ABSTRACT Ultrasonic Damage Imaging. (Under the direction

ABSTRACT

CHANG, CHE-YUAN. Chirp Signal for Wave Dispersion Relationships and Nonlinear Ultrasonic Damage Imaging. (Under the direction of Dr. Fuh-Gwo Yuan).

The aim of this research is to investigate the feasibility of utilizing chirp signal for

examining wave dispersion relationships and ultrasonic damage imaging based on linear and

nonlinear phenomenon upon isotropic and carbon fiber reinforced polymer (CFRP) plates

with a system of piezoelectric transducer (PZT) and laser Doppler vibrometer (LDV).

Accurate determination of the dispersion curves is a prerequisite to optimize the wave mode

selection in ultrasonic wave-based nondestructive inspection technology. If exact material

properties are not known in priori, elastic equations are only ideal models.

Guided wave dispersion curves in isotropic and anisotropic materials can be extracted

experimentally by matrix pencil (MP) method. A transducer emits a broadband excitation,

linear chirp signal to generate guided waves in the plate. The propagating waves are

measured at discrete locations along the lines for LDV. The measurements are first Fourier

transformed into either wavenumber-time k-t domain or space-frequency x-ω domain. The

matrix pencil method is then employed to extract the dispersion curves in different wave

modes simultaneously. In this research, the inspections for dispersion relations on aluminum

plates with different thickness are demonstrated and compared by two-dimensional Fourier

transform (2-D FFT). Another experiment on composite plate is analyzed by MP method.

The results are confirmed by three-dimensional (3-D) theoretical curves computed

numerically. It shows that the MP method is not only accuracy for distinguishing the

dispersion curves on isotropic material, but also on anisotropic and laminated materials.

Another approach is presented for determining a dispersion curve of a thin plate

continuously by using chirplet transform (CT). Time-frequency representations are widely

used to characterize dispersive waves. Wavelet transform (WT) as one of them has been

employed to narrowband excitation signals for mapping a single point on the dispersion

curve under each excitation. For linear chirp signals, although the response to the linear

medium can be considered as a linear superposition of the excitation narrowband signals, the

WT cannot apply to analyze each response signal as narrowband signal and then

superimposed. In this research, the peak of the magnitude of the CT in time-frequency

domain is related to the time of arrival time of the group velocity. Experiments are performed

by a PZT and LDV at two locations on thin plates comprising of an aluminum plate and a

laminated composite. The study demonstrates that the dispersion curve can be effectively

deduced by only one excitation with a broadband chirp signal.

The other purpose is to examine vibration patterns and imaging delamination using

zero-lag cross-correlation (ZLCC) imaging condition on a carbon fiber reinforced polymer

(CFRP) plate with a barely visible impact damage (BVID). Based on linear and nonlinear

behaviors at local defect resonance (LDR) frequency, images of damage and delamination

are enhanced. LDR as resonant ultrasound spectroscopy of defects opens up an opportunity

to detect and visualize damages in material. A linear chirp provides a linear relation of

frequency and a wideband signal for excitation, which enables analyzing vibration patterns

and wave propagating as sinusoidal waves in multiple frequencies. A simple case of an

aluminum plate with a flat bottom hole (FBH) is investigated first in experiment and later a

CFRP plate with a BVID is examined. LDR on a laminated CFRP plate with defects not only

shows linear behavior but also induces nonlinearity due to nonlinear mechanism in the

material. The experimental investigation shows good agreements from the ZLCC imaging

condition and LDR methods, demonstrating a strong capability of using chirp signal

excitation to locate and map the geometry of BVID in thin composite structures.

© Copyright 2017 Che-Yuan Chang

All Rights Reserved

Chirp Signal for Wave Dispersion Relationships and Nonlinear Ultrasonic Damage Imaging

byChe-Yuan Chang

A dissertation submitted to the Graduate Faculty ofNorth Carolina State University

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

Mechanical Engineering

Raleigh, North Carolina

2017

APPROVED BY:

_______________________________ _______________________________Dr. Fuh-Gwo Yuan Dr. Xiaoning JiangCommittee Chair

_______________________________ _______________________________Dr. Yun Jing Dr. Larry Silverberg

ii

DEDICATION

To my beloved wife Mu-Shan Hu

Two sons Frederick and Everett

My parents who have many supports

And

All those who inspired me

iii

BIOGRAPHY

Che-Yuan attended the National Taiwan University and graduated in 2006 with a BS

in mechanical engineering. Following he decided to keep pursuing his graduate level

research at NTU and obtained his Master of Science in Manufacturing Group of Mechanical

Engineering near the middle of 2008. After graduation, he started to work in industry as a

Research and Mechanical Design Engineer for hinges design of laptops and PC about 3

years. In August 2012, Che-Yuan joined the research group of Dr. Fuh-Gwo Yuan and

began his Ph.D. study in Raleigh, NC. One year later, he moved to Hampton, Virginia where

he and his labmate initiated a new lab for Integrated Structure Health Management

Laboratory in National Institute of Aerospace. His research interests includes nondestructive

inspection, structure health monitoring, signal processing, dispersion curves and wave

propagation on isotropic and complex materials. More specifically, he is applying wideband

signal for signal processing to examine the feasibility for dispersion relationships and

nonlinear phenomena on complex structures with nondestructive inspection methods. He is

also trying to obtain the ultrasonic imaging of barely visible impact damage through

measured data from noncontact sensing system.

iv

ACKNOWLEDGMENTS

In the beginning, I would like to express my gratitude to my adviser Dr. Fuh-Gwo

Yuan for his patient guidance, profound insight, professional attitude and dedication towards

researches. Thank you for your support, countless advice and timely response to all my

questions. I would also like to thank my committee members: Dr. Xiaoning Jiang, Dr. Yun

Jing, and Dr. Larry Silverberg for your time, advises and helps in completing my dissertation.

Thanks to National Institute of Aerospace for all supports in financial of scholarship, courses,

and life helps. I spend meaningful time in NIA, Virginia.

Additionally, I would also like to thank the members, graduates and friends in our

research group at North Carolina State University: Dr. Yan, Dr. He, Dr. Harb, Donato, Chao

Wan, Ni Sui, Howuk, Karthik, and Sakib; Labmate in NIA, Tyler, Patrick, Dongwon, Abel,

Huan-Yu, and Yu-Sheng. Thank you for all the helpful supports I had with you.

My gratitude also goes to my dear parents and sisters, who has unconditionally loved

and supported me. You are always supporting and encouraging me with your best wishes.

Finally, my deepest gratitude goes to my beloved wife, Mu-Shan Hu, who stands by

me all the time with her endless love and takes cares of me and two naughty sons, Frederick

and Everett, days and nights. With her smiles and happiness, I am able to go through good

times and bad in Ph.D. life.

Che-Yuan ChangOctober 2017 in Hampton, VA

v

TABLE OF CONTENTS

LIST OF TABLES.................................................................................................................viiLIST OF FIGURES..............................................................................................................viiiCHAPTER 1 Introduction ...................................................................................................1

1.1 Purpose of Research..................................................................................................11.2 Lamb Waves in Plates...............................................................................................91.3 Chirp Signal ............................................................................................................161.4 Nonlinear Ultrasonic...............................................................................................21

1.4.1 Nonlinear Properties of Damage....................................................................231.4.2 Nonlinear Ultrasonic Methods for Damage Detection ..................................30

1.5 Thesis Outlines .......................................................................................................38CHAPTER 2 Lamb Wave Propagation Analysis ............................................................41

2.1 Waves in Isotropic Materials ..................................................................................412.1.1 Dispersion relationship representation...........................................................412.1.2 Phase and Group Velocities ...........................................................................44

2.2 Waves in Composites..............................................................................................482.2.1 Elasticity equations for wave propagation .....................................................502.2.2 Lamb wave in a composite lamina.................................................................532.2.3 Lamb waves in a composite laminate ............................................................562.2.4 Computation process for obtaining dispersion relation in composite............592.2.5 Velocity dispersions and characteristic wave curves .....................................60

2.3 Dispersion Relationship Analysis...........................................................................642.3.1 Fourier Transform and Dispersion Curve ......................................................642.3.2 Dispersion Curve Analysis.............................................................................662.3.3 2-D Fourier Transform...................................................................................672.3.4 Time-frequency Analysis ...............................................................................69

CHAPTER 3 Lamb Wave Dispersion Analysis by Matrix Pencil Method ...................743.1 Matrix Pencil (MP) Method....................................................................................74

3.1.1 Matrix Pencil Method To A Sum Of Complex Exponentials ........................743.1.2 Matrix Pencil Algorithm ................................................................................763.1.3 Matrix Pencil Implementation .......................................................................80

3.2 Experimental Setup And Dispersion Curves By 2-D FFT In Aluminum Plate......823.2.1 Experiment on an aluminum plate in thickness 2.29 mm ..............................843.2.2 2-D Fourier Transform...................................................................................86

3.3 Matrix Pencil Method for Measurements ...............................................................893.4 Experiment And Results In Thicker Aluminum Plate ............................................913.5 Experiment With Higher Frequency For More Modes...........................................933.6 Measurements By Matrix Pencil Data And Reconstruction For Dispersion Curves

................................................................................................................................97

vi

3.7 Experiment Setup And Results In Composites Plate............................................1023.8 Summary...............................................................................................................110

CHAPTER 4 Chirplet Transform for Group Velocity .................................................1134.1 Signal Propagation and Group Velocity ...............................................................1134.2 Chirp Signal Analyzed by Chirplet Transform.....................................................1184.3 Determination of Group Velocity .........................................................................1214.4 The Formula of Chirplet .......................................................................................1224.5 Experimental Setup for Chirplet Transform .........................................................123

4.5.1 Dispersion Relation Of Group Velocity on Aluminum Plate ......................1254.5.2 Dispersion Relation of Group Velocity on Composite Plate .......................130

4.6 Summary...............................................................................................................132CHAPTER 5 Chirp-coded Ultrasonic Wave for Damage Imaging .............................133

5.1 Linear and Nonlinear Local Defect Resonance (LDR) ........................................1335.1.1 Concept of LDR and Linearity.....................................................................1335.1.2 Nonlinear LDR.............................................................................................137

5.2 Chirp Signal on Aluminum Plate with Flat Bottom Hole (FBH) .........................1405.2.1 Experimental Setup ......................................................................................1405.2.2 Response on FBH.........................................................................................1425.2.3 LDR for FBH ...............................................................................................144

5.3 Imaging Processing for Linear LDR.....................................................................1475.3.1 Zero-lag Cross-Correlation (ZLCC) Imaging Condition .............................1475.3.2 ZLCC Imaging without Bandwidth Filters ..................................................1495.3.3 ZLCC Imaging at LDR ................................................................................150

5.4 Chirp Signal on Composite Plate with Barely Visible Impact Damage (BVID)..1525.4.1 Specimen Description and Experimental Setup...........................................1545.4.2 Spectrum on Impact Area and LDR.............................................................1565.4.3 Vibration Pattern on at LDR ........................................................................1575.4.4 ZLCC Imaging without Filtering on Composite Plate.................................1605.4.5 ZLCC Imaging at LDR on Composite Plate................................................1625.4.6 Nonlinearity at LDR and Imaging on Delamination....................................162

5.5 Summary...............................................................................................................167CHAPTER 6 Conclusions and Future Works ...............................................................170

6.1 Dispersion Relationships with Chirp Signal.........................................................1706.2 Chirp signal for Nonlinear Ultrasonic Imaging ....................................................1726.3 Future Works ........................................................................................................173

REFERENCES ....................................................................................................................175APPENDICES......................................................................................................................183

vii

LIST OF TABLES

Table 1.1 The harmonic amplitude dependence from plots of strain spectrum.......................29

Table 3.1 Material properties of AS4/3502 composite lamina..............................................103

Table 5.1 Material properties of T800/3900-2 composite lamina [89] .................................156

viii

LIST OF FIGURES

Figure 1.1 The formation of Lamb wave from incident and reflected pressure (P) and shear

(S) waves and the propagation wave as incident wave from excitation source into the sensing

point in a plate-like structure with thickness h. .......................................................................10

Figure 1.2 Waveforms and displacements of Lamb wave modes in the plate for (a) anti-

symmetric (b) symmetric .........................................................................................................10

Figure 1.3 Fresnel Integrals of C(y) and S(y). .........................................................................19

Figure 1.4 Spectrum of chirp signal for different numbers of ripples when D = 20 in red dash

line and D = 200 in blue line. ..................................................................................................20

Figure 1.5 Spectrum of chirp signal (blue line) is approached by approximate spectrum (red

dash line)..................................................................................................................................20

Figure 1.6 Clapping mechanism and stress-strain bi-linear effect ..........................................26

Figure 1.7 A defect with rough contact interfaces. Example plot for stress-strain hysteresis

and end-point memory.............................................................................................................27

Figure 1.8 A sinusoidal waveform interacting with defect in the test object causes harmonic

distortion. .................................................................................................................................31

Figure 1.9 Spectrum for linear response with no defect and nonlinear response with defect of

hysteretic material....................................................................................................................32

ix

Figure 1.10 (a) Excited load A2 > A1 (b) Stress-strain curve for two different load amplitude.

Black line for A1 and red line for A2. .......................................................................................33

Figure 1.11 Resonance frequency for different driven amplitude. (a) undamaged sample. (b)

damaged sample.......................................................................................................................33

Figure 1.12 Nonlinear vibro-acoustic wave modulation technique. (a) Scheme of the method.

(b) Spectrum of response signal with undamaged material. (c) Spectrum of response signal

with damaged material. HF and LF denote high frequency and low frequency respectively. 35

Figure 2.1 Theoretical curves on aluminum plate of thickness 4.72 mm for phase velocity cp

and group velocity cg. ..............................................................................................................45

Figure 2.2 Non-dimensional group velocity calculated from the equation of non-dimensional

ω and k. ( = 0.33) ...................................................................................................................48

Figure 2.3 The computation process for obtaining the dispersion relation in composite plate.

.................................................................................................................................................60

Figure 2.4 Excited signal in temporal sequence and spectrum. (a) received signal with 70 kHz

in time domain (b) spectrum of received signal. .....................................................................67

Figure 2.5 A ω-k contour which plots for an aluminum plate (thickness 4.72 mm) shows a

peak with 70kHz actuated frequency in the position relative to the wavenumber. .................68

Figure 2.6 A ω-k map in 3D on aluminum plate (thickness 4.72 mm) shows a peak with

70kHz actuated frequency relative to the wavenumber...........................................................69

x

Figure 2.7 Scheme of the experimental setup with two measuring locations using laser

Doppler vibrometer and the data management system............................................................71

Figure 2.8 Actuating Hanning window five-peaked toneburst as input signal........................72

Figure 2.9 Determining the arrival time b1 and b2 at position S1 and S2..................................72

Figure 2.10 Dispersion relation for group velocity (cg) via frequency ( f ) by Gabor wavelet.

.................................................................................................................................................73

Figure 3.1 Experiment setup for aluminum plate Al 6061-T6 with a PZT actuator and LDV

mounting on the 2-axis translation stage sensing in an array. .................................................83


mounting on the 2-axis translation stage sensing in an array. .................................................84

Figure 3.3 Excitation and response (a) Excited source with chirp signal from 50 to 350 kHz.

(b) Spectrum of input signal after Fourier transform. (c) Response signal to linear chirp

excitation from laser Doppler vibrometer................................................................................86

Figure 3.4 Analytical non-dimensional Lamb wave dispersion curves of isotropic plates for

frequency-wavenumber, phase velocity and group velocity. The black lines mark the

excitation normalized frequency range 50-350 kHz (h = 2.29 mm) and the dots represents the

modes obtained from excitation frequency range in this study. ..............................................87

Figure 3.5 A ω-k contour which plots for aluminum plate shows peaks with chirp signal in

multiple frequencies and dispersion plots based on the local maxima points. (a) Contour plot

xi

for 2-D FFT result. (b) Peak values from 2-D FFT amplitude. (c) Phase velocity cp of first

anti-symmetry A0 mode. (d) Group velocity cg of first anti-symmetry A0 mode.....................88

Figure 3.6 Extracted points show on (a) ω-k, (b) cp-ω, and (c) cg-ω diagrams where measured

data use Fourier transform in time ω and MP method in space x on h = 2.29 mm Al plate....90

Figure 3.7 Extracted points show on (a) ω-k, (b) cp-ω, and (c)cg-ω diagrams where measured

data use Fourier transform in space k and MP method in time t on h = 2.29 mm Al plate......91

Figure 3.8 Experimental data by MP method and analytical dispersion curves for h = 6.35

mm aluminum plate, extracting from x-ω domain. Dispersion maps for (a) ω-k, (b) cp-ω, and

(c) cg-ω diagrams are shown from left to right respectively....................................................93


mm aluminum plate, extracting from k-t domain. Dispersion maps for (a) ω-k, (b) cp-ω, and



(b) Spectrum of input signal after Fourier transformed. (c) Response of first location from

LDV. ........................................................................................................................................95


(a) frequency-wavenumber ω-k, (b) phase velocity cp-ω and (c) group velocity cg-ω. The

black lines mark the excitation normalized frequency range 5-1000 kHz and the dots

represents the modes obtained from excitation frequency range in this study. .......................95

xii

Figure 3.12 Transformed plots for aluminum plate (h = 6.35 mm) show peaks with chirp

signal in excited frequency range and dispersion relationship plots by the local maxima

points. (a) Contour plot for 2-D FFT results. (b) Peak values from 2-D FFT amplitudes. (c)

Phase velocity cp relation (d) Group velocity cg relation.........................................................97






(c) cg-ω diagrams are shown from left to right respectively..................................................100

Figure 3.15 Coefficients of determination R2 for each mode are shown to percentage of

experimental data matching to analytical solutions of dispersion curves on h = 6.35 mm

aluminum plate. .....................................................................................................................101

Figure 3.16 Reconstruction curves from analysis of MP method on h = 6.35 mm aluminum

plate, extracting from x-ω domain. Dispersion maps for (a) ω-k, (b) cp-ω, and (c) cg-ω

diagrams are shown from left to right respectively. ..............................................................102

Figure 3.17 Experiment setup for composite plate laminate [±45/02]2s AS4/3502 Gr/Ep with

PZT actuator and LDV sensor. Retro-reflective foil covers an area for three scanning

directions................................................................................................................................103

xiii

Figure 3.18 Dispersion curves (ω-k, cp-ω and cg-ω diagrams) of laminate [±45/02]2s

AS4/3502 Gr/Ep extracted from x-ω domain. Propagation direction (I) 0 degree (II) 45

degree (III) 90 degree. ...........................................................................................................108


AS4/3502 Gr/Ep extracted from k-t domain. Propagation direction (I) 0 degree (II) 45 degree

(III) 90 degree. .......................................................................................................................109

Figure 3.20 SH0 mode for composite plate laminate [±45/02]2s AS4/3502 Gr/Ep is analyzed

from x-ω domain and (a) ω-k (b) cp-ω and (c) cg-ω diagrams shown in the propagation

direction 90° degree. ..............................................................................................................110

Figure 3.21 Amplitude allocation in each scanning direction for (a) A0 mode (b) S0 mode and

(c) SH0 mode. For (b) S0 mode, there are many points in each direction, but for (c) SH0 mode,

only 90° degree is existed and 0° and 45° degree barely see amplitude for most frequencies.

...............................................................................................................................................110

Figure 4.1 Schematic of the experimental setup with two measuring locations using laser

Doppler vibrometer and the data management system..........................................................124

Figure 4.2 The real (line) and image (dash line) part of mother chirplet with chirp rate π/4 at

center frequency ω0 = 2π. ......................................................................................................125

Figure 4.3 Picture of experiment setup for aluminum plate with piezoelectric actuator and

LDV sensor. ...........................................................................................................................127

xiv

Figure 4.4 The responses received by LDV and magnitude of CT transform at point s1 (a)

waveform of detected signal (b) magnitude of time-frequency distribution (c) peak value

showing in the contour plot at time 106 μs in frequency 300 kHz . ......................................128

Figure 4.5 The responses received by LDV and magnitude of CT transform at point S2 (a)


showing in the contour plot at time 124 μs in frequency 300 kHz. .......................................129

Figure 4.6 Group velocity decomposition toward different frequencies extracted by transform

from responses 1 and 2. (a) Chirplet transform (b) Gabor wavelet. ......................................130

Figure 4.7 Measured group velocities and theoretical curves along (a) 0° degree (b) 45°

degree (c) 90° degree (d) all directions..................................................................................132

Figure 5.1 Fundamental LDR frequency for linear and higher harmonics for nonlinearity..133

Figure 5.2 (a) Spectrum of the out-of-plane velocity at center of defect. (b) Showing the

contour plot corresponding to LDR in space-frequency domain...........................................136

Figure 5.3 Nonlinear oscillator model for local delamination region in a laminated composite

plate with excitation signal have LDR and nonlinearity due to out-of-plane motion. ..........138

Figure 5.4 A schematic picture for experimental setup with a system by PZT as excitation

and LDV as sensor.................................................................................................................141

Figure 5.5 Scheme for aluminum plate Al 6061-T6 sized by 304 mm × 304 mm × 2.29 mm

with FBH D25 mm and depth 0.9 mm (grey dash line) underneath scanning area 40 mm × 40

mm (red box). A PZT is attached outside of scanning area for generating signals. ..............143

xv

Figure 5.6 Excitation and response at the center of FBH show the waveforms and their

spectrums. (a) Excitation waveform (b) excitation spectrum (c) response waveform and (d)

response spectrum shows LDR and geometric resonance (GR) at the location of FBH center.

