup
All the simulations were modelled in 3D and solved using ABAQUS. In addition, further post processing
is carried out in MATLAB to extract and analyse the strain field obtained from the simulations. All the
thin plate structures were modelled using aluminium which has the following material properties as
shown in Table 5.
48 | P a g e
Table 5: Material properties of Aluminium plate
Plate Thickness 2mm
Young’s Modulus, E 73 GPa
Poisson’s Ratio, v 0.34
Density, ρ 2700 kg/m3
In order to understand the behavior of Lamb waves clearly with the help of Finite Element (FE), a
relatively simple geometry is modelled first. The same 2mm thick Aluminium plate model was used
for all three excitations, with S0 mode only, A0 mode only and combination of S0 and A0.
It is not necessary to simulate all the possible modes, thus, the simulation was set up to be limited to
those modes that are of interest to this research. A few criteria in setting up FE simulations for Lamb
wave propagation are being observed such as the computational time step and the node spacing. The
computational time step is influenced by the longitudinal velocity of the wave. On the other hand, the
slowest phase velocity and the shortest wavelength dictate the maximum possible node spacing or
the mesh size to avoid spatial aliasing.
Both these criteria determine the stability and accuracy of the simulation of Lamb wave propagation.
Node spacing of 1mm is used to have a compromise between speed and accuracy of the simulations.
Both the temporal and spatial resolution of the FE model is important for the convergence of these
numerical results. Although the accuracy of the modelling results increases as the integration time
step decreases, time steps that are too small will be a waste of time. On the other hand, a large time
step will prevent the higher frequency components from being accurately resolved.
The element used in all the ABAQUS simulation is a C3D8R which represents a continuum (solid) 8-
node linear brick that utilises reduced integration elements with hourglass control. This minimizes the
computational expense of element calculations. This model was meshed with a structured three
dimensional square shaped element hexahedral mesh with a default hourglass control of a stiffness-
49 | P a g e
viscous weight factor of 0.5 A few constraints were implemented on the model based on the
expression below, where le is the element length and λmin is the shortest wavelength of interest.
Equation 13
𝑙𝑒 =𝜆𝑚𝑖𝑛
20
This is well above the threshold of eight (8) elements per wavelength which other researchers have
discovered to be a good limit for accurate modelling and to avoid aliasing. [41] The minimum
wavelength is of interest and determines the minimum mesh size because it corresponds to a
maximum wave number.
Hence, an optimum time step is required to be able to provide an accurate result without wasting too
much time on unnecessary calculations. Based on the Newmark time integration scheme [42], a
method to obtain a compromise between accuracy and efficiency will be using 20 points per cycle of
the highest frequency results of interest, fmax. This yields the formula as:
Equation 14
∆𝑡 =1
20𝑓𝑚𝑎𝑥
In addition, the stability criterion for an explicit algorithm was also considered to ensure that the time
step chosen satisfies that requirement. [41, 43]
Equation 15
∆𝑡 ≤ 0.8∆𝑠
𝑐
By utilizing and satisfying all the criteria above, an integration time step, Δt = 0.1µs and an element
length, le = 1mm was chosen. The model was solved for a real time of 100µs which means 1000 solution
steps were calculated. Depending on the excitation input or simulation, 1000 to 3000 time steps may
be required to fully capture all the necessary interactions of Lamb waves.
50 | P a g e
During the initial stages, a total of 32 sensors were used for the FE model. These sensors were
placed in the middle of the Aluminium plate lined up in a straight array with 7mm apart between
each sensor. A range of dimensions were explored for the square sensors that vary between 3mm to
7mm with a constant thickness of 10microns. However, a final dimension of 3mm was chosen to be
used for all the FE simulation as smaller dimensions allow more sensors to be lined up within the
same area of interest. An example of the model is as shown in Figure 20.
Besides the mesh size and computational time step, the input parameter also influences the
simulation. The input controls the frequency and modes of the wave that is excited. For instance, the
direction that the input is being excited is able to adjudicate the mode of the wave. The excitation is
a point node excitation being loaded out of the plane to generate (a) Symmetrical, S0 (b) Asymmetrical,
A0 & (c) combination of both S0 & A0 Lamb waves in three different configurations. The other factor
that affects the simulation is the excitation pulse which determines the frequency of the wave. For a
symmetrical wave mode, S0, the excitation used is a Hanning windowed 5 cycle sine wave with centre
frequency of 200 kHz. A 200 kHz centre frequency was used for the combination input of S0 & A0
combined as well. On the other hand, a Hanning windowed 5 cycle sine wave with centre frequency
of 50 kHz is used as an excitation for the A0.
Figure 20: Example of Preliminary FE Model
51 | P a g e
After the initial steps of assembling the model and applying the simulation parameters, the interested
results which are the stress values in each direction, namely, S11 & S22 from each of the 32
piezoelectric sensors are being collected. With the stress values, these results are being exported and
extracted using MATLAB. The extracted data are being post processed in two stages. In Stage 1 of post
processing, the objective is to yield a voltage-time response in order to clearly identify the wave packet
captured by the sensors that is of interest to be analysed in Stage 2. The stress values are applied on
to Equation 3 so that the voltage values can be obtained. An example of the voltage-time response
after undergoing Stage 1 of post processing of the results obtained from the ABAQUS FE simulation is
as in Figure 22.
Figure 21: a) Excitation to generate S0 mode
b) Excitation to generate A0 mode
c) Excitation to generate both S0 & A0 mode
52 | P a g e
After identifying the wave that is of interest, Stage 2 of post processing involves conducting a 2D FFT
on the interested range of data and subsequently, plotting it on a dispersion curve. The dispersion
curves are obtained from a program called “DISPERSE” which was developed by Lowe in 2001 at
Imperial College in United Kingdom. It solves the Lamb wave roots and provides quick theoretical
solutions for Lamb wave behaviour. An illustration of the dispersion curve is as shown in Figure 23.
Voltage-time response of the piezoelectric sensors
0 0.5 1 1.5
x 10-4
-1.5
-1
-0.5
0
0.5
1
1.5
Time,sec
Tota
l V
oltage,
V
Figure 22: Voltage-Time response recorded by a piezoelectric sensor
0 100 200 300 400 500 600 700 800 900 10000
100
200
300
400
500
600
700
800
900
1000
Frequency (kHz)
Wave n
um
ber
(rad/m
)
A0
S0
Figure 23: Example of a dispersion curve
53 | P a g e
To obtain the 2D FFT dispersion curves, the sensors are used as representation of spatial coordinates.
In this situation, there are 32 points which will be able to provide the data required for 2D FFT. This
2D FFT dispersion curve is obtained using the 2D Fast Fourier Transform analysis on the time and
spatial history of the wave propagation. This transformation produces a frequency-wavenumber
representation for the analytical and numerical solutions as shown in Figure 24.
3.4 Validation Model
Prior to carrying out any simulations, the model is first validated. Three different excitations were
carried out to validate if the excitation or the model was accurate. The model was excited by having
an out of the plane point excitation on one node. The excitation is windowed by Hanning window as
discussed earlier. An example of the validation model is shown in Figure 25.
Frequency (kHz)
Wave n
um
ber
(rad/m
)
2D FFT results compared with the dispersion curve
0 50 100 150 200 250 300 350 400 450 5000
100
200
300
400
500
600
700
800
900
1000
Results
A0
S0
Figure 24: Example of comparing 2D FFT results with dispersion curve after Stage 2 of Post Processing
54 | P a g e
For the analysis of the validation model, a series of nodes from the plate which are represented by
the red coloured dots in Figure 25 is taken in a straight line along the excitation location. A 2D FFT is
then carried out to determine the propagating mode of the excited wave and also verify the frequency
content of the wave. The data and results obtained from the validation simulation are discussed in the
subchapter below.
3.4.1 Validation of S0 , A0 and S0A0
The plots on Figure 26 to Figure 28 are results obtained from the 2D FFT and being superimposed onto
the dispersion curve for comparison. It can be clearly seen that the 2D FFT results coincide with the
green line of the dispersion curve at 200 kHz which signifies a S0 mode. This agreement of the results
from Figure 26 indicates that the excitation is excited at 200 kHz and it is a S0 mode. Similar to the S0
mode, the 2D FFT results from the antisymmetrical excitation rests in the region of the red line which
is the A0 line of the dispersion curve. Unlike the S0 mode, the A0 mode was excited at 50 kHz instead
and this is observed in Figure 27. As expected from the combination of both S0 & A0 mode, the 2D FFT
results indicate the presence of both modes propagating along the plate. This is represented by the
existence of contours on both the red A0 line and the green S0 line shown in Figure 28.
Figure 25: Validation FE Model
55 | P a g e
These dispersion curves are a good indication on the accuracy of the simulation model. The results
obtained from the dispersion curves suggest that all the modes that were intended are able to be
excited.
All three different excitation configurations are in accordance with the dispersion curve. This indicates
that the validation model is accurate and can be used for further modelling simulations.
Frequency (kHz)
Wa
ve
nu
mb
er
(ra
d/m
)
Validation of So
0 100 200 300 400 500 600 700 800 900 10000
100
200
300
400
500
600
700
800
900
1000
Result
A0
S0
Figure 26: Validation of S0 wave
56 | P a g e
Figure 27: Validation of A0 wave mode
Frequency (kHz)
Wa
ve
nu
mb
er
(ra
d/m
)
Validation of SoAo
0 100 200 300 400 500 600 700 800 900 10000
100
200
300
400
500
600
700
800
900
1000
Result
A0
S0
Figure 28: Validation of combination of S0 & A0 wave mode
57 | P a g e
3.4.2 Problem Identification
The model was assembled such that two different input locations were used to excite the propagating
waves. The first simulation is with the location of excitation straight in-line with the sensors. On the
other hand, the second simulation is with the location placed such that the loading is at an angle
towards the sensors. For instance, the excitation will either be excited at location A (in-line) or location
B (at an angle) as depicted in Figure 29 below.
Two separate simulations of the two different excitation locations are carried out to compare the
outcome of the 2D FFT results. As can be seen in Figure 30 below, it is obvious that if the source of
excitation is located at Location A, the 2D FFT result clearly indicates that the S0 mode is being excited.
On the contrary, if the source of excitation is located at Location B, the 2D FFT result indicates
otherwise. The result falls short below the S0 dispersion curve as shown in Figure 31. As the distance
between the point of origin and sensor increases, the inaccuracy in the result indicating that the 2D
FFT result is much lower than the dispersion curve will be a lot more evident. This is due to the skewing
of the wave propagation with respect to the direction of measurement.
Location A
Location B
Direction of 2D FFT line
Figure 29: Two different point of origin
58 | P a g e
Frequency (kHz)
Wave n
um
ber
(rad/m
)2D FFT result from Location A
0 50 100 150 200 250 300 350 400 450 5000
100
200
300
400
500
600
700
800
900
1000
Results
A0
S0
Figure 30: Dispersion curve for Location A with the 2D FFT results superimposed
Frequency (kHz)
Wave n
um
ber
(rad/m
)
2D FFT result from Location B
0 50 100 150 200 250 300 350 400 450 5000
100
200
300
400
500
600
700
800
900
1000
Results
A0
S0
Figure 31: Dispersion curve for Location B with the 2D FFT results superimposed indicating skewed results
59 | P a g e
The skewing of the results becomes more prominent as the angle between the point of origin of the
source and the sensors increases. This issue is present when the propagating waves are not
propagating in-line with the direction of the FFT line of the sensors (ie. waves propagating at an angle
towards the sensor array arrangement). As the direction of measurement is not the same as the
direction of propagation, this leads to inaccuracy when revealing the spectral content of a propagating
wave. This skewing of results is contributed by an incorrect wavelength that is measured and as a
result, an incorrect wavenumber is obtained. This then caused the inability to determine the wave
mode of the propagating wave due to the inaccuracy of the wavenumber which yields the wrong
wavelength.
As previously mentioned in the report, the mode of a propagating wave can be determined by both
the frequency and wavenumber of the wave. However, there is a relationship between the angle from
source to sensor and the apparent wavelength of the wave. As the angle between the source and
sensors increases, the apparent wavelength measured increases. Thus, a correction of the
wavenumber is needed to shift the 2D FFT back to its dispersion curve. The representation of shifting
the result is as depicted in Figure 32 below.
This leads to defining the problem and suggesting an approach to identify the origin of the wave source
in order to distinguish the various wave modes.
60 | P a g e
3.5 Conclusion
Chapter 3 concentrates on the application of computational modelling in this research. By using FE
simulations, it enabled the propagation of Lamb waves to be predicted before it is actually being
applied in the application of real life. Prior to setting up the FE simulations, there were some key
information that were looked into such as applying Hanning window and understanding the concept
of 2D FFT.
0 50 100 150 200 250 300 350 400 450 5000
100
200
300
400
500
600
700
800
900
1000
Frequency (kHz)
Wave n
um
ber
(rad/m
)
A0
S0
Kmeasured
Kactual
Figure 32: Shifting the wavenumber, K values back to the dispersion curve
61 | P a g e
A few fundamental theories about FE analysis were also looked into. This was to ensure the simulation
that was being set up satisfies the modelling requirements such as the stability criterion, temporal and
spatial resolution necessary for the simulation of Lamb wave propagation. Thus, the accuracy and
stability of the FE simulations can be ascertained once all those criterions are satisfied.
Validation models were also set up to show that the FE models can be used for further analysis.
Besides that, this chapter also discussed about the two stages of post processing. The first is importing
the data into MATLAB from ABAQUS and by applying Equation 3 to convert the stress values to
voltages, this produces a voltage-time response of the wave propagation. The second stage involves
applying 2D FFT on the time signal recorded. The results obtained from the 2D FFT are then plotted in
a frequency-wavenumber graph. This led to the introduction of the dispersion curve which acts as a
reference for our results.
This chapter also identified the issue that is encountered in the application of Lamb waves. The
problem that was discussed is that if the 2D FFT measurements are not carried out in the direction of
the wave propagation, the results obtained on the dispersion curve will be skewed. This restricts the
decomposition of Lamb waves to be carried out accurately. As a result of that, the solution to that
issue would be to introduce the usage of multiple array sensors in hopes that it will provide more
information about the propagating wave. Consequently, it will be able to decompose the wave to
identify the point of origin of the wave and reveal the modal content of the wave. Therefore, the next
chapter addresses all these points that were raised in Chapter 3 to provide a better understanding of
Lamb wave propagation.
62 | P a g e
4.0 Mode Identification & Source Location
The objective of this research is to be able to identify the wave mode and its point of origin using 2D
Fast Fourier Transform (2D FFT). This chapter focuses on the use of an array of sensors to facilitate the
decomposition of the wave modes present in the incident wave packet and to identify the point of
origin of these wave modes. The method that is applied to decompose the Lamb waves is by carrying
out 2D FFT in three different directions, namely, vertical, horizontal and diagonal direction. By
gathering information from multiple directions, sufficient information will be obtained to analyse and
accurately decompose the wave.
4.1 Multiple Array Sensors
The setup of the model with multiple array sensors is as shown in Figure 33. The simulation is carried
out by having individual excitations originated at different regions, namely in Region A, Region B and
Region C. The three regions are used to simulate possible locations of where the origin of a wave
could be positioned at. Upon exciting the wave, the analysis and post processing are carried out by
conducting 2D FFT in three different directions which will be along the vertical, horizontal and diagonal
direction.
The three directions that the measurements are carried out along the array sensors as represented by
the red arrows in Figure 33 would be able to provide an indication of where the point of origin of the
wave is potentially located at. By observing and analysing the results gathered from the 2D FFT plots,
the origin of the wave can be established to be in either of the three regions mentioned which are
Region A, Region B or Region C. It should also be noted that the naming and numbering of the array
sensors begin with the origin which is Column 1 & Row 1 located at the bottom left of the array.
Therefore, as the sensors moves to the right, the column number increases and as the sensors move
upwards, the row number increases. The naming and numbering of the sensors are shown in Figure
33.
63 | P a g e
4.2 Strategy proposed for the identification of wave mode and its
location of origin
A method to estimate and identify the location of the point of origin of the incident wave is proposed.
The proposed method involves performing 2D FFT on the results obtained from the array sensors to
produce a wavenumber-frequency plot so that the region that the wave may be propagating from can
be identified. The steps taken by this approach are discussed and the definition of some terms that
are introduced is also highlighted. Figures are also included to provide a better representation of how
this strategy proposed is going to locate the origin of the incident wave and the corresponding wave
mode.
Figure 33: Multiple Array Sensors
Region A
Region
B
Region
C
Column 1 to Column 32
Ro
w 1
to
Ro
w 3
2
64 | P a g e
4.2.1 Guide for Source Location Identification
A FE simulation is performed to further examine the effects of using multiple array sensors instead of
a linear array. The excitation was carried out at the centre of Region A of the plate as shown in Figure
34. This is also to ensure that the wave propagates symmetrically across the array sensors. A guide for
the source location that incorporates the application of 2D FFT is proposed as a method to locate and
predict the point of origin of the wave.
The data from this simulation were then being post processed using MATLAB to obtain the 2D FFT
results. Three sets of 2D FFT results that were measured along the vertical, horizontal and diagonal
direction were collected. These 2D FFT results are used to assist in mode decomposition and origin of
the wave. There are multiple parts or steps to this analysis:
Step 1: Identify the region that the wave is propagating from.
Step 2: Determine the column or row that the wave is propagating from using the array sensors
Step 3: Identify the location of the origin of the wave and its propagating wave mode
Figure 34: S0 simulation used as guide for source location
65 | P a g e
This systematic approach will be used in analysing all of the results obtained from the simulations. By
utilising all of the 2D FFT graphs, the mode and region of excitation of the wave were able to be
determined. The simulation carried out with the excitation being along Column 16 as shown in Figure
34 will be used as the baseline or guide for source location for the other S0 simulations.
All of the 2D FFT results measured from various directions are plotted in Figure 35 to Figure 37. The
plots are showing measurements from each array and the wavenumber recorded along that particular
array. Due to the vast information or plot, it may be difficult to the view the points on the graph.
However, as can be seen from Figure 35 to Figure 37, there is a significantly higher concentration of
results near the S0 dispersion curve in Figure 35 compared to in Figure 36 or Figure 37. The results
measured along the vertical direction as represented in Figure 35 showed a wavenumber that is
consistent with a particular theoretical wave mode. These observations made on the plots are able to
allow some inference to be made about the wave.