...............................................................................................................................................144

Figure 5.7 Vibration patterns are shown according to the spectrum at the center of damage

with its region depicted in white circle..................................................................................146

Figure 5.8 ZLCC imaging for linear chirp as wideband signal and damage region depicted in

white circle.............................................................................................................................150

Figure 5.9 ZLCC image with vibration pattern at LDR frequency. ......................................151

Figure 5.10 Experimental setup for damage imaging on BVID in a composite plate...........155

Figure 5.11 Damage region (dash red) and delamination (dash white) of different layer are

shown in the C-scan. The size of C-scan matches the scanning area. ...................................156

Figure 5.12 Spectrum and C-scan for damage and delamination regions are in the scanning

area. (a) The detcting point at the center of damage region. (b) Peaks in spectrum at detecting

point. ......................................................................................................................................157

Figure 5.13 The corresponding three-dimensional vibration pattern for the peaks of spectrum

at center of damage region. (a) 29.4 kHz (b) 41.9 kHz (c) 55.6 kHz (d) 69.4 kHz.................159

Figure 5.14 The corresponding vibration pattern in top view for the peaks of spectrum at

center of damage region. (a) 29.4 kHz (b) 41.9 kHz (c) 55.6 kHz (d) 69.4 kHz ....................160

Figure 5.15 ZLCC imaging cumulating all frequencies on composite plate with BVID. .....161

xvi

Figure 5.16 ZLCC imaging at LDR for vibration pattern with dicption from C-scan. (damage

region for red dash line and delamination for white dash lines)............................................162


area. (a) The detcting point at location of delamination. (b) Peaks in spectrum at detecting

point. ......................................................................................................................................164

Figure 5.18 Vibration pattern on the delamination at (a) 56.3 kHz (b) in 3-D view and at (c)

68.4 kHz (d) in 3-D view. ......................................................................................................165

Figure 5.19 C-scan imaging in (a) layer 4 and (b) layer 5 shows different size delamination in

the same location as cross-section. ........................................................................................166

Figure 5.20 ZLCC imaging with vibration patterns at LDR frequency (a) 56.3 kHz (b) 68.4

kHz. ........................................................................................................................................166

Figure 5.21 Vibration patterns for nonlinearity happened at LDR for frequency (a) 115.6 kHz

(b) in 3-D view and (c) 137.8 kHz (d) in 3-D view. ..............................................................167

1

CHAPTER 1

Introduction

1.1 Purpose of Research

Structure Health Monitoring (SHM) offers the solution to the damages, based on sensors that

are integrated with structures and computational systems. Many of them have been

developed for efficient damage detection. Methods based on ultrasonic wave propagation are

particularly attractive and have been exploited for many years [1, 2].

Integrity of structural components is evaluated by nondestructive testing with

ultrasonic guided waves to analyze wave propagation in plate-like isotropic or layered

materials. Much information about the generation and propagation of elastic signals is

needed before utilizing the evaluation. When the specimen is plate-like, wave propagation

away from the excited source will be govern by Lamb’s homogeneous equation [3]. With a

high susceptibility to interference on a propagation path, Lamb waves can travel a long

distance [4]. Lamb wave as one kind of guided waves can be made sensitive to defects of

materials, in terms of shapes and locations, by careful control of testing parameters. These

evaluations can be quantified based on the ultrasonic wave speed inferred from network of

actuators and sensors on the surface. The use of guided wave yields a frequency-dependent

behavior that can also be used in verse identification procedure for material characterization.

In a thin plate, S0 and A0 are two basic modes that show different velocities and dispersive

characteristics. Measurements of the velocities are useful for determining material properties

and evaluating healthy state of materials [5, 6].

2

Guided wave propagation in plate-like structures made either isotropic or anisotropic

materials is critical in ultrasonic wave-based NDI technology. Because of the inherent

dispersion and multi-modal characteristics, a clear understanding of wave propagation

characteristics is important. Accurate determination of the dispersion curves is a prerequisite

to optimize the wave mode selection. For well-known materials, dispersion curves can be

predicted and computed based on material properties and geometries from the elastic wave

equation [7, 8]. If exact material properties are not known in priori, elastic equations are only

ideal models. This is important by experimental extraction of dispersion curves for guided

wave propagation.

Many conventional detection methods for determination of dispersion curves are

based on time of flight (ToF) measurements [5, 9]. Wave propagation travels a time through

a medium and the distance between actuation and signal receiver is already known while

setting up. Time-frequency representations (TFR) are used for determining the arrival time of

transient waves propagating in a medium with less measurement. Fourier based transform to

provide spatial information in dealing with multiple modes. Dispersion relationships can be

approached by either method.

The dispersion relations in a frequency-wavenumber plane can be seen as the

dispersion curves as ridges in the diagrams through two-dimensional Fourier transform (2-D

FFT). Alleyne and Cawley [10] analyzed propagating multi-mode of Lamb wave on different

thickness plates for frequency-wavenumber domain and phase velocities. Costley and

Berthelot [11] used laser excitation to examine the dispersion of Lamb wave on the plate by

3

applying 2-D FFT. However, 2-D FFT would be rather hard to identify these ridges even the

exact values are known for frequencies.

The Matrix Pencil (MP) method provides a means which can automatically extract

dispersion curves from laser Doppler vibrometer (LDV) measurement data in an easy and

robust manner [12]. Dispersion analysis can be seen as a modal set of measurements that

vary with time. There exist several approaches for estimating the modal content of a time

varying waveform, such as Fourier transform based method, Prony based method etc. [13-

15]. Prony method is sensitive to signal to ratio (SNR) in the system. MP’s inherent ability to

accurately analyze noise signals makes it a promising technique [16]. It works by applying

one-dimensional Fourier transform of the data into wavenumber domain or frequency

domain from spatial or temporal resolution respectively, and the applying the MP method to

extract the wavenumber-dependent frequencies, or frequency-dependent wavenumbers vise

versa. Chang and Yuan [17] used MP method for different thickness metallic plates and

demonstrated abilities to gain dispersion relationships up to A1 mode with broadband signal.

The theoretical results to experimental results for isotropic aluminum plates are compared

and the comparison is confirmed as expected.

Two-dimensional fast Fourier transform (2-D FFT) [10] operates on multiple, equally

spaced waveforms and indicates time consuming. In contrast, time-frequency representations

(TFR) are used for determining the dispersion relationships of transient waves propagating in

a medium with less measurement. Kishimoto et al. [18] utilized wavelet transform with

Gabor wavelet for investigating the dispersion character of structure waves. Niethammer et

al.[19] applied short time Fourier transform (STFT) to develop the dispersion curves for

4

multimode Lamb waves in an aluminum plate. Jeong [20] applied the wavelet transform

using a Gabor wavelet to determine group velocities of plate wave propagation in anisotropic

laminates. Kuttig [21] and Kerber et al. [22] refined approach by using chirplet transform

basis as a generalized TFR with more flexibility to adjust the window function to group

velocity.

In order to efficiently acquire guided wave data, broadband excitation is applied to

evaluate wave responses for a range of frequencies. For such approach, a coded excited

signal with broadband such as chirp, white noise signal or rectangular burst is used to get

good response [21, 23]. A coded, swept continuously over pre-determined range chirp signal

as an excitation source is used in this paper for the efficient implementation of guided wave

data acquisition. Using chirp excitations is not only as a broadband signal for multiple

frequencies, but also a designed chirp signal can provoke vibration patterns in detecting

damages.

Current acoustic imaging and non-destructive testing systems are most based on

linear theory, like a linear acoustic image field identical to that of the acoustic wave injected

into medium. Various methods have been developed for damage detection on plate-like

structures over the last few decades. These include the different non-destructive testing (NDT)

technologies based on visual inspection, ultrasonic testing, acoustic emission, X-rays or

vibrothermography [1, 24]. Early detection of material damage in nondestructive evaluation

has been one of demanding issue in recent years [25]. Early detection means to identify the

micro-cracks or sub-structural material damages before visible defects happen. For this

5

purpose, the most promising technique is nonlinear ultrasonic technique. Nonlinear ultrasonic

research is recently developed in the effort to diagnose materials and predict life.

The nonlinear approach to nonlinear nondestructive technology (NNDT) developed

for decades is concerned with the nonlinearity of defects and applied with extreme frequency

changes of input signal. These spectrum changes are related to nonlinearity of micro- or

macro- scale defects. Besides, the behavior of wave propagation is linear for the intact or

undamaged parts outside defects. Thus, a sub-structural material damage behaves as an active

source of new frequency components. This makes the NNDT unique and based on defect-

selective instrument.

Nonlinearity of damage detection can be divided for two groups, classical and non-

classical nonlinearities. Classical nonlinearities are related to variations of wave velocity

resulting from different strain characteristics for static and dynamics loading. They are in

form of higher harmonics or quadratic frequency shifts. Another group is called non-classical

nonlinearities. These nonlinearities are linked to dissipative behaviors and models, including

contact acoustic nonlinearity (CAN), hysteresis, Hertzian contact, and thermo-elasticity [26].

Based on these models, many methods like harmonic distortion, vibro-acoustic wave

modulation, second-harmonic generation or higher harmonic generation, are applied to

analyze nonlinearity of different types sub-structural material damages. These cases are

manifested by mixing frequencies, linear resonant frequency shifts or amplitude dependent

Q-factor.

Local defect resonance (LDR) frequencies can be used to provoke nonlinear elastic

wave effects by activating the defect into finite amplitude nonlinear regime [27, 28]. LDR to

6

detect various types of damage represents one of latest additions to the family of ultrasonic

NDT method, having interesting effect that strongly resembles the typical resonance behavior

of solid structures [29, 30]. This way as traditional vibration analysis makes the local

vibration amplitude of a defect increase significantly when the excitation frequency matches

the LDR. It provides good contrast between damaged and intact areas of specimen. The

incensement of local amplitude as vibration pattern can be measured by means of laser

Doppler vibrometer (LDV) as easy detection nowadays.

LDR can be not only observed for nonlinearity in delamination or disbonds of

composite but also illustrated for different types of damages such as flat bottom hole (FBH),

cracks, or fatigue in isotropic materials [31]. LDR also can be used to enhance detection [32]

and observed by heat generation due to induction of strong vibration by the defect [31].

Therefore, the detection and analysis of LDR become more interesting for a broad range of

NDT application.

Different imaging processing methods have been proposed for the detection of

damage in isotropic and composite structures. Most studies are based on frequency-

wavenumber filtering as the first step to analyze data, such as instantaneous wavenumbers

[33], local wavenumbers [34], or Fourier-based filtering [35]. Using filtered data to identify

locations and characterization of damage was developed in various theories, such as

cumulative total wave energy (CTWE) and cumulative standing wave energy (CSWE) [36],

standing waves [37], zero-lag cross-correlation (ZLCC) [38], enhanced zero-lag cross-

correlation (E-ZLCC) [39] techniques. In these, studies in frequency-wavenumber domain

7

for detection of damages in plate-like structures using flexural waves and cross-correlation

were obtained for rapid imaging [40].

Cross-correlation imaging is based on the concept that damage exists at the location

where forward waves interact in phase with backward waves. The imaging condition for

reconstructing wavefields is formulated in time domain or in frequency domain as cross

correlation as zero-lag in phase. In the case, forward and backward waves are based on the

excitation source for certain bandwidth. ZLCC imaging condition is integrated by the input

signal such as five-peaked toneburst signal in narrowband [38]. However, it is time-

consuming to try different frequencies for unknown damage sizes due to tuning of Lamb

wave mode. Pulse signal excited by laser overcame the difficulty but it is uncontrollable in

desired bandwidth. As wideband signal, pulse signal showed the ability to detect damage by

using CSWE technique which is also on Fourier-based filtering [36]. Chirp as wideband

signal can be controllable and excited linearly as sinusoidal wave. ZLCC imaging condition

can cumulates all frequencies by using linear chirp signal.

LDR can be excited in certain frequency with a broadband signal. Through wideband

excitation such as chirp signal, damage is vibrated for higher amplitude in contrast to other

regions [29, 30]. Since chirp signal is not only for wave propagation analysis but also for

detecting LDR, ZLCC imaging condition analyze and cumulates all the frequencies of

excitation through vibration patterns by LDR. Therefore, observing data at the local defect

resonance is used to enhance the imaging of defect by means of measurements with PZT and

LDV system.

8

The aim of this research is to investigate the feasibility of utilizing chirp signal for

examining wave dispersion relationships and ultrasonic damage imaging based on linear and

nonlinear phenomenon upon isotropic and carbon fiber reinforced polymer (CFRP) plates

with a system of piezoelectric transducer (PZT) and laser Doppler vibrometer (LDV). In

order to reach the goals, this research is investigated and discussed following below:

1. Chirp signal is excited and analyzed by Matrix Pencil method for dispersion

curves from frequency-wavenumber domain to group velocity.

2. Dispersion curves from Matrix Pencil method are reconstructed for higher modes

in different thickness of aluminum plate.

3. Chirp signal applied to a composite plate and analyzed by Matrix Pencil method

is investigated and discussed.

4. To obtain the dispersion curves of group velocity directly, time-frequency

representative as chirplet transform with chirp signal is analyzed and discussed on

aluminum and composite plate respectively.

5. Chirp signal is not only excited as wave propagation but also can excite vibration

for local defect region for local defect resonance.

6. Vibration patterns for simple case in aluminum plate with flat bottom hole are

discussed with imaging in frequency domain and ZLCC imaging conditions.

7. A barely visible impact damage in a composite plate is imaged by LDR method

and ZLCC imaging condition.

8. Delamination due to BVID in composite plate is highlighted by LDR method for

nonlinear phenomena.

9

Background theories for Lamb waves in plates and nonlinear ultrasonic are introduced later.

The structure of this research will be listed and in the end of this chapter.

1.2 Lamb Waves in Plates

Signal transmitting into a plate-like medium vibrates the inside particles and make them as

wave propagation. When the particles of the medium are displaced from their equilibrium

positions, internal restoring forces arise. It is these elastic restoring forces between particles,

combined with the inertia of the particles, which lead to oscillatory motions of the medium.

When propagating through plate-like structures (e.g. solid plates) and in the hypothesis of

linear elasticity, these waves are classified as Lamb waves, known to be multimodal and

dispersive. The formation of Lamb wave from longitude (P) and shear (S) waves is shown in

the Figure 1.1.

The number of modes of Lamb wave propagation, as mathematically derived in

following, depends on the wave frequency, f, and the plate thickness, h. There are at least two

fundamental modes, known as anti-symmetric, A0, and symmetric, S0, existing alone if the

product fh ranges between 0-1MHz·mm. Higher vibration modes, such as A1 and S1, appear

by increasing the frequency and/or thickness. The different vibration modes are characterized

by different propagation velocities. Additionally, to make things even more complicated, the

propagation velocity of each vibration mode is a function of the frequency. These

characteristics make the interpretation of Lamb wave signals very difficult. As result, Lamb

waves have been used in real applications only below the first cut-off frequency, in order to

have only the two fundamental modes of vibration. Waveforms of Lamb wave in different

10

modes are shown in the Figure 1.2. The waves have anti-symmetric or symmetric through

neutral line of the plate.

Figure 1.1 The formation of Lamb wave from incident and reflected pressure (P) and shear

(S) waves and the propagation wave as incident wave from excitation source into the sensing

point in a plate-like structure with thickness h.

(

a)

(

b)

11

Figure 1.2 Waveforms and displacements of Lamb wave modes in the plate for (a) anti-

symmetric (b) symmetric

In the absence of body forces the components of the displacement vector in a

homogeneous, isotropic, linearly elastic medium are governed by the following equation of

motion:

(1.2.1)

where u = (u1, u2, u3) is the displacement in the medium at the location x = (x1, x2, x3) and

time t, ρ is the density, λ and µ are the Lame’s constant and shear modulus respectively.

The displacement vector can be expressed by using Helmoltz decomposition

(1.2.2)

where ϕ is the scalar potential representing longitudinal waves traveling with a wave speed cL

while ψ is the vector potential representing transverse waves traveling with a wave speed cT.

Scalar and vector potentials satisfy the uncoupled wave equations:

(1.2.3)

where

(1.2.4)

12

cL ( 2) / E(1 )

(1 )(1 2 )(1.2.5)

cT / E

2(1 )(1.2.6)

are the function of the material properties only.

Applying the previous general equations to a plate bounded by the surfaces z = ± h/2

and of infinite extent in the x and y directions, by assuming plane strain motions, the

displacement vector components become

(1.2.7)

uy 0 (1.2.8)

(1.2.9)

while the wave equations from Eq. (1.2.1) become

(1.2.10)

(1.2.11)

Equations show that Lamb waves have the dual characteristics of being standing

waves across the thickness and thus traveling waves only in the x1 direction. The potentials

13

can be considered with the complex form and then taken into the equations for strains and

stresses. The complex forms for the potential representing can be considered as

(1.2.12)

(1.2.13)

where

(1.2.14)

(1.2.15)

The Lamb waves travels as harmonic waves in positive x direction. A harmonic wave

has a single frequency and transmits in t > 0. Before applying the potential scales into the

displacement equations, the stress components for consideration of boundary conditions

should be discussed from Hooke’s law, which are expressed as

(1.2.16)

(1.2.18)

The potential scales are applied into the displacements and stress components as

(1.2.19)

(1.2.20)

(1.2.17)

14

xx (k2 p2 2k2 )[A1 sin( pz) A2 cos( pz)]i2kq[B1 cos(qz) B2 sin(qz)]

(1.2.21)

zz (k2 p2 2 p2 )[A1 sin( pz) A2 cos( pz)]i2kq[B1 cos(qz) B2 sin(qz)]

(1.2.22)

zx {2ikp[A1cos( pz) A2 sin( pz)]

(k2 q2 )[B1 sin(qz) B2 cos(qz)]}(1.2.23)

where the term exp[i(kx−ωt)] is dropped for simplification. By inspecting the displacements

in Eq. (1.2.19) and Eq. (1.2.20) with potential scales, the results indicate the motions can be

distinguished as two modes and showed in the Figure 1.2. The names of modes are named

from the particle movements. The motions can be regarded as symmetric and anti-symmetric

modes.

The coefficients can be separated for different modes. For the displacement ux of

symmetric mode in x direction, the coefficients A2 and B1 are relative to even functions. From

the Figure 1.2, the displacements along x axel can be seen as symmetric according to neutral

line by those two terms. Moreover, for the displacement uz of symmetric mode in z direction,

the coefficients A2 and B1 are associated with odd functions, which show movements as

symmetric along z axel according to neutral line. Therefore, symmetric mode (S) is

associated with A2 and B1 while assuming A1 = B2 = 0. On the other hand, anti-symmetric

mode (A) is associated with A1 and B2 and has opposite ways that it has anti-symmetric

movement with odd functions in ux displacement, while with even function in uz direction.

Consequently the displacement is corresponding to symmetric mode (S) with A1 = B2

= 0 as

15

ux [ikA2 cos( pz) qB1 cos(qz)]ei(kx t ) (1.2.24)

uz [ pA2 sin(pz) ikB1 sin(qz)]ei(kx t ) (1.2.25)

The displacement corresponding to anti-symmetric mode (A) with A2 = B1 = 0 is

ux [ikA1 sin( pz) qB2 sin(qz)]ei(kxt ) (1.2.26)

uy [ pA1 cos( pz) ikB2 cos(qz)]ei(kxt ) (1.2.27)

Lamb waves are two-dimensional vibrations propagating in plates. The displacement

on the plate surface or in the plate may be symmetric or anti-symmetric with respect to the

middle plane. Lamb waves are guided and dispersive with displacement in both x1- and x3-

direction as the bending waves. They are derived as eigen-solutions of characteristics as the

dispersion equations which describe the transitions between these types of waves. By

imposing the traction free boundary conditions at z = ± h/2, τzz = τxz = 0 and substituting into

the equations of stress components, the Rayleigh-Lamb dispersion equations are presented as

below [3]

tanqh/2tan ph/2

4 pqk2

(k2 q2 )2

1

(1.2.28)

where ± represents symmetric and anti-symmetric waves respectively, and

and p2 2

cL2 k2 q2

2

cT2 k 2 (1.2.29)

k is the wavenumber. ω is the angular velocity. cL and cT are the longitudinal and shear phase

velocity respectively. h presents the total thickness of the plate.

16

From above, the particle displacement and velocity can be simply expressed and

represented as

u(x,t) A(x, )ei(kxt ) (1.2.30)

where A(x,ω) is the amplitude of waveform and wavenumber k consists the function of

frequency as k = k(ω) which presents the coefficients of amplitudes of symmetric and anti-

symmetric modes.

1.3 Chirp Signal

Guided waves propagating in the plate are dispersive and depending on the frequencies of

excitation signals. The mode tuning is done empirically by exciting a variety of central

frequency tone burst signals and selecting one to show the best mode purity [41]. This

method is often desirable to use multiple narrowband excitations for acquiring data, which is

time consuming and inconvenient. Exploiting narrowband signal requires multiple operations

to obtain final dispersion curves of interest and suffers the problem of aliasing for being less

robust to noise. Lower SNR affects the identification of dispersion relationship. To achieve

higher SNR, an alternative technique is employed for other ultrasonic applications by pulse

compression. Pulse compression uses a broadband signal with long time to increase

excitation energy. Therefore, in order to efficiently acquire guided wave data, broadband

excitation is applied to evaluate wave responses for a range of frequencies. A coded chirp

signal as excitation source is used in this research for the efficient implementation of guided

wave data acquisition [23]. Chirp signal types includes various types of chirp signals, Barker

17

code or Golay complementary code [42]. Linear chirp is the optimal selection and widely

used for applications [43]. Chirp signal is excited for a continuous frequency range not only

to provide better resolutions in evaluating the characterization of interest but also provide

complementary information against noise.

The general chirp signal can be expressed as

(t) a(t)exp i( 0t BT

t 2 )

, T2

t T2

(1.3.1)

where a(t) is the amplitude, ω0 is the initial frequency, B is the bandwidth and T is the signal

duration. Thus, chirp signal sweeps frequency from ω0 to instantaneous frequency ωi by

i d(t)

dt

d( 0t BT

t 2 )

dt 0 2 B

Tt (1.3.2)

and the time t is associated with bandwidth during the duration.

When chirp signal propagates in dispersive medium, the signal uc(x,t) will become

uc (x,t) Ac ( )ei(kx0tt2 ) (1.3.3)

where Ac means the amplitude of propagating signal and α presents the chirp rate of

bandwidth and duration. In order to observe the spectrum in frequency domain, 2-D Fourier

transform is used for transferring to frequency-wavenumber domain by

Uc (k, ) 1

4 2 u(x,t)ei(kx t ) dx dt

Ac ( ) (k k0 ) 1

2ei( 0 )te it2

dtT /2

T /2

(1.3.4)

18

The last term is an integral with quadratic term so that the integral becomes as called as

Fresnel Integrals as

12

ei( 0 )teit2

dtT /2

T /2

2

exp i ( 0 )2

4

cos(2

y2 )dy iY1

Y2

sin(2

y2 )dyY1

Y2

(1.3.5)

In order to solve the equation, Eq. (1.3.5) is simplified to

2

exp i ( 0 )2

4

C(Y1) C(Y2 ) iS(Y1) iS(Y2 ) (1.3.6)

where C and S present the integrals of cos and sin function with quadratic terms as

and C(z) cos y2

20

z

dy S(z) sin y2

20

z

dy (1.3.7)

The z means the integration limitation. The plot for Fresnel Integral is shown in the Figure

1.3. The integral limitations Y1 and Y2 in the Eq. (1.3.6) are associated with bandwidth and

duration as

and Y1 D2

(1 0

B) Y2

D2

(1 0

B) (1.3.8)

where D is equal to the multination of duration T and bandwidth B, D = TB. Since the Fresnel

Integrals cause that the signal has many upside down as “ripples”, the spectrum of chirp

signal having many ripples is expected and the number of ripples is associated with D due to

the integral limitation. When D increase, the number of ripples is proportional to D shown in

the Figure 1.4. In the figure, when the D value increase from 20 (red dash line) to 200 (blue

line), the ripples increases and the amplitude in each frequency approaches a certain value.

19

Therefore, an approximate spectrum can be approached from the Eq. (1.3.6) by its magnitude

as

Uc (k, )

Ac( ) (k k0 ) rect( 0

2 B)exp i ( 0 )2

4 i

4

(1.3.9)

The approximate magnitude is like a rectangular window in spectrum, which is plotted in the

Figure 1.5.

C(y)

S(y)

Figure 1.3 Fresnel Integrals of C(y) and S(y).

20

D = 200

D = 20

Figure 1.4 Spectrum of chirp signal for different numbers of ripples when D = 20 in red dash

line and D = 200 in blue line.

Spectrum

ApproximateSpectrum

Figure 1.5 Spectrum of chirp signal (blue line) is approached by approximate spectrum (red

dash line).

21

1.4 Nonlinear Ultrasonic

Current acoustic imaging and non-destructive testing systems are most based on linear theory,

like a linear acoustic image field identical to that of the acoustic wave injected into medium.

Various methods have been developed for damage detection on plate-like structures over the

last few decades. These include the different non-destructive testing technologies based on

visual inspection, ultrasonic testing, acoustic emission, X-rays or vibro-thermography [1, 24].

Most of these wave propagation methods rely on various linear phenomena of ultrasonic

wave propagation. Wave reflections or scattering are often used for damage detection. Only

few technologies described the nonlinear ultrasound.

Early detection of material damage in nondestructive evaluation has been one of

demanding issue in recent years [25]. Early detection means to identify the micro-cracks or

sub-structural material damages before visible defects happen. For this purpose, the most

promising technique is nonlinear ultrasonic technique. Nonlinear ultrasonic research is

recently developed in the effort to diagnose materials and predict life.

Nonlinear phenomena can be observed when ultrasonic waves have large amplitude

or travel long distances. In these cases, sinusoidal waves are often modified and components

of smaller amplitudes are dispersed. Physically, these phenomena are caused by different

speeds of sound waves in regions of compression and rarefaction. It is also well known that

in a nonlinear medium the superposition principle is broken, leading to interaction of waves

of different frequencies.

22

Nonlinearity of damage detection can be divided for two groups, classical and non-

classical nonlinearities. Classical nonlinearities are related to variations of wave velocity

resulting from different strain characteristics for static and dynamics loading. They are in

form of higher harmonics or quadratic frequency shifts. Another group is called non-classical

nonlinearities. These nonlinearities are linked to dissipative behaviors and models, including

contact acoustic nonlinearity (CAN), hysteresis, Hertzian contact, and thermo-elasticity [26].

Based on these models, many methods like harmonic distortion, vibro-acoustic wave

modulation, second-harmonic generation or higher harmonic generation, are applied to

analyze nonlinearity of different types substructural material damages. These cases are

manifested by mixing frequencies, linear resonant frequency shifts or amplitude dependent

Q-factor.

The nonlinear approach to nonlinear nondestructive technology (NNDT) developed

for decades is concerned with the nonlinearity of defects and applied with extreme frequency

changes of input signal. These spectrum changes are related to nonlinearity of micro- or

macro- scale defects. Besides, the behavior of wave propagation is linear for the intact or

undamaged parts outside defects. Thus, a sub-structural material damage behaves as an active

source of new frequency components. This makes the NNDT unique and based on defect-

selective instrument. Later on, nonlinear properties and models of damage is introduced and

nonlinear acoustic methods for damage detection is described.

23

1.4.1 Nonlinear Properties of Damage

Numerous experimental and studies show strong nonlinearities observed for defects having

contact interfaces such as cracks, delamination, disbands. For detecting these defects as early

examination, classic nonlinear acoustic theory has been formulated with observations of

higher harmonics in the late 19th century. Later on, the physical phenomenon of a crack’s

behavior is not likely classical nonlinearity and being discussed widely. It is well known that

cracks can open and close if medium undergoes certain levels of tension and compression.

This motion is related to clapping, kissing, friction, adhesion, slow-dynamic and various

wave interactions. The different physical mechanism can lead to similar nonlinear model or

vice versa. Therefore, the models for these mechanisms are discussed in this section.

Classical nonlinear elasticity is going first and later non-classical nonlinear including CAN

theory and hysteretic are described.

1.4.1.1 Classical Nonlinear Elasticity

The classical nonlinear theory of elasticity has higher-order elastic terms in the Hooke’s law.

The Hooke’s law is based on the expansion to the second order by the relationship between

stress and strain [44, 45]. One generally introduces nonlinearity in the theoretical model by

expressing the elastic moduli in power of the strain. This is equivalent to accounting for a

strain dependency of the energy density. The free energy in power series for Hooke’s law:

F F0 12

uii2 uik

2 (1.4.1)

24

where F0 is the initial value, λ and μ are Lame constants, uik is the deformation tensor that can

be described as

uik 12

(ui

xk

uk

xi

ul

xi

ul

xk

) (1.4.2)

The linear wave propagation can be derived using the classical Hooke’s law without

the last term. When nonlinear wave propagation is considered, the strain tensor in this case

becomes leading to quadratic terms. The elastic energy density as an analytic function of the

strain field can be conducted to third-order equation [46, 47]:

E uik2 (

2

3

)ull2

A3

uikullukl Buik2 ull

C3

ull3 (1.4.3)

where μ, κ, A, B, and C are the constants that can be found in principle from experiment. The

equation is also called the five-constant theory of nonlinear elasticity. It can be noted that the

second order of nonlinearity is equivalent to a linear relationship.

The nonlinear wave equation with second and third order terms can be expressed as

[48]

2ut2 c2 2u

x2(1.4.4)

where c is nonlinear wave speed

c2 c02 1 u

x (u

x)2

(1.4.5)

25

and c0 is the linear elastic wave speed, du/dx is the strain, and β and δ are higher-order

contributions to the nonlinear wave speed. β is the quadratic nonlinear coefficient which

produces both odd and even higher frequencies harmonics in the spectrum. Respectively, δ is

the cubic term for the occurrence of odd harmonics only.

The classical nonlinear elasticity is not able to explain some physical phenomena

observed in rocks. Some studies uses stress-strain hysteresis to build the model [26, 46]. The

hysteresis will be introduced later.

1.4.1.2 Contact Acoustic Nonlinearity (CAN) Theory

Strong nonlinearity is observed for defects from the contact interfaces, leading to internal

motion of crack faces as opening or closing. When the crack is open, the stiffness is reduced

and material discontinuity. When crack is close, the stiffness is not changed. The physical

nature can be considered as a planar interface separates two elastic materials wit surfaces but

no traction forces across the interface. This type of nonlinear model is called contact acoustic

nonlinearity (CAN) with bi-linear stiffness for breathing or clapping mechanism.

For 1-D case, the bi-linear model treats the crack as a spring with a nonlinear

stiffness coefficient consisting of two different stiffness values in strain level as

(1.4.6)

where

26

k(q) kc

kt

if q q0

if q q0

(1.4.7)

and q denotes the crack response, q0 is the value of the response when the crack opens or

close, kc and kt present stiffness of the compression and tension respectively. This model can

be described in the Figure 1.6 as pressed together during compression and as separated

during tensile when elastic wave is applied to medium.

Figure 1.6 Clapping mechanism and stress-strain bi-linear effect

1.4.1.3 Hysteresis

Stress-strain hysteresis is one of non-classical nonlinearity in materials, also accompanied

called “discrete memory” [44]. The hysteresis in metals and rocks was observed

experimentally under rapid loading or unloading. In the metal, hysteresis is very weak and

often omitted in the equation of state (EOS), whereas it is considered to be as one of material

behavior in rocks [49].

27

A material behavior is called “hysteresis” if it follows different stress-strain curve

when loading or unloading happens. An example for hysteresis and discrete memory is

shown in the Figure 1.7. Material behavior follows the relation A-B-C when applying to

loading. Normally, a classical nonlinear material follows the original curve C-B-A to the

initial point, but the hysteresis materials go through point D to E in compression. When the

loading is applied at the second times, it will follow E-F-C curve and ending at C. If a

smaller stress variation happens during loading, another small loop in the stress-strain plot

happens and follows G-H-G. It is noted the start and finish point at the same. This effect is

called “end point” memory [26].

Figure 1.7 A defect with rough contact interfaces. Example plot for stress-strain hysteresis

and end-point memory

Some experiments show that micro-inhomogeneous like crack, voids and contact

have complex compliance and local nonlinear forces dominates the atomic nonlinearity

leading to a nonlinear behavior that cannot be explained by classical nonlinearity. Another

28

way with explanation is to get the theoretical description of nonlinear mesoscopic that can

describe not only classical nonlinearity but also hysteresis and discrete memory [27, 45, 50].

The process can be explained by using the Hooke’s law describing the stress-strain

relationship. The general form can be expressed as:

(1.4.8)

where K is the parameter of elasticity modulus for nonlinearities, ε is strain and σ is stress.

Theoretical description of non-linear mesoscopic elastic materials based on Preisach-

Mayergoyz (PM) space representation can be approached by a formulation for elasticity

modulus K that relates stress not only to strains and deviations but also the time deviation of

the strains, given as [50]:

(1.4.9)

where K0 is the linear modulus, β and δ classical nonlinear coefficients, and α material

hysteresis measure. Δε is the local strain amplitude change over the last period, Δε = (εmax −

εmin)/2 for a simple continuous sine excitation. is the strain rate. And is

defined as:

(1.4.10)

For comparison, the acoustic contribution of the material model coefficients can be

summarized for classical nonlinearity and hysteresis behavior in Table 1.1. In linear acoustics,

harmonic amplitude is independent. When it gets the first order, second or following

29

harmonic frequencies would show up. Continuously, there is no even harmonic when the

formulation gets to second order.

It is noted that the third harmonic for the hysteresis nonlinearity is quadratic in the

fundamental strain amplitude whereas classical nonlinearity predicts the cubic dependence.

Similarly, two excited frequencies f1 and f2 generate a second-order sideband f2±2f1 with

amplitude proportional to αA1A2, whereas classical theory predict a higher-order dependence

Cβδ(A1)2A2, Cβδ a constant combination with β and δ. Instead, the first-order intermodulation

frequencies at f2±f1 sideband is predicted and proportional to βA1A2.

Table 1.1 The harmonic amplitude dependence from plots of strain spectrum

30

1.4.2 Nonlinear Ultrasonic Methods for Damage Detection

Among numbers of studies for nonlinear methods on damage detection, there are three

methods practically and widely used: harmonic distortion, vibro-acoustic modulation and

time-reversal method. These methods are replying on the nonlinear responses which are

proportional to the degree of damage and are based on a particular mechanism to the

observations.

1.4.2.1 Harmonic distortion method

One of detective methods to characterize acoustic nonlinearity is to measure the harmonic

distortion of a vibration signal. This method is applied for many fields like fluids, biological

media, electromechanical systems, or material nonlinearity of solids. The essence of this

method is described as following.

An input signal is a sinusoidal waveform with a frequency f1 which has an amplitude

A1. When the waveform propagates in the medium with defects, the nonlinearity distorts the

waveforms so that the spectrum of response contains additional harmonics. Typically, the

higher harmonics are times of excited frequency as 2f1, 3f1, ∙∙∙ and diminishing the amplitudes

A1 > A2 > A3 >∙∙∙ respectively. Stronger nonlinearity coupled with larger excited acoustics

may lead to more distortion with higher-order harmonics and intermodulation. The scheme is

illustrated in the Figure 1.8.

31

Figure 1.8 A sinusoidal waveform interacting with defect in the test object causes harmonic

distortion.

For some cases with complex compliance and nonlinear behaviors which cannot be

explained in classical nonlinear models, composite laminated materials are structural

heterogeneous material exhibiting linear behavior while undamaged and nonlinear hysteretic

behavior while damaged. In this scene, if the nonlinearity is strong enough, only odd

harmonics appear and form a complex spectrum as Figure 1.9. In addition, this effect leads to

use resonance and analyze the dependence between resonance and strain amplitude. The

spectrum and amplitude of driven frequency would shift and reveal the presence of defect.

This method is usually referred as nonlinear resonance ultrasonic spectroscopy (NRUS).

32

Figure 1.9 Spectrum for linear response with no defect and nonlinear response with defect of

hysteretic material.

For NRUS, material is examined by a sine wave with different amplitudes and

analyzed for a different hysteresis loop in stress-strain curve and dependence of spectrum. If

the material is excited with a sine wave with a certain amplitude A1, the stress-strain curve

follows a loop having an average modulus k1. When increasing the driven amplitude to A2 (A2

> A1), the material follows a different loop with another modulus k2. This causes modulus

reduction and incensement of attenuation as Figure 1.10. This effect causes that the

resonance and amplitude would shift as figure. The nonlinear hysteretic parameter α is

predominant and evaluated as

f0 fi

f0

(1.4.11)

where Δε is the average strain amplitude, α comes from hysteretic parameter, f0 is the natural

frequency of intact frequency, and fi is the resonance frequency for an increasing driving

amplitude as Figure 1.11. The evaluation of relative shift dependence as function of the strain

amplitude can be used for indication of defect presence.

Linear

Response

Nonlinear

Response

33

Figure 1.10 (a) Excited load A2 > A1 (b) Stress-strain curve for two different load amplitude.

Black line for A1 and red line for A2.

Figure 1.11 Resonance frequency for different driven amplitude. (a) undamaged sample. (b)

damaged sample.

Meo et al. [51] used NRUS to examine the laminated composite plate with different

sizes of barely visible impact damages (BVID) cased by low impact (<12 J) . The results

showed high sensitivity to the presence of damage with higher values of nonlinear parameter

α when damage size grows bigger. Cheong [52] used the same way to evaluate the crack and

34

fatigue process in a material for nuclear reactor pressure vessel. Nonlinear parameter α is

associated with the fatigue cycles and crack lengths.

1.4.2.2 Vibro-acoustic Modulation

The modulation method utilizes the effect of nonlinear interaction of acoustic waves in the

presence of the defects. This method combines vibration (pumping) and acoustic waves

(probing) for the interaction of low frequency vibration excitation and high frequency

ultrasonic wave introduced simultaneously as Figure 1.12(a). When the medium is intact or

undamaged, spectrum of responses exhibit only two major frequency components. When it

has damages, spectrum of responses not only have original two ones but also additional

sidebands around major ultrasonic components. The illustration is showed in the Figure

1.12(b) and (c). The number of sidebands and their amplitudes are dependent on intensity of

modulation and the sidebands are

fsn fH nfL (1.4.12)

where fsn means the nth sideband, n = 1, 2, 3, ∙∙∙,n, fH and fL presents the high frequency of

acoustic wave and low frequency of vibration.

35

Figure 1.12 Nonlinear vibro-acoustic wave modulation technique. (a) Scheme of the method.

(b) Spectrum of response signal with undamaged material. (c) Spectrum of response signal

with damaged material. HF and LF denote high frequency and low frequency respectively.

Then the intensity of modulation can be described by the parameter R, approaching

from the amplitudes of two major sidebands A+ and A- and high frequency ultrasonic

components A0 as

R A A

A0

(1.4.13)

In recent developments, nonlinear ultrasonic techniques need to overcome the

technical hurdles before applying to real cases. There are two particular issues for nonlinear

modulation [53]. First, spectral sidebands generation are altered by environmental and

operational conditions such as temperature or loading, which depends on the dynamic

36

characteristics of target structures. Second, the exiting nonlinear ultrasonic modulation

technologies depend on comparing the amplitudes of the spectral sidebands obtained from

non-damaged specimen as baseline and damage condition, but these technologies are

susceptible to false alarms due to signal variations unrelated to the defect.

Many studies are based on this method to examine many materials including

aluminum, composite panel or sandwich structure. For aluminum plate, the studies focus on

the micro-crack or fatigue issues. Parsons et al. [54] used Low profile piezoceramic actuators

with low frequency excitations in nonlinear acoustics for detecting a fatigue crack in an

aluminum plate. The result showed the piezoelectric excitation with surface-bonded low-

profile piezoceramic stack transducer was suitable for crack detection. Ryles et al. [55]

studied nonlinear acoustic and Lamb wave techniques for fatigue crack detection was done in

an aluminum plate. The plate was instrumented with two low-profile piezoceramic

transducers and a low-profile piezoceramic stack actuator. The stack actuator was actuated

by a low frequency sine wave when nonlinear acoustics was applied. For the case

investigated, the study indicated that nonlinear acoustic method has ability to detect fatigue

cracks of the same order as the Lamb wave-based method. H.F. Hu et al. [56] used the theory

of amplitude and angular modulation conducted in the study with a piezoceramic sensor and

two piezoceramic actuators for the undamaged and cracked aluminum plate. Klepka et al. [57]

used Numerical analysis for panel vibration modals and nonlinear vibro-acoustic experiments

tested on an aluminum plate with a fatigue crack. The results presented provide experimental

evidence to the theoretical models, i.e. Nonlinear mechanical elastic (NME) [44, 58] and the

37

Luxembourg-Gorki effect [59, 60] based on thermo-elastic coupling for fatigue cracks in

metals.

Complex structures raise attentions in recent years. Low velocity impact which is

barely invisible on the composite panel (barely invisible impact damage, BVID) is a serious

issue and hard to be identified on the composite panels. Therefore, NNDT is used for early

detection on these micro-structural damages. Aymerich et al. [61] demonstrated the

application nonlinear acoustic for impact damage detection in composite laminates. The plate

was instrumented with an electromagnetic shaker and two bonded low profile piezoceramic

transducers, one as exciter and another as receiver. Pieczonka et al. [32] used local defect

resonance (LDR) and the second harmonic imaging technique (SEHIT) implemented in the

study with one piezoceramic actuator and LDV on a laminated composite plate. The LDR

method is that a structural defect has associated resonant frequencies, being a function of

damage size and geometry. The assumption is that the amplitude level of higher harmonic in

measured response spectra increases dramatically near the location of damage. The second

harmonic imaging technique (SEHIT) is an alternative approach used for imaging. The

method is based on the spatial mapping of higher harmonic amplitudes generated by damage.

The assumption is that the spatial distribution of the ratio between the amplitude of a second

harmonic over the amplitude at excitation frequency reveals the location of damage. Kelpka

et al. [45] used similar way to do an arbitrary frequency and frequency corresponding to local

defect resonance for examining the composite panels. LDR enhanced the intensity of

modulation. Lim et al. [53] presented a reference-free detection by using multiple signal

processes for nonlinear ultrasonic modulation. When the low frequency (LF) and high

38

frequency (HF) are excited by two surface-mounted PZTs respectively, the fatigue crack can

provide a nonlinear mechanism. The crack-induced spectral sidebands are isolated by

combination of linear response subtraction (LRS), synchronous demodulation (SD) and

continuous wavelet (CWT) filtering through individually applying the LF and HF. P. Liu et

al. [62] utilized the impulse laser as broadband signals on fatigue issues of aluminum plate.