0 50 100 150 200 250 300 350 4000
50
100
150
200
250
300
350
400
Frequency (kHz)
Wave n
um
ber
(rad/m
)
2D FFT Results measured vertically from all array sensors
Figure 35: 2D FFT measured in the vertical direction
66 | P a g e
0 50 100 150 200 250 300 350 4000
50
100
150
200
250
300
350
400
Frequency (kHz)
Wave n
um
ber
(rad/m
)
2D FFT Results measured Horizontally from all array sensors
Figure 36: 2D FFT measured in the horizontal direction
0 100 200 300 400 500 600 700 800 900 10000
100
200
300
400
500
600
700
800
900
1000
Frequency (kHz)
Wave n
um
ber
(rad/m
)
2D FFT Results measured diagonally
Results from Sensors (Diagonally)
A0
S0
Figure 37: 2D FFT measured in the diagonal direction
67 | P a g e
The first step of the analysis is to identify the region that the wave is propagating from. By observing
the results from the 2D FFT plots, the location or the region of the origin of the wave can be estimated.
This is achieved with the information that we have which are the wavenumbers measured by each
array in a set of directions. By comparing all of the 2D FFT results from Figure 35 to Figure 37, it is
evident that the results are concentrated in the graph that is measured vertically and the peak
wavenumber is observed from the graph that is measured vertically. The computed wavenumber is
also closer to the theoretical. However, the range of wavenumber computed is due to curvature of
the wave front. The results indicate that the point of origin of the wave is expected to originate from
Region A.
The second step of the analysis is to determine the column or row that the wave is propagating along.
Since the region of excitation has been specified to be in Region A, only the results in that region are
replotted with respect to its column number. By plotting just the wavenumbers of each column
obtained from Region A, the trend in Figure 38 indicates that the peak wavenumber is along Column
16 or in other words, the lowest measured wavelength occurs along Column 16. It should be noted
that regardless of the location of the source, the highest wavenumber measured should always
represent the correct solution. This is due to the fact that it is measuring the shortest wavelength of
the propagating wave. Any inaccurate measurement of the wavelength will yield a larger wavelength
and never a shorter wavelength compared to the actual wavelength. Therefore, the shortest
wavelength measured which translates to a largest wavenumber will be interpreted as the correct
solution. As a result, this suggests that the excited wave is along Column 16 in Region A.
68 | P a g e
Based on the results obtained, it is established that the origin of the wave is propagating along Column
16 which is consistent with the simulation. Upon identifying the column that it is propagating from,
the last step of the analysis can be achieved as the mode of the wave will be determined by carrying
out a 2D FFT on Column 16. In addition to that, the plot in the dispersion curve as depicted in Figure
39 clearly indicates that the propagating wave is a S0 wave. Both the plots in Figure 38 and Figure 39
are able to verify the accuracy of the 2D FFT results that the propagating wave lies along Column 16
and is a S0 mode. Apart from that, the plots and the systematic approach adopted are able to reassert
that this method is able to be used to locate the location of the source. Therefore, this can be used to
establish a source location guide for the rest of the S0 simulations.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1718 19 20 21 22 23 24 25 26 27 28 29 30 31 32190
195
200
205
210
215
220
225
230
235
Trend of wavenumber across all array measured vertically
Column
Wavenum
ber
Figure 38: Wavenumbers obtained from measuring vertically indicating Region A
69 | P a g e
The guide for source location is made up of the wave propagation angle, θ vs difference in
wavenumber, ΔK with respect to the source of excitation which is along Column 16. The purpose of
this guide is to suggest that each difference in wavenumber measured by each array sensor
corresponds to a change in propagation angle, θ. As can be seen from Figure 38, the trend of the wave
number is symmetrical, therefore, only one half of the analysis is used to produce the guide for source
location. All of the wave propagation angles, θ are measured towards the array that is predicted and
a representation of the angle is shown in Figure 40. The graph in Figure 41 depicts the wave
propagation angle with respect to the change in wavenumber. There are only three curves in Figure
41 and these curves represent the array sensor that is being used as a guide towards the solution
which is the location of the source. Despite using only three arrays, the idea is that any array sensor
can be used as a guide to the origin of the wave. Row 1, 16 and 32 were chosen for this analysis purely
for brevity as it represents the front, middle and end of the array sensor.
Frequency (kHz)
Wave n
um
ber
(rad/m
)2D FFT to verify the mode of the predicted Array which is found to be along Column 16
0 50 100 150 200 250 300 350 400 450 5000
100
200
300
400
500
600
700
800
900
1000
Results
A0
S0
Figure 39: Plot on dispersion curve to verify the mode of the propagating wave
70 | P a g e
By doing so, it allows the observation on how well these arrays can accurately guide the results to the
location of the source.
From the earlier analysis, the array and region of the wave origin can already be narrowed down. With
the inclusion of the propagation angle of the wave, the point of origin can then be located. Any array
of sensor can be used as a measurement point to locate the source.
An example of the usage of the source location guide is briefly explained and described using Figure
40. Presuming that preliminary analysis has already been performed, thus, the initial investigation of
the 2D FFT results allows the location of the wave origin to be narrowed down to the red dashed line
in Figure 40 which represents the array that the wave is propagating from. However, the point of
origin could lie anywhere along the red dashed line. Thus, once the region that the wave is propagating
from is identified, any array of sensors adjacent to the predicted array can be used as a guide to the
correct location of the origin of the wave with the assistance of the differences in wavenumber
recorded by each array.
Figure 40: Representation of the prediction of the point of origin of the wave
x
y
θ
+5 Column -5 Column
Row 1
Row 8
Source
θ
71 | P a g e
To help with the process of locating the source, a few key points should be noted. One of them is that
due to the reason that any sensors can actually be used to locate the source, it will be kept to using
the arrays that are within a ± 5 array from the source.
Assuming that the results indicate that the wave is located along the array represented by the red
dashed line, the origin is set to be along that array which means that the change in wavenumber, ΔK
is zero along this array. On the other hand, the boundary of the analysis will be confined to using just
the results obtained within the ± 5 array in the same direction of measurement from the wave which
are represented by the black vertical dashed lines. From those 2D FFT results, the difference in
wavenumber with respect to the predicted array can then be used to direct to the point of origin of
the wave. As can be seen from the guide for source location in Figure 41, every row represents a guide.
Any of the row can be utilised to locate the origin and an example is such as using Row 1 or Row 8 as
depicted in Figure 40. Prior to carrying out the analysis, the guide for source location has to be
produced first similarly to how it is created in Figure 41.
There are three curves that are presented to be used as guides as shown in Figure 41 which are Row
1, Row 16 and Row 32. Each of these curves provide approximation to the location of the origin of the
wave. Based on the ± 5 array approach and the three curves, it should be able to provide up to thirty
(30) estimations of the location of the point of origin as each curve is able to provide up to ten
estimations. If more estimations are required, additional curves can be generated to assist in locating
the origin of the wave.
72 | P a g e
It is important to understand the limitations or restrictions to the accuracy of the guide. Since the
relationship between the change in wavenumber, ΔK and the propagation angle, θ is only plotted up
to a difference of less than 40 in its ΔK value, it is recommended that the array that is chosen to locate
the origin of the wave does not have a difference of wavenumber greater than the maximum ΔK value.
Therefore, the recommendation of using the ± 5 array distance as the guide would assist in ensuring
that the ΔK value is within the range of the curve. By doing so, it helps in narrowing the scope of the
analysis to just a specific region but yet it is accurate rather than having a wide scope but has the
potential to be inaccurate.
0 5 10 15 20 25 30 35 400
10
20
30
40
50
60
Del K
Angle
Wave Propagation Angle vs Change in Wavenumber, Del K
Row1
Fit1
Row16
Fit16
Row32
Fit32
Figure 41: Relationship between the change in wavenumber and the propagation angle
, θ
, ΔK
73 | P a g e
4.3 Conclusion
This chapter discussed about the usage of multiple array sensors to assist in the identification of the
point of origin of wave and its propagating wave modes. This is possible with the help of carrying out
2D FFT measurements in various directions namely, vertical, horizontal and diagonal direction.
The beginning of Chapter 4 involves naming and numbering of the array sensors so that an
understanding is first established. The point of excitation of the wave source is also separated to
different regions which are called Region A, Region B and Region C. This is so that the location of the
origin of the wave can be established and narrowed down easier by just suggesting the region that
the origin of the wave is most likely to be propagating from.
The next part of this chapter introduces the strategy adopted to achieve the aim of identifying the
location of the point of origin of the wave and its propagating wave mode. The idea is to use a guide
as means to locate the source location of the incident wave. This guide is produced with the use of 2D
FFT measured in the directions mentioned earlier to obtain the wavenumber values from every array
sensors. As shown in this chapter, different array sensors provide a different wavenumber reading as
a result of the skewing of the wave with respect to the 2D FFT measurement direction. Therefore, a
relationship was established between the wavenumbers obtained and the wave propagation angle.
This strategy suggests that each difference in wavenumber corresponds to a change in the
propagation angle towards the origin of the wave. It was noted that every array of sensors can be used
as a guide to locate the origin of the incident wave. Apart from that, the approach on using the guide
to locate the source of the wave was also discussed. This chapter was wrapped up by showing the
representation of the terms used and the strategy proposed to facilitate in the predication of the point
of origin of the wave.
74 | P a g e
5.0 Case Studies using S0 Input
Chapter 5 looks at case studies that were performed by carrying out simulations in Region A as well
as Region B to test out the guide for source location that was produced in the previous Chapter 4.2.
The approach discussed in the earlier chapter is applied in all the simulations and its applicability is
tested. Thus, the 2D FFT results obtained from both these simulations are used to predict the origin
of the wave and its mode with the assistance of the guide. The initial step of locating the point of
origin is always from observing the 2D FFT results from those set regions which are Region A, Region
B and Region C. These 2D FFT results are firstly obtained from analysing the voltage response derived
from the stress values using Equation 3. An example of the response recorded is in Figure 42 where
the incident wave packet is identified and extracted for post processing.
As can be seen from Figure 42, the initial wave packet detected originates from the source. On the
other hand, the subsequent wave packets detected could potentially be from reflections. Therefore,
the first wave packet is isolated and this wave packet is being used for post processing.
0 0.5 1 1.5
x 10-4
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Time,sec
Tota
l V
oltage,
V
Voltage response recorded by array sensors
Figure 42: Example of a voltage response recorded by the array sensors
75 | P a g e
The isolated incident wave packet shown in Figure 43 is then post processed using 2D FFT. Each voltage
response recorded by each array sensor contains vital information about the spectral content such as
the propagating frequency and wavenumber. By carrying out 2D FFT and analysing those results, it
will be able to give a clear indication as to which region the excitation or the location that the wave
originates from. The focus when analysing the 2D FFT results is where the concentration of the results
are and also which graph has the peak wavenumber.
This process is carried out for all of the case studies presented in this thesis. By implementing the
strategy proposed, the effectiveness of the guide for source location identification produced can be
quantified. Various excitation location is presented to study and ensure that the strategy proposed is
able to provide a good estimation of the location of the point of origin of the wave source.
0 0.5 1 1.5
x 10-4
-1.5
-1
-0.5
0
0.5
1
1.5
Time,sec
Tota
l V
oltage,
V
Extracted wave packet of voltage response
Figure 43: Example of an incident wave packet extracted from its original response
76 | P a g e
5.1 Source of Excitation from Region A
The first simulation is excited along Column 8 in Region A and the other one is excited along Column
32 in Region A as well. The numbering of the column arrays are from the left to the right. The objective
of carrying out two different simulations with two different excitation location is to study the
effectiveness of the guide when the point of origin of the wave either gets closer or further away from
the array of sensors.
5.1.1 S0 Simulation along Column 8
The simulation for an excitation input along Column 8 from Region A is shown in Figure 44.
The results from this analysis are represented in Figure 45 to Figure 51. The graphs in Figure 45 to
Figure 47 display the 2D FFT results obtained from the post processing of the data gathered from the
array of sensors. On the other hand, the figures from Figure 48 to Figure 51 represents the approach
where the predictions are made to determine the location of origin of the incident wave and its
corresponding propagating wave mode.
Column 8
Figure 44: Excitation along Column 8 from Region A
77 | P a g e
0 50 100 150 200 250 300 350 4000
50
100
150
200
250
300
350
400
Frequency (kHz)
Wave n
um
ber
(rad/m
)
2D FFT Results measured Horizontally from all array sensors
Figure 46: 2D FFT measured horizontally with origin of wave located along Column 8 in the Region A
0 50 100 150 200 250 300 350 4000
50
100
150
200
250
300
350
400
Frequency (kHz)
Wave n
um
ber
(rad/m
)
2D FFT Results measured vertically from all array sensors
Figure 45: 2D FFT measured vertically with origin of wave located along Column 8 in the Region A
78 | P a g e
The purpose is to decompose the wave and identify its mode and origin of the wave. Therefore, the
observation in the 2D FFT results will be the basis for the initial step in analysing and interpreting the
data. Based on observations, the region that the wave is excited from can be identified and its
approximate location can be determined based on the wavenumber values. Once the region and array
is identified, with that information, the mode of the propagating wave can be verified. Subsequently,
the exact location of the wave origin can be predicted by inputting the wavenumbers from the 2D FFT
results into the curve in Figure 41. Through observation of all the 2D FFT results, it is obvious that the
results are concentrated in Region A and the results measured from Region A has the highest
wavenumber. This signifies that the region of the excitation is situated in Region A. By just focusing
at the results measured along Region A, the trend of the wavenumber can be deduced based on Figure
48.
0 100 200 300 400 500 600 700 800 900 10000
100
200
300
400
500
600
700
800
900
1000
Frequency (kHz)
Wave n
um
ber
(rad/m
)
2D FFT Results measured diagonally
Results from Sensors (Diagonally)
A0
S0
Figure 47: 2D FFT measured diagonally with origin of wave located along Column 8 in the Region A
79 | P a g e
In addition, through observation of the trend, it can be specifically inferred that the wavenumber
measured from Column 8 has the peak wavenumber. This allows the deduction that the point of origin
is located somewhere along Column 8 in Region A because the measured peak wavenumber is along
Column 8 which indicates that it has the shortest measured wavelength. Since a straight line
measurement that is in line with the direction of data processing will always have the shortest length,
therefore it can safely conclude that the shortest measured wavelength corresponds to the actual
wavelength of the propagating wave. As aforementioned that for simplicity, the prediction of the
location of the origin of the wave is set to be carried out using ± 5 arrays away from the predicted
array which in this situation the predicted array is Column 8.
Figure 48: Trend observation of wavenumber obtained when 2D FFT measured vertically which indicates Region A
3 4 5 6 7 8 9 10 11 12 13224
226
228
230
232
234
236
Trend of wavenumber across all array measured vertically
Array
Wavenum
ber
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1718 19 20 21 22 23 24 25 26 27 28 29 30 31 32190
195
200
205
210
215
220
225
230
235
Trend of wavenumber across all array measured vertically
Column
Wavenum
ber
80 | P a g e
Apart from just the observation in wavenumber, the wave propagation is also able to provide some
insight on the origin of the excitation wave. By plotting the time series recorded from each array
sensor, a rough estimate of the direction of wave propagation can be perceived as the wave front is
able to be replicated. The wavefront for this simulation is reproduced using the voltage-time signal
recorded as shown in Figure 49.
The wavefront that is plotted in Figure 49 suggests that the wave is first encountered at a time step
of approximately t=200 and this occurs somewhere in the region between Column 7 to Column 10.
This reproduced wavefront is based on the voltage-time signal that is being measured by using just
the first row of array sensors. The information that can be deduced from this is that the plot can be
used as means to verify the accuracy of the prediction obtained from Figure 48 is accurate. Based on
Figure 48, the trend clearly indicates that the source of excitation lies somewhere along Column 8.
This deduction conforms to the observation and analysis obtained from Figure 49, as the wavefront
plotted clearly shows that Column 8 seems to be an accurate prediction.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
200
400
600
800
1000
1200
1400
Wave Propagation
Array
Tim
e
Figure 49: Wavefront reproduced using the voltage-time signal showing propagation along Column 8 from Region A
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1718 19 20 21 22 23 24 25 26 27 28 29 30 31 32190
195
200
205
210
215
220
225
230
235
Trend of wavenumber across all array measured vertically
Column
Wavenum
ber
81 | P a g e
The plot in Figure 50 is able to track the movement of the wave propagation as it moves across the
entire array sensor. As can be observed, the same wave that is measured using Row 1 exhibits a more
circular wavefront compared to the wave that is measured using Row 32 which looks a lot more like a
planar wave. As the wave propagates over a considerable amount of distance, the wave front will
eventually have a wavefront that is a lot more planar rather than a circular wavefront. This leads to a
potential for future work which is the need to analyse wave decomposition in multiple scenarios
namely, whether if a far-field source or a near-field source is present.
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3132
200
400
600
800
1000
1200
1400
Array
Wave Propagation
Tim
e
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Figure 50: Wavefront reproduced using array sensors from Row 1 and 32
Using Row 1
Using Row 32
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1718 19 20 21 22 23 24 25 26 27 28 29 30 31 32190
195
200
205
210
215
220
225
230
235
Trend of wavenumber across all array measured vertically
Column
Wavenum
ber
82 | P a g e
Considering that there is sufficient information to deduce that the origin of the wave is propagating
along Column 8, the mode of the wave can then be affirmed with the dispersion curve by carrying out
a 2D FFT using only the results obtained from the sensors along Column 8. This will then reveal the
spectral content of the propagating wave. Based on the result shown in Figure 51, it is evident that
the propagating mode of the wave is a symmetrical, S0 mode. Therefore, by collating all the solutions
and reasoning made so far, it is found that the wave is propagating from somewhere along Column 8
and it is a S0 wave. The next step of the analysis will be to narrow down even further as to where the
exact location of the point of excitation.
Frequency (kHz)
Wave n
um
ber
(rad/m
)
2D FFT to verify the mode of the predicted Array which is found to be along Array 8
0 50 100 150 200 250 300 350 400 450 5000
100
200
300
400
500
600
700
800
900
1000
Results
A0
S0
Figure 51: 2D FFT revealing the spectral component of the wave
Frequency (kHz)
Wave n
um
ber
(rad/m
)
2D FFT to verify the mode of the predicted Array which is found to be along Column 8
0 50 100 150 200 250 300 350 400 450 5000
100
200
300
400
500
600
700
800
900
1000
Results
A0
S0
83 | P a g e
The wavenumbers obtained from Column 3 to Column 13 will then be used to predict the location of
the wave origin. These wavenumber values are put into the guide for source location which will then
be used to predict the exact point of origin of the wave. By applying the guide on to the 2D FFT results,
the calculated angle is then used to locate or predict the location of the excitation point.