Laser nonlinear wave modulation spectroscopy (LNWMS) was used one excitation and then

measured multiple peaks in spectrum. By using impulse laser, a noncontact nonlinear

ultrasonic system was built with Q-switch Nd:YAG laser for ultrasonic wave generation and

a laser Doppler vibrometer (LDV) for detection.

1.5 Thesis Outlines

In this research, the structure of dissertation follows the sequence. First, the purpose of this

research is introduced. Then the theory of Lamb waves, chirp signal, nonlinear ultrasonic and

related field are included in CHAPTER 1. Before analyzing the feasibility of analyzing chirp

signal on dispersion relationship, Lamb wave propagation analysis is introduced in

CHAPTER 2. This chapter includes discussions of waves in isotropic, wave in composite

plate, and dispersion relation analysis.

In CHAPTER 3, MP method is used to aim at automatically extracting dispersion

curves from LDV measurement for maximizing mode purity with an efficient way by a

broadband signal. MP method algorithm and implementation are introduced in first sections.

In second section, measurements and analysis of 2-D FFT are discussed. The following

works measurements by Fourier transforms in wavenumber domain with applying MP

39

method to extract wavenumber-dependent frequencies. Experiment setup and result in

aluminum plate with different thickness are discussed in third and sixth section. The

experiment is also extended to composite plate as follow as seventh section. The result is as

expected and confirmed by theoretical values computed numerically from composite plate.

In CHAPTER 4, the study presents the signal process based on chirplet transform for

chirp signal. First, the relation between signal propagation and group velocity is discussed.

The chirplet transform is briefly introduced and the application to dispersive and transmitted

signal is explained in the following second section. It will be shown on using the peak of

magnitude of transform to attain the time of arrival with a specific frequency in the section 3.

The experiments are implemented as similar setup by using a piezoelectric transducer as an

excitation source and laser Doppler vibrometer (LDV) as a sensor. In the Section 5,

experimental results of group velocity undertaking an aluminum plate as specimen are shown

and discussed. The experimental group velocities of measurement are compared with

theoretical predictions. This study also extends to a laminated plate undertaking same method

and comparing with theoretical curves based on the 3-D elasticity theory. The short summary

in the final section show agreements to apply chirplet transform for chirp signal with less

measurement.

In CHAPTER 5, the organization of this study follows as below. Linear and nonlinear

behavior concept of local defect resonance will be described in the section one. Chirp signal

is implemented to examine an aluminum plate with FBH for LDR as following in second

section. Through the vibration patterns at LDR, ZLCC imaging condition at LDR is

40

discussed in the third section. In the forth section, ZLCC imaging and LDR methods are

employed to a case of a composite plate with a barely invisible impact damage (BVID) for

observing linear and nonlinear behaviors.

In the final chapter, CHAPTER 6, discussions for the chirp signal on dispersion

relationships and ultrasonic imaging are organized. In the last section, futures works inspired

and extended from this research are described. Related materials in this research are also

attached in APPENDICES.

41

CHAPTER 2

Lamb Wave Propagation Analysis

2.1 Waves in Isotropic Materials

2.1.1 Dispersion relationship representation

According to Eq. (1.2.28), two equations of dispersion relation for symmetric and anti-

symmetric modes can be combined into one equation

tan(qh / 2 )tan( ph / 2 )

4k 2 pq

(k 2 q2 )2

(k 2 q2 )2 4k 2q2 4k 2q2 4k 2 pq tan( ph / 2 )tan(qh / 2 )

(k 2 q2 )2 4k 2q2 1pq

tan( ph / 2 )tan(qh / 2 )

(2.1.1)

where γ represents the phase as 0 or π/2 as symmetric or anti-symmetric mode respectively.

Moreover, dispersion relations could be represented into the following forms

1) Original form represented by (ω, k, p, q)

Substituting k2 + q2 = ω2 / cT 2 into Eq. (2.1.1) to obtain an unified dispersion relation

in terms of (ω, k, p, q) shown as

42 2

4tan( / 2 )4 1tan( / 2 )T

p phk qc q qh

(2.1.2)

Eq. (2.1.2) is commonly known as Rayleigh-Lamb dispersion relation and is the overall h

thickness of the plate. Dimensional variables are defined by

and 2 f 2 / / pk c (2.1.3)

42

and2 2 2 2/ Lp c k 2 2 2 2/ Tq c k (2.1.4)

where cT is the transverse wave velocity, cL is the longitudinal wave velocity and cp is phase

velocity. Implicitly, for symmetric modes

2 2 2 2( , , ) ( ) tan( / 2) 4 tan( / 2)S k p q k q qh k pq ph (2.1.5)

and for anti-symmetric modes,

2 2 2 2( , , ) 4 tan( / 2) ( ) tan( / 2)A k p q k pq qh k q ph (2.1.6)

equations are represented by above.

2) Non-dimensional form represented by ( , , , )k p q

By introducing the non-dimensional variables, the non-dimensional form of the

dispersion relation in terms of gives( , , , )k p q

4 2 2 tan( / 2 )4 1tan( / 2 )

p pk qq q

(2.1.7)

where the non-dimensional variables are defined by

and h / cT k kh (2.1.8)

2 2 2 2 2( )p ph k 2 2 2 2( )q qh k T Lc c (2.1.9)

Therefore, for symmetric modes

2 2 2 2( , , ) ( ) tan( / 2) 4 tan( / 2)S k p q k q q k pq p (2.1.10)

and for anti-symmetric modes,

2 2 2 2( , , ) 4 tan( / 2) ( ) tan( / 2)A k p q k pq q k q p (2.1.11)

equations are represented as above.

43

3) Non-dimensional form represented by (Ω, K)

As the Lamb wave dispersion relation has been given in Eq. (2.1.2), the dispersion

curves which give the relationship between frequency and wavenumber (ω, k) can be

obtained. There are four parameters (ω, k, p, q) as discussed before and only two of them are

independent parameters which infer dispersion relation (p, q) could be further represented

within two selected parameters (ω, k).

To plot the dispersion curves, it is convenient to introduce the non-dimensional

variables. The non-dimensional form of the dispersion relation in terms of (Ω, K) gives

4 4K 2 2 K 2 1 22 K 2

2 K 2

tan 2 K

22

tan 1K

22

(2.1.12)

where non-dimensional variables are defined by

, , T

hc

K kh (1 2 )2(1 )

T

L

cc

(2.1.13)

and transverse wavenumber in terms of (Ω, K) are

2

2 2 2 2 22

12 2ph Kp K

h

(2.1.14)

2

2 2 22

1 12 2

qh Kq Kh

(2.1.15)

Similarly, for symmetric modes

44

22 2

2 2 2 2 2 2

( , ) 1 2( / ) tan( 1 ( / ) 2)

4( / ) 1 ( / ) ( / ) tan( ( / ) 2)

S K K K

K K K K

(2.1.16)

and for anti-symmetric modes

2 2 2 2 2

22 2 2

( , ) 4( / ) 1 ( / ) ( / ) tan( 1 ( / ) 2)

1 2( / ) tan( ( / ) 2)

A K K K K K

K K

(2.1.17)

equations are represented by above.

2.1.2 Phase and Group Velocities

In section 2.1.1, the curves describe the relation between wavenumber k and angular velocity

ω. In theory, the ratio of angular frequency to wavenumber is the phase velocity cp and the

slope of the dispersion curve is the group velocity cg. Phase velocity cp and group velocity cg

can be conducted by simple relation from frequency and wavenumber as

and cp k

cg ddk

(2.1.18)

where wavenumber k is found by the peak magnitude relative to frequency. When the

relative frequency and wavenumber are known, the relationships of phase and group

velocities are obtained as Figure 2.1 which draws for theoretical curves of thickness 4.72 mm

plate.

45

A0

S0

A1 S1

A0

S0

A1

S1

Figure 2.1 Theoretical curves on aluminum plate of thickness 4.72 mm for phase velocity cp

and group velocity cg.

Group velocity relation can be represented by a closed form

4 4 k2q2 1pq

tan( p / 2 )tan(q / 2 )

(2.1.19)

Or

(2.1.20)

where G = 0 and represent symmetric and anti-symmetric modes, respectively and the p / 2

non-dimensional variables are defined by

, h / cT k kh /

, p2 ( ph / )2 2 2 k 2 q2 (qh / )2 2 k 2(2.1.21)

46

where , , cL ( 2) / [E(1 )] / [(1 )(1 2 )] cT / E / [2(1 )]

and its ratio squared is defined as . cL / cT ( 2) / 2(1 ) / (1 2 )

The group velocity can be deduced from the implicit form of the dispersion relation,

Eq. (2.1.20), as follows (bar on the symbol is neglected):

(2.1.22)

With this definition, any dispersion relation linking frequency and wavenumber is suitable to

calculate group velocity. From implicit differentiating, the dispersion function G can be

considered as dG(ω,k) = 0. Since the function G is zero, it can be rewritten as

(2.1.23)

The non-dimensional group velocity is obtained as

cg k

AB

(2.1.24)

where

A 4[2 p2q2 k 2( p2 q2 )]tan( p / 2 ) 8 pq(2k 2 2 ) tan(q / 2 )

p2

4q2k 2

cos2( p / 2 )

(2k 2 2 )2

cos2(q / 2 )

(2.1.25)

B 4k 2( p2 2q2 ) tan( p / 2 ) 4 pq(2k 2 2 ) tan(q / 2 )

p2

2 4q2k 2

cos2( p / 2 )

(2k 2 2 )2

cos2(q / 2 )

(2.1.26)

Or the equation can be expressed as

47

cg k

AB

(2.1.27)

where

A 4

2 [2 p2q2 k 2( p2 q2 )]tan( p / 2 )

8 pq(2k 2 / 2 1) tan(q / 2 )

p2

4q2k 2 / 2

cos2( p / 2 )

(2k 2 / 2 1)2

cos2(q / 2 )

(2.1.28)

B 4 k 2

2 ( p2 2q2 ) tan( p / 2 )

4 pq(2k 2 / 2 1) tan(q / 2 )

p2

2 4q2k 2 / 2

cos2( p / 2 ) 2 (2k 2 / 2 1)2

cos2(q / 2 )

(2.1.29)

Based on equations, non-dimensional group velocity relations are plotted in the Figure 2.2 as

an example.

48

Figure 2.2 Non-dimensional group velocity calculated from the equation of non-dimensional

ω and k. ( = 0.33)

2.2 Waves in Composites

For active diagnosis utilizing ultrasonic transient waves for damage detection, localization

and assessment, understanding the wave propagation characteristics in composite is essential

for successful application of these techniques. The wave propagation in composite is

complex due to the nature of heterogeneity of the constituents, inherent material anisotropy

and the multi-layered construction, which leads to the velocity of wave mode depended

macroscopically on the laminate layup, direction, and interface condition.

49

When waves propagate in isotropic plate-like structures, repeated reflections at the

top and bottom surfaces alternately are experienced. The resulting waves from their mutual

interference are guided by the plate surface and formed. The guided wave can be modeled by

imposing surface boundary conditions on the equation of motion as previous section

introduced. This approach introduces the dispersion phenomenon, associating that the

velocity of propagation of the guided wave along the plate being a function of frequency and

wavenumber. The dispersion relations for an elastic isotropic plate with infinite extent in

plane strain state with generalization in traction-free boundaries were first derived Lamb [5]

so called Lamb wave as previous section introduced.

For wave propagating in multi-layered composites, the wave interactions depend on

the constituent properties, geometry, direction of propagation, frequency, and interfacial

conditions. If wavelengths are significantly longer than the sizes of the constituents of

composites, each lamina can treated as an equivalent homogeneous orthotropic or

transversely isotropic material with symmetry axis parallel to the fibers. Considering the

laminates composed of macroscopically homogeneous layers, the wave interactions involve

not only the reflection on the surfaces but also the reflection and refraction between layers,

manifested themselves in the form of waves propagating along plane. In addition, the

velocity dependent on the direction of propagation, the other significant consequence of

elastic anisotropy is the loss of pure wave modes. The dependence of wave velocity on the

direction of propagation implies that the direction of group velocity does not generally

coincide with the wave vector. The distinction between wave modes in composites is

somehow artificial since three types of wave modes are generally coupled. For symmetric

50

modes, one type is designated as quasi-extensional (qSn), where the dominant component of

the polarization vector is along the propagation direction. For anti-symmetric modes, the

quasi-flexural (qAn) are generated. In both types of modes, quasi-horizontal shear for

symmetric (qSH2n) and anti-symmetric (qSH2n-1) are the other type, where the polarization

vector is mainly parallel to the plane of plate.

In general, there are two theoretical approaches to investigate Lamb wave in

composites, one for exact solution by 3-D elasticity theory and another is approximate

solution by plate theory. Although the exact solutions provide accurate results, the

computation for dispersion characteristics of multi-layered composites is intensive because of

the transcendental equations. To make the solutions tractable, approximate solutions by

laminated plate theories are strived. Here the formulation of Lamb waves in composite by 3-

D elasticity theory is introduced.

2.2.1 Elasticity equations for wave propagation

A Cartesian coordinate system is used with z-axis normal to the mid-plane of a composite

laminate spanned by x and y axes. Two outer surfaces of the laminate are at z = ± h/2 where h

is the thickness of composite plate. Each layer of the composite laminate with an arbitrary

orientation in the global coordinate system (x, y, z) is considered as a monoclinic material

having x–y as a plane of symmetry, the stress–strain relations therefore take the following

matrix form:

51

x

y

z

yz

xz

xy

C11 C12 C13 0 0 C16

C12 C22 C23 0 0 C26

C13 C23 C33 0 0 C36

0 0 0 C44 C45 0

0 0 0 C45 C55 0

C16 C26 C36 0 0 C66

x

x

x

yz

xz

xy

(2.2.1)

When the global coordinate system (x, y, z) does not coincide with the principal material

coordinate system ( , , z) of each layer but makes an angle ϕ with the x-axis shown, the 𝑥' 𝑦'

stiffness matrix Cij (i, j = 1, 2, 3,…, 6) in (x, y, z) system can be obtained from the lamina

stiffness matrix C ij in ( , , z) system by using a transformation matrix method [21]. The ' 𝑥' 𝑦'

lamina is orthotropic or transversely isotropic with respect to the principal material axes in (

, , z) and its lamina stiffness matrix C ij can be calculated from the lamina engineering 𝑥' 𝑦' '

material properties Ek, vkl, and Gkl (k, l = 1, 2, 3) [22].

The linear engineering strain–displacement relations are

and

x u,x

y v,y

z w,z

yz v,z w,y

xz u,z w,x

xy u,y v,x

(2.2.2)

where subscript comma indicates partial differential; u, v, and w are the displacements in the

x, y, and z-directions respectively.

The equations of motion in the absence of body forces are governed by:

(2.2.3)

52

where ρ is the mass density of the lamina, and dot denotes time derivative. The traction-free

boundary conditions on the top and bottom surfaces are

z xz yz 0, at z h / 2 (2.2.4)

Since the Lamb waves travel along the plane of a plate with traction-free boundaries but are

standing waves in the z-direction of the plate, the wave motion may be expressed by

superposition of plane harmonic waves. Eachplane harmonic wave traveling in the direction

of wavenormal k is represented by

u,v, w U (z),V (z),W (z) ei(kxt ) (2.2.5)

where k = [kx, ky]T and its magnitude is |k| = . Note that k points the direction of 𝑘2𝑥 + 𝑘2

𝑦

propagation. In the x–y plane, k = k[cosθ, sinθ]T where θ is the direction of wave

propagation.

Substituting Eq. (2.2.5) and (2.2.2) in (2.2.1) gives the expressions of stresses in each

layer

x

y

z

yz

xz

xy

C11 C12 C13 0 0 C16

C12 C22 C23 0 0 C26

C13 C23 C33 0 0 C36

0 0 0 C44 C45 0

0 0 0 C45 C55 0

C16 C26 C36 0 0 C66

ikxU

ikyV

WV ikyW

U ikxW

ikyU ikxV

ei[(kxxky y ) t ](2.2.6)

where the prime denotes the derivative with respect to z. Substituting Eq. (2.2.6) into (2.2.3),

the equations of motion of each layer become

53

C55 U C45 V (C11kx2 2C16kxky C66ky

2 2 )U

[C16kx2 (C12 C66 )kxky C26ky

2]V

i[(C13 C55)kx (C36 C45)ky ] W 0

(2.2.7)

C45 U C44 V [C16kx2 (C12 C66 )kxky C26ky

2]U

(C66kx2 2C26kxky C22ky

2 2 )V

i[(C36 C45)kx (C23 C44 )ky ] W 0

(2.2.8)

i[(C13 C55)kx (C36 C45)ky ] U

i[(C36 C45)kx (C23 C44 )ky ] V

C33 W (C55kx2 2C45kxky C44ky

2 2 )W 0(2.2.9)

2.2.2 Lamb wave in a composite lamina

In an off-axis lamina, the solutions of Eq. (2.2.5) can be simply separated into symmetric and

anti-symmetric wave modes, which render the analytical representation particularly simple:

and

U s

Vs

Ws

As cosz

Bs cosz

Cs sinz

Ua

Va

Wa

Aa sinz

Ba sinz

Ca cosz

(2.2.10)

where ξ is a unknown variable to be determined later. Moreover, the subscripts s and a

represent symmetric and anti-symmetric modes respectively.

First symmetric modes are considered. Substitution of Eq. (2.2.10)into equations of

motion, Eq. (2.2.6) leads to an expression in a matrix form

54

11 2 12 13

12 22 2 23

13 23 33 2

As

Bs

Cs

0 (2.2.11)

where the bar indicates complex conjugate. For anti-symmetric modes,

11 2 12 13

12 22 2 23

13 23 33 2

As

Bs

Cs

0 (2.2.12)

The elements in the above matrix defined by (Γ−ρω2I) are listed as follows

11 C11kx2 2C16kxky C66ky

2 C552 (2.2.13)

12 C16kx2 (C12 C66 )kxky C26ky

2 C452 (2.2.14)

13 i[(C13 C55)kx (C36 C45)ky ] (2.2.15)


2 C442 (2.2.16)

23 i[(C36 C45)kx (C23 C44 )ky ] (2.2.17)


2 C332 (2.2.18)

where I is a 3 by 3 identity matrix.

Eq. (2.2.11) and (2.2.12) are standard linear eigenvalue problems of Hermitian matrix

Γ. If the matrix is positive definite, it can be shown that the eigenvalues ρω2 of Γ are positive

and nonzero, furthermore the right eigenvectors follow the orthogonality relation [21]. The

similar process is followed by the anti-symmetric mode.

55

For nontrivial solutions of As, Bs, and Cs in Eq. (2.2.11), the determinant of the matrix

(Γ−ρω2I) yields the following sixth-order polynomial in ξ:

16 2

4 32 4 0 (2.2.19)

where αi (i = 1, 2, 3) are real-valued coefficients of Cij, k, and ρω2. If variable ξ is redefined

as kξ, the coefficients αi are functions of Cij, θ, and ρω2. Roots of this equation can be

obtained explicitly from known formulas since the equation can be reduced to a cubic

polynomial in terms of ξ2. It is a similar process for Eq. (2.2.12).

In general there are three positive, nonzero, and discrete ξj (j = 1, 2, 3). For each ξj in

symmetric modes, Bs and Cs related to symmetric modes can be expressed in terms of As via

Eq. (2.2.11) as

Bs (11 2 )23 1213

13(22 2 ) 1223

As RAs (2.2.20)

Cs 12

2 (11 2 )(22 2 )23

13(22 2 ) 1223

As iSAs (2.2.21)

and similarly for anti-symmetric modes, Ba = RAa and Ca = −iSAa. With the above equations,

the polarization displacement vectors are determined from the three roots. Consequently, the

general solution of Eq. (2.2.10) is

and

U s

Vs

Ws

Asj

coszR j cosz

iS j sinz

j1

3

Ua

Va

Wa

Aaj

sinzR j sinz

iS j cosz

j1

3

(2.2.22)

Substituting of Eq. (2.2.22) in Eq. (2.2.6) and considering Eq. (2.2.4), σz, τyz, and τxz

are rearranged as

56

z

yz

xz

z h/2

H1 j sin( jz )

H2 j cos( jz )

H3 j cos( jz )

j1

3

Aj 0 (2.2.23)

where φ = 0 and π/2 represent anti-symmetric and symmetric lamb wave modes respectively.

H1 j

H2 j

H3 j

C13kx C23ky R j C33 jS j C36(ky kx R j )

C44( j R j kyS j ) C45( j kxS j )

C45( j R j kyS j ) C55( j kxS j )

(2.2.24)

The existence of a nontrivial solution of Eq. (2.2.24) leads to closed-form dispersion relations

as

H11(H22 H33 H23H32 ) tan(1h / 2 )H12(H23H31 H21H33) tan(2h / 2 )H13(H21H32 H22H31) tan(3h / 2 ) 0

(2.2.25)

Eq. (2.2.25) is a transcendental equation implicitly relating ω to k. For a fixed h, a numerical

iterative root-finding method is employed to compute the admissible ω for a range of k

values, leading to dispersion relations of Lamb wave modes in the direction of propagation.

Furthermore in general the frequency ω of each mode is single-valued function of k.

2.2.3 Lamb waves in a composite laminate

In formulating Lamb waves in a laminate, the interfaces between layers are assumed to be

perfectly bonded. The displacement components of each layer in the z-axis need to be

modified in exponential forms to accommodate the inhomogeneity of the multi-layered

laminates.