By collating the change in wavenumber, ΔK, the predicted propagation angles, θ can be calculated
using the curve fit in Figure 41. This is carried out by utilising the array of sensors as means to steer
the solution back to its origin which is ideally the point of origin of the wave. If every array is being
used as a guide to the excitation source, it just provides more information which is essentially the
prediction as to where the origin of the wave is expected to be. However, instead of using every row
to predict the location of the excitation wave, only Rows 1, 16 and 32 are used to predict the location
via the estimated propagation angle, θ. This allows the prediction to be limited to just a reasonable
amount of prediction points as that will allow the amount of data handling to be maintained at a
manageable level.
However, it should be noted that the analysis is not restricted to just these three rows. The reason
that these three rows were chosen is because of its strategic location being front of the array, middle
and end of the array. The predicted values obtained are tabulated in a table such as in Table 6, so that
it can be compared to the actual values. There are slight differences between the actual angles
compared to the predicted angles because the predictions are carried out based on a curve fit.
However, as can be seen in Table 6, the differences between the angles are not huge and therefore,
it should still provide a substantial indication on the location of the excitation. Those results were then
plotted in a graph as depicted in Figure 52 so that the comparison can be made easier.
84 | P a g e
Table 6: Predicted and actual propagation angles for Column 8 in Region A
Guiding Row
Row 1 Row 16 Row 32
Propagation Angle
(°)
Predicted (°)
Actual (°)
Predicted (°)
Actual (°)
Predicted (°)
Actual (°)
Column 3 35.898 37.598 14.227 13.096 8.723 7.596
Column 4 27.990 31.635 11.222 10.542 6.854 6.09
Column 5 21.215 24.798 8.177 7.946 4.981 4.575
Column 6 12.797 17.12 4.761 5.316 2.892 3.053
Column 7 6.496 8.755 2.451 2.664 1.484 1.528
Column 8 4.249 0 1.775 0 1.072 0
Column 9 6.929 8.755 2.616 2.664 1.584 1.528
Column 10 13.843 17.12 5.045 5.316 3.065 3.053
Column 11 21.715 24.798 8.508 7.946 5.184 4.575
Column 12 28.681 31.635 11.562 10.542 7.064 6.09
Column 13 36.301 37.598 14.487 13.096 8.886 7.596
0
5
10
15
20
25
30
35
40
3 4 5 6 7 8 9 10 11 12 13
An
gle
(°)
Column Number
Comparison between the Predicted and Actual Angles
Predicted using Row 1 Predicted using Row 16 Predicted using Row 32
Actual using Row 1 Actual using Row 16 Actual using Row 32
Figure 52: Comparison between predicted and actual propagation angles of Column 8 in Region A
85 | P a g e
As can be seen in Figure 52, each row that is used for comparison has its own discrepancy between
the results. It is obvious that the discrepancies between the results in Row 1 is significantly larger
compared to Row 16 and 32. Despite that, the differences recorded between the predicted angles and
the actual angles in all three of the rows are not large. All the predicted results are still able to direct
and point out the predicted location of the excitation. All of the answers showed that the predicted
point of origin of the wave do converge towards a certain vicinity. This region definitely coincides with
the actual location of the wave origin.
However, if the predicted angle values are calculated in terms of distances on the aluminium plate, a
few observations were noticed. The distances calculated are measured along Column 8, therefore,
Column 8 was omitted from the prediction as this is ideally where the wave is from. All the calculated
values are tabulated in Table 7 for comparison. Despite the angle discrepancies from Row 1 being
larger than the Row 16 and 32, the prediction of the origin of the wave is actually fairly precise. On
the other hand, the predicted point of origin using Row 32 produced a more inconsistent prediction.
A graph is plotted in Figure 53 to provide a representation of the data tabulated in Table 7. The
predicted distances are plotted with respect to the actual distance so that the trend from the
predictions can be observed. It is noted that the results obtained from Row 32 had a larger fluctuation
of results with respect to the actual distance. This fluctuation contributes to more inconsistencies in
the results as the error tolerance is a lot wider. However, the fluctuations decreases as the row of
sensors that is being used as the guide gets closer to the source. In spite of the fluctuations, there
were more accurate predictions from Row 32 compared to Row 1 where the predictions were more
precise with less fluctuations but the point of origin were consistently over-estimated.
86 | P a g e
Table 7: Predicted and actual distances measured along Column 8 in Region A
Guiding Row
Row 1 Row 16 Row 32
Distance (cm)
Predicted (cm)
Actual (cm)
Predicted (cm)
Actual (cm)
Predicted (cm)
Actual (cm)
Column 3 6.908
6.493
19.720
21.493
32.589
37.493
Column 4 7.526 20.161 33.279
Column 5 7.728 20.878 34.422
Column 6 8.805 24.011 39.585
Column 7 8.782 23.363 38.612
Column 9 8.229 21.889 36.163
Column 10 8.116 22.655 37.347
Column 11 7.533 20.055 33.069
Column 12 7.312 19.552 32.278
Column 13 6.807 19.352 31.982
0
5
10
15
20
25
30
35
40
45
3 4 5 6 7 8 9 10 11 12 13
Dis
tan
ce (
cm)
Column Number
Comparison between the Predicted and Actual Distances
Predicted using Row 1 Predicted using Row 16 Predicted using Row 32
Actual using Row 1 Actual using Row 16 Actual using Row 32
Figure 53: Comparison between predicted and actual point of origin measured along Column 8 in Region A
87 | P a g e
5.1.2 S0 Simulation along Column 32
The simulation for the setup of this case study with the origin of the wave being excited along Column
32 is represented in Figure 54. The point of excitation for this simulation is situated a lot closer to the
array sensors compared to the simulation in Chapter 5.1.1.
The 2D FFT results obtained from the excitation along Column 32 are displayed from Figure 55 to
Figure 57. As expected, the results measured from vertically clearly indicates that the source of
excitation is from Region A. This is because the wavenumber measured vertically has a significant
concentration in the S0 dispersion curve. On the other hand, the wavenumbers measured from the
horizontally and diagonally are far off from the S0 dispersion curve.
By analysing the results from measuring horizontally in Figure 56, it shows that all the results from
that region recorded wavenumbers that are extremely close to zero. Since the wavenumber recorded
is so small, it shows that the recorded wavelength is very large which also means that the excitation
is almost perpendicular to the direction of the measurement of 2D FFT. Therefore, it can further verify
that the source of excitation is not from Region B. Since it has been established that the excitation is
propagating from Region A, the focus is then shifted to the results obtained in Figure 55.
Column 32
Figure 54: Excitation along Column 32 from Region A
88 | P a g e
It is obvious that the wavenumber is moving from the black section towards the red section. By
implementing the ± 5 array rule and actually magnifying the results or perhaps plotting the
wavenumber values against column number, it shows that the excitation is along Column 32 in Region
A. This is described in Figure 58 where Column 32 showed the highest wavenumber then as it moves
further away from Column 32, a steady decline in the wavenumber is observed. This explains why the
wavenumber was shifting from the black section towards the red section because the measured
wavenumber is decreasing as the array moves further away from the point of origin.
0 50 100 150 200 250 300 350 4000
50
100
150
200
250
300
350
400
Frequency (kHz)
Wave n
um
ber
(rad/m
)
2D FFT Results measured vertically from all array sensors
Figure 55: 2D FFT measured vertically with origin of wave located along Column 32 in the Region A
89 | P a g e
0 100 200 300 400 500 600 700 800 900 10000
100
200
300
400
500
600
700
800
900
1000
Frequency (kHz)
Wave n
um
ber
(rad/m
)
2D FFT Results measured diagonally
Results from Sensors (Diagonally)
A0
S0
Figure 57: 2D FFT measured diagonally with origin of wave located along Column 32 in the Region A
0 50 100 150 200 250 300 350 4000
50
100
150
200
250
300
350
400
Frequency (kHz)
Wave n
um
ber
(rad/m
)
2D FFT Results measured Horizontally from all array sensors
Figure 56: 2D FFT measured horizontally with origin of wave located along Column 32 in the Region A
90 | P a g e
As stated earlier that the prediction of the origin of the wave will be carried using ±5 array distance,
however, due to the excitation being excited at the end of the array, only the 5 columns to the left of
Column 32 were taken into account. Hence, only the wavenumbers from Column 27 to Column 32 are
taken into consideration and from Figure 58, it can be determined that the wave is propagating along
Column 32.
Similarly to the previous simulation, the wavefront is recreated to view the movement of the wave.
The wavefront shown in Figure 59 proposes that the wave is propagating along the end of the array
sensor which is along Column 32. By the utilising the prior deduction based on the 2D FFT results and
subsequently, the plot of the wave propagation, these information provide us with reasonable amount
of confidence to infer that the excitation of the wave originates from somewhere along Column 32.
27 28 29 30 31 32218
220
222
224
226
228
230
232
234
Trend of wavenumber across all array measured vertically
Array
Wavenum
ber
Figure 58: Trend observation of wavenumber obtained when 2D FFT measured vertically which indicates Region A
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1718 19 20 21 22 23 24 25 26 27 28 29 30 31 32190
195
200
205
210
215
220
225
230
235
Trend of wavenumber across all array measured vertically
Column
Wavenum
ber
91 | P a g e
After confirming that the excitation is propagating along Column 32, the mode of the propagating
wave is then verified as shown in Figure 60. The data from Column 32 is analysed using 2D FFT to
produce the results in Figure 60. This suggests that the mode of the wave is a S0 mode which will then
allow us to determine the location of the point of excitation through various ways. An important
observation from Figure 55 to Figure 57 that should be noted is that none of the measured
wavenumbers are beyond the S0 dispersion curve. This indicates that it is definite that the propagating
wave is a S0 wave as the peak wavenumber recorded falls on the S0 dispersion curve as well.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
200
400
600
800
1000
1200
1400
Wave Propagation
Array
Tim
e
Figure 59: Wavefront reproduced using the voltage-time signal showing propagation along Column 32 from Region A
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1718 19 20 21 22 23 24 25 26 27 28 29 30 31 32190
195
200
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220
225
230
235
Trend of wavenumber across all array measured vertically
Column
Wavenum
ber
92 | P a g e
Similar to the excitation along Column 8, the exact method is implemented on this simulation to study
how well it can estimate the source of the excitation at different locations. The propagation angles as
predicted by the curve are tabulated in Table 8 and compared with the actual propagation angles. The
differences between the angles predicted by the sensors and the actual angles are plotted in Figure
61 for a visual representation on the discrepancies if there are any.
Frequency (kHz)
Wave n
um
ber
(rad/m
)2D FFT to verify the mode of the predicted Array which is found to be along Array 32
0 50 100 150 200 250 300 350 400 450 5000
100
200
300
400
500
600
700
800
900
1000
Results
A0
S0
Figure 60: 2D FFT revealing the spectral component of the wave
Frequency (kHz)
Wave n
um
ber
(rad/m
)
2D FFT to verify the mode of the predicted Array which is found to be along Column 32
0 50 100 150 200 250 300 350 400 450 5000
100
200
300
400
500
600
700
800
900
1000
Results
A0
S0
93 | P a g e
Table 8: Predicted and actual propagation angles for Column 32 in Region A
Guiding Row
Row 1 Row 16 Row 32
Propagation Angle (°)
Predicted (°)
Actual (°)
Predicted (°)
Actual (°)
Predicted (°)
Actual (°)
Column 27 40.023 66.275 18.107 16.211 11.191 8.565
Column 28 36.583 61.217 15.985 13.094 9.831 6.871
Column 29 30.524 53.778 12.753 9.895 7.803 5.164
Column 30 21.068 42.307 8.398 6.633 5.117 3.448
Column 31 10.355 24.469 4.028 3.328 2.445 1.725
Column 32 3.980 0.000 1.669 0.000 1.008 0.000
From Figure 61, it is interesting to note that the predictions from Row 1 has a significantly larger
discrepancy compared to the ones predicted using Row 16 and Row 32. This situation is observed in
the earlier simulation as well. It is consistent throughout both simulations that the predictions using
Row 1 yield a larger difference compared to using Row 16 or Row 32. This could be due to the circular
wavefront that is potentially more susceptible to the change in angle with respect to wavenumber.
0
10
20
30
40
50
60
70
27 28 29 30 31 32
An
gle
(°)
Column Number
Comparison between the Predicted and Actual Angles
Predicted using Row 1 Predicted using Row 16 Predicted using Row 32
Actual using Row 1 Actual using Row 16 Actual using Row 32
Figure 61: Comparison between predicted and actual propagation angles of Column 32 in Region A
94 | P a g e
By actually focusing at the differences in Row 16 and 32, it is noticeable that the differences between
the predicted values and the actual values are not huge. This leads to believe that this method is still
able to reasonably predict the location of the origin of the wave fairly accurately. It should also be
highlighted that the differences observed in the earlier simulation as shown in Figure 52 is a lot lower
than the differences in this simulation.
To obtain a better understanding of the relevance of these angles, the estimated point of origin were
deduced from the predicted angles and compared with the actual point of origin. The predicted
distances measured along Column 32 using the other sensors as a guide are tabulated in Table 9 and
are also plotted in Figure 63 to examine the trend in those predictions. These predicted distances are
compared with the actual distance to have a better understanding of the accuracy of the predictions.
02468
101214161820
27 28 29 30 31 32
An
gle
(°)
Column Number
Comparison between the Predicted and Actual Angles
Predicted using Row 1 Predicted using Row 16 Predicted using Row 32
Actual using Row 1 Actual using Row 16 Actual using Row 32
Figure 62: Magnification of the comparison in Row 16 and 32
95 | P a g e
Table 9: Predicted and actual distances measured along Column 32 in Region A
Guiding Row
Row 1 Row 16 Row 32
Distance (cm)
Predicted (cm)
Actual (cm)
Predicted (cm)
Actual (cm)
Predicted (cm)
Actual (cm)
Column 27 5.957
2.197
15.291
17.197
25.272
33.197
Column 28 5.406 13.963 23.082
Column 29 5.105 13.255 21.893
Column 30 5.221 13.547 22.336
Column 31 5.560 14.199 23.418
All of the predicted point of origins are fairly different from the location of the actual origin of the
wave. It is worth noting that the predicted distance using Row 1 has once again produced a result that
is significantly more consistent and precise compared to Row 32 which has more variation in its
predicted values. In comparison to the prior simulation, the predictions from this simulation are less
accurate as the error radius is larger but the fluctuations in the values are not as prominent. Based on
the graphs shown in Figure 53 and Figure 63, it seems that using Row 16 as a guide is a safe option in
the event that it is unsure which row of sensors would provide a better approximation. All 32 rows of
array sensors can also be used as means to provide an estimate of the location of origin.
0
5
10
15
20
25
30
35
27 28 29 30 31
Dis
tan
ce (
cm)
Column Number
Comparison between the Predicted and Actual Distances
Predicted using Row 1 Predicted using Row 16 Predicted using Row 32
Actual using Row 1 Actual using Row 16 Actual using Row 32
Figure 63: Comparison between predicted and actual point of origin measured along Column 32 in Region A
96 | P a g e
5.2 Source of Excitation from Region B
A similar set up to the earlier simulation was used to conduct a new set of simulation. The difference
is that the wave is excited from Region B instead of Region A. The purpose of this is to ensure that the
results still hold true despite the excitation being at another region. With the results and analysis from
these simulations, it allows a clearer understanding and verification of the approach proposed in this
thesis. As mentioned previously, the arrays are numbered from bottom to top with the bottom of the
array labelled as Row 1 and the top array as Row 32. The two simulations carried out have the
excitation along Row 16 and Row 25 in Region B respectively and its analysis will be discussed in the
subchapters below.
5.2.1 S0 Simulation along Row 16
The first simulation set up has the origin of the wave located along Row 16 in Region B as shown in
Figure 64. This set of data collected from Row 16 is being analysed in this section. It is expected that
the results and analysis will yield a similar outcome if not identical to the previous simulation in the
previous subchapter.
Figure 64: Excitation along Row 16 from Region B
Row 16
97 | P a g e
The measurement points are still the same where results are analysed vertically, horizontally and
diagonally. As expected, the first step will be to observe the 2D FFT results obtained after post
processing of the voltage-time signals recorded by each array sensors. These results will once again
allow us to narrow down the location of the source of excitation to a specific region. The 2D FFT results
from all directions are displayed in Figure 65 to Figure 67. These results clearly identifies that the
excitation is from Region B as there is a significant concentration of results within a small region as
recorded in Figure 66. Apart from that, the highest wavenumber is recorded from measuring
horizontally as well which is shown in Figure 66. Since the highest wavenumber corresponds to the
shortest wavelength which is the actual wavelength of a propagating wave, it can be deduced that the
excitation is located in Region B.
0 50 100 150 200 250 300 350 4000
50
100
150
200
250
300
350
400
Frequency (kHz)
Wave n
um
ber
(rad/m
)
2D FFT Results measured vertically from all array sensors
Figure 65: 2D FFT measured vertically with origin of wave located along Row 16 in Region B
98 | P a g e
0 100 200 300 400 500 600 700 800 900 10000
100
200
300
400
500
600
700
800
900
1000
Frequency (kHz)
Wave n
um
ber
(rad/m
)
2D FFT Results measured diagonally
Results from Sensors (Diagonally)
A0
S0
Figure 67: 2D FFT measured diagonally with origin of wave located along Row 16 in Region B
0 50 100 150 200 250 300 350 4000
50
100
150
200
250
300
350
400
Frequency (kHz)
Wave n
um
ber
(rad/m
)
2D FFT Results measured Horizontally from all array sensors
Figure 66: 2D FFT measured horizontally with origin of wave located along Row 16 in Region B
99 | P a g e
In order to confirm that the waves are propagating from Region B as suggested by the 2D FFT results,
the wave propagation recorded by the array sensors can be of assistance. The voltage-time signal
recorded is recreated and an example of the wave propagation is as shown in Figure 68 below.
The wavefront observed in Figure 68 is able to describe the movement and propagation of the wave.
The reproduced wavefront is obtained using the first array sensors which is ideally perpendicular to
the direction of propagation of the wave. It is distinct from the wavefront that the waves are
propagating from Region B. Apart from that, the plot shows that the wave is first encountered at the
time step of close to t=200 and the area of the sensor that first detected the waves are around the
middle area of the array sensors. This gives us enough confidence to predict that the wave is
propagating from Region B and somewhere between Row 14 and Row 18 is where the origin of the
wave could possibly be located at.
123456789
1011121314151617181920212223242526272829303132
200400600800100012001400
Time Step
Wave Propagation from the Horizontal Region
Arr
ay
Figure 68: Wavefront reproduced using the voltage-time signal showing propagation along Row 16 from Region B
12
34
56
78
910
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
190
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215
220
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235
Tre
nd
of
wave
num
be
r acro
ss a
ll arr
ay m
easure
d v
ert
ically
Ro
w
Wavenumber
100 | P a g e
This deduction can be supported by the wavenumbers plotted in Figure 69. Upon focusing onto Figure
66, it was found that the peak wavenumber is recorded along Row 16 which is in accordance to the
simulation carried out. Therefore, the trend in the change of wavenumbers is monitored such as in
Figure 69. Similarly to prior simulations, the graph is plotted by utilising the ± 5 array rule and plotting
the wavenumbers of Row 11 to Row 21. The trend of the wavenumber indicates that the peak
wavenumber is recorded along Row 16 while the wavenumbers from the adjacent rows decrease in a
symmetrical manner. As a result of that, the wavefront shown in Figure 68 and the wavenumber plot
in Figure 69 provide more evidence that the origin of the wave is located along Row 16 in Region B.
Thus far, this whole approach is on the right path in predicting the point of origin as all the reasoning
made matches the simulation that was carried out.
11 12 13 14 15 16 17 18 19 20 21229
230
231
232
233
234
235
236
237
Trend of wavenumber across all array measured horizontally
Array
Wavenum
ber
Figure 69: Trend observation of wavenumber obtained when 2D FFT measured horizontally which indicates Region B
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1718 19 20 21 22 23 24 25 26 27 28 29 30 31 32190
195
200
205
210
215
220
225
230
235
Trend of wavenumber across all array measured vertically
Row
Wavenum
ber
101 | P a g e
Considering that the preliminary analysis reveal that Row 16 in Region B is most likely to be the array
that the point of origin is located along, the mode of the propagating wave can then be verified by
performing 2D FFT onto the data recorded by Row 16. Based on Figure 70, the 2D FFT result of Row
16 clearly reveals that the propagating mode is a S0 mode. This provides the clarification that the
excited wave has a S0 wave mode and is originated from somewhere along Row 16.
Frequency (kHz)
Wave n
um
ber
(rad/m
)2D FFT to verify the mode of the predicted Array which is found to be along Array 16
0 50 100 150 200 250 300 350 400 450 5000
100
200
300
400
500
600
700
800
900
1000
Results
A0
S0
Figure 70: 2D FFT revealing the spectral component of the wave
Frequency (kHz)
Wave n
um
ber
(rad/m
)2D FFT to verify the mode of the predicted Array which is found to be along Row 16
0 50 100 150 200 250 300 350 400 450 5000
100
200
300
400
500
600
700
800
900
1000
Results
A0
S0
102 | P a g e
Table 10: Predicted and actual propagation angles for Row 16 in Region B
Guiding Column
Column 1 Column 16 Column 32
Propagation Angle (°)
Predicted (°)
Actual (°)
Predicted (°)
Actual (°)
Predicted (°)
Actual (°)
Row 11 27.770 26.795 11.458 11.354 7.000 6.970
Row 12 21.587 22.000 8.668 9.126 5.282 5.586
Row 13 16.232 16.858 6.423 6.870 3.907 4.195
Row 14 9.855 11.421 3.897 4.592 2.365 2.799
Row 15 5.628 5.768 2.289 2.300 1.385 1.401
Row 16 4.249 0.000 1.775 0.000 1.072 0.000
Row 17 6.075 5.768 2.457 2.300 1.487 1.401
Row 18 10.249 11.421 4.049 4.592 2.458 2.799
Row 19 16.232 16.858 6.423 6.870 3.907 4.195
Row 20 22.805 22.000 9.198 9.126 5.608 5.586
Row 21 28.303 26.795 11.712 11.354 7.157 6.970
0
5
10
15
20
25
30
11 12 13 14 15 16 17 18 19 20 21
An
gle
(°)
Row Number
Comparison between the Predicted and Actual Angles
Predicted using Column 1 Predicted using Column 16 Predicted using Column 32
Actual using Column 1 Actual using Column 16 Actual using Column 32
Figure 71: Comparison between predicted and actual propagation angles of Row 16 in Region B
103 | P a g e
The work presented in this section includes three different columns that are chosen instead of rows
due to the different excitation region. However, the analysis and approach still remains the same. The
values tabulated in Table 10 shows the predicted angles obtained when different arrays and sensors
are used to locate the point of origin of the wave. These predicted values are compared with the actual
angles and the differences between them has found to be very minor. Despite these small differences,
the source of excitation can still be determined with very high confidence. This is primarily because
once the region of excitation is determined and with the help of these predicted angles, all these angle
values will then be able to trace back to the point of origin.
A graph is plotted in Figure 71 to provide a different view of the comparison. By observing the trend
in three different arrays, it is established that predictions using Column 1 has a slightly larger
disagreement compared to Column 16 and Column 32. However, the differences are too small that it
may be insignificant in predicting the location of the wave origin. In fact, as shown in Figure 72 later,
the predictions using Column 1 has less fluctuations in its predictions. This is due to the fact that
regardless of the differences, all these angles are going to be directed towards the same localised area
or point where the location of the origin of the wave is.
Similarly, the predicted angles are converted to distances as shown in Table 11 so that a real
representation of the estimated point of origins can be compared with the actual location of the origin.
The same step is taken where these values are plotted in a graph as depicted in Figure 72 to obtain a
visual representation of the comparison. A few observations can be made about the trend that these
predictions propose. As expected, there is an obvious pattern where the predictions from Column 32
exhibits more fluctuations and these fluctuations gradually decreases as it progresses from towards
Column 1. The approximation of the location of the origin in this simulation are significantly more
accurate compared to the ones that were discussed in the previous subchapter.
This is as expected due to the fact that the predicted angles matches a lot closer to the actual
propagation angles.
104 | P a g e
Table 11: Predicted and actual distances measured along Row 16 in Region B
Guiding Column
Column 1 Column 16 Column 32
Distance (cm)
Predicted (cm)
Actual (cm)
Predicted (cm)
Actual (cm)
Predicted (cm)
Actual (cm)
Row 11 9.495
9.901
24.668
24.901
40.722
40.901
Row 12 10.110 26.238 43.267
Row 13 10.305 26.649 43.924
Row 14 11.512 29.363 48.430
Row 15 10.148 25.014 41.358
Row 17 11.061 28.255 46.599
Row 18 10.305 26.649 43.924
Row 19 9.513 24.701 40.739
Row 20 9.285 24.120 39.821
Row 21 9.495 24.668 40.722
0
5
10
15
20
25
30
35
40
45
50
55
11 12 13 14 15 16 17 18 19 20 21
Dis
tan
ce (
cm)
Row Number
Comparison between the Predicted and Actual Distances
Predicted using Column 1 Predicted using Column 16 Predicted using Column 32
Actual using Column 1 Actual using Column 16 Actual using Column 32
Figure 72: Comparison between predicted and actual point of origin measured along Row 16 in Region B
105 | P a g e
5.2.2 S0 Simulation along Row 25
As previously mentioned, in order to confirm the method and approach, the exact same simulation as
the excitation in Region A is carried out but this time the excitation is from Region B. This simulation
shown in Figure 73 is carried out in Region B as well but it is along Row 25 to observe if there are any
variations in results because of the excitation location that is further away from the array sensors. Due
to the consistency of previous results and the similarity of the simulation, it is expected that the 2D
FFT results will still be able to reveal the location of the origin of the wave. The only difference is that
the 2D FFT results measured along the horizontal direction will have a concentration of results near
the S0 dispersion curve instead of the results from the vertical direction and the predicted array that
the wave is propagating from will be different too.
It is not surprising that the results for this simulation to be similar to the simulation in Region A. This
is because the propagation characteristic of the wave should still be the same regardless of the
location of the excitation. The only difference that can be observed from the results between the
excitation from Region A and Region B is which graph will actually reveal the location of the origin of
excitation. For instance, in this specific simulation, it is expected that the 2D FFT plot that is measured
horizontally along the array sensors will have a concentration of results near the S0 dispersion line
with the result from Row 25 records the peak wavenumber.
Row 25
Figure 73: Excitation along Row 25 from Region B
106 | P a g e
0 50 100 150 200 250 300 350 4000
50
100
150
200
250
300
350
400
Frequency (kHz)
Wave n
um
ber
(rad/m
)
2D FFT Results measured Horizontally from all array sensors
Figure 75: 2D FFT measured horizontally with origin of wave located along Row 25 in Region B
0 50 100 150 200 250 300 350 4000
50
100
150
200
250
300
350
400
Frequency (kHz)
Wave n
um
ber
(rad/m
)
2D FFT Results measured vertically from all array sensors
Figure 74: 2D FFT measured vertically with origin of wave located along Row 25 in Region B
107 | P a g e
The same approach and steps are taken in analysing the results obtained to produce a logical
estimation of the point of origin. Based on Figure 74 & Figure 75 the results exhibit is as predicted
where a similar trend is observed as the earlier simulations. However, the only difference is that the
results are now focused in the graph where the 2D FFT is measured horizontally. This outcome
matches the prediction that was formed earlier and also allows the verification that conducting
multiple 2D FFTs in various directions enable the location of the origin to be narrowed down. The
results obtained not only matches well with the predictions but it also indicates that the results are in
agreement with the actual simulation carried out.
0 100 200 300 400 500 600 700 800 900 10000
100
200
300
400
500
600
700
800
900
1000
Frequency (kHz)
Wave n
um
ber
(rad/m
)2D FFT Results measured diagonally
Results from Sensors (Diagonally)
A0
S0
Figure 76: 2D FFT measured diagonally with origin of wave located along Row 25 in Region B
108 | P a g e
Since the region of excitation has been established to be from Region B, a plot of the collective voltage-
time series from every array will be able to reveal and assert the region that the wave is propagating
from. The wave propagation shown in Figure 77 provides an indication that the wave is propagating
somewhere in between Row 23 to Row 27 from Region B. This once again conforms to the results
suggested by the 2D FFT plots and provides sufficient evidence that the wave is indeed propagating
from Region B. In order to further validate these predictions and by searching deeper into the results,
the trend of the wavenumber plot recorded from measuring horizontally is produced as shown in
Figure 78. Only the ± 5 row from the peak wavenumber is included in the plot and it is evident that
the peak wavenumber occurs along Row 25 in Region B. This proves that the results obtained conforms
to the simulation as the results clearly identified the region and array that the wave is propagating
from.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
02004006008001000120014001500
Wave Propagation
Time Step
Array
Figure 77: Wavefront reproduced using the voltage-time signal showing propagation along Row 25 from Region B
12
34
56
78
910
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
190
195
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205
210
215
220
225
230
235
Tre
nd
of
wave
num
be
r acro
ss a
ll arr
ay m
easure
d v
ert
ically
Ro
w
Wavenumber
109 | P a g e
This steady symmetrical decrease in wavenumbers that are adjacent to Row 25 is as expected. This is
due to the fact that the only changes made was the location of excitation while the propagating wave
remained the same. Therefore, it is only logical that the propagation characteristics that the wave
exhibits will produce the same result.
Given that now there is a reasonable amount of supporting evidence that the location of the origin of
the wave is along Row 25 from Region B, the mode of the wave can be checked. Upon conducting a
2D FFT on just the sensors along Row 25 to reveal the modal content of the wave, it was found that
the propagating wave lies along the green line as depicted in Figure 79 which indicates a symmetric,
S0 mode. This is true and conforms to the excitation input of the simulation carried out.
20 21 22 23 24 25 26 27 28 29 30231.5
232
232.5
233
233.5
234
234.5
235
235.5
236
236.5
Trend of wavenumber across all array measured horizontally
Array
Wavenum
ber
Figure 78: Trend observation of wavenumber obtained when 2D FFT measured horizontally which indicates Region B
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1718 19 20 21 22 23 24 25 26 27 28 29 30 31 32190
195
200
205
210
215
220
225
230
235
Trend of wavenumber across all array measured vertically
Row
Wavenum
ber
110 | P a g e
Since the region, mode and array of the point of origin that the wave is propagating from has already
been found and established, the next step is to estimate or find an approximation of the location of
the point of origin. By utilising the change in wavenumber, ΔK of the adjacent arrays with respect to
Row 25, prediction of propagation angles can be made. All these predicted angles are measured based
off from Row 25 and the values obtained are tabulated in Table 12. Once again, the values obtained
gave very good approximations as to where the exact point of origin of the wave is. These anticipated
angles allowed the possibility of locating the wave origin through the usage of wavenumber and angles
to drive the solution back to the exact location of the origin. The graph in Figure 80 provides a visual
representation of how close the predictions were compared to the actual angles. That plot offers
tangible evidence that this approach can be used as a method to locate the point of origin of the
propagating wave.
Frequency (kHz)
Wave n
um
ber
(rad/m
)2D FFT to verify the mode of the predicted Array which is found to be along Array 25
0 50 100 150 200 250 300 350 400 450 5000
100
200
300
400
500
600
700
800
900
1000
Results
A0
S0
SH0
Figure 79: 2D FFT revealing the spectral component of the wave
Frequency (kHz)
Wave n
um
ber
(rad/m
)2D FFT to verify the mode of the predicted Array which is found to be along Row 25
0 50 100 150 200 250 300 350 400 450 5000
100
200
300
400
500
600
700
800
900
1000
Results
A0
S0
111 | P a g e
Table 12: Predicted and actual propagation angles for Row 25 in Region B
Guiding Column
Column 1 Column 16 Column 32
Propagation Angle (°)
Predicted (°)
Actual (°)
Predicted (°)
Actual (°)
Predicted (°)
Actual (°)
Row 20 23.272 20.891 9.404 10.089 5.734 6.468
Row 21 18.059 16.980 7.174 8.102 4.367 5.183
Row 22 12.869 12.899 5.073 6.094 3.083 3.892
Row 23 8.229 8.680 3.273 4.071 1.984 2.597
Row 24 5.174 4.365 2.120 2.038 1.282 1.299
Row 25 4.249 0.000 1.775 0.000 1.072 0.000
Row 26 5.628 4.365 2.289 2.038 1.385 1.299
Row 27 9.053 8.680 3.588 4.071 2.176 2.597
Row 28 13.224 12.899 5.214 6.094 3.169 3.892
Row 29 18.638 16.980 7.416 8.102 4.514 5.183
Row 30 23.729 20.891 9.607 10.089 5.859 6.468
0
5
10
15
20
25
20 21 22 23 24 25 26 27 28 29 30
An
gle
(°)
Row Number
Comparison between the Predicted and Actual Angles
Predicted using Column 1 Predicted using Column 16 Predicted using Column 32
Actual using Column 1 Actual using Column 16 Actual using Column 32
Figure 80: Comparison between predicted and actual propagation angles of Row 25 in Region B
112 | P a g e
Due to the fairly matching results between the predicted and actual propagation angles, it is expected
that the estimated point of origin of the waves is going to conform reasonably well to the actual
location of the origin. The estimated values of the distance measured with respect to Row 25 are
tabulated in Table 13. As expected, the values predicted using Column 32 will yield a wider range of
predicted values compared to the ones predicted using Column 1 or Column 16. This has been
observed across all the simulations thus far and this shows the consistency of the approach that is
introduced in this research.
By observing the trend that is plotted from the comparison of predicted and actual point of origin of
the wave as displayed in Figure 81, it is evident that the predictions across each row reveals the same
uniform pattern. It is true that fluctuations do occur for all three array sensors that were used as guide,
however, the amplitude of fluctuations decrease significantly for Column 1 compared to using Column
32. This will once again showcase the precision and accuracy that the prediction using the sensors
closest to the origin of the wave provides.
It is apparent that using Column 1, which is ideally using sensors that are closer to the origin of the
wave compared to using sensors that are further away will provide a better approximation of the point
of origin of the wave. This reasoning has been true throughout all of the simulations presented in this
research and thesis.
113 | P a g e
Table 13: Predicted and actual distances measured along Row 25 in Region B
Guiding Column
Column 1 Column 16 Column 32
Distance (cm)
Predicted (cm)
Actual (cm)
Distance (cm)
Predicted (cm)
Actual (cm)
Distance (cm)
Row 20 11.625
13.100
30.189
28.100
49.794
44.100
Row 21 12.268 31.778 52.383
Row 22 13.132 33.791 55.703
Row 23 13.829 34.975 57.723
Row 24 11.044 27.019 44.696
Row 26 12.553 31.898 52.626
Row 27 12.766 32.874 54.189
Row 28 11.860 30.733 50.662
Row 29 11.375 29.541 48.729
Row 30 11.625 30.189 49.794
05
101520253035404550556065
20 21 22 23 24 25 26 27 28 29 30
Dis
tan
ce (
cm)
Row Number
Comparison between the Predicted and Actual Distance
Predicted using Column 1 Predicted using Column 16 Predicted using Column 32
Actual using Column 1 Actual using Column 16 Actual using Column 32
Figure 81: Comparison between predicted and actual point of origin measured along Row 25 in Region B
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5.3 Conclusion
Chapter 5 is mainly consist of achieving the aim of decomposing Lamb waves in terms of mode
identification and locating the point of origin of Lamb wave. This chapter present case studies that
includes FE simulations which are used to simulate or model the propagation of Lamb waves. Various
excitation locations were considered in order to verify the capabilities of the proposed method of
decomposing Lamb waves. The suggested approach in this thesis involves utilising 2D FFT to provide
substantial information about the propagating Lamb waves. By gathering 2D FFT results obtained
through the usage of multiple array sensors, a different dimension is exposed in revealing the content
of the propagating Lamb waves because more information can be acquired via 2D FFT in multiple
directions.
In lieu of using just 2D FFT as a method to locate the origin of the wave, the 2D FFT results are
combined with angles to provide an approximation of the whereabouts of the wave source. It was
discovered from these simulations that the method and approach proposed showed significant
coherency between the predicted angles compared to the actual propagation angles. These angles
obtained were then calculated so that the predicted distance measured with respect to the array that
the wave is propagating from can be determined. These distances were essentially approximations of
the point of origin of the propagating wave. It was found that majority of the estimation did provide
a reasonably accurate and precise prediction of the origin of the wave. Despite Row 1 and Column 1
having a larger discrepancy in its prediction of the propagation angle of the wave for Region A and
Region B respectively, the estimation of the point of origin of the wave is actually more accurate and
consistent compared to using rows or columns that are further such as Row 16 or Column 16 and Row
32 or Column 32. It can then be suggested that by utilising the sensors that are closer to the predicted
origin of the wave, it will yield a better approximation of the location of the origin of the wave.