57

UVW

Aeiz

Beiz

iCeiz

(2.2.26)

Substituting these expressions into the equations of motion, Eq. (2.2.6) may be rearranged in

a matrix form

11 2 12 13

12 22 2 23

13 23 33 2

ABC

0 (2.2.27)

The nontrivial solution for A, B, and C yields the sixth order polynomial in terms of ξ

shown in Eq. (2.2.19). The six roots for ξ can be arranged in three pairs as ξj+1 = − ξj (j = 1, 3,

5). For each ξj, B and C can be expressed in terms of A via Eq. (2.2.27) as Bi = RiAi and Ci =

−SiAi (i = 1, 2, 3,…, 6). Further, Rj+1 = Rj and Sj+1 = Sj, (j = 1, 3, 5).

Consequently, the general solution of Eq. (2.2.26) in each lamina is

UVW

ei(kxt ) Aj

1R j

S j

j1

6

ei jz (2.2.28)

The interlaminar stress components, σz, τyz, and τxz in each lamina may be expressed as

z , yz , xz ikei(kx t ) H1 j , H2 j , H3 j j1

6

Ajei jz 0 (2.2.29)

and

H1( j1) H1 j , H2( j1) H2 j , H3( j1) H3 j (2.2.30)

where j = (1, 3, 5).

58

Generally, there are two methods named transfer matrix method (TMM) and

assemble matrix method (AMM) for obtaining the dispersion relations in laminates [8,23].

Although the procedures of these two methods seem different, they are identical in principle

by both satisfying traction-free boundary conditions on the outer surfaces of the laminate and

continuity of interface conditions between two adjacent laminas. Both methods can calculate

dispersion curves in a general laminate with an arbitrary stacking sequence.

Using Eq. (2.2.29), it may be observed that symmetric and anti-symmetric wave

modes in general laminates cannot be decoupled. However, symmetric laminates are

practically used in designing the composite structures. A robust method is to separate the two

types of wave modes by imposing boundary conditions at both top and mid-plane surface.

Traction-free boundary conditions on the top surface of the laminate are given by

z xz yz 0, at z h / 2 (2.2.31)

Because of the symmetric geometry and symmetric material property of the laminate, only

half of the laminate needs to be considered and then the following conditions on the stress

and displacement components at the mid-plane for symmetric modes are imposed

w yz xz 0, at z 0 (2.2.32)

Likewise, the boundary conditions of anti-symmetric modes at the mid-plane are

u v z 0, at z 0 (2.2.33)

By imposing displacement and stress continuity conditions along the interfaces of half layup

of an N-layered laminate, total 3N equations are constructed if the assemble matrix method is

used. Then set the determinant of the 3N equations to zero, and numerically solve the

59

resulting transcendental equation for the dispersion relations of Lamb waves in symmetric

laminates. In this paper, the transfer matrix method (TMM) is adopted due to its stable nature

of the numerical implementation [8].

2.2.4 Computation process for obtaining dispersion relation in composite

The process to obtain the dispersion relation in laminated composite plate is shown in the

Figure 2.3. In the beginning, the material properties of lamina are known in prior. Material

properties are applied into principal material coordinate and then employed to global

coordinate with fiber directions. Before taking into equations of motion to solve the sixth

order polynomial equation, the normalized wavenumbers and phase velocity with

propagation direction are defined manually as numerical step. When the solutions of ξj are

defined, P and Q as matrix as ξj and R, S, H respectively are taken into equations for upper

and lower layers with boundary conditions. After calculating with determinant, the roots

associated with frequencies are found. In the sorting process, associated modes are rendered

in order.

60

Material Properties

Solving six ξj for each layer

Stiffness Matrix

Each Layer Orientation φ

Equations for upper layerEquations for lower layer

E, G, ρ, h

C

Cφ

Define [kx, ky] = [cosθ, sinθ], cp

Determinant

Boundary ConditionsS: w = σyz = σxz = 0A: u = v = σz = 0SH: u = w = σz =0

Mode Sorting

P ξj, Q R, S, H

Figure 2.3 The computation process for obtaining the dispersion relation in composite plate.

2.2.5 Velocity dispersions and characteristic wave curves

The dispersion relation between ω and k can be symbolically represented by an implicit

functional form G(ω, k) = 0 or G(ω, k, θ) = 0. It is assumed that this relation may be

explicitly solved in the form of real roots of ω = W (k). There are an infinite number of

possible solutions, in general, with different functions W. Such solutions correspond to

different wave modes. For plane waves, the phase velocity vector is defined as cp = (ω

/k)(k/|k|) = (ω/k2)k and its magnitude is cp = ω/k. A curve generated by all choices of k from

61

the origin for cp at a given frequency is called velocity curve. The radius vectors of velocity

curves in the direction of a given k represent the admissible phase velocity dispersion of

different wave modes.

The group velocity, which can be measured by tracking envelopes of a wave packet,

is defined by cg = gradkW. If the closed form of implicit function G can be obtained, the

group velocity may be also calculated by

(2.2.34)

In a polar coordinate system, gradkW has a radial component in the direction of k and an

angular component perpendicular to k. Using coordinate transformation, the group velocity

in a Cartesian coordinate can be attained as

(2.2.35)

where the subscripts x and y represent the components in x- and y-axes respectively. The

magnitude of group velocity cg and the angle θg from the x-axis are given by

and cg cgx2 cgy

2 g tan1 cgy

cgx

(2.2.36)

The skew angle ϑ or steering angle [10] is defined as

g (2.2.37)

In isotropic plates since ω is only function of the magnitude of the wave vector, the

direction of group velocity coincides with the direction of wave vector (i.e., ϑ = 0). The

62

magnitude of the group velocity is cg = ∂W/∂k. However for Lamb waves in composites,

∂W/∂θ does not vanish in general; thus the direction of group velocity is not parallel to k, i.e.,

the skew angle ϑ may not be zero.

The locus of group velocity vector along all choices of cg from the origin at a given

frequency is referred to as wave curve (or wave front curve). It is worth of noting that the

radius vector joining the origin (or source point) to a point on a wave curve represents the

distance traveled by the elastic disturbance in unit time. The wave curve therefore gives the

locus of wave front, at a unit time, by the disturbance emitted by a point source acting

through the origin at time t = 0. Thus wave curves are of great importance in damage

detection of SHM or NDE.

An important geometric relation between slowness curve and group velocity direction

is introduced for computing wave curves. The dispersion relation of each Lamb wave mode

can be expressed as an explicit function of W (k, θ) which may be viewed as a conical-like

surface in 3-D domain. Furthermore, the slowness curve is geometrically the level surface of

W (k, θ) at W (k, θ) = ω0. Differentiating both sides of the equation with respect to h yields

(2.2.38)

In polar coordinates, the tangent vector of slowness curve W (k, θ) = ω0 is

proportional to dk/dθ and group velocity vector in polar coordinates. It can be concluded

from Eq. (2.2.38) that the group velocity vector cg is perpendicular to the tangent vector of

slowness curve; that is, the group velocity vector cg is parallel to the normal direction of

63

slowness curve. Similarly it can be proved that the wave normal k is parallel to the normal

direction of wave curve. Based on Eq. (2.2.38), Eq. (2.2.35) can be rearranged as

(2.2.39)

Although group velocity expressions in Eq. (2.2.35) and (2.2.39) are physically equivalent,

they are suitable for different numerical implementations: Eq. (2.2.35) is conveniently

employed to compute group velocity dispersions along a given wave propagation direction;

while Eq. (2.2.39) is more suited to calculating wave curves at a given frequency.

To obtain the group velocity dispersion at a given wave propagation direction θ1, two

dispersion relations W (k, θ) having slightly different directions of propagation θ1 ± Δθ/2 are

required. Then ∂W/∂k in Eq. (2.2.35) is calculated from one of the two obtained dispersion

relations, and the derivative term ∂W/∂θ in Eq. (2.2.35) can be approximated by finite central

difference:

(2.2.40)

For computation of a wave curve at a given frequency, a semi-exact method is developed by

performing finite difference on the exact solutions of two slowness curves corresponding to

two very close frequencies ω1 ≈ ω2. ∂W/∂k in Eq. (2.2.41) can be calculated by

(2.2.41)

64

where k1 and k2 are both function of θ. In addition, dk/dθ can be computed from one of the

two known slowness curves.

Note that the geometric relation above is also valid for bulk (non-dispersive) waves.

In addition, the polar reciprocal of the slowness curve is the wave curve (i.e., cg = 1) and cp =

cg cosϑ [24]. However, these two relations break down for Lamb waves in lamina and

laminates because of the dispersive behavior [10].

2.3 Dispersion Relationship Analysis

2.3.1 Fourier Transform and Dispersion Curve

2-D Fourier transform overcomes the problems of multiple modes and dispersion by

transforming the received amplitude-space-time record to amplitude-frequency-wavenumber

ones, where individual Lamb waves may be resolved and their amplitude showed. Since the

Fourier transform is a reversible transformation, this application could potentially be used to

generate a synthetic surface-wave profile from characteristics of propagation specified in the

x-t plane from ω-k domain after filtering. For intuition, 2-D Fourier transform provides a tool

to describe the dispersion relation of harmonic waves as derived equation.

Obtaining the image of the dispersion curve relies on the spectral decomposition of

the wave field. To image the dispersion relation from the wave field determines the

transform. The harmonic wave can be represented from frequency-wavenumber [63, 64]

u(x,t) 1

4 2 dk dei(kxt ) N(k, )D(k, ) (2.3.1)

65

where x is seen as offset, t is travel-time of propagation waves. k is the wavenumber, N(k,ω)

is a function of the source of excitation, and D(k,ω) is the dispersion relation. Now taking the

travel-time t to be a function via x with a coefficient p starting time τ integrates as linearly

summation of harmonic propagation waves [64].

U(p, ) u(x,t)dx u(x, px)dx dx dk dei(kx ( px ))

N (k, )D(k, )

dx dk de i ei(k p)xN(k, )D(k, )

dk de iN (k, )D(k, )

(k p)

(2.3.2)

This equation describes a transformation of the original wave field. The integration of

δ(k-ωp) has value and the equation can be represented when k = ωp.

U(p, ) deiN( p, )D( p, )

N( p, )D( p, )

e i d(2.3.3)

The equation is seen as an inverse Fourier transform. Then result of Fourier transform over τ

is

U(p, ) N ( p, )D( p, ) (2.3.4)

In the field Ū(p,ω) will satisfy the dispersion relation. The values in the Ū(p,ω) wave field is

exactly the dispersion curve that is seeked for.

66

The spectral peak of the 2-D Fourier transform of the wave field gives the frequency

associated with phase velocity. Therefore, the wave field is linearly transformed from space-

time domain to frequency-wavenumber domain which dispersion curves are imaged. From

Eq. (2.3.4), the transformation can be seen as inverse 2-D Fourier transform. In other words,

the result of measured signals by Fourier transform represents the dispersion curve in ω-k

map.

2.3.2 Dispersion Curve Analysis

From Eq. (2.2.1), wavenumbers are functions of frequency ω as k(ω) which contains various

wavenumbers caused by symmetric or anti-symmetric modes. It can be also derived as

k( ) 2

2qcT2

tan(qh / 2 )

tan(qh / 2 ) pq

tan(ph / 2 )

1/2

(2.3.5)

where k(ω) depends on γ=0 or π/2 to be symmetric or anti-symmetric mode respectively.

This function leads the 2-D Fourier transform as

U(k, ) A( ) ( 0 ) (k k0 ) (2.3.6)

when excited signal has mono-frequency ω0, the received responds as propagating as Lamb

waves have different wavenumbers generated by theory of Lamb wave dispersion relation.

2.3.3 2-D Fourier Transform

The two-dimensional Fourier transform method is applied by carrying out a Fourier

transform of the time history of the response at each position monitored to obtain a frequency

67

spectrum for each position. At this stage, an array with the spectral information for each

position is obtained. The data is put into a matrix in its respective column. A spatial Fourier

transform of the vector formed by the components at a given frequency then amplitude

wavenumber frequency information. In practice a two-dimension fast Fourier transform

algorithm (2-D FFT) may be used.

(a) (b)

Figure 2.4 Excited signal in temporal sequence and spectrum. (a) received signal with 70 kHz

in time domain (b) spectrum of received signal.

In an array, spectrum information for each position in column and for time history in

row can be transformed by 2-D FFT into frequency-wavenumber matrix as Figure 2.5 and

Figure 2.6. The figures clearly can be seen in the position with maximum amplitude when

appropriate actuated frequency meets relative wavenumber to present the dispersion property

of wave propagation. While more measured data in different frequencies is showed on the

68

relative position in frequency-wavenumber domain, the peaks are connected as dispersion

curves as well.

Figure 2.5 A ω-k contour which plots for an aluminum plate (thickness 4.72 mm) shows a

peak with 70kHz actuated frequency in the position relative to the wavenumber.

69

Figure 2.6 A ω-k map in 3D on aluminum plate (thickness 4.72 mm) shows a peak with

70kHz actuated frequency relative to the wavenumber.

2.3.4 Time-frequency Analysis

In contrast to 2-D FFT, time-frequency representations (TFR) are used for determining the

dispersion relationships of transient waves propagating in a medium with less measurement.

Wavelet transform (WT) using a Gabor wavelet to determine group velocities of plate wave

propagation is one of TFR method and introduced as following.

The continuous WT (CWT) of a function f(t) is defined as

WT f (a,b) 1a

f (t)*( t ba

)dt

(2.3.7)

where a > 0 and the superscript * denotes a complex conjugation. The analysis function as a

kernel for the WT can also be expressed as

70

a ,b(t) 1a

( t ba

) (2.3.8)

Then the WT is obtained as

WT f (a,b) f (t)a ,b* (t)dt

(2.3.9)

The WT elements are generated by shifting b and scaling a a basic wavelet function

Ψ(t). Ψa,b(t) is presented for centering at b with a spread proportional to a. The function

Ψa,b(t) is considered as a windowed function both in time and frequency domains. The size of

the time window is controlled by the scale a. It is possible to change the window size either

in time or in frequency domain.

By using wavelet transform with Gabor function to determine the dispersion relation

of group velocity cg and frequency f, experiment is set up with two fixed points and five

peaks tone burst signal actuation. The Gabor function for the kernel is expressed as

g (t) 14

0

ei0te

(0 / )2 t2

2 (2.3.10)

In the experiments, the group velocity was obtained and calculated by collecting

signals from sensors at point S1 and S2 where the practical signals simultaneously delayed to

the time when flexural waves were excited. The experiment shown in the Figure 2.7 is set up

with a piezoelectric actuator as excitation source and using laser Doppler vibrometer as a

sensor. Excited signals shown in the Figure 2.8 are controlled by the function generator

connecting to data management system (DMS). The LDV is also manipulated through

controller and the computer storing the data. The data will be collected and analyzed by the

71

commercial computational software MATLAB. The detecting points S1 and S2 are arranged

on the surface of fixed specimen with distances d1 and d2 respectively. Not all frequency

components appear at the same time due to time lag happened between different frequencies.

Therefore, using two points S1 and S2 to measure responses can effectively side effects.

Figure 2.7 Scheme of the experimental setup with two measuring locations using laser

Doppler vibrometer and the data management system.

The procedure of evaluating group velocities follows the below orders at each

frequency by the scale a where central angular frequency is ωc = ω0 / a. First, responses are

computed for the coefficients of wavelet at point S1 and S2. Determine the arrive time b1 of

response 1 and b2 of response 2 at each scale a from finding the maximum magnitude of

|WTf(a,b)| as in the Figure 2.9 for the responses. From the definition of group velocity cg = x /

b, the velocity is considered as the traveling distance over the different duration. The formula

is transformed as

72

cg db

d2 d1

b2 b1

(2.3.11)

where Δd means the traveling distance from S1 to S2, Δb means the traveling time from S1 to

S2, and d1 and d2 can be found as designed location as setting up the experiment.

Figure 2.8 Actuating Hanning window five-peaked toneburst as input signal.

Figure 2.9 Determining the arrival time b1 and b2 at position S1 and S2.

b

1

b

2

@ S1 @ S2

73

The responses received by LDV at S1 and S2 locations were analyzed by the wavelet.

The detecting locations are apart for certain distances Δd away. In the plots, the peak of

response means the arrival time of a wave traveling with the group velocity. According to

equation, the corresponding group velocity for each value f = 1/a can be identified at each

frequency. The red line presents the theoretical dispersion curve of A0 modes based on plate

theory. The frequencies were alternative from 100 kHz by 10 kHz increment to 300 kHz to

get the relationship in the experiment. Analyzed data was compared with the theoretical

dispersion curves. The result matched the theoretical curve well in the frequency range of

100 – 300 kHz.

Figure 2.10 Dispersion relation for group velocity (cg) via frequency ( f ) by Gabor wavelet.

74

CHAPTER 3

Lamb Wave Dispersion Analysis by Matrix Pencil Method

3.1 Matrix Pencil (MP) Method

3.1.1 Matrix Pencil Method To A Sum Of Complex Exponentials

Let u(x,t) be the out-of-plane velocity of a wave travelling in the plate under certain

frequency and can be transformed from the frequency-wavenumber domain by

u(x,t) 1

(2 )2 V ( )G(k, )ei(kTxt ) dkd (3.1.1)

where x is the distance between the actuator and sensor as a vector x = (x1, x2)T indicating

directions on the plant, k is wave-vector k = (k1, k2)T, V(ω) is the Fourier transform of the

excitation voltage, and G(k, ω) is the two-dimensional Fourier transform of Green function

of the medium. The poles and singularities of

u(x,t) 1

2V ( )

2G(k, )eikT x dk

e it d

12

U(x, )e i t d (3.1.2)

as its Fourier transform in one dimension into the frequency domain U(x,ω) with space

variable x. In Eq. (3.1.2), k is a function of ω. After the initial excitation, there are supposed

to be no external force and fixed boundaries. u(x,t) as velocity vectors are obtained from the

receiving signals of laser Doppler vibrometer. Therefore, the transformed signal U(x,ω) as

propagating wave modes is decomposed to be a sum of complex exponentials equation,

U(x, ) V ( )

2G(k, )eikTx dk am ( )eikm

T ( )x

m (3.1.3)

75

where the sum is all signals km(ω) depending on excited frequency and amplitudes am(ω) for

different frequencies.

In theory, the sum of the complex exponentials may be infinite, but in practice, there

are only a few finite numbers M of primary frequencies making major contributions to the

signal. The signal by finite sum is approximated with some additive noises when the

measurement from detection points is received by sensors, un(x,t) = u(x,t) + ñ(x,t) where

ñ(x,t) means the noise for each detecting location. The Fourier transform of received

measurement is derived as

U(x, ) am ( )eikmT ( )x

m1

M

(x, ) (3.1.4)

From the equation, the dispersion relations are obtained by estimating wavenumbers km(ω)

under specific frequency. The process can be easily done by the MP method which is

described as next section.

Note that the ñ(x,t) described in above approximation is not always avoidable, but the

discretization error would be small due to sufficiently high temporal resolution measured by

the laser Doppler vibrometer. On the other hand, it means that there will be introduced a

cutoff error due to a restricted and finite measurement. The error can be made relatively

small especially it will dissolve due to dispersion property after some time.

76

3.1.2 Matrix Pencil Algorithm

Two of popular approximating functions by a sum of complex exponential equation are the

“polynomial” method and the “matrix pencil” method, applied in many areas of

electromagnetics. The basic difference between two is that the polynomial method is a two-

step process and the MP method is a one-step process in finding the poles. The MP method

finds poles by solving the generalized eigenvalue problem instead of two-step process where

the first step involves the solution of a matrix equation, and the second step entails finding

the roots of a polynomial, as Prony’s method.

The MP algorithm can be stated as constructing two matrices in a way that the desired

poles are the rank reducing numbers from the generated eigenvalues of the corresponding

function. In particular, the MP method is robust to noise in sampled data. The MP approach

has a lower variance of the estimates of the parameters of interest than a polynomial-type

method and is also more efficient computationally.

The approach can be studied by either fixed wavenumber or frequency. For

convenience, the MP method is first introduced in frequency domain. For each fixed

frequency ω, a data sequence ul is stated as a simple notation.

(3.1.5)

where N is the number of locations receiving signals which is supposed to be larger than a

finite number M and xl refers to different sensing location. M is the finite number of primary

wavenumbers making major contributions to the signal. Since a plate-like material is

77

examined, all detection points are measured as one line as one-dimensional location xl and

wavenumber km.

For simplicity with sampling location variable, the estimating equation can be

rewritten as below:

(3.1.6)

where

zm eikm ( )x (3.1.7)

Δx presents the sensing location as discrete space. By receiving data of wave propagation in

different location, the next objective is to find the best estimates of M, am(ω) and zms from

the measurements. In general, the estimation is essentially a nonlinear problem. The MP

approach is not only more efficient computationally, but also it is better with statistical

properties for the estimate of zm. From a finite number M data, the few signals with larger

amplitudes am have primary contributions in signals and can be extracted from zm for

imaginary part km as dispersion modes related to the excited frequency ω.

For noiseless data, a Hankel matrix H1 and a shifted Hankel matrix H2 can be formed

and are defined from the data sequence ul:

(3.1.8)

78

and

(3.1.9)

where L is referred to the Pencil parameter [65]. H1 and H2 are both (N − L) × L matrices but

H2 shifts a data sequence and start with u1 rather than u0 which is the beginning in the H1.