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The findings from the case study using FE simulations provide another perspective when it comes to
decomposing Lamb waves by locating the point of origin of the wave and identifying its propagating
wave mode. This approach can be widen and stretched further with either more array of sensors or it
may be possible to have sensors that are arranged closer to one another. One other possible method
to provide better estimations of the origin of the wave would be to utilise more array sensors when
the estimation is carried out instead of using just the three array sensors that were shown in this thesis.
Apart from that, there are various other possibilities and variations to this set up that could be
explored. However, the essence of the whole approach proposed is as shown in Chapter 4 and the
analysis has shown a huge agreement between the predicted and actual results. As a result of that,
this approach can be utilised as an alternative method to decompose Lamb waves but the limitations
and boundaries of this approach should be noted.
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6.0 Conclusion & Future Work
6.1 Conclusion
The application of Lamb waves in Non-Destructive Testing has a growing usage and there are various
different techniques available for Structural Health Monitoring (SHM). This thesis is intended to
contribute to a fundamental area of this body of work. The work presented in this thesis seek to use
an array of sensors arranged in a square matrix to determine the point of origin of the Lamb wave
mode and the type of wave mode.
Chapter 1 introduces about the importance of Structural Health Monitoring (SHM). The applicability
of SHM with the usage of sensors in the field of Non-Destructive Testing (NDT) and Non-Destructive
Evaluation (NDE) was also discussed in this chapter. This chapter also provided an overview of
ultrasonic waves for on-board SHM specifically Lamb waves was also included in this chapter. The
approach that was going to be analysed in the subsequent chapters was also briefly mentioned to
provide an insight to what the problem is and the method that is utilised as a solution to the problem.
The chapter was concluded with a list of aims that are looking to be achieved at the end of this
research.
Chapter 2 of this thesis focuses more on the theories and literature that are relevant to this research.
These background information are discussed comprehensively to provide a more detailed
understanding of the literature that encompasses the entire research. The literature review was
conducted from a wider perspective which is the entire SHM field and being narrowed down all the
way to the mathematical solution of Lamb waves as well as the formula behind piezoelectric sensors.
This chapter was concluded with the linkage between the theories and the Finite Element modelling
to showcase the significance of this research. By the end of this chapter, a good grasp of the knowledge
and understanding of the literature behind the motivation of this research should be acquired.
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In Chapter 3, an evaluation of the validation and fundamental concepts of Lamb waves were also
discussed prior to the application and analysis of Lamb waves. The usage of 2D FFT as a technique to
decompose Lamb waves was examined and the applicability of the method is then broaden to allow
for a more meaningful understanding of the propagating Lamb waves. The problem that is of interest
to this thesis was highlighted and reported in this chapter and this then leads on to the proposition of
the systematic approach in addressing this issue. The idea is to utilise a technique that is already
known and develop an extension to that technique. The method to analyse the 2D FFT results which
includes using multiple array sensors, frequency-wavenumber contour plots, wave propagation plots
and relationship between angle and wavenumbers described in this chapter.
In Chapter 4, the final design model which is using array of sensors arranged in a 32x32 square matrix
array were included in the writing of this thesis. Multiple array sensors coupled with 2D FFT formed
the cornerstone of this research and have been leading the analysis in many areas of this thesis. A
strategy was proposed for the identification of the propagating wave mode and its location of origin.
The frequency-wavenumber contour plots obtained from 2D FFTs are then able to be post processed
to provide a deeper understanding of the propagating Lamb waves. Along with the wave propagation
plots, these provides an alternative perspective to examine and study Lamb waves. Furthermore,
these plots also serve as a verification and confirmation to the deduction formed from the 2D FFT
results. With these information acquired from those post processing techniques, an approximation of
the position of the origin of the propagating Lamb waves can be narrowed down towards a specific
region and the wave mode that is propagating can be determined as well. Subsequently, the next step
would be to be able to physically pinpoint the origin of the excitation wave and locate it. This is
achieved through the usage of the wavenumbers obtained from the 2D FFT analysis matched with the
wave propagation angles to create a relationship between them. This relationship formed is used as a
guide to locate the point of origin of the wave source.
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Chapter 5 describes a series of case studies carried out to apply and examine the effectiveness of the
guide that was created as well as the strategy proposed in Chapter 4. The case studies were divided
into two sub-chapters which are excitation input from Region A and Region B. From all the simulations,
the strategy proposed are able to provide a good estimate of the location of origin of the wave. The
predicted propagation angles agree with the actual propagation angles. This result was used to
provide a good prediction of the origin of the excitation wave. This is done from the distance between
the sensors that were used as guide. Majority of the distances calculated did provide a good
approximation of the location of the origin of the propagating wave. It was consistent throughout all
the results that the predictions using the furthest array of sensor produces the largest inaccuracy in
its estimation.
The first two chapters gave a comprehensive understanding in the field of SHM that includes the
theory as well as the literature behind this research which ultimately led on to the motivation and
drive of this study. The two subsequent chapters focused on the FE simulations which encompassed
validating the simulation models and proposing a strategy to be adopted to achieve the aim of the
thesis. The strategy proposed utilises a guide to locate the point of origin of the wave and
subsequently, identify the mode of the propagating wave. The next chapter then presents case studies
using various FE simulations to demonstrate a study on the usage of array sensors as means to
decompose Lamb waves. The results are based on FE simulations and existing theories or knowledge.
Hence, the real-time data acquisition and analysis needs to be carried out in order to provide more
tangible results besides acting as a verification tool to the method proposed.
In the following Chapter 6.2, there are a few recommendations and suggestions on future work that
can be carried out. All these recommended future work could potentially produce a more refined and
accurate estimation or solution. It is hoped that further development of this method will yield a more
significant breakthrough.
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6.2 Future Work
Based on the preceding chapters, the systematic approach proposed does render results and
predictions that are in accordance to the actual values which showcases its accuracy. However,
despite that it is imperative to understand the importance and emphasis for accurate results, it is just
as important to understand the limitations and the boundaries of the study. As with every research,
there is a potential for it to be developed further in the future and this research is no different than
that.
One of the many possibilities is that this study can be widen into different propagating modes to affirm
its efficacy as well. The excitation of the incident wave can be tested with A0 wave and also the
combination of both S0 and A0 wave. In doing so, it allows this approach to be tested in various
different situation and its applicability across various propagating modes can be confirmed as well. A
similar set up as the ones presented in this thesis can be carried out but instead, the wave input is
excited by a point excitation that generates A0 wave or a combination of both S0 and A0 wave.
There are several other possibilities that could be explored in the future such as improving the
resolution of the results by having more sensors. A higher spatial resolution or in other words, just
having a smaller spacing between sensors allow the waves with larger wavelengths to be measured
more accurately. This can perhaps be achieved by having smaller sensors or have the sensors arranged
closer to one another so that a larger area can be covered if there is a need to. The resolution of the
recording of data is very much dependent on the excitation frequency and mode of the wave,
therefore, it is ideal to have sensors with smaller spacing as it will be able to satisfy most of the
propagating wavelengths of the wave. By doing so, the change in wavenumber between arrays are
smaller and can perhaps provide a better estimation to the point of origin of the wave.
Apart from altering the setup of the simulations, there are improvements that can be made on the
existing study presented in this thesis. One of the improvement is, a better curve fit can also be used
in producing the relationship between change in wavenumber, ΔK and the propagation angle, θ.
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By having a better curve fit line, it provides a better approximation of the propagation angle which
then yields a more accurate estimation of the point of origin of the wave. Besides that, instead of using
just three rows or arrays of sensor as a guide to locate the point of origin of the wave, every possible
row and array can be used. By doing so, it provides more estimation of the position of the origin of
the wave and perhaps the outliers will then be able to be identified and removed as there should be
a concentration of predictions surrounding the actual point of origin.
In addition, analytical solutions can be derived to allow the results obtained in Chapter 5 to be
compared with some equations as a form of verification of its accuracy. In doing so, this provides
assurance that this approach is a good method to decompose Lamb waves. With the support of
mathematical solutions, a more concrete evidence and the significance of the strategy proposed can
then be verified. Perhaps, the analytical or mathematical approach should look into providing a
solution such as near-field and far-field solution for wave propagation. While still maintaining the
same approach and without varying the analysis too much, another possibility of future work could
be to carry out more 2D FFT measurements in the diagonal direction. This can be done by performing
the post processing at various diagonal angles instead of just one along 45° as shown in this thesis. By
acquiring more information at an increment of angles between 0° to 90° along the diagonal direction,
it provides more knowledge about the origin of the wave if the point of origin of the wave is detected
to be in Region C. If this can be carried out, it will improve the understanding of the wave as this
provides a wider coverage and a more thorough overview of the propagating wave. This will then yield
a greater accuracy in determining the origin of the propagating wave.
All of the recommendations for improvement and future work will help to provide a deeper knowledge
of the SHM field. The suggestions above will assist in offering a more comprehensive understanding
of Lamb wave propagation and the effectiveness of the strategy proposed in identifying the location
of the origin and the wave mode of the incident wave.
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7.0 References
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on Ultrsonics, Ferroelectrics and Frequency Control, vol. 39, no. 3, pp. 381-397, 1992.
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circular piezoelectric transducers,” Wave Motion, vol. 48, no. 4, pp. 358-370, 2011.
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of Propagating Multimode Signals,” The Journal of Acoustical Society of America, vol. 89, pp.
1159-1168, 1991.
[9] W. H. Ong, Lamb wave based in-situ structural health monitoring approach for future metallic
structures, Monash University, Department of Mechanical Engineering, 2012.
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Toledo, 2014, p. 98.
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C. Duke Jr., Ed., Springer US, 1988, pp. 1-21.
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fuselage structure,” IEE Proceedings Science. Measurement & Technology, vol. 148, no. 4, pp.
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[27] K. F. Graff, Wave Motion in Elastic Solids, Clarendon Press, 1975.
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Journal of Nondestructive Testing , vol. 1, pp. 251-283, 1969.
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ultrasonic guided waves,” NDT&E International, vol. 34, pp. 1-9, 2001.
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Mathematical Modelling, pp. 550-562, 2011.
[31] F. B. Cegla, A. Rohde and M. Veidt, “Analytical prediction and experimental measurement for
mode conversion and scattering of pate waves at non-symmetric circular blind holes in
isotropic plates,” Wave Motion, vol. 45, pp. 162-177, 2008.
[32] L. Yu, C. A. C. Leckey and Z. Tian, “Study on crack scattering in aluminum plates with Lamb
wave frequency-wavenumber analysis,” Smart Materials and Structures, no. 22, 2013.
[33] Y. Cho, “Estimation of Ultrasonic Guided Wave Mode Conversion in a Plate with Thickness
Variation,” IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, vol. 47, no.
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America, no. 4, pp. 131-136, 1958.
[36] C. Valle, M. Niethammer, J. Qu and L. Jacobs, “Crack characterization using guided
circumferential waves,” Journal of the Acoustical Society of America, pp. 1282-1290, 2001.
[37] Y. Cho, J.-C. Park, J. L. Rose and D. D. Hongerholt, “A Study on the Guided Wave Mode
Conversion Using Self-calibrating Technique,” Journal of the Korean society for nondestructive
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Waves,” in 17th World Conference of Nondestructive Testing, Shanghai, China, 2008.
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[42] F. Moser, L. J. Jacobs and J. Qu, “Modeling Elastic Wave Propagation in Waveguides with the
Finite Element Method,” NDT&E International, vol. 32, pp. 225-234, 1999.
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8.0 Appendices
Scripting was done for ABAQUS and MATLAB. The entire script was not included in this thesis, however,
some of the script that were written are included in this chapter as an example.
8.1 Example of ABAQUS Script
from abaqus import * from abaqusConstants import * session.Viewport(name='Viewport: 1', origin=(0.0, 0.0), width=199.926559448242, height=264.936126708984) session.viewports['Viewport: 1'].makeCurrent() session.viewports['Viewport: 1'].maximize() from caeModules import * from driverUtils import executeOnCaeStartup executeOnCaeStartup() openMdb('S_AluminiumPlate.cae') #: The model database "C:\Users\pseo\ABAQUS\Test Plate\S_new\S_AluminiumPlate.cae" has been opened. session.viewports['Viewport: 1'].setValues(displayedObject=None) session.viewports['Viewport: 1'].partDisplay.geometryOptions.setValues( referenceRepresentation=ON) p = mdb.models['3 Patches'].parts['Aluminium Plate'] session.viewports['Viewport: 1'].setValues(displayedObject=p) session.viewports['Viewport: 1'].setValues(displayedObject=None) o1 = session.openOdb(name='C:/Users/pseo/ABAQUS/Test Plate/S_new/S0_32x32_Vertical.odb') session.viewports['Viewport: 1'].setValues(displayedObject=o1) #: Model: C:/Users/pseo/ABAQUS/Test Plate/S_new/S0_32x32_Vertical.odb #: Number of Assemblies: 1 #: Number of Assembly instances: 0 #: Number of Part instances: 1025 #: Number of Meshes: 1025 #: Number of Element Sets: 0 #: Number of Node Sets: 1027 #: Number of Steps: 1 session.viewports['Viewport: 1'].view.setValues(cameraPosition=(-0.0395533, 0.0599148, 2.3038), cameraUpVector=(0, 1, 0)) session.viewports['Viewport: 1'].odbDisplay.commonOptions.setValues( visibleEdges=FREE) NodeSets=['SENSOR-1.BOTTOM RESPONSE','SENSOR-1-LIN-1-2.BOTTOM RESPONSE','SENSOR-1-LIN-1-3.BOTTOM RESPONSE','SENSOR-1-LIN-1-4.BOTTOM RESPONSE','SENSOR-1-LIN-1-5.BOTTOM RESPONSE','SENSOR-1-LIN-1-6.BOTTOM RESPONSE','SENSOR-1-LIN-1-7.BOTTOM RESPONSE','SENSOR-1-LIN-1-8.BOTTOM RESPONSE','SENSOR-1-LIN-1-9.BOTTOM RESPONSE','SENSOR-1-LIN-1-10.BOTTOM RESPONSE','SENSOR-1-LIN-1-11.BOTTOM RESPONSE','SENSOR-1-LIN-1-12.BOTTOM RESPONSE','SENSOR-1-LIN-1-13.BOTTOM RESPONSE','SENSOR-1-LIN-1-14.BOTTOM RESPONSE','SENSOR-1-LIN-1-15.BOTTOM RESPONSE','SENSOR-1-LIN-1-16.BOTTOM RESPONSE','SENSOR-1-LIN-1-17.BOTTOM RESPONSE','SENSOR-1-LIN-1-18.BOTTOM RESPONSE','SENSOR-1-LIN-1-19.BOTTOM
126 | P a g e
RESPONSE','SENSOR-1-LIN-1-20.BOTTOM RESPONSE','SENSOR-1-LIN-1-21.BOTTOM RESPONSE','SENSOR-1-LIN-1-22.BOTTOM RESPONSE','SENSOR-1-LIN-1-23.BOTTOM RESPONSE','SENSOR-1-LIN-1-24.BOTTOM RESPONSE','SENSOR-1-LIN-1-25.BOTTOM RESPONSE','SENSOR-1-LIN-1-26.BOTTOM RESPONSE','SENSOR-1-LIN-1-27.BOTTOM RESPONSE','SENSOR-1-LIN-1-28.BOTTOM RESPONSE','SENSOR-1-LIN-1-29.BOTTOM RESPONSE','SENSOR-1-LIN-1-30.BOTTOM RESPONSE','SENSOR-1-LIN-1-31.BOTTOM RESPONSE','SENSOR-1-LIN-1-32.BOTTOM RESPONSE', 'SENSOR-1-LIN-2-1.BOTTOM RESPONSE','SENSOR-1-LIN-2-2.BOTTOM RESPONSE','SENSOR-1-LIN-2-3.BOTTOM RESPONSE','SENSOR-1-LIN-2-4.BOTTOM RESPONSE','SENSOR-1-LIN-2-5.BOTTOM RESPONSE','SENSOR-1-LIN-2-6.BOTTOM RESPONSE','SENSOR-1-LIN-2-7.BOTTOM RESPONSE','SENSOR-1-LIN-2-8.BOTTOM RESPONSE','SENSOR-1-LIN-2-9.BOTTOM RESPONSE','SENSOR-1-LIN-2-10.BOTTOM RESPONSE','SENSOR-1-LIN-2-11.BOTTOM RESPONSE','SENSOR-1-LIN-2-12.BOTTOM RESPONSE','SENSOR-1-LIN-2-13.BOTTOM RESPONSE','SENSOR-1-LIN-2-14.BOTTOM RESPONSE','SENSOR-1-LIN-2-15.BOTTOM RESPONSE','SENSOR-1-LIN-2-16.BOTTOM RESPONSE','SENSOR-1-LIN-2-17.BOTTOM RESPONSE','SENSOR-1-LIN-2-18.BOTTOM RESPONSE','SENSOR-1-LIN-2-19.BOTTOM RESPONSE','SENSOR-1-LIN-2-20.BOTTOM RESPONSE','SENSOR-1-LIN-2-21.BOTTOM RESPONSE','SENSOR-1-LIN-2-22.BOTTOM RESPONSE','SENSOR-1-LIN-2-23.BOTTOM RESPONSE','SENSOR-1-LIN-2-24.BOTTOM RESPONSE','SENSOR-1-LIN-2-25.BOTTOM RESPONSE','SENSOR-1-LIN-2-26.BOTTOM RESPONSE','SENSOR-1-LIN-2-27.BOTTOM RESPONSE','SENSOR-1-LIN-2-28.BOTTOM RESPONSE','SENSOR-1-LIN-2-29.BOTTOM RESPONSE','SENSOR-1-LIN-2-30.BOTTOM RESPONSE','SENSOR-1-LIN-2-31.BOTTOM RESPONSE','SENSOR-1-LIN-2-32.BOTTOM RESPONSE', 'SENSOR-1-LIN-3-1.BOTTOM RESPONSE','SENSOR-1-LIN-3-2.BOTTOM RESPONSE','SENSOR-1-LIN-3-3.BOTTOM RESPONSE','SENSOR-1-LIN-3-4.BOTTOM RESPONSE','SENSOR-1-LIN-3-5.BOTTOM RESPONSE','SENSOR-1-LIN-3-6.BOTTOM RESPONSE','SENSOR-1-LIN-3-7.BOTTOM RESPONSE','SENSOR-1-LIN-3-8.BOTTOM RESPONSE','SENSOR-1-LIN-3-9.BOTTOM RESPONSE','SENSOR-1-LIN-3-10.BOTTOM RESPONSE','SENSOR-1-LIN-3-11.BOTTOM RESPONSE','SENSOR-1-LIN-3-12.BOTTOM RESPONSE','SENSOR-1-LIN-3-13.BOTTOM RESPONSE','SENSOR-1-LIN-3-14.BOTTOM RESPONSE','SENSOR-1-LIN-3-15.BOTTOM RESPONSE','SENSOR-1-LIN-3-16.BOTTOM RESPONSE','SENSOR-1-LIN-3-17.BOTTOM RESPONSE','SENSOR-1-LIN-3-18.BOTTOM RESPONSE','SENSOR-1-LIN-3-19.BOTTOM RESPONSE','SENSOR-1-LIN-3-20.BOTTOM RESPONSE','SENSOR-1-LIN-3-21.BOTTOM RESPONSE','SENSOR-1-LIN-3-22.BOTTOM RESPONSE','SENSOR-1-LIN-3-23.BOTTOM RESPONSE','SENSOR-1-LIN-3-24.BOTTOM RESPONSE','SENSOR-1-LIN-3-25.BOTTOM RESPONSE','SENSOR-1-LIN-3-26.BOTTOM RESPONSE','SENSOR-1-LIN-3-27.BOTTOM RESPONSE','SENSOR-1-LIN-3-28.BOTTOM RESPONSE','SENSOR-1-LIN-3-29.BOTTOM RESPONSE','SENSOR-1-LIN-3-30.BOTTOM RESPONSE','SENSOR-1-LIN-3-31.BOTTOM RESPONSE','SENSOR-1-LIN-3-32.BOTTOM RESPONSE', 'SENSOR-1-LIN-4-1.BOTTOM RESPONSE','SENSOR-1-LIN-4-2.BOTTOM RESPONSE','SENSOR-1-LIN-4-3.BOTTOM RESPONSE','SENSOR-1-LIN-4-4.BOTTOM RESPONSE','SENSOR-1-LIN-4-5.BOTTOM RESPONSE','SENSOR-1-LIN-4-6.BOTTOM RESPONSE','SENSOR-1-LIN-4-7.BOTTOM RESPONSE','SENSOR-1-LIN-4-8.BOTTOM RESPONSE','SENSOR-1-LIN-4-9.BOTTOM RESPONSE','SENSOR-1-LIN-4-10.BOTTOM RESPONSE','SENSOR-1-LIN-4-11.BOTTOM RESPONSE','SENSOR-1-LIN-4-12.BOTTOM RESPONSE','SENSOR-1-LIN-4-13.BOTTOM RESPONSE','SENSOR-1-LIN-4-14.BOTTOM RESPONSE','SENSOR-1-LIN-4-15.BOTTOM RESPONSE','SENSOR-1-LIN-4-16.BOTTOM RESPONSE','SENSOR-1-LIN-4-17.BOTTOM RESPONSE','SENSOR-1-LIN-4-18.BOTTOM RESPONSE','SENSOR-1-LIN-4-19.BOTTOM RESPONSE','SENSOR-1-LIN-4-20.BOTTOM RESPONSE','SENSOR-1-LIN-4-21.BOTTOM RESPONSE','SENSOR-1-LIN-4-22.BOTTOM RESPONSE','SENSOR-1-LIN-4-23.BOTTOM RESPONSE','SENSOR-1-LIN-4-24.BOTTOM
127 | P a g e
RESPONSE','SENSOR-1-LIN-4-25.BOTTOM RESPONSE','SENSOR-1-LIN-4-26.BOTTOM RESPONSE','SENSOR-1-LIN-4-27.BOTTOM RESPONSE','SENSOR-1-LIN-4-28.BOTTOM RESPONSE','SENSOR-1-LIN-4-29.BOTTOM RESPONSE','SENSOR-1-LIN-4-30.BOTTOM RESPONSE','SENSOR-1-LIN-4-31.BOTTOM RESPONSE','SENSOR-1-LIN-4-32.BOTTOM RESPONSE', 'SENSOR-1-LIN-5-1.BOTTOM RESPONSE','SENSOR-1-LIN-5-2.BOTTOM RESPONSE','SENSOR-1-LIN-5-3.BOTTOM RESPONSE','SENSOR-1-LIN-5-4.BOTTOM RESPONSE','SENSOR-1-LIN-5-5.BOTTOM RESPONSE','SENSOR-1-LIN-5-6.BOTTOM RESPONSE','SENSOR-1-LIN-5-7.BOTTOM RESPONSE','SENSOR-1-LIN-5-8.BOTTOM RESPONSE','SENSOR-1-LIN-5-9.BOTTOM RESPONSE','SENSOR-1-LIN-5-10.BOTTOM RESPONSE','SENSOR-1-LIN-5-11.BOTTOM RESPONSE','SENSOR-1-LIN-5-12.BOTTOM RESPONSE','SENSOR-1-LIN-5-13.BOTTOM RESPONSE','SENSOR-1-LIN-5-14.BOTTOM RESPONSE','SENSOR-1-LIN-5-15.BOTTOM RESPONSE','SENSOR-1-LIN-5-16.BOTTOM RESPONSE','SENSOR-1-LIN-5-17.BOTTOM RESPONSE','SENSOR-1-LIN-5-18.BOTTOM RESPONSE','SENSOR-1-LIN-5-19.BOTTOM RESPONSE','SENSOR-1-LIN-5-20.BOTTOM RESPONSE','SENSOR-1-LIN-5-21.BOTTOM RESPONSE','SENSOR-1-LIN-5-22.BOTTOM RESPONSE','SENSOR-1-LIN-5-23.BOTTOM RESPONSE','SENSOR-1-LIN-5-24.BOTTOM RESPONSE','SENSOR-1-LIN-5-25.BOTTOM RESPONSE','SENSOR-1-LIN-5-26.BOTTOM RESPONSE','SENSOR-1-LIN-5-27.BOTTOM RESPONSE','SENSOR-1-LIN-5-28.BOTTOM RESPONSE','SENSOR-1-LIN-5-29.BOTTOM RESPONSE','SENSOR-1-LIN-5-30.BOTTOM RESPONSE','SENSOR-1-LIN-5-31.BOTTOM RESPONSE','SENSOR-1-LIN-5-32.BOTTOM RESPONSE', 'SENSOR-1-LIN-6-1.BOTTOM RESPONSE','SENSOR-1-LIN-6-2.BOTTOM RESPONSE','SENSOR-1-LIN-6-3.BOTTOM RESPONSE','SENSOR-1-LIN-6-4.BOTTOM RESPONSE','SENSOR-1-LIN-6-5.BOTTOM RESPONSE','SENSOR-1-LIN-6-6.BOTTOM RESPONSE','SENSOR-1-LIN-6-7.BOTTOM RESPONSE','SENSOR-1-LIN-6-8.BOTTOM RESPONSE','SENSOR-1-LIN-6-9.BOTTOM RESPONSE','SENSOR-1-LIN-6-10.BOTTOM RESPONSE','SENSOR-1-LIN-6-11.BOTTOM RESPONSE','SENSOR-1-LIN-6-12.BOTTOM RESPONSE','SENSOR-1-LIN-6-13.BOTTOM RESPONSE','SENSOR-1-LIN-6-14.BOTTOM RESPONSE','SENSOR-1-LIN-6-15.BOTTOM RESPONSE','SENSOR-1-LIN-6-16.BOTTOM RESPONSE','SENSOR-1-LIN-6-17.BOTTOM RESPONSE','SENSOR-1-LIN-6-18.BOTTOM RESPONSE','SENSOR-1-LIN-6-19.BOTTOM RESPONSE','SENSOR-1-LIN-6-20.BOTTOM RESPONSE','SENSOR-1-LIN-6-21.BOTTOM RESPONSE','SENSOR-1-LIN-6-22.BOTTOM RESPONSE','SENSOR-1-LIN-6-23.BOTTOM RESPONSE','SENSOR-1-LIN-6-24.BOTTOM RESPONSE','SENSOR-1-LIN-6-25.BOTTOM RESPONSE','SENSOR-1-LIN-6-26.BOTTOM RESPONSE','SENSOR-1-LIN-6-27.BOTTOM RESPONSE','SENSOR-1-LIN-6-28.BOTTOM RESPONSE','SENSOR-1-LIN-6-29.BOTTOM RESPONSE','SENSOR-1-LIN-6-30.BOTTOM RESPONSE','SENSOR-1-LIN-6-31.BOTTOM RESPONSE','SENSOR-1-LIN-6-32.BOTTOM RESPONSE', 'SENSOR-1-LIN-7-1.BOTTOM RESPONSE','SENSOR-1-LIN-7-2.BOTTOM RESPONSE','SENSOR-1-LIN-7-3.BOTTOM RESPONSE','SENSOR-1-LIN-7-4.BOTTOM RESPONSE','SENSOR-1-LIN-7-5.BOTTOM RESPONSE','SENSOR-1-LIN-7-6.BOTTOM RESPONSE','SENSOR-1-LIN-7-7.BOTTOM RESPONSE','SENSOR-1-LIN-7-8.BOTTOM RESPONSE','SENSOR-1-LIN-7-9.BOTTOM RESPONSE','SENSOR-1-LIN-7-10.BOTTOM RESPONSE','SENSOR-1-LIN-7-11.BOTTOM RESPONSE','SENSOR-1-LIN-7-12.BOTTOM RESPONSE','SENSOR-1-LIN-7-13.BOTTOM RESPONSE','SENSOR-1-LIN-7-14.BOTTOM RESPONSE','SENSOR-1-LIN-7-15.BOTTOM RESPONSE','SENSOR-1-LIN-7-16.BOTTOM RESPONSE','SENSOR-1-LIN-7-17.BOTTOM RESPONSE','SENSOR-1-LIN-7-18.BOTTOM RESPONSE','SENSOR-1-LIN-7-19.BOTTOM RESPONSE','SENSOR-1-LIN-7-20.BOTTOM RESPONSE','SENSOR-1-LIN-7-21.BOTTOM RESPONSE','SENSOR-1-LIN-7-22.BOTTOM RESPONSE','SENSOR-1-LIN-7-23.BOTTOM RESPONSE','SENSOR-1-LIN-7-24.BOTTOM RESPONSE','SENSOR-1-LIN-7-25.BOTTOM RESPONSE','SENSOR-1-LIN-7-26.BOTTOM RESPONSE','SENSOR-1-LIN-7-27.BOTTOM RESPONSE','SENSOR-1-LIN-7-28.BOTTOM
128 | P a g e
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129 | P a g e
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130 | P a g e
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131 | P a g e
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132 | P a g e
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133 | P a g e
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134 | P a g e
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135 | P a g e
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136 | P a g e
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137 | P a g e
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138 | P a g e
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139 | P a g e
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140 | P a g e
25','SENSOR-1-LIN-26-26','SENSOR-1-LIN-26-27','SENSOR-1-LIN-26-28','SENSOR-1-LIN-26-29','SENSOR-1-LIN-26-30','SENSOR-1-LIN-26-31','SENSOR-1-LIN-26-32', 'SENSOR-1-LIN-27-1','SENSOR-1-LIN-27-2','SENSOR-1-LIN-27-3','SENSOR-1-LIN-27-4','SENSOR-1-LIN-27-5','SENSOR-1-LIN-27-6','SENSOR-1-LIN-27-7','SENSOR-1-LIN-27-8','SENSOR-1-LIN-27-9','SENSOR-1-LIN-27-10','SENSOR-1-LIN-27-11','SENSOR-1-LIN-27-12','SENSOR-1-LIN-27-13','SENSOR-1-LIN-27-14','SENSOR-1-LIN-27-15','SENSOR-1-LIN-27-16','SENSOR-1-LIN-27-17','SENSOR-1-LIN-27-18','SENSOR-1-LIN-27-19','SENSOR-1-LIN-27-20','SENSOR-1-LIN-27-21','SENSOR-1-LIN-27-22','SENSOR-1-LIN-27-23','SENSOR-1-LIN-27-24','SENSOR-1-LIN-27-25','SENSOR-1-LIN-27-26','SENSOR-1-LIN-27-27','SENSOR-1-LIN-27-28','SENSOR-1-LIN-27-29','SENSOR-1-LIN-27-30','SENSOR-1-LIN-27-31','SENSOR-1-LIN-27-32', 'SENSOR-1-LIN-28-1','SENSOR-1-LIN-28-2','SENSOR-1-LIN-28-3','SENSOR-1-LIN-28-4','SENSOR-1-LIN-28-5','SENSOR-1-LIN-28-6','SENSOR-1-LIN-28-7','SENSOR-1-LIN-28-8','SENSOR-1-LIN-28-9','SENSOR-1-LIN-28-10','SENSOR-1-LIN-28-11','SENSOR-1-LIN-28-12','SENSOR-1-LIN-28-13','SENSOR-1-LIN-28-14','SENSOR-1-LIN-28-15','SENSOR-1-LIN-28-16','SENSOR-1-LIN-28-17','SENSOR-1-LIN-28-18','SENSOR-1-LIN-28-19','SENSOR-1-LIN-28-20','SENSOR-1-LIN-28-21','SENSOR-1-LIN-28-22','SENSOR-1-LIN-28-23','SENSOR-1-LIN-28-24','SENSOR-1-LIN-28-25','SENSOR-1-LIN-28-26','SENSOR-1-LIN-28-27','SENSOR-1-LIN-28-28','SENSOR-1-LIN-28-29','SENSOR-1-LIN-28-30','SENSOR-1-LIN-28-31','SENSOR-1-LIN-28-32', 'SENSOR-1-LIN-29-1','SENSOR-1-LIN-29-2','SENSOR-1-LIN-29-3','SENSOR-1-LIN-29-4','SENSOR-1-LIN-29-5','SENSOR-1-LIN-29-6','SENSOR-1-LIN-29-7','SENSOR-1-LIN-29-8','SENSOR-1-LIN-29-9','SENSOR-1-LIN-29-10','SENSOR-1-LIN-29-11','SENSOR-1-LIN-29-12','SENSOR-1-LIN-29-13','SENSOR-1-LIN-29-14','SENSOR-1-LIN-29-15','SENSOR-1-LIN-29-16','SENSOR-1-LIN-29-17','SENSOR-1-LIN-29-18','SENSOR-1-LIN-29-19','SENSOR-1-LIN-29-20','SENSOR-1-LIN-29-21','SENSOR-1-LIN-29-22','SENSOR-1-LIN-29-23','SENSOR-1-LIN-29-24','SENSOR-1-LIN-29-25','SENSOR-1-LIN-29-26','SENSOR-1-LIN-29-27','SENSOR-1-LIN-29-28','SENSOR-1-LIN-29-29','SENSOR-1-LIN-29-30','SENSOR-1-LIN-29-31','SENSOR-1-LIN-29-32', 'SENSOR-1-LIN-30-1','SENSOR-1-LIN-30-2','SENSOR-1-LIN-30-3','SENSOR-1-LIN-30-4','SENSOR-1-LIN-30-5','SENSOR-1-LIN-30-6','SENSOR-1-LIN-30-7','SENSOR-1-LIN-30-8','SENSOR-1-LIN-30-9','SENSOR-1-LIN-30-10','SENSOR-1-LIN-30-11','SENSOR-1-LIN-30-12','SENSOR-1-LIN-30-13','SENSOR-1-LIN-30-14','SENSOR-1-LIN-30-15','SENSOR-1-LIN-30-16','SENSOR-1-LIN-30-17','SENSOR-1-LIN-30-18','SENSOR-1-LIN-30-19','SENSOR-1-LIN-30-20','SENSOR-1-LIN-30-21','SENSOR-1-LIN-30-22','SENSOR-1-LIN-30-23','SENSOR-1-LIN-30-24','SENSOR-1-LIN-30-25','SENSOR-1-LIN-30-26','SENSOR-1-LIN-30-27','SENSOR-1-LIN-30-28','SENSOR-1-LIN-30-29','SENSOR-1-LIN-30-30','SENSOR-1-LIN-30-31','SENSOR-1-LIN-30-32', 'SENSOR-1-LIN-31-1','SENSOR-1-LIN-31-2','SENSOR-1-LIN-31-3','SENSOR-1-LIN-31-4','SENSOR-1-LIN-31-5','SENSOR-1-LIN-31-6','SENSOR-1-LIN-31-7','SENSOR-1-LIN-31-8','SENSOR-1-LIN-31-9','SENSOR-1-LIN-31-10','SENSOR-1-LIN-31-11','SENSOR-1-LIN-31-12','SENSOR-1-LIN-31-13','SENSOR-1-LIN-31-14','SENSOR-1-LIN-31-15','SENSOR-1-LIN-31-16','SENSOR-1-LIN-31-17','SENSOR-1-LIN-31-18','SENSOR-1-LIN-31-19','SENSOR-1-LIN-31-20','SENSOR-1-LIN-31-21','SENSOR-1-LIN-31-22','SENSOR-1-LIN-31-23','SENSOR-1-LIN-31-24','SENSOR-1-LIN-31-25','SENSOR-1-LIN-31-26','SENSOR-1-LIN-31-27','SENSOR-1-LIN-31-28','SENSOR-1-LIN-31-29','SENSOR-1-LIN-31-30','SENSOR-1-LIN-31-31','SENSOR-1-LIN-31-32', 'SENSOR-1-LIN-32-1','SENSOR-1-LIN-32-2','SENSOR-1-LIN-32-3','SENSOR-1-LIN-32-4','SENSOR-1-LIN-32-5','SENSOR-1-LIN-32-6','SENSOR-1-LIN-32-7','SENSOR-1-LIN-32-8','SENSOR-1-LIN-32-9','SENSOR-1-LIN-32-10','SENSOR-1-LIN-32-11','SENSOR-1-LIN-32-12','SENSOR-1-LIN-32-13','SENSOR-1-LIN-32-14','SENSOR-1-LIN-32-15','SENSOR-1-LIN-32-16','SENSOR-1-LIN-32-17','SENSOR-1-LIN-32-18','SENSOR-1-LIN-32-19','SENSOR-1-LIN-32-20','SENSOR-1-LIN-32-21','SENSOR-1-LIN-32-22','SENSOR-1-LIN-32-23','SENSOR-1-LIN-32-24','SENSOR-1-LIN-32-25','SENSOR-1-LIN-32-26','SENSOR-1-LIN-32-27','SENSOR-1-LIN-32-28','SENSOR-1-LIN-32-29','SENSOR-1-LIN-32-30','SENSOR-1-LIN-32-31','SENSOR-1-LIN-32-32']
141 | P a g e
x_s=['x0','x1','x2','x3','x4','x5','x6','x7','x8','x9','x10','x11','x12','x13','x14','x15'] length = len(NodeSets) #counter = range(length+1) counter = range(length+1) for count in counter: odb = session.odbs['C:/Users/pseo/ABAQUS/Test Plate/S_new/S0_32x32_Vertical.odb'] session.xyDataListFromField(odb=odb, outputPosition=NODAL, variable=(('S', INTEGRATION_POINT, ((COMPONENT, 'S11'), )), ), nodeSets=( NodeSets[count], )) x0 = session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 5'.format(PatchSets[count])] x1 = session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 6'.format(PatchSets[count])] x2 = session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 7'.format(PatchSets[count])] x3 = session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 8'.format(PatchSets[count])] x4 = session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 13'.format(PatchSets[count])] x5 = session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 14'.format(PatchSets[count])] x6 = session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 15'.format(PatchSets[count])] x7 = session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 16'.format(PatchSets[count])] x8 = session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 21'.format(PatchSets[count])] x9 = session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 22'.format(PatchSets[count])] x10 = session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 23'.format(PatchSets[count])] x11 = session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 24'.format(PatchSets[count])] x12 = session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 29'.format(PatchSets[count])] x13 = session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 30'.format(PatchSets[count])] x14 = session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 31'.format(PatchSets[count])] x15 = session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 32'.format(PatchSets[count])] session.writeXYReport( fileName='C:/Users/pseo/Dropbox/PhD/2013/Microsensors_Waves/ABAQUS/S_new Post Processing/S0_32x32_Vertical/{0}_S11.txt'.format(PatchSets[count]), xyData=(x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15)) del session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 5'.format(PatchSets[count])] del session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 6'.format(PatchSets[count])] del session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 7'.format(PatchSets[count])] del session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 8'.format(PatchSets[count])] del session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 13'.format(PatchSets[count])] del session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 14'.format(PatchSets[count])] del session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 15'.format(PatchSets[count])] del session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 16'.format(PatchSets[count])] del session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 21'.format(PatchSets[count])] del session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 22'.format(PatchSets[count])] del session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 23'.format(PatchSets[count])] del session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 24'.format(PatchSets[count])] del session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 29'.format(PatchSets[count])] del session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 30'.format(PatchSets[count])] del session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 31'.format(PatchSets[count])] del session.xyDataObjects['S:S11 (Avg: 75%) PI: {0} N: 32'.format(PatchSets[count])] for count in counter:
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odb = session.odbs['C:/Users/pseo/ABAQUS/Test Plate/S_new/S0_32x32_Vertical.odb'] session.xyDataListFromField(odb=odb, outputPosition=NODAL, variable=(('S', INTEGRATION_POINT, ((COMPONENT, 'S22'), )), ), nodeSets=( NodeSets[count], )) x0 = session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 5'.format(PatchSets[count])] x1 = session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 6'.format(PatchSets[count])] x2 = session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 7'.format(PatchSets[count])] x3 = session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 8'.format(PatchSets[count])] x4 = session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 13'.format(PatchSets[count])] x5 = session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 14'.format(PatchSets[count])] x6 = session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 15'.format(PatchSets[count])] x7 = session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 16'.format(PatchSets[count])] x8 = session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 21'.format(PatchSets[count])] x9 = session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 22'.format(PatchSets[count])] x10 = session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 23'.format(PatchSets[count])] x11 = session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 24'.format(PatchSets[count])] x12 = session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 29'.format(PatchSets[count])] x13 = session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 30'.format(PatchSets[count])] x14 = session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 31'.format(PatchSets[count])] x15 = session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 32'.format(PatchSets[count])] session.writeXYReport( fileName='C:/Users/pseo/Dropbox/PhD/2013/Microsensors_Waves/ABAQUS/S_new Post Processing/S0_32x32_Vertical/{0}_S22.txt'.format(PatchSets[count]), xyData=(x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15)) del session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 5'.format(PatchSets[count])] del session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 6'.format(PatchSets[count])] del session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 7'.format(PatchSets[count])] del session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 8'.format(PatchSets[count])] del session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 13'.format(PatchSets[count])] del session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 14'.format(PatchSets[count])] del session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 15'.format(PatchSets[count])] del session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 16'.format(PatchSets[count])] del session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 21'.format(PatchSets[count])] del session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 22'.format(PatchSets[count])] del session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 23'.format(PatchSets[count])] del session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 24'.format(PatchSets[count])] del session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 29'.format(PatchSets[count])] del session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 30'.format(PatchSets[count])] del session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 31'.format(PatchSets[count])] del session.xyDataObjects['S:S22 (Avg: 75%) PI: {0} N: 32'.format(PatchSets[count])]
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8.2 Example of MATLAB Script
8.2.1 Acquiring voltage-time signal from sensors
%% Written by Posearn Seo, 2015
% This MATLAB file is used to extract data from ABAQUS and save it as a % variable for post processing.