The parameter L is useful in eliminating some effects of noise in the data. It also efficiently

reduces the calculating numbers of matrices size. The parameter L is chosen such that M ≤ L

≤ N − M. When the pencil parameter L is chosen to be close to L = N/2, the MP method

reaches the approximate Cramer-Rao bound [66]. The basic theory behind the phenomenon

states the dimension of the column subspace of matrix H1 or H2, indicating that more noise

can be filtered out by eliminating the largest noise subspace in decomposition of matrices.

Both matrices H1 and H2 have rank M and each zm is a rank-reducing number for the MP

method.

Due to the characteristic of Hankel matrices, H2 has a time shifting corresponding to

H1. In other words, H2 is equal to H1 multiplying to matrix Z0 containing of zms. H1 and H2

can be rewritten as

H2 Z1AZ0Z2 (3.1.10)

H1 Z1AZ2 (3.1.11)

79

where

(3.1.12)

(3.1.13)

Z0 diag z1,z2 ,..., zM (3.1.14)

A diag A1, A2 ,..., AM (3.1.15)

The matrix A is residues or complex amplitudes and diag represents a M × M diagonal

matrix.

Now the matrix pencil function is considered as:

H2 H1 Z1A(Z0 I)Z2 (3.1.16)

where I is the M × M identity matrix. In general, the rank of H2 − λH1 should be M.

However, the rank will be M − 1 if λ = zm, m = 1,2,.., M and the nth row of Z0 − λI is zero .

Therefore, the complex numbers zn can be found as the generalized eigenvalues of the matrix

pair H1+H2 as an ordinary eigenvalue problem.

H2 H1 H1H2 I (3.1.17)

where H1+H1 = I and H1

+ is the Moore-Penrose pseudoinverse of H1 [67]. In turn, H1+ is

defined as:

80

H1 (H1

*H1)1H1* (3.1.18)

where the superscript “*” denotes the conjugate transpose.

In contrast to fix the frequency ω getting estimate wavenumbers, approaching with

the same procedure by fixing wavenumber k along one direction from Eq. (3.1.6) can be

represented as

(3.1.19)

where

zm e im (k )t and t 1/ fs (3.1.20)

zm has another form in finite primary frequencies for fixing wavenumber k and discrete time

ts, amplitudes bm(k) for different wavenumbers, fs is the sampling rate and the extracted

angular frequency has relationship with extracted frequency ωm = 2πfm.

3.1.3 Matrix Pencil Implementation

For the MP method implementation, it has an easier manner to perform computation.

This process provides some pre-filtering to noise reduction from data [65]. One can form the

data matrix H from the noise contaminated data ul by combining H1 and H2 as:

81

(3.1.21)

H1 is obtained from H by deleting the last column and H2 is obtained by deleting the first

column. For efficient noise filtering, the Pencil parameter L is chosen between N/3 and N/2

[68]. By using these values of parameter L, the variance of noise in parameters zm has been

found to be minimum. From the matrix H, all the N data samples are utilized even if L is

smaller than N.

By properly selecting the parameter L, the matrix Y is decomposed through a singular

value decomposition (SVD):

H UV* (3.1.22)

where U and V are unitary matrices having orthonormal columns and Σ is a diagonal matrix

containing the non-negative singular values of Y in decreasing order.

(3.1.23)

The parameter M for the largest integer is selected such that

j

1

, j 1,2,..., M (3.1.24)

where δ is a threshold so that relatively small singular values may be attributed to noise and

then discarded. On the other hand, the first few singular values are important as primary

82

factors for reconstruction of the data. They are as primary frequencies in the sum of complex

exponentials.

Next the filtered matrix V’ which only contains M dominant right singular vectors

V is represented as:

V ' [v1,v2 ,...,vM ] (3.1.25)

where v1, v2, …, vm are vectors. The right singular vector from M + 1 to L are corresponding

to small singular values and discarded. Therefore, H1 and H2 can be formed as:

H1 U 'V1'* (3.1.26)

H2 U 'V2' * (3.1.27)

where V1’ is obtained from V’ without the last row and V2

’ is obtained from V’ without the

first row. Σ’ is also formed from the M dominant singular values.

For noiseless cases, the eigenvalues of the following matrix can be found:

{H2 H1}LM {H1H2 I}M M (3.1.28)

Determining the λ to solve for zm provides the primary frequencies in the data sequence.

3.2 Experimental Setup And Dispersion Curves By 2-D FFT In Aluminum Plate

Experiment is set up with actuator and sensor for picking up the out-of-plane velocity on the

surface of plate. The PZT actuator (APC International, Ltd.) is a circular piezoelectric disc

which is glued to the plate surface. The sensor as a laser Doppler vibrometer (Polytec OFV

505) is mounted on a 2-axis translation stage and moving along horizontal direction with

83

multiple points for detection. The interval for detected points is equally spaced from the

excitation source. The scanning path is attached with a strip of reflector foil to improve the

receiving signals. Actuated signal in this study is a linear chirp for wideband signal. The

input signal is generated by a Tektronix AFG 3022C single channel arbitrary function

generator and is amplified with wideband power amplifier Krohn-Hite KH model 7602. The

amplified voltage is applied to the piezoelectric disc attached on the surface of specimen.

Function generator and LDV are connected by workstation for synchronization. The setup

scheme is referred to Figure 3.1 for the similar setup in following experiments in different

thickness.


mounting on the 2-axis translation stage sensing in an array.

84

3.2.1 Experiment on an aluminum plate in thickness 2.29 mm

The experiment is carried out for measuring the out-of-plane velocity at the surface of plate.

Experiment arrangement is set up as Figure 3.2. The first study is on an aluminum plate Al

6061-T6 with size 1219 mm × 609 mm × 2.29 mm (48” × 24” × 1/11”). The PZT actuator

(Steiner & Martins, Inc.) is a 7.0 mm diameter and 0.3 mm thickness circular piezoelectric

disc which is glued to the plate surface. The sensor as a laser Doppler vibrometer (Polytec

OFV 505) is moving along horizontal direction with 300 points for detection. The interval for

detected points is equally spaced 0.5 mm from the excitation source. The scanning path is

attached with a strip of reflector foil to improve the receiving signals. For enhancing SNR,

every received signal is averaged for 25 measurements.


mounting on the 2-axis translation stage sensing in an array.

85

Actuated signal is a chirp signal as input for multiple frequencies. The wide range of

excited frequencies is from 50 to 350 kHz (About ω = 315 – 2200 krad/s) over a 250 μs

window. The input is generated by a Tektronix AFG 3022C single channel arbitrary function

generator and is amplified with wideband power amplifier Krohn-Hite KH model 7602.

The input signal is referred to “chirp” as a sinusoidal signal for which the frequency

is swept as a function of time. For a typical chirp, the frequency is swept from minimum to

maximum value within time duration. The equation for this is:

s(t) w(t)sin (0 12

t)t

(3.2.1)

where ω0 is the starting frequency and β is the designed chirp bandwidth β = (ω1 − ω0)/Td.

For the bandwidth, ω1 is the cut-off frequency and Td is the duration of the chirp signal. The

function w(t) is a voltage amplitude rectangular window starting at t = 0 until the duration Td.

The function generator makes a transient burst chirp signal and the frequency of the signal

increases linearly from ω0 to ω1 along with time t going. For this study, linear chirp signal

from 50 to 350 kHz and the spectrum of Fourier transform are used and plotted in Figure

3.3(a) and (b). The measured response to the chirp from an excited source by a piezoelectric

disk mounted on aluminum plate is shown in Figure 3.3(c).

86


(b) Spectrum of input signal after Fourier transform. (c) Response signal to linear chirp

excitation from laser Doppler vibrometer.

3.2.2 2-D Fourier Transform

All points in one line are measured and hence the two-dimensional Fourier transform in

space and time is applied. The laser Doppler vibrometer detects velocity values for N = 300

equal distance points in space and T = 1024 times where the duration of measurement 400 μs.

The raw data forms a 2-D matrix with u(x,t) which includes the information of 1-D location

and transient signals. The data needs to be transformed and then be compared to analytical

curves. Analytical dispersion curves are descripted in the Figure 3.4 based on Al 6061-T6

material properties (ν = 0.33) in non-dimensional axes. After 2-D Fourier transform, the ω-k

contour is shown on Figure 3.5(a). In the figure, the white lines present A0 mode and S0 mode

respectively. The data after 2-D Fourier transform (2-D FFT) has peaks along A0 mode curve,

which can be found out and seen in the figure clearly. By using local maxima method [11],

all points between frequencies and wavenumbers are examined and the peak values of ω-k

contour are identified and picked for the Figure 3.5(b) along with analytical S0 and A0 curves.

(a) (b) (c)

87

Fewer peaks are obtained along S0 curve when analyzing the data by each wavenumber. It

means that the magnitudes of extracted S0 signals are relatively weak so that they are barely

seen in the contour plot but shown in finding maxima.

Traveling waves are propagating with simple relations for phase velocity cp = ω/k and

group velocity cg = dω/dk [69, 70]. From the local maxima points of the ω-k contour, the

relationships of phase velocity cp and group velocity cg with frequencies are obtained and

shown in the Figure 3.5 (c) and (d) respectively. From the plots, most obtained points are

along A0 curve since A0 mode dominates in the selected frequency range.


frequency-wavenumber, phase velocity and group velocity. The black lines mark the

excitation normalized frequency range 50-350 kHz (h = 2.29 mm) and the dots represents the

modes obtained from excitation frequency range in this study.

(a) (b) (c)

88

Figure 3.5 A ω-k contour which plots for aluminum plate shows peaks with chirp signal in

multiple frequencies and dispersion plots based on the local maxima points. (a) Contour plot

for 2-D FFT result. (b) Peak values from 2-D FFT amplitude. (c) Phase velocity cp of first

anti-symmetry A0 mode. (d) Group velocity cg of first anti-symmetry A0 mode.

(c) (d)

(a) (b)

89

3.3 Matrix Pencil Method for Measurements

The dispersion relations may be guessed by calculating the wavenumber k and frequency ω

with two dimensional Fourier transform of u(x,t). Fixing on one direction of k and plotting

the Fourier coefficient in a frequency-wavenumber-diagram (ω-k domain) can see the

dispersion curves as ridges in the Figure 3.5 (a). It is easy to see the contour and maximum

points virtually, but the exact values from figure are hard to be extracted and interpreted.

Moreover, the desired mode of received response is hard to be extracted.

The MP method can be approached from two sides. Either the discrete Fourier

transform in time is calculated first and then the MP is used to approach wavenumbers with a

specific frequency, or the discrete Fourier transform in space is calculated first and then the

data row is applied to the MP algorithm to estimate the frequencies for each wavenumber, In

ideal conditions, both approaching would show the same result, but in practice, one is better

than another since the resolutions of dataset in time or in space are different.

Both figures containing dispersion curves on ω-k domain are easily interpreted to the

relations of phase velocity cp-ω and group velocity cg-ω, which are shown together. Red

circles indicate the extracted points. Black lines refer to analytical A0 and S0 curves marked

adjacent to curves. These analytical curves based on Rayleigh-Lamb dispersion equation are

computed numerically by conventional root-finding algorithm. Figure 3.6 and Figure 3.7 are

compared that the two approaches namely use the discrete Fourier transform in space and MP

method in time (k-t domain), or the discrete Fourier transform in time and MP method in

space (x-ω domain) respectively to attain the primary mode as A0 in this case. Both figures

show the same approach along with analytical result, but apparently one approach on x-ω

90

domain shows more exact result than another on k-t domain visually. The difference comes

from the resolution of wavenumber or frequency. The resolution of wavenumber is 6.67 with

0.5 mm spacing and 300 measurements. It only has 17 points within the range of excited

frequency. In contrast, the resolution of frequency is 32.17 × 103 rad/s with discrete time 3.9

× 10-7 s and 1024 times. There are 59 points within excited range. Due to the axes with

limited length, one is visually more intense than another. In (b) and (c) of Figure 3.6 and

Figure 3.7, the dispersion velocities follow the relations from ω-k domain for phase velocity

cp = ω/k and group velocity cg = dω/dk. The result by MP method shows more accurate than

by 2-D FFT comparing to theoretical curves of A0 mode. It is noted that the extracted points

for group velocity are more diverging due to the derivation of differential calculation which

is sensitive to relative values between points.

Figure 3.6 Extracted points show on (a) ω-k, (b) cp-ω, and (c) cg-ω diagrams where measured

data use Fourier transform in time ω and MP method in space x on h = 2.29 mm Al plate.

(a) (b) (c)

91

Figure 3.7 Extracted points show on (a) ω-k, (b) cp-ω, and (c)cg-ω diagrams where measured

data use Fourier transform in space k and MP method in time t on h = 2.29 mm Al plate.

3.4 Experiment And Results In Thicker Aluminum Plate

In order to examine other higher wave modes by using MP method, a thicker plate is selected

with the same actuation. Experiment is set up as Figure 3.2 for a thicker aluminum plate Al

6061-T6 with size 609 mm × 609 mm × 6.35 mm (24” × 24” × 1/4”). The actuator

piezoelectric disc is an excitation source and the sensor LDV was moving along horizontal

direction with 200 points equally spacing 1 mm for detection. The scanning path is attached

with a strip of reflector foil to improve the receiving signals. For enhancing SNR, every

received signal is averaged for 25 measurements.

Experimental data is analyzed by MP method and extracted points are matched with

analytical dispersion curves shown in the Figure 3.8 and Figure 3.9. Two approaches from x-

ω domain and k-t domain are shown respectively. Due to the limitation of detection point and

spacing, the resolution relative to wavenumber k is far lower than frequency ω as mentioned

previously. In the Figure 3.9, the extracted points are shown and matched A0, S0 and A1

modes. For A1 mode, there are fewer points obtained from k-t domain. In contrast, intense

(a) (b) (c)

92

points from x-ω domain are shown within the exciting bandwidth for A0, S0 and A1 modes. In

both figures, experimental data has good agreement with theoretical curves. The phase

velocity follows the simple relations cp = ω/k from ω-k domain. For the dispersion map of cg-

ω domain, extracted points are sensitive to the derivation between wavenumber and

frequency and they are a little away from prediction but still close to analytical results. It

should be noted that the points from k-t domain shows data for relatively lower wavenumbers

for S0 and A1 mode in ω-k diagram and the points from x-ω domain are shown in relatively

higher frequencies. That means that two approaches can complement each other and

complete the dispersion curves.

In each mode, numbers of extracted points are different due to decomposition of

primary frequencies or wavenumbers. After analyzing signals, M data would be obtained for

fixed frequency or wavenumber. In M, primary frequencies are related to dispersion

relationship with fixed wavenumber. However, some primary frequencies are hard to be

detected under low amplitudes that are cased by lower displacement for the mode while

detecting. When getting more modes and analyzing data, primary frequencies are shown for

few mode. As experiment setup, LDV is sensitive to the out-of-plane displacement, which

causes that A0 mode is easy to be detected. Other modes are in the signals with lower

detected energy and hard to be extracted. In most cases, S0 mode is hard to be detected since

waves propagate in the plane so that 1-D LDV barely detects. Therefore, MP method has

abilities to decompose signals for contained primary frequencies in fixed wavenumber and

vise versa while modes with sufficient detected energies even the mode propagating in the

plane.

93



(c) cg-ω diagrams are shown from left to right respectively.




(a) (b) (c)

(a) (b) (c)

94

3.5 Experiment With Higher Frequency For More Modes

Another experiment is set up for more modes with different excited frequency range. For this

experiment, linear chirp signal from 5 to 1000 kHz and the spectrum of Fourier transform are

used and plotted in Figure 3.10(a) and (b). The measured response to the chirp from an

excited source by a piezoelectric disk mounted on aluminum plate is shown in Figure 3.10(c).

This study is also on an aluminum plate Al 6061-T6 with size 609 mm × 609 mm × 6.35 mm

(24” × 24” × 1/4”). LDV is moving along horizontal direction with 300 points for detection.

The interval Δx for detected points is equally spaced 0.5 mm from the excitation source. For

enhancing SNR, every received signal is averaged for 25 measurements.

All points in one line assuming Lamb waves propagate are measured. LDV consists

velocity values measured at N = 300 equal distance points in space and T = 8192 times where

the duration of measurement 800 μs. The raw data forms a 2-D matrix as u(x,t) which

includes the information of 1-D location and transient signals. The data is transformed by

two-dimensional in space x and time t. In the next, results are compared to analytical curves

which are computed numerically. Analytical dispersion curves are descripted in Figure 3.11

based on Al 6061-T6 material properties (ν = 0.33) in non-dimensional axes. The observing

range in gray area is assigned by excitation frequency with bandwidth of spectrum. In this

range, there are multiple modes of dispersion curves up to A2 mode, which can be predicted

and compared.

95


(b) Spectrum of input signal after Fourier transformed. (c) Response of first location from

LDV.


(a) frequency-wavenumber ω-k, (b) phase velocity cp-ω and (c) group velocity cg-ω. The

black lines mark the excitation normalized frequency range 5-1000 kHz and the dots

represents the modes obtained from excitation frequency range in this study.

(a) (b) (c)

(a) (b) (c)

96

The result of 2-D FFT is shown on Figure 3.12 after data is transformed. The ω-k

contour is shown on Figure 3.12(a). In the figure, the black lines represent A0 mode and black

dash lines represent S0 mode respectively. Contours of results as ridges represent the result of

transformation, showing maximum area along theoretical curves. Therefore, the transformed

data having peaks along each mode curve can be found out and seen in the figure clearly. By

using local maxima method [11], all points between frequencies and wavenumbers are

examined and the peak values of ω-k contour are identified and picked for the Figure 3.12(b).

Points are clearly seen for modes A0, A1, A2, S0, S1, and S2. Few peaks are obtained along A3

and S4 curve when analyzing the data by each wavenumber. It means that the magnitudes of

extracted signals of modes A3 and S4 are relatively weak so that they are barely seen in the

contour plot but shown in finding maxima. Traveling waves are propagating with simple

relations for phase velocity cp = ω/k and group velocity cg = dω/dk [69, 70]. From the local

maxima points of the ω-k contour, the relationships of phase velocity cp and group velocity cg

with frequencies are shown in the Figure 3.12(c) and (d) respectively.

97

Figure 3.12 Transformed plots for aluminum plate (h = 6.35 mm) show peaks with chirp

signal in excited frequency range and dispersion relationship plots by the local maxima

points. (a) Contour plot for 2-D FFT results. (b) Peak values from 2-D FFT amplitudes. (c)

Phase velocity cp relation (d) Group velocity cg relation.

3.6 Measurements By Matrix Pencil Data And Reconstruction For Dispersion Curves

The dispersion relations may be guessed by calculating the wavenumber k and frequency ω

with two dimensional Fourier transform of u(x,t). Fixing on one direction of k and plotting

the Fourier coefficient in a frequency-wavenumber-diagram (ω-k domain) can see the

(c) (d)

(a) (b)

98

dispersion curves as ridges in the Figure 3.12(a). It is easy to see the contour and maximum

points virtually, but the exact values from figure are hard to be extracted and interpreted.

The MP method can be approached from two sides. Either the discrete Fourier

transform in time is calculated first and hence the MP is used to approach wavenumbers with

a specific frequency, or the discrete Fourier transform in space is calculated first and then the

data row is applied to the MP algorithm to estimate the frequencies for each wavenumber.

According to ideal conditions, both approaching would show the same result, but in practice,

one is better than another since the resolutions of dataset in time or in space are different.

Figures containing dispersion curves on ω-k domain are easily interpreted to the

relations of phase velocity cp-ω and group velocity cg-ω and shown together. Colorful circles

indicate the extracted points. Black lines refer to analytical A mode and dash lines to S mode

curves respectively, modes marked adjacent to curves. These analytical curves based on

Rayleigh-Lamb dispersion equation are computed numerically by conventional root-finding

algorithm. Figure 3.13 and Figure 3.14 are compared that the two approaches namely use the

discrete Fourier transform in time and MP method in space (x-ω domain) or the discrete

Fourier transform in space and MP method in time (k-t domain) respectively. Both figures

show the same approaching along with analytical result, but apparently one approach on x-ω

domain shows more results than another on k-t domain visually. The difference comes from

the resolution of wavenumber or frequency. The resolution of wavenumber is 6.67 with 0.5

mm spacing and 300 measurements. It only has 51 points within the range of excited

frequency. In contrast, the resolution of frequency is 7.85 × 103 rad/s with discrete time 9.76

× 10-8 s and 8192 times. There are 796 points within excited range. Due to the axes with

99

limited length, one is visually more intense than another. In (b) and (c) of Figure 3.13 and

Figure 3.14, the dispersion velocities follow the relations from ω-k domain for phase velocity

cp = ω/k and group velocity cg = dω/dk. The results by MP method show more intense than

by 2-D FFT to theoretical curves. It is noted that the extracted points for group velocity are

more diverging due to the derivation of differential calculations which are sensitive to

relative values between points.




(a) (b) (c)

100




In each mode, numbers of extracted points are different due to decomposition of

primary frequencies or wavenumbers. After signal analysis, M data would be obtained for

fixed frequency or wavenumber. For M, primary frequencies are related to dispersion

relationship with fixed wavenumber. However, some primary frequencies are hard to be

detected under low amplitudes that are caused by lower displacement for the mode while

detecting. When data is analyzed to obtain more modes, primary frequencies are shown for

few modes. As experiment setup, LDV is sensitive to the out-of-plane displacement, which

causes that A0 mode is easy to be detected. Other modes are in the signals with lower

detected energy and difficult to be extracted. In most cases, S mode is hard to be detected

since waves propagate in the plane so that 1-D LDV barely detects. Therefore, MP method

has abilities to decompose signals for contained primary frequencies in fixed wavenumber

and vise versa while modes with sufficient detected energies even the mode propagating in

the plane.