% The file imports the "Stress (S11 & S22) vs. Time" data from the
Piezoelectric Sensor % as an input and output the "Voltage vs Time" data
clear; close ALL HIDDEN; clc; format longg; %% Step 1: Importing Stress Data from ABAQUS % Extracting data from txt files.
% Input path PathName1='C:\Users\pseo\ABAQUS\ABAQUS_Dropbox\S_new Post
Processing\S0_32x32_Horizontal_Array8\Array 32\'; % Output path PathName2='C:\Users\pseo\ABAQUS\ABAQUS_Dropbox\S_new Post Processing\Data
Collation Results\';
% Defining the number of nodes per sensor sensor_node=16; %% D1=dir(PathName1); dirIndex = [D1.isdir]; %# Find the index for directories Listfile = {D1(~dirIndex).name}'; [fileList,nn]=sort_nat(Listfile,'ascend'); %% for mm=1:length(fileList)/2
filetext1=fileList(2*mm-1); filetext2=fileList(2*mm); FileName1=char(filetext1); FileName2=char(filetext2); [FID1,xxx] = fopen([PathName1,FileName1]); FID2 = fopen([PathName1,FileName2]); S_Total_S11 = textscan(FID1, '%s','Headerlines',4); %skips 3 lines of
headings S_Total_S11 =
str2double(reshape(S_Total_S11{1},sensor_node+1,[]))'; %reshapes data
similar to the text file & converts it to number S_Total_S22 = textscan(FID2, '%s','Headerlines',4); %skips 3 lines of
headings S_Total_S22 =
str2double(reshape(S_Total_S22{1},sensor_node+1,[]))'; %reshapes data
similar to the text file & converts it to number
%% % Extracting the Time Values check1=S_Total_S11(:,1); check2=S_Total_S22(:,1);
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% For Loop to identify the repeated time values in S11 data1(1,1)=1; data2(1,1)=1; for num_1=1:length(check1)-1; if check1(num_1+1)==check1(num_1) data1(num_1+1,1)=0; else data1(num_1+1,1)=1; end end
% For Loop to identify the repeated time values in S22 for num_2=1:length(check2)-1; if check2(num_2+1)==check2(num_2) data2(num_2+1,1)=0; else data2(num_2+1,1)=1; end end
% Finding the nonzero elements and storing it in another variable new_data2=find(data2); new_data1=find(data1);
% For Loops to replace all the identified time values wanted into another
variable for num_1_1=1:length(new_data1) Modified_total_S11(num_1_1,:)=S_Total_S11(new_data1(num_1_1),:); end for num_2_1=1:length(new_data2) Modified_total_S22(num_2_1,:)=S_Total_S22(new_data2(num_2_1),:); end
% Substituting the Original S11 Time Series with the Modified Time Series clear Total_S11 Total_S22 S_Total_S11=Modified_total_S11; S_Total_S22=Modified_total_S22; clear data1 data2 check1 check2 new_data1 new_data2 num_1 num_2 num_1_1
num_2_1 Modified_total_S11 Modified_total_S22 %% Step 2: Inporting key parameters which were pre-defined
load('S_data.mat')
%% Step 3: Sum total strain, S11 & S22 % This analysis is carried out for the total response of the nodes
available on the sensor
% Summation of stress values from each time step % Do not sum the time (ie. column 1) Sum_S11 = sum(S_Total_S11(:,2:sensor_node+1),2); Sum_S22 = sum(S_Total_S22(:,2:sensor_node+1),2); Combined=S_Total_S11(:,2:sensor_node+1)+S_Total_S22(:,2:sensor_node+1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % STRAIN % % Converting Stress to Strain % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Converting stress to strain sum_strain1= Sum_S11./YM; sum_strain2= Sum_S22./YM; combined_strain=Combined./YM;
% Extracting the Time data into an array Time_total=S_Total_S11(:,1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % VOLTAGE % % Converting Strain to Voltage % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Calculating the Voltage response from the summation of strains V_total = ((d*t*YM)/(e*l*b))*(l*b)*(sum_strain1+sum_strain2); V_combined = ((d*t*YM)/(e*l*b))*(l*b)*(combined_strain);
%% Step 4: Naming & saving the variable
cc=sprintf(['V' num2str(mm)]); dd='_3mm_32of32Array_S0_32x32_Horizontal_Array8'; ee=[cc,dd]; eval([ee '=V_total;']); ff=[PathName2,ee];
save (ff,ee); fclose('all'); end
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8.2.2 Extracting wave packet of interest
%% Locating peaks so that the wave packet can be captured wave_size=input('What is the size of the wave packet?\n')/2; for i_count=1:numberofarray % Array number env_name=sprintf(['env' num2str(i_count)]); env=eval(env_name); for j_count=1:patches_in_an_array % Patch number peak_time1 = peakfinder(env(:,j_count));% Array 1, sensor 1 if peak_time1(1) < wave_size if length(peak_time1)==1 peak_time1=peak_time1(1); else peak_time1=peak_time1(2); end
else peak_time1=peak_time1(1); end peak_time(i_count,j_count)=peak_time1; % Array-row, % Patch-column end end
%% Extracting the wave packet that is of interest
for counter_j=1:numberofarray; aaak1=eval(['V_pig' num2str(counter_j)]); [rr1,cc1]=size(aaak1); ak1=zeros(rr1,cc1); [rr2,cc2]=size(ak1);
for counter_i=1:cc1 yye1=aaak1(:,counter_i); sizing=size(yye1); yye1_max=peak_time(counter_j,counter_i);
if yye1_max<=wave_size yye1_capture_max=yye1(1:yye1_max+wave_size); ak1(1:length(yye1_capture_max)-1,counter_i)=yye1_capture_max; elseif yye1_max>=sizing(1)-wave_size yye1_capture_max=yye1(yye1_max-wave_size:end); ak1(yye1_max-wave_size:yye1_max-wave_size+length(yye1_capture_max)-
1,counter_i)=yye1_capture_max; else yye1_capture_max=yye1(yye1_max-wave_size:yye1_max+wave_size); ak1(yye1_max-wave_size:yye1_max-wave_size+length(yye1_capture_max)-
1,counter_i)=yye1_capture_max; end de1=round(rr2/2-length(yye1_capture_max)/2); end magic_str = ['V_pig',int2str(counter_j),' = ak1;']; eval(magic_str); end clear aaak1 ak1 yye1 %% Plotting the time series from all arrays
Time_total=Time_total(1:rr2);
% Peak across different arrays figure(2)
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% Array 1 subplot(8,1,1), plot(Time_total,V_pig1(:,1),'b'); hold on; subplot(8,1,1),plot(Time_total,env1(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 1st Array')
% Array 2 subplot(8,1,2), plot(Time_total,V_pig2(:,1),'b'); hold on; subplot(8,1,2),plot(Time_total,env2(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 2nd Array')
% Array 3 subplot(8,1,3), plot(Time_total,V_pig3(:,1),'b'); hold on; subplot(8,1,3),plot(Time_total,env3(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 3rd Array') % Array 4 subplot(8,1,4), plot(Time_total,V_pig4(:,1),'b'); hold on; subplot(8,1,4),plot(Time_total,env4(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 4th Array')
% Array 5 subplot(8,1,5), plot(Time_total,V_pig5(:,1),'b'); hold on; subplot(8,1,5),plot(Time_total,env5(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 5th Array')
% Array 6 subplot(8,1,6), plot(Time_total,V_pig6(:,1),'b'); hold on; subplot(8,1,6),plot(Time_total,env6(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 6th Array')
% Array 7 subplot(8,1,7), plot(Time_total,V_pig7(:,1),'b'); hold on; subplot(8,1,7),plot(Time_total,env7(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 7th Array')
% Array 8 subplot(8,1,8), plot(Time_total,V_pig8(:,1),'b'); hold on; subplot(8,1,8),plot(Time_total,env8(:,1),'r','linewidth',2); grid minor
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xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 8th Array')
suptitle('Total response node for 1st Sensor and Last sensor on the 1st
Array')
figure(3) % Array 9 subplot(8,1,1), plot(Time_total,V_pig9(:,1),'b'); hold on; subplot(8,1,1),plot(Time_total,env9(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 9th Array')
% Array 10 subplot(8,1,2), plot(Time_total,V_pig10(:,1),'b'); hold on; subplot(8,1,2),plot(Time_total,env10(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 10th Array')
% Array 11 subplot(8,1,3), plot(Time_total,V_pig11(:,1),'b'); hold on; subplot(8,1,3),plot(Time_total,env11(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 11th Array')
% Array 12 subplot(8,1,4), plot(Time_total,V_pig12(:,1),'b'); hold on; subplot(8,1,4),plot(Time_total,env12(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 12th Array')
% Array 13 subplot(8,1,5), plot(Time_total,V_pig13(:,1),'b'); hold on; subplot(8,1,5),plot(Time_total,env13(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 13th Array')
% Array 14 subplot(8,1,6), plot(Time_total,V_pig14(:,1),'b'); hold on; subplot(8,1,6),plot(Time_total,env14(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 14th Array')
% Array 15 subplot(8,1,7), plot(Time_total,V_pig15(:,1),'b'); hold on; subplot(8,1,7),plot(Time_total,env15(:,1),'r','linewidth',2); grid minor
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xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 15th Array')
% Array 16 subplot(8,1,8), plot(Time_total,V_pig16(:,1),'b'); hold on; subplot(8,1,8),plot(Time_total,env16(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 16th Array') suptitle('Response only on the 1st sensor of each Array')
figure(4) % Array 17 subplot(8,1,1), plot(Time_total,V_pig17(:,1),'b'); hold on; subplot(8,1,1),plot(Time_total,env17(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 17th Array')
% Array 18 subplot(8,1,2), plot(Time_total,V_pig18(:,1),'b'); hold on; subplot(8,1,2),plot(Time_total,env18(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 18th Array')
% Array 19 subplot(8,1,3), plot(Time_total,V_pig19(:,1),'b'); hold on; subplot(8,1,3),plot(Time_total,env19(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 19th Array')
% Array 20 subplot(8,1,4), plot(Time_total,V_pig20(:,1),'b'); hold on; subplot(8,1,4),plot(Time_total,env20(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 20th Array')
% Array 21 subplot(8,1,5), plot(Time_total,V_pig21(:,1),'b'); hold on; subplot(8,1,5),plot(Time_total,env21(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 21th Array')
% Array 22 subplot(8,1,6), plot(Time_total,V_pig22(:,1),'b'); hold on; subplot(8,1,6),plot(Time_total,env22(:,1),'r','linewidth',2); grid minor xlabel('Time,sec')
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ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 22th Array')
% Array 23 subplot(8,1,7), plot(Time_total,V_pig23(:,1),'b'); hold on; subplot(8,1,7),plot(Time_total,env23(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 23th Array')
% Array 24 subplot(8,1,8), plot(Time_total,V_pig24(:,1),'b'); hold on; subplot(8,1,8),plot(Time_total,env24(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 24th Array') suptitle('Response only on the 1st sensor of each Array')
figure(5) % Array 25 subplot(8,1,1), plot(Time_total,V_pig25(:,1),'b'); hold on; subplot(8,1,1),plot(Time_total,env25(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 25th Array')
% Array 26 subplot(8,1,2), plot(Time_total,V_pig26(:,1),'b'); hold on; subplot(8,1,2),plot(Time_total,env26(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 26th Array')
% Array 27 subplot(8,1,3), plot(Time_total,V_pig27(:,1),'b'); hold on; subplot(8,1,3),plot(Time_total,env27(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 27th Array')
% Array 28 subplot(8,1,4), plot(Time_total,V_pig28(:,1),'b'); hold on; subplot(8,1,4),plot(Time_total,env28(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 28th Array')
% Array 29 subplot(8,1,5), plot(Time_total,V_pig29(:,1),'b'); hold on; subplot(8,1,5),plot(Time_total,env29(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 29th Array')
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% Array 30 subplot(8,1,6), plot(Time_total,V_pig30(:,1),'b'); hold on; subplot(8,1,6),plot(Time_total,env30(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 30th Array')
% Array 31 subplot(8,1,7), plot(Time_total,V_pig31(:,1),'b'); hold on; subplot(8,1,7),plot(Time_total,env31(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 31th Array')
% Array 32 subplot(8,1,8), plot(Time_total,V_pig32(:,1),'b'); hold on; subplot(8,1,8),plot(Time_total,env32(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for 1st Sensor on the 32th Array') suptitle('Response only on the 1st sensor of each Array') end
% Peak across same array but different sensors figure(21) subplot(2,1,1), plot(Time_total,V_pig32(:,1),'b'); hold on; subplot(2,1,1),plot(Time_total,env32(:,1),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V')
subplot(2,1,2), plot(Time_total,V_pig32(:,nod),'b'); hold on; subplot(2,1,2),plot(Time_total,env32(:,nod),'r','linewidth',2); grid minor xlabel('Time,sec') ylabel('Total Voltage, V') suptitle('Total response node for 1st Sensor and Last sensor on the 1st
Array')
else figure(2) plot(Time_total,sum(V_pig,2),'r') xlabel('Time,sec') ylabel('Total Voltage, V') title('Total response node for all sensors') grid minor suptitle('Comparing the Voltages between analysing 1 node and all nodes
[Symmetrical Loading: Center of Plate]') hold off end
warning=msgbox('Next Analysis: Analysing the difference in response between
2 different location of load. Press OK to continue analysis.','Click to
continue...','help') ; uiwait(warning);
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8.2.3 2D Fast Fourier Transform
%% Grouping the matrices into one big matrix for 2D FFT
if array==2 && numberofarray==3 UU=cat(3,u1,u2,u3); elseif array ==2 && numberofarray==5 UU=cat(3,u1,u2,u3,u4,u5);
elseif array ==2 && numberofarray==8 UU=cat(3,u1,u2,u3,u4,u5,u6,u7,u8);
elseif array ==2 && numberofarray==32
UU=cat(3,u1,u2,u3,u4,u5,u6,u7,u8,u9,u10,u11,u12,u13,u14,u15,u16,u17,u18,u19
,u20,u21,u22,u23,u24,u25,u26,u27,u28,u29,u30,u31,u32); end
Nt=(2^13); Nx=(2^13); Fs=1/(t_rang(3)-t_rang(2)); Freq = Fs/2*linspace(0,1,Nt/2+1);