(a) (b) (c)

101

Since the results analyzed by MP method have high accuracy to match analytic data,

reconstructions of dispersion curves based on ω-k domain can be approximated by high-order

polynomial fitting lines. Extracted points from x-ω domain to establish relationships by MP

method are compared to analytical data. For each mode, the coefficient of determination R2 is

defined and is above 97% as shown in the Figure 3.15. It indicates that analysis of MP

method has high accuracy for dispersion curves in excited frequency range. Therefore, fifth

order polynomial fitting curves are applied for each mode based on higher resolution data on

x-ω domain of MP method. Smoother curves can be seen on Figure 3.16. The fitting curves

are along with analytical curves in ω-k domain. From the simple relation for phase cp and

group cg velocity, fitting curves are also approximated along with analytical ones in cp-ω and

cg-ω domains. This indicates that dispersion curves can be reconstructed by MP method for

unknown materials.

99.93% 99.92% 99.87% 98.84% 99.59%97.46%

R-square

A0 S0 A1 S1 S2 A2

Figure 3.15 Coefficients of determination R2 for each mode are shown to percentage of

experimental data matching to analytical solutions of dispersion curves on h = 6.35 mm

aluminum plate.

102

Figure 3.16 Reconstruction curves from analysis of MP method on h = 6.35 mm aluminum

plate, extracting from x-ω domain. Dispersion maps for (a) ω-k, (b) cp-ω, and (c) cg-ω

diagrams are shown from left to right respectively.

3.7 Experiment Setup And Results In Composites Plate

The experiment is extended to composite material with MP method. The experiment is

carried out for measuring the out-of-plane velocity at the face of plate and retro-reflective foil

is attached on the surface along the scanning paths for increasing the SNR due to low

reflection of composite surfaces. Experiment is set up as shown in the Figure 3.17 on a

composite plate AS4/3502 graphite/epoxy with size 685 mm × 685 mm × 2.15 mm (27” × 27”

× 1/12”) and layup [±45/02]2s. The PZT actuator is glued to the plate surface. The LDV is

sensing and moving along horizontal direction 0°, inclined direction 45° and vertical

direction 90° with each 150 points for detections. The interval for detected points is equally

spaced 1 mm for horizontal and vertical direction from the excitation source and keep the

same spacing for inclined direction. For enhancing SNR, every received signal is also

averaged for 25 measurements.

(a) (b) (c)

103

Figure 3.17 Experiment setup for composite plate laminate [±45/02]2s AS4/3502 Gr/Ep with

PZT actuator and LDV sensor. Retro-reflective foil covers an area for three scanning

directions.

Table 3.1 Material properties of AS4/3502 composite lamina

E1 (GPa)

E2 (GPa)

E3 (GPa)

G12 (GPa)

G13 (GPa)

G23 (GPa) v12 v13 v23

ρ(kg/m3)

127.6 11.3 11.3 5.97 5.97 3.75 0.3 0.3 0.34 1578

As same processes as above description, extracted points for dispersion curves are

compared to analytical curves. In the Figure 3.18 and Figure 3.19, black line represents the

analytical curves of S0 and A0 modes of composite panel. Analytical curves are computed

numerically by 3-D elasticity theory [71] and material property of plate is shown in the Table

3.1.

104

Analytical result and experiment result are compared in different propagation

directions shown in the Figure 3.18 and Figure 3.19. Figure 3.18 shows that experimental

results of frequency-wavenumber ω-k diagram, phase cp and group cg velocities versus

frequencies in different scanning directions are extracted by MP method from fixed

frequency x-ω domain and Figure 3.19 shows results from fixed wavenumber k-t domain. In

the figures, extracted points of S0 and A0 modes from experiment show good agreement with

analytical dispersion curves. For all propagation directions, some experimental points of S0

mode are extracted and approaching to theoretical curve. Results in Figure 3.20 show more

approaches in modes due to higher resolution of frequency domain, which match well in ω-k

domain and cp-ω domain.

Group velocity cg is calculated differently for extracted points in different domains.

For ω-k and cp-ω diagrams, experimental points from k-t domain and x-ω domain match well

with analytical dispersion curves. Phase velocity can be obtained by simple relation cp = ω/k

for both domains. However, the calculations of group velocity have different ways due to the

fixing frequency or wavenumber. The group velocity is affected by the dispersion behavior

for Lamb waves in composites, having different forms and attaining as [71, 72]

cgx

cgy

cos sin

sin cos

kk

(3.7.1)

where subscripts x and y represent the components in the Cartesian coordinate system for x-

and y- axes respectively. The angle θ is the direction of wave propagation. For isotropic

105

plates, the wavenumber doesn’t change with propagation direction . In other ∂ω/∂θ = 0

word, the direction of group velocity coincides with the direction of wave vector. But for

anisotropic plates, does not vanish. The magnitude of group velocity is obtained by∂ω/∂θ

cg cgx2 cgy

2 (3.7.2)

Group velocity for composite plate has different expressions with equivalent physical

meaning. Calculation of group velocity for composite plates is corrected, however, the

expressions of group velocity are different in numerical implementations. Due to the

propagation angle θ, the Eq. (3.7.1) is conveniently employed to compute group velocity

dispersions along a given propagation direction. To obtain the group velocity at a given wave

propagation direction θ1, two dispersion relations having slightly different directions of θ1 ±

Δθ/2 are required for Eq. (3.7.3). The derivative term can be approximated by finite ∂ω/∂θ

central difference

1

(k) 1 /2 (k) 1 /2

(3.7.3)

Respectively, due to geometric relation between slowness curve and group velocity

direction for computing wave curves, expression of group velocity according to Eq. (3.7.1)

can be rearranged as

cgx

cgy

cos sin

sin cos

k

k k

dkd

(3.7.4)

The Eq. (3.7.4) is suited to calculating wave curves at a given frequency.

106

For fixing wavenumber k-t domain, group velocity of composite plate is affected by

layup orientation and obtained by Eq. (3.7.1). Extracted frequencies by MP method are

functions of wavenumber and angle ). Because the calculation needs two dispersion ω(k, θ

relations having slightly different directions of propagation θ1 ± Δθ/2 by Eq. (3.7.3), the

difference Δθ is set up for 2° degree as ±1° of θ1. The results are shown in the figures of cg-ω

domain of Figure 3.19. For fixing frequency x-ω domain, extracted wavenumbers by MP

method are used for dk/dθ according to Eq. (3.7.4) as

dkd 1

k( ) 1 /2 k( ) 1 /2

(3.7.5)

Due to the same data set from experiment, the extracted wavenumbers of slightly

different angle can be also yielded through MP method. The results are shown in the figures

of cg-ω domain of Figure 3.18. The group velocities of A0 mode have good agreements with

analytical dispersion curves. For the group velocity of S0 mode, calculated points from

extracted data show diversity and discontinuity. It can be explained by the sensitivity of

double derivation of dispersion relation and between these few points.∂ω/∂k dk/dθ

For 1-D laser Doppler vibrometer, out-of-plane velocity is measured on the surface of

plate and perpendicular to plate, which mostly contains anti-symmetric waves due to particle

motion. In the all figures of Figure 3.18 and Figure 3.19, comparisons can be easily seen with

A0 mode due to pureness which is good to be identified according to excited frequency range,

especially in looking for phase and group velocities of excitation. S0 mode has weaker

amplitude but it can be also analyzed through MP method with less points. Therefore, A0

mode has good responses and some points for S0 mode are mis-matched in composite plate.

107

To be noticed, it is interesting that SH0 mode shows on while analyzing data along

90° degree and it is hard to get it in others directions. Figure 3.20 shows that the SH0 mode is

extracted with A0 and S0 mode from x-ω domain. It can be recognized that SH0 mode can be

obtained in 90 degree direction because wave energy is separated into each mode for no-fiber

orientation as shown in the Figure 3.21. Energy distribution for A0 mode has biggest

amplitudes and has demonstrated S0 mode in each scanning direction. For SH0 mode, only

90° degree direction has energy distributed. It also can be noticed that the amplitudes of S0

and SH0 are very close, which means that wave propagation along no-fiber direction makes

LDV sense the vibrations of both modes.

108

(I) wave propagation along 0 degree direction

(II) wave propagation along 45 degree direction

(III) wave propagation along 90 degree direction


AS4/3502 Gr/Ep extracted from x-ω domain. Propagation direction (I) 0 degree (II) 45

degree (III) 90 degree.

109

(I) wave propagation along 0 degree direction

(II) wave propagation along 45 degree direction

(III) wave propagation along 90 degree direction


AS4/3502 Gr/Ep extracted from k-t domain. Propagation direction (I) 0 degree (II) 45 degree

(III) 90 degree.

110

Figure 3.20 SH0 mode for composite plate laminate [±45/02]2s AS4/3502 Gr/Ep is analyzed

from x-ω domain and (a) ω-k (b) cp-ω and (c) cg-ω diagrams shown in the propagation

direction 90° degree.

Figure 3.21 Amplitude allocation in each scanning direction for (a) A0 mode (b) S0 mode and

(c) SH0 mode. For (b) S0 mode, there are many points in each direction, but for (c) SH0 mode,

only 90° degree is existed and 0° and 45° degree barely see amplitude for most frequencies.

3.8 Summary

Applying the MP method in aluminum plate and composite plate showed that it is an easy

and robust method to determine the dispersion curves of plate-like structures. This method is

(a) (b) (c)

(a) (b) (c)

111

compared and confirmed first on the two thicknesses aluminum plates by theoretical curves

which are numerically computed by the dispersion equations of Lamb waves. The thicker

aluminum plate of 6.35 mm shows other modes S0 and A1 within identically certain

bandwidth 50-350 kHz excitation. Experiment is also extended with higher frequency 5-1000

kHz for more up to A2 and S2 modes on thickness 6.35 mm aluminum plate. The results show

accuracy and dispersion curves from the results can be reconstructed comparing to theoretical

ones. MP method shows the ability to examine unknown material for dispersion

relationships.

Experiments are also on a layered composite plate for more complicated structures.

Inspections in three different directions 0°, 45°, and 90° on the composite plate show

compromised results with theoretical curves by 3-D elasticity theory. For MP method, there

are two approaches from k-t domain or x-ω domain to determine the relationship of

dispersion curves. Both approaches show accurate results but different points and they can

complement each other to complete dispersion curves. Moreover, for k-t domain, it has lower

resolution due to much less detection points x relative to transient time t. The analysis from

k-t domain still matches to theoretical curves and has indications for relatively low

frequencies range.

The dispersion relationship of group velocity is calculated in this study. MP method

obtains the relations of wavenumber and frequency from signal directly and phase velocity cp

has a simple relationship from them. Group velocity for isotropic plate is derived from

wavenumber and frequency as gradients of curves. But group velocity for composite plate is

112

affected by the propagation angle for dispersion behavior and has different expressions with

different numerical implementations. For analysis of composite plate from k-t domain, the

Eq. (3.7.1) and Eq. (3.7.3) are suited to calculating group velocities at a fixed wavenumber.

Meanwhile, the Eq. (3.7.4) and Eq. (3.7.5) are best to wave curves at a given frequency

which analyzes signals from x-ω domain. The equations have lots of derivations that make

results sensitive due to slightly changes so that the diagrams of group velocities show

diversity or discontinuity.

113

CHAPTER 4

Chirplet Transform for Group Velocity

4.1 Signal Propagation and Group Velocity

The concept of group velocity cg is associated with the wave envelope and transmitted signal

whose spectrum is concentrated in ω0. Dispersion phenomenon has significant effect on

received signals in the plate. For simplicity of presentation, the nature of dispersion effect of

single mode wave is made in 1-D dispersive medium with using Fourier analysis. Wave

signals with center frequency ω0 are propagating from excitation source where is usually

marked as x = 0 to location for receiver. At location x = 0, the modulation property of Fourier

transforms for wave envelope Eg(ω) and transmitted signal F(ω) in frequency and time

domain are shown

F(0, ) Eg (0, 0 ) f (0,t) eg (0,t)e i0t (4.1.1)

When the signal is launched into a dispersive medium with wavenumber k(ω), the

propagated signal to distance x will be given by

f (x,t) eg (x,t)ei(k0x0t ) (4.1.2)

where defining k0 = k(ω0) as the form of a modulated plane wave. For the propagated

envelope eg(x,t), it can be represented as

eg (x,t) 1

2ei[(kk0 )x( 0 )t ]Eg (0, 0 )d

(4.1.3)

114

The Eg(0, ω−ω0) restricts the integration about ω0. For the dispersion of a propagating

wave, the wavenumber can be expanded in a Taylor series in the neighborhood of the

frequency ω0 as

(4.1.4)

where

k0 k( 0 ), k0 dkd 0

, k0 d 2kd 2

0

(4.1.5)

If k(ω) is real, the first derivative is the inverse of group velocity

k0 dkd 0

1cg

(4.1.6)

The second derivative is referred to as the “dispersion coefficient” and is responsible for the

spreading and chirping the wave package as following.

Keeping the quadratic term in the quantity k(ω) − k0 and changing ϖ = ω − ω0 in the

integration, the approximation of the equation yields

eg (x,t) 1

2e

i[( k0 12

k02 )x t ]

Eg (0, )d

1

2ei[ k0x 2 /2 (t k0x )]Eg (0, )d

(4.1.7)

In the linear approximation, the second derivative can be ignored, but in quadratic

approximation, the first and second terms are both kept. For linear case, the plane wave with

the envelope is obtained as

115

f (x,t) eg (x,t k0x)ei(k0x0t ) eg (x,t x / cg )ei(k0x0t ) (4.1.8)

This indicates that the transmitted signal travels a distance with group velocity cg by tracking

the wave envelope. The frequency carrier suffers a phase change k0x.

However, for quadratic term, the wave envelope in frequency domain becomes

Eg (x, ) ei[( k0 k02 /2)x ]Eg (0, ) (4.1.9)

Meanwhile, from the Fourier integration, the quadratic can be integrated as

12

ei[ k0x 2 /2 (t k0x )] d

1

2i k0xexp

(t k0x)2

2i k0x

(4.1.10)

Therefore, the wave envelope is obtained with linear case ad following the convolution

expression in the quadratic one in the time domain by replacing ϖ to ω as

eg (x,t) 1

2i k0xexp

( t k0x)2

2i k0x

eg (0,t t )d t

(4.1.11)

Now a Gaussian-windowed chirp signal is modulated a chirped sinusoid by a

Gaussian envelope at the first location as

fc(0,t) exp( t 2

2 2 )e i(0tt2 /2) egc (0,t)ei (t ) (4.1.12)

where α means the chirp rate of change of the instantaneous frequency and σ is the standard

deviation of Gaussian distribution. The phase θ(t) and instantaneous frequency dθ(t)/dt can

be obtained

(4.1.13)

116

The parameter α can be positive or negative resulting in increasing or decreasing frequency.

From Eq. (4.1.12), the chirped signal can be represented as

fc(0,t) egc (0,t)ei0t exp(1 i 2

2 2 t 2 )ei0t exp( t 2

2 c2 )e i0t (4.1.14)

where σc is an equivalent complex-valued standard deviation of Gaussian distribution due to

chirp rate as

c2

2

1 i 2 (1 i 2 ) 2

1 2 4 (4.1.15)

In frequency domain, the chirped signal at the initial location is transformed as

Fc(0, ) 2 c2 e c

2 ( 0 )2 /2 (4.1.16)

Thus, the chirped envelope Egc(x,ω) at initial location can be considered as

Egc (0, ) 2 c2 e c

2 2 /2 (4.1.17)

For propagated signal Egc(x,ω), it can be determined by

Egc (x, ) ei[( k0 k02 /2)x ]Egc (0, ) 2 c

2 ei[( k0 k02 /2)x ]e c

2 2 /2

2 c2 ei k0 xe( c

2 i k0x ) 2 /2(4.1.18)

Then the Fourier integral gives the propagated envelope in time domain as

egc (x,t) c

2

c2 i k0x

exp (t k0x)2

2( c2 i k0x)

(4.1.19)

Therefore, the propagated chirp signal for dispersive medium can be approached as

117

fc(x,t) egc (x,t)ei(k0x0t )

c

2

c2 i k0x

exp (t k0x)2

2( c2 i k0x)

ei(k0x0t ) (4.1.20)

In order to observe the propagated signal, the magnitude is found for propagated

envelope while assuming the wavenumbers are real as

egc (x,t) c

4

c4 ( k0x)2

1/4

exp (t k0x)2 c

2

2( c4 ( k0x)2 )

(4.1.21)

Thus, the peak maximum occurs at the group delay when t = x/cg under the instantaneous

frequency ω0 and the value of maximum is

egc max

c4

c4 ( k0x)2

1/4

(4.1.22)

The effective complex-valued standard deviation will be

c2 i k0x

(1 i 2 ) 2

1 2 4 i k0x 2

1 2 4 i( 4

1 2 4 k0x) (4.1.23)

When the chirp rate α is selected as

4

1 2 4 k0xi 2 1k0xi

1

4 0 (4.1.24)

At the location xi for positive distance, the standard deviation can be written a

c2 i k0x

2

1 2 4 i[ k0 (x xi )] (4.1.25)

Thus, when x increases in the interval 0 ≤ x ≤ x0, the signal width is getting narrower and

becoming the narrowest while x = x0. Then the width increases again beyond x > x0.

118

Moreover, comparing chirp envelope egc(0,t) at initial location with propagated envelope

egc(x,t) with special case when the chirp rate is zero α = 0 as a Gaussian pulse envelope eg(x,t)

egc (0,t) exp t 2

2 2 (1 i 2 )

(4.1.26)

eg (x,t) 2

2 i k0xexp

(t k0x)2

2( 2 i k0x)

2

2 i k0xexp

(t k0x)2

2( 4 ( k0x)2 )( 4 i k0x)

(4.1.27)

they have similarity and the chirp rate α can be identified due to propagation by

k0x

4 ( k0x)2 (4.1.28)

The input chirp rate has relationship with complex-valued standard deviation and the

dispersive parameter. In other words, if a chirped Gaussian signal is input into a dispersive

medium, the dispersive parameter due to propagation will combine with the chirp rate.

Therefore, the initial chirp rate and the dispersive parameter can cancel each other at x = x0.

This phenomena is able to be used some dispersion compensation methods based on this

effect.

4.2 Chirp Signal Analyzed by Chirplet Transform

Gaussian chirplet which is sparse and energy preserving is introduced into chirp signal

decomposition to represent chirp-type signals. The sparseness property provides a compact

representation of the complex signal by decomposing it into a limited number of chirp

119

components. Based on the chirplet transform (CT) of the signal, the algorithm identifies the

location and duration of the most dominant chirp component in time-frequency domain.

In most application cases, a single chirp signal can be represented and modeled as [23,

73]

fc(t) exp[1(t )2 i 0(t ) i 2(t )2] (4.2.1)

where β is the amplitude, τ is the time of arrival, α1is the bandwidth factor, α2 denotes the

chirp rate and ω0 is used to present the center frequency described above section 4.1.

In order to analyze chirp signal, the CT of fc(t) with respect to a mother chirplet Ψc(t)

can be defined as

CT fc (t)c*(t)dt

(4.2.2)

The mother chirplet Ψc(t) is expressed as

c exp 1(t b)2 i c ( t ba

) 2(t b)2

(4.2.3)

where (*) denotes the conjugate of Ψc(t) and where η presents the normalized energy, γ1 is

the bandwidth parameter for modulated window, γ2 is the coefficients for the chirp rate in the

kernel, and ωc is the center frequency of the kernel. a and b are the operation parameters for

the kernel. Hence, the CT of input signal fc(t) can be expressed as

CT

exp ( 0 c / a)2

42 ig c / a g 0

2 (b ) g g

2 (b )2

(4.2.4)

120

where Φ = [α1+γ1−i(α2−γ2)]1/2 for the denominator, gα = α1−iα2 and gγ = γ1−iγ2. The maximum

similarity between input signal fc(t) and chirplet kernel Ψc(t) can correct estimate the

parameters. Therefore, the peak of CT is used to accomplish the goal, which is given by

CT 1/2 exp

( 0 c / a)2

42 g

2 (0 c

a)(b )

h

2 (b )2

(4.2.5)

where Θ = [(α1+γ1)2+(α2−γ2)2]1/2 presents the denominator, gαγ = α1γ2+α2γ1 and hαγ = |gα|2

γ1+|gγ|2 α1 for simplification.

The maximum can be obtained by taking partial derivatives in respect to operation

parameters a and b,

CTa

CT22

c

a2 g (b ) c

a2 (1 1)( c

a 0 )

0 (4.2.6)

CTb

CT22 ( g

2 1 g

21)(b )

12

g ( c

a 0 )

0 (4.2.7)

The solutions of the equations occurs when

and b c

a 0 (4.2.8)

The estimation of the peak position of |CT| can lead to obtain the center frequency ω0 and

arrival time τ by setting up the coefficient a and b.

Based on the estimations of a and b, the chirp rate γ2 is estimated by taking derivation

of |CT| respective with γ2 as same way and it can be obtained

121

CT 2

CT2 [

( 2 2 )2

1( 0 c

a)(b ) 21 2(b )2

( 2 2 )

2 g ( 0 c

a)(b )

( 2 2 )2 h (b )2]

0

(4.2.9)

when b = τ and ω0 = ωc / a, the derivation yields

CT 2 b ,0

ca

CT2 (

2 2

2) 0 (4.2.10)

The solution indicates that the maximum of |CT| is reached when γ2 = α2. In other words,

looking for the maximum can get the arrival time τ and center frequency ωc if γ2 is identical

to α2.