% 2D FFT in the vertical direction array_number=input('Number of arrays=\n'); Freq_reference=input('What is the excitation frequency? (Hz) \n='); hold on for counter=1:array_number
% Spatial resolution if patch_size == 3 Fx=(32/0.313); % 32 patches @ 31.3cm apart (3mm sensor) elseif patch_size == 4 Fx=(32/0.314); % 32 patches @ 31.4cm apart (4mm sensor) elseif patch_size == 5 Fx=(32/0.315); % 32 patches @ 31.5cm apart (5mm sensor) elseif patch_size == 3 && numberofarray==8 Fx=(16/0.153); % 16 patches @ 15.3cm apart (3mm sensor) elseif patch_size == 3 && numberofarray==32 Fx=(32/0.31); % 32 patches @ 1cm apart each(3mm sensor)
else error('Re-check MATLAB code') end
K=Fx*2*pi/2*linspace(0,1,Nx/2+1); Y_UU =fftshift(abs(fftn(UU(:,:,counter),[Nt Nx]))/(numel(UU))); Yr_UU=Y_UU((end/2):end,1:end/2+1); Yrr4_UU=fliplr(Yr_UU);
figscl=1.2; factor_c=1.6;
if array==2,
% Locating only Centre Frequency
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Freq_left=Freq(Freq<=Freq_reference); Freq_right=Freq(Freq>=Freq_reference); Freq_test=[Freq_left(end),Freq_right(1)]; Freq_diff=Freq_test-Freq_reference; Freq_min=min(Freq_diff); if Freq_test(1)==Freq_test(2) ggggg(counter,1)=Freq(length(Freq_left)); [xa wavenumber] = max(Yrr4_UU(:,length(Freq_left))); ggggg(counter,2)=K(wavenumber); elseif Freq_min==1 ggggg(counter,1)=Freq(length(Freq_left)); [xa wavenumber] = max(Yrr4_UU(:,length(Freq_left))); ggggg(counter,2)=K(wavenumber); else ggggg(counter,1)=Freq(length(Freq_left)+1); [xa wavenumber] = max(Yrr4_UU(:,length(Freq_left)+1)); ggggg(counter,2)=K(wavenumber); end
else
% If only one array is considered Y =fftshift(abs(fftn(u1,[Nt Nx]))/(numel(u1))); Yr=Y((end/2):end,1:end/2+1); Yrr4=fliplr(Yr);
end end
% Displaying results for iiiii=1:length(ggggg) fprintf('Array %d has wavenumber of %f \n',iiiii,ggggg(iiiii,2)) fprintf('Array %d has wavelength of %0.2fcm \n',iiiii,
(2*pi*100)/(ggggg(iiiii,2))) end
figure(10) legend_name=cell(length(ggggg),1); hold on plotStyle =
{'r*','ro','rh','rx','r.','rs','rd','r^','rv','r>','r<','b*','bo','bh','bx'
,'b.','bs','bd','b^','bv','b>','b<','k*','ko','kh','kx','k.','ks','kd','k^'
,'kv','k>'}; for iiiii=1:length(ggggg) random_colour=[{rand rand rand}]; plot(ggggg(iiiii,1)/1000,ggggg(iiiii,2),plotStyle{iiiii}, 'MarkerSize', 8); legend_name{iiiii}=['Result from Array',num2str(iiiii)];
end xlabel('Frequency (kHz)'); ylabel('Wavenumber (rad/m)'); c_map=[1 1 1;0 0 0.5;0 0 0.8;0 0 1;1 1 0;0.9 1 0;1 1 0;1
0 0;1 0 0;1 0 0;1 0 0;0.9 0 0;0.7 0 0;0.5 0 0;0.5 0 0;0.5 0 0]; colormap(c_map)
%Importing known data r_data=importdata('C:\Users\pseo\ABAQUS\ABAQUS_Dropbox\Dispersion
Curves\2mm.txt','',10000000);
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jj=4; for ii=jj:jj+333 kk=r_data(ii); s_0_x(ii-jj+1,:)=str2num(char(kk)); end
jj=342; for ii=jj:jj+330 kk=r_data(ii); a_0_x(ii-jj+1,:)=str2num(char(kk)); end
hold on p_1=plot(a_0_x(:,1)/1000,a_0_x(:,2)*pi*2); set(p_1,'Color','red','LineWidth',2); p_2=plot(s_0_x(:,1)/1000,s_0_x(:,2)*pi*2); set(p_2,'Color','green','LineWidth',2) ;
xsam=linspace(0,5000,1000); ysam=2*pi*xsam/3.2; p_3=plot(xsam,ysam); set(p_3,'Color','blue','LineWidth',2) ;
hold all
legend_line1=['A0']; legend_line2=['S0']; legend_line3=['SH0']; legend_name=vertcat(legend_name,legend_line1,legend_line2,legend_line3); legend(legend_name) ylim([0 1000]); xlim([0 500]); xlabel('Frequency (kHz)','FontSize',16) ylabel('Wave number (rad/m)','FontSize',16) set(gca,'fontsize',12)
title(['2D FFT Results measured vertically from all array sensors (',
waves,'-Excited from the ', loc, ' Region along Array ',
num2str(line),' )'],'FontSize',14) axis([0 1000 0 1000])
shading flat
%% Verifying wave mode if locate==1 expect=input('Which is the expected array? = \n'); Expect_2D=(['u' num2str(expect)]); Expect_2D=eval(Expect_2D);
Y_Expect =fftshift(abs(fftn(Expect_2D,[Nt Nx]))/(numel(UU))); Yr_Expect=Y_Expect((end/2):end,1:end/2+1); Yrr4_Expect=fliplr(Yr_Expect);
figscl=1.2; factor_c=1.6;
hold on
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figure contourf(Freq/1000,K,abs(Yrr4_Expect)/numel(Expect_2D)); xlabel('Frequency (kHz)'); ylabel('Wavenumber (rad/m)'); c_map=[1 1 1;0 0 0.5;0 0 0.8;0 0 1;1 1 0;0.9 1 0;1 1 0;1 0 0;1 0 0;1 0
0;1 0 0;0.9 0 0;0.7 0 0;0.5 0 0;0.5 0 0;0.5 0 0]; axis([0 1000 0 1000]) colormap(flipud(gray)); shading flat set(gca,'YDir','normal');
%Importing known data r_data=importdata('C:\Users\pseo\ABAQUS\ABAQUS_Dropbox\Dispersion
Curves\2mm.txt','',10000000);
jj=4; for ii=jj:jj+333 kk=r_data(ii); s_0_x(ii-jj+1,:)=str2num(char(kk)); end
jj=342; for ii=jj:jj+330 kk=r_data(ii); a_0_x(ii-jj+1,:)=str2num(char(kk)); end
hold on p_1=plot(a_0_x(:,1)/1000,a_0_x(:,2)*pi*2); set(p_1,'Color','red','LineWidth',2); p_2=plot(s_0_x(:,1)/1000,s_0_x(:,2)*pi*2); set(p_2,'Color','green','LineWidth',2) ;
xsam=linspace(0,5000,1000); ysam=2*pi*xsam/3.2; p_3=plot(xsam,ysam); set(p_3,'Color','blue','LineWidth',2) ;
hold all
legend('Results','A0','S0','SH0') ylim([0 1000]); xlim([0 500]); title(['2D FFT to verify the mode of the predicted Array which is found
to be along Array ',num2str(expect)],'FontSize',14) xlabel('Frequency (kHz)','FontSize',16) ylabel('Wave number (rad/m)','FontSize',16) set(gca,'fontsize',12) end %% clear u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 u11 u12 u13 u14 u15 u16 u17 u18 u19
u20 u21 u22 u23 u24 u25 u26 u27 u28 u29 u30 u31 u32; clear Y_UU Yr_UU Yrr4_UU ysam xsam
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8.2.4 Guide for Source Location & Prediction
%% Calibration Region1={'Vertical Region','Horizontal Region','Diagonal Region'}; [Region,OK1]=listdlg('PromptString','Expected Source
Region','SelectionMode','single','ListString',Region1);
if Region==1 [K_max,Loc_K_max]=max(ggggg(:,2)); figure; % plot((expect-5:expect+5),ggggg(expect-5:expect+5,2),'b-o') plot((1:32),ggggg(1:32,2),'b-o') set(gca,'XGrid','on','XTick',[1:1:length(ggggg(:,2))]) title('Trend of wavenumber across all array measured
vertically','FontSize',14 ) xlabel('Array','FontSize',16) ylabel('Wavenumber','FontSize',16) elseif Region==2 [K_max,Loc_K_max]=max(ggggg_hor(:,2)); figure; plot((expect-5:expect+5),ggggg_hor(expect-5:expect+5,2),'b-o') % plot((1:32),ggggg_hor(1:32,2),'b-o') set(gca,'XGrid','on','XTick',[1:1:length(ggggg_hor(:,2))]) title('Trend of wavenumber across all array measured
horizontally','FontSize',14) xlabel('Array','FontSize',16) ylabel('Wavenumber','FontSize',16) elseif Region==3 [K_max,Loc_K_max]=max(ggggg_2(:,2)); figure; plot((1:32),ggggg_2(1:32,2),'b-o') set(gca,'XGrid','on','XTick',[1:1:length(ggggg_2(:,2))]) title('Trend of wavenumber across all array measured
diagonally','FontSize',14) xlabel('Array','FontSize',16) ylabel('Wavenumber','FontSize',16) else disp('Error') end
if line==16 cal_del_K=abs(ggggg(Loc_K_max,2)-ggggg(:,2)); dist_to_first_row=input('What is the distance of source to first
sensor?'); % Using Array16 as Calibration for cal_i=1:32 angle16_Row1(cal_i,1)=atand(abs(cal_i-16)/dist_to_first_row); angle16_Row16(cal_i,1)=atand(abs(cal_i-16)/dist_to_first_row+15); angle16_Row32(cal_i,1)=atand(abs(cal_i-16)/dist_to_first_row+31); end
calibration_Row1=fit(cal_del_K,angle16_Row1,'exp2'); calibration_Row16=fit(cal_del_K,angle16_Row16,'exp2'); calibration_Row32=fit(cal_del_K,angle16_Row32,'exp2');
else del_K=abs(ggggg(Loc_K_max,2)-ggggg(:,2)); end load('CalibrationNEW200000_using_Array16.mat')
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figure; hold on plot(calibration_Row1,'b',cal_del_K(1:16),angle16_Row1,'bo') plot(calibration_Row16,'r',cal_del_K(1:16),angle16_Row16,'ro') plot(calibration_Row32,'k',cal_del_K(1:16),angle16_Row32,'ko') xlabel('Del K', 'FontSize', 16) ylabel('Angle', 'FontSize', 16) title('Wave Propagation Angle vs Change in Wavenumber, Del K', 'FontSize',
14) legend('Row1','Fit1','Row16','Fit16','Row32','Fit32') set(gca,'XGrid','on','YGrid','on') hold off
predict_with_Row1=calibration_Row1(del_K); predict_with_Row16=calibration_Row16(del_K); predict_with_Row32=calibration_Row32(del_K); cal_row=[1,16,32]; predict=cat(2,predict_with_Row1,predict_with_Row16,predict_with_Row32); prediction1=cat(2,predict_with_Row1(expect-
5:expect+5),predict_with_Row16(expect-
5:expect+5),predict_with_Row32(expect-5:expect+5)); prediction=cat(1,cal_row,prediction1); %% Source Location prediction
% Plate is rotated 90deg CCW compared to ABAQUS simulation
es_r=1; % element size in mm row es_c=1; % element size in mm column l_r=800; l_c=800;
gaps_patch=10; size_patch=3; no_patch=32;
% Defining (1,1): measured from bottom & left side of plate % Values shift the plate r_first_array=240; c_first_array=75;
col_scale_array=50; col_scale_source=70; col_scale_others=10; col_scale_pre_source=100;
%Source location
% Reference(1,1) is top left with plate rotated 90deg CCW pos_x=input('Location of source measured from vertical region (Array
number): For referencing \n='); pos_y=input('Location of source measured from horizontal region (Array
number): For referencing \n='); source_reference_coordinate_on_array=[pos_x,pos_y]; %Which row and which
column as reference
% Measurement Reference source_r_distance_from_reference=0; %+ve number go up, -ve number go down source_ref_dist=input('Distance of source measured from reference(mm)\n=');
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source_c_distrance_from_reference=source_ref_dist; %+ve number go right, -
ve number go left
% Using det K vs angle to locate source % First row = location of reference array (eg. locating source using which % array - 1,2,16 and etc) det_K_matrix=prediction; %Load your angle matrix after finding it using
det K matrix and keep the first line as the column or row away from the
source det_K_row=det_K_matrix(1,:); det_K_result=det_K_matrix(2:end,:);
% [det_K_min_r,det_K_min_c]=find(det_K_result==min(min(det_K_result))); [det_K_min_r,det_K_min_c]=find(predict==min(min(predict)));
c_array=zeros(no_patch*gaps_patch-
gaps_patch+size_patch,no_patch*gaps_patch-gaps_patch+size_patch); counter_ii=1; counter_jj=1; while counter_jj<=no_patch*gaps_patch-gaps_patch+size_patch; while counter_ii<=no_patch*gaps_patch-gaps_patch+size_patch
c_array(counter_ii:counter_ii+size_patch-
1,counter_jj:counter_jj+size_patch-1)=col_scale_array; counter_ii=counter_ii+gaps_patch; end counter_jj=counter_jj+gaps_patch; counter_ii=1; end A_plate=zeros(l_r,l_c); sizing_c_array=size(c_array); A_plate(r_first_array:r_first_array+sizing_c_array(1)-
1,c_first_array:c_first_array+sizing_c_array(2)-1)=c_array;
[ar1,ac1]=find(A_plate); det_ar1=ar1(2:end)-ar1(1:end-1); det_ac1=ac1(2:end)-ac1(1:end-1); number_ar1=1; counter_ii=1; for counter_kk=2:length(ar1); if det_ar1(counter_kk-1)==1 number_ar1(counter_kk)=counter_ii; else counter_ii=counter_ii+1; number_ar1(counter_kk)=counter_ii; end end counter_ii=1; number_ac1=1; for counter_kk=2:length(ac1) if det_ac1(counter_kk-1)<2 number_ac1(counter_kk)=counter_ii; else counter_ii=counter_ii+1; number_ac1(counter_kk)=counter_ii; end end % A_plate(500:500+size_patch,500:500+size_patch)=15; number_ar1(number_ar1>no_patch)=mod(number_ar1(number_ar1>no_patch),no_patc
h);
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% number_ac1(number_ac1>32)=mod(number_ac1(number_ac1>32),32); number_ar1(number_ar1==0)=no_patch; % number_ac1(number_ac1==0)=32; coordinate_setting=[number_ar1',number_ac1']; % array number in row and in
column cc_coordinate_setting=[ar1,ac1]; %real coordinate on plate
kk1=find(coordinate_setting(:,1)==source_reference_coordinate_on_array(1)&
coordinate_setting(:,2)==no_patch+1-
source_reference_coordinate_on_array(2)); % kk2=find(coordinate_setting(:,2)==source_coordinate(2)); kk2=median(kk1); refer_point=cc_coordinate_setting(kk2,:);
%source point A_plate(refer_point(1)+source_r_distance_from_reference-
2:refer_point(1)+source_r_distance_from_reference+2,refer_point(2)+source_c
_distrance_from_reference-
2:refer_point(2)+source_c_distrance_from_reference+2)=col_scale_source; A_plate(A_plate==0)=col_scale_others;
% Find Predicted Source for counter_jj=1:length(det_K_row) magic_strr=['A_plate_',int2str(counter_jj),'=A_plate;']; eval(magic_strr); magic_strr=['array_1=det_K_result(:,counter_jj);']; eval(magic_strr); af1=eval(['A_plate_' num2str(counter_jj)]); af2=array_1; % counter_ii=1; for counter_ii=1:length(af2) kk1=find(coordinate_setting(:,1)==counter_ii&
coordinate_setting(:,2)==no_patch+1-det_K_row(counter_jj)); kk2=median(kk1); refer1=cc_coordinate_setting(kk2,:); kk3=find(coordinate_setting(:,1)==det_K_min_r&
coordinate_setting(:,2)==no_patch+1-det_K_row(counter_jj)); kk4=median(kk3); refer0=cc_coordinate_setting(kk4,:); diff_distance=sqrt((refer1(1)-refer0(1))^2+(refer1(2)-refer0(2))^2); pred_distance=round(diff_distance/tand(array_1(counter_ii)));
magic_strr=['pred_coordinate',int2str(det_K_row(counter_jj)),'(counter_ii,:
)=refer0;'];
magic_strr1=['pred_coordinate',int2str(det_K_row(counter_jj)),'(counter_ii,
2)=refer0(2)+pred_distance;']; eval(magic_strr);eval(magic_strr1); af1(refer0(1)-1:refer0(1)+1,refer0(2)+pred_distance-
1:refer0(2)+pred_distance+1)=col_scale_pre_source; end magic_strr=['A_plate_',int2str(det_K_row(counter_jj)),'=af1;']; eval(magic_strr); figure;surf(eval(['A_plate_',int2str(det_K_row(counter_jj))]),'LineStyle','
none');az = 0;el = 90; view(az, el); alpha(0.7) title(['Prediction using Row '
int2str(det_K_row(counter_jj))],'FontSize',14) end