4.3 Determination of Group Velocity

The group velocity can be determined by the derivative of |CT| with respect to b from Eq.

(4.2.7). If the derivative equals to zero, the term b−τ takes the determination to be CT b

zero. Moreover, τ presents t where magnitude reaches maximum when t = k’0x from Eq.

(4.1.21). Therefore, the analyzed result for determined parameter b and the group delay k’0

has a relation as

k0 bx

(4.3.1)

To be noted, the result is under the k’(ω0) condition. This equation Eq. (4.3.1)

indicates that the maximum magnitude of the CT happens at the center frequency ω0.

According to the Section 4.2, center frequency ω0 and arrival time τ can be obtained by

122

locating the peak value of magnitude after transformation by alternating shift time b and

scale a. In other words, the arrival time τ of propagating waves in the group velocity at each

local frequency ω0 can be determined by a and b. By the definition of group velocity cg

cg ddk

1k0

xb (4.3.2)

In here, the coefficient b can be seen as the arrival time because of the definition of group

velocity cg = x / b, x as propagation distance.

The corresponding group velocity can be calculated by each frequency if the distance

x is known by the actuator and sensor between designed locations, and the arrival time b

obtained from the maximum magnitude of CT. The group velocity will be obtained.

4.4 The Formula of Chirplet

Both the STFT and the wavelet transform result from the inner multiplication of the time

function,

CTw wx (t)c*(t)dt

(4.4.1)

where (*) denotes the conjugation of function Ψ(t). Ψ(t) is a time-frequency kernel, obtained

from the window function either shifting in time and frequency directions (comparing to

STFT), or in time shifting and scaling (comparing to wavelet transform).

Now Mann and Haykin [74] and Kuttig et al. [21, 22] uses his framework and

expend the idea by using two additional operators, shifting in time and frequency, to

123

formulate the chirp let transform, frequency and time shear. Using the five operators, time-

frequency kernel is translated, scaled, and sheared in the time-frequency domain. The

operations are discussed in the appendix. The kernel function becomes:

t1 ,1 ,s,q ,p (t) Tt1F1

SsQqPph(t) (4.4.2)

To represent the formula, the kernel can be re-defined as

c 14

exp 12

t2 i ct i c2

t2

(4.4.3)

where c presents the

factor of linear chirp rate. To be noted, the operators are not

commutative and interchanging two of them generally result in phase shift of chirplet.

4.5 Experimental Setup for Chirplet Transform

In the experiments, the group velocity was obtained and calculated by collecting signals from

sensors at point S1 and S2 where the practical signals simultaneously delayed to the time

when flexural waves were excited as Figure 4.1. The experiment is set up with a

piezoelectric actuator as excitation source and using laser Doppler vibrometer as a sensor.

Excited signals are controlled by the function generator connecting to data management

system (DMS). The LDV is also manipulated through controller and the computer storing the

data. The data will be collected and analyzed by the commercial computational software

MatLab. The detecting points S1 and S2 are arranged on the surface of fixed specimen with

distances d1 and d2 respectively. Not all frequency components appear at the same time due

124

to time lag happened between different frequencies. Therefore, using two points S1 and S2 to

measure responses can effectively side effects.

Figure 4.1 Schematic of the experimental setup with two measuring locations using laser

Doppler vibrometer and the data management system.

The procedure of evaluating group velocities follows the below orders at each

frequency by the scale a where central angular frequency is ω0 = ωc/a. First, responses are

computed for the coefficients of chirplet transform at point S1 and S2. The chirplet transform

is displayed as Figure 4.2. Determine the arrive time b1 of response 1 and b2 of response 2 at

each scale a from finding the maximum magnitude of |CT|. From the definition of group

velocity cg = x/b, the velocity is considered as the traveling distance over the different

duration. The formula is transformed as

cg db

d2 d1

b2 b1(4.5.1)

125

where Δd means the traveling distance from S1 to S2, Δb means the traveling time from S1 to

S2, and d1 and d2 can be found as designed location as setting up the experiment.

Figure 4.2 The real (line) and image (dash line) part of mother chirplet with chirp rate π/4 at

center frequency ω0 = 2π.

4.5.1 Dispersion Relation Of Group Velocity on Aluminum Plate

The experiment is carried out for measuring the out-of-plane velocity on the surface of plate.

Experiment is set up as shown in Figure 4.3, operating on an aluminum plate Al6061-T6 with

size 1200 mm × 609 mm × 2.286 mm. The actuator is a 7.0 mm diameter and 0.3 mm

thickness circular piezoelectric disc which is bonded to the plate surface. The sensor as a

laser Doppler vibrometer (Polytec OFV 505) is moving along horizontal direction with two

locations for detection. The distances for detected points are 25 mm and 75 mm for d1 and d2

respectively from the excitation source. Excited signal has a linear chirp signal in the

frequency range from 50 – 350 kHz. The detecting locations are attached with strips of

126

reflector foil to improve the receiving signals. For enhancing SNR, every received signal is

averaged for 25 measurements.

The responses received by LDV at S1 and S2 locations as Figure 4.4 and Figure 4.5

are analyzed by the chirplet transform. The detecting locations are apart for 50 mm distances

away. In the plots, the peak of response means the arrival time of a wave traveling with the

group velocity. According to Eq. (4.2.8), the corresponding group velocity for each value f =

1/a can be identified at each frequency. Figure 4.6(a) shows the analyzed group velocities of

A0 flexural mode between two points by chirplet transform. The red lines presents the

theoretical dispersion curve of A0 modes based on plate theory. Analyzed data is compared

with the theoretical dispersion curves. The result matches the theoretical curve well in the

frequency range of 50 – 350 kHz. Comparing to Figure 4.6(b) which was analyzed by Gabor

wavelet, result by chriplet transform has better agreement to theoretical curve in relatively

higher frequency of 250 – 300 kHz. In lower frequency range 50 – 200 kHz, the group

velocities in Figure 4.6(a) shows more continuous points along the theoretical curve. The

coefficient of determination between result and theoretical curve rises up to 73% (chirplet)

from 51% (Gabor). This indicates that chirplet transform was good for a chirp signal with a

wideband frequency by one measurement at two points.

127

Figure 4.3 Picture of experiment setup for aluminum plate with piezoelectric actuator and

LDV sensor.

128

Figure 4.4 The responses received by LDV and magnitude of CT transform at point s1 (a)


showing in the contour plot at time 106 μs in frequency 300 kHz .

(a) (b)

(c)

129

Figure 4.5 The responses received by LDV and magnitude of CT transform at point S2 (a)


showing in the contour plot at time 124 μs in frequency 300 kHz.

(a) (b)

(c)

130

(a) (b)

Figure 4.6 Group velocity decomposition toward different frequencies extracted by transform

from responses 1 and 2. (a) Chirplet transform (b) Gabor wavelet.

4.5.2 Dispersion Relation of Group Velocity on Composite Plate

The experiment is extended to composite material for dispersion curves. The experiment is

carried out for measuring the out-of-plane velocity at the face of plate but retro-reflective foil

is attached on the surface along the scanning paths for increasing the SNR due to low

reflection of composite surfaces. Experiment is set up as similar as Figure 4.3, but working

on a composite plate AS4/3502 graphite/epoxy with size 685 mm × 685 mm × 2.15 mm and

layup [±45/02]2s. Material properties of composite lamina is listed in the Table 3.1. The

actuator is a 7.0 mm diameter and 0.3 mm thickness circular piezoelectric disc which is glued

to the plate surface. The laser Doppler vibrometer (Polytec OFV 505) is used to sense out-of-

plane velocity. The LDV travels along horizontal direction 0°, inclined direction 45° and

131

vertical direction 90° with each two points for detections. The distance for each detected

point is 25 mm and 75 mm as the setup of aluminum plate for horizontal and vertical

direction from the excitation source but 1.414 times for inclined direction. Every received

signal is averaged for 25 measurements for enhancing signal-to-noise ratio (SNR).

Figure 4.7 shows the measured group velocities of A0 mode in the 0°, 45°, and 90°

degree directions of unidirectional laminate. Also in the Figure 4.7, the theoretical curves

based on the 3-D elasticity theory via the relationship prediction [71] are shown in the plots.

For Figure 4.7(a) to 7(c), the group velocities along directions respectively matches the

theoretical curves well after about 60 kHz. In the Figure 4.7(d) for comparison in all

directions, the measures and predicts dispersion curves had similar pattern that velocity had

highest speed in 0° direction, and velocity in 90° direction has lowest speed. Again, there are

excellent agreements for all directions between measurements and theory.

(a) (b)

132

(c) (d)

Figure 4.7 Measured group velocities and theoretical curves along (a) 0° degree (b) 45°

degree (c) 90° degree (d) all directions.

4.6 Summary

This application of chirplet transform (CT) to the time-frequency analysis on group velocity

of wave propagation has shown good agreement with theoretical dispersion curves. A linear

chirp signal with a broadband frequency provides robust excitation to carry rich information

for the relationship of dispersion. It is found that CT is an effective tool for linear chirp signal

excitation on experimental analysis of dispersion curves in solid media. In addition, only two

measurements for one chirp signal excitation are received and analyzed. The peak of

magnitude determined the arrival time of each frequency component in the group velocity

computation. For isotropic (aluminum plate) or laminated materials (composite plate), the

group velocities of the flexural plate mode are measured in all directions and had good

agreements with theoretical curves.

133

CHAPTER 5

Chirp-coded Ultrasonic Wave for Damage Imaging

5.1 Linear and Nonlinear Local Defect Resonance (LDR)

5.1.1 Concept of LDR and Linearity

The concept based on assumption is that the damage causes a local change in stiffness for a

certain mass of the structure so that a particular resonant frequency manifests the vibration

for the local region. The theoretical frequency can be conducted by a function of depth and

size of defect. Moreover, when the local region at LDR, the magnitudes of higher harmonics

in the spectrum of signal response increase for nonlinearity while other different values of

frequencies behave closely to linear. As imaging on the plate as whole field, the vibration

amplitude at defect location increases significantly when excitation matches LDR. This

strongly localizes the defect region and provides contrast between damaged and intact parts

of specimen.

Figure 5.1 Fundamental LDR frequency for linear and higher harmonics for nonlinearity

134

The theoretical function of local defect resonance is based on the classical plate

theory which has been proposed for the case of a circular flat bottom hole [28, 31]. For

estimating the frequency of local region, the fundamental frequency f0 is approximated as

f0 1.6ha2

E12(1 2 )

(5.1.1)

where h = H − d is the residual thickness and a is the radius of the plate at the location of flat

bottom hole, H is the thickness of the plate, and d is the depth of hole; E is the effective

elastic modulus, ρ is the material density and ν is the Poisson ratio. Note the formula is valid

for a thin flat-bottomed hole (1 << a/h) [30]. As expected, the analytical formula faces the

difficulty in the case of defect with low a/h ratio in the stiffer materials. The same formula

can be adapted to a simple delamination case in which h presents the remaining thickness of

the layers above delamination. In the condition of this equation, the fundamental f0 is

independent to the boundary conditions as well as the excitation location, which means that

the resonance frequency will not be affected by free or clamped boundary conditions.

The concept of a local defect resonance is based on the fact that a cracked defect in

the material diminishes the stiffness locally to make a certain mass of materials around the

defect vibrate at local nature frequency. It is often observed in laminated composite materials

[31, 32, 75] or the interfaces of sandwich structure [29]. The resonance frequency for the

consistence between the defect eigen-frequency and the probing ultrasonic waves causes

increase of the vibration amplitude on the defect. Therefore, a local stiffness and an effective

mass of the defect are considered in a spring-mass LDR model.

135

Similar to a conventional vibration, an equation of the forced vibration in the

presence of damping is a sum of the free and forced terms for zero initial conditions [76].

(5.1.2)

The solution is

u(t) Aet sin( pt ) A0 cos( dt ) (5.1.3)

where ζ is damping ratio, ω is undamped natural frequency, ωp is the damped natural

frequency and ωd is the driving frequency. A and θ are homogeneous solutions depending on

initial conditions; A0 and ϕ are the coefficients of particular solutions. When driving

frequency close to natural frequency ωd ≈ ω takes the form of the transient vibration, the

displacement is

u u0(1 e t )cos dt (5.1.4)

with the steady state amplitude u0 ≈ F0/2ζω2. It can be altered to u0 ≈ (F0/ω2)Q times the

vibration amplitude and Q = 1/2ζ. While reaching to steady state, the amplification of

vibration is only u0 left.

A way to observe LDR is experimentally measuring individual contribution of each

point on the specimen for its overall frequency response. For the purpose, a wideband PZT is

attached on the aluminum plate 610 mm × 610 mm × 2.29 mm with flat bottom hole (FBH)

D25 mm and depth 0.9 mm. By combining with laser Doppler vibrometer to scan the

specimen, it is capable to probe and indicate all possible resonances in the spectrum of

scanning area. Then the maximum amplitude in imaging vibration pattern at the

corresponding frequency can be verified.

136

Figure 5.2 shows an example of the LDR frequency for responses and vibration

pattern. A strong enhancement is observed locally in the defect area at the fundamental

frequency 24.7 kHz as local defect resonance. In the conducted experiment, the time domain

response u(x, y, t) is measured in scanning area on by means of LDV. The recorded data are

transformed to the frequency domain using Fast Fourier Transform (FFT) as post-processing

in MATLAB.

(5.1.5)

At the LDR frequency, the contrast between the vibration amplitude of the defect region and

the averaged amplitudes in rest of scanning area can be clearly observed.

Figure 5.2 (a) Spectrum of the out-of-plane velocity at center of defect. (b) Showing the

contour plot corresponding to LDR in space-frequency domain.

LDR

137

5.1.2 Nonlinear LDR

Material with imperfections or cracked defects causes nonlinear phenomena, which is an

important role played by internal boundaries of the defects. Local stiffness changes due to

intermittent contact between the fragments of defect and thus causes nonlinearity as bi-

modular vibration, which is related to Contact Acoustic Nonlinearity (CAN). The

mechanism of CAN is “clapping” or “kissing” of crack interfaces and nonlinear friction

between surfaces. Due to CAN, the spectrum of responses acquires multiple higher

harmonics, which are related to nonlinear NDE signatures of crack defects observed in

experiments [77, 78], e.g. metals [79, 80], composites [25, 61, 81, 82], nonhomogeneous

materials [31, 44] etc. To interpret non-classical effects, the damaged area can be identified

as a nonlinear oscillator and exhibit both nonlinear and resonance properties. The model for

LDR is therefore introduced.

Since LDR is an efficient resonant amplifier of local vibrations, one would expect

to use it for detecting nonlinearity. Multiple higher harmonics generations and

thermodynamics are observed at moderate input [83]. However, the description of the

resonance in nonlinear system is complicated and lack of exact solution to the nonlinear

partial differential equation. Unlike the forced response of linear system, the forced response

of a nonlinear system does not just consist of a simple sum of homogeneous and particular

solutions. Using perturbation solutions to approximate the features of the nonlinear

resonance phenomena can describe the generations of higher harmonics and sub-harmonics

[84]. Here is only discussing the higher harmonics or called super-harmonic resonances.

138

Figure 5.3 Nonlinear oscillator model for local delamination region in a laminated composite

plate with excitation signal have LDR and nonlinearity due to out-of-plane motion.

Assume that a steady state solution to the equation of driven motion for a nonlinear

oscillator [84]

(5.1.6)

where second β and third δ nonlinear terms are added. The solution may be thought as a

harmonic function u(t) = A(ωd)cos(ωdt) when F0 is small. By substituting the harmonic

function into equation, the solution is obtained as

A( d ) F0

2m [( d ) 3 A2( d ) / 8 ]2 2 2 (5.1.7)

Unlike linear case, the resonance is not happened at ωd = ω when damping ratio is

zero. The resonance of nonlinear oscillator becomes amplitude A dependent. Since the elastic

materials usually has negative coefficient δ < 0 as soft spring system, the resonance

139

frequency reduces when the driving amplitude increases. This model can also explain the

phenomena of nonlinear resonance ultrasonic spectroscopy (NRUS) [51].

Another feature of the nonlinear oscillator is the resonance generation of higher

harmonics. For super-harmonics, the driving frequency is considered as ωd ≈ ω/n and

converted into ω via the nth order nonlinearity. The example given as second harmonics

generation corresponding to n = 2 uses the perturbation method to approach the solution of

nonlinear equation. For driving frequency ωd = ω/2 + ε and very small damping ratio, the

first approximation is a linear driven vibration

u1 4F0

3m 2 cos[( / 2 )t] (5.1.8)

The resonance driving force is developed directly to quadratic nonlinearity so that

only keeping resonance terms in the right hand side after substituting the first approximation

into equation obtains

(5.1.9)

A solution could be found if neglecting the second order nonlinear term in the left hand side

u2 4 F0

2

9m2 5 4 2 2 2cos[( 2 )t] (5.1.10)

This solution indicates the resonant generation of the second or higher harmonics for

nth order. Since a step-wise stiffness modulation by CAN enhances its higher order

parameters [77], one can expect a strong development of higher harmonics in resonant

defect.

140

5.2 Chirp Signal on Aluminum Plate with Flat Bottom Hole (FBH)

LDR behavior can be discovered on different cases, such as flat bottom hole (FBH), cracks,

and also delaminations, disbonds in complex structures [84]. The detection and analysis of

LDR has strong qualification that makes it interesting for NDR in broadband signal. Linear

chirp as a sweep frequency excitation is applied for broadband frequency to get the local

resonance. In this study, an aluminum plate with FBH is examined first to see the responses

and vibration pattern at local resonance.

5.2.1 Experimental Setup

The experimental setup for measurements on the surfaces of plate is depicted in the Figure

5.4 to detect out-of-plane velocity with a system. The system consists of a PZT actuator, a

function generator, an amplifier, 2-axis translation stage, laser Doppler vibrometer and

workstation. PZT actuator (APC International, Ltd.) is a D14 mm diameter and H0.7 mm

thickness circular piezoelectric disc which is glued to the plate surface. Laser Doppler

vibrometer (Polytec OFV 505) as a sensor is mounted on a 2-axis translation stage and

moving with multiple points in scanning area for detection. The interval for detected points is

equally spaced for Δx = Δy = 1 mm. The scanning path is attached with a strip of reflector

foil to improve the SNR of receiving signals. Excited signal is a linear chirp generated by a

single channel arbitrary waveform function generator (Tektronix AFG 3022C) and amplified

with a wideband power amplifier (Krohn-Hite KH model 7602). The amplified voltage is

applied to the piezoelectric disc. Function generator and LDV are connected by workstation

with velocity decoder and controller (Polytec OFV 5000) for synchronization.

141

A direct way to experimentally obverse a local defect at resonance is to measure each

individual contribution in the scanning area in a broadband frequency range. For this

purpose, the input signal is referred to “linear chirp” as a wideband sinusoidal signal for

which the frequency is swept as a function of time. According to Eq. (3.2.1), the linear chirp

is excited in the system.

Figure 5.4 A schematic picture for experimental setup with a system by PZT as excitation

and LDV as sensor.

The received responses from LDV as one-dimensional time domain signal at each

scanning point form a three-dimensional matrix (x, y, t) and are analyzed by a developed

MATLAB script for propagating wavefield and post-processing imaging. The script contains

two sections, one for observing vibration patterns in spatial-frequency domain (x, y, ω) and

another for imaging of damage. The part of imaging in the script converts measured spatial-

142

time data into frequency-wavenumber domain and uses the wavenumber filtering which

assists in separating the forward waves and backward waves. By applying cross-correlation

imaging method discussed later, wave separation due to any damage, discontinuity, or

boundary in specimen helps to identify them and makes an imaging.

5.2.2 Response on FBH

A simple case for an aluminum plate with FBH is examined for LDR in experiment. The

aluminum plate Al 6061-T6 by 304 mm × 304 mm × 2.29 mm has a FBH with a = 12.5 mm

and depth d = 0.9 mm underneath scanning surface. The material properties are elastic

modulus E = 68.9 GPa, Poisson’s ratio v = 0.33, and density 2700 kg/m3. A piezoelectric disc

is attached on outside of top-right corner of scanning area. The scanning area sensed by LDV

is 40 mm × 40 mm for equally spacing 1 mm and total are 1,681 points. After scanning, data

is transformed to frequency domain according to Eq. (5.1.5) for observing vibration patterns.

Measurements were conducted by LDV and PZT along scanning path with excitation.

Excitation waveform comprised of a linear chirp signal with a starting frequency at 0 kHz

and bandwidth 100 kHz. Input signal is created by function generator and amplified to peak-

to-peak voltage of Vpp = 50 V. The duration for excitation and sensing as one burst was 3.2

ms containing 16,384 samples. Excitation signal is shown in Figure 5.6(a) and its spectrum in

Figure 5.6(b). The response captured by LDV at the center of FBH is shown in Figure 5.6(c)

and its spectrum in Figure 5.6(d) has a peak amplitude at 24.7 kHz as LDR. According to Eq.

(5.1.1), the theoretical LDR is about 22 kHz for approximately 12.3% error percentage due to

difference between exact shape of FBH and fabricated hole. From Figure 5.6(d), other peaks

143

in spectrum are also shown but they are geometric resonance (GR) corresponding to natural

frequencies of specimen responded in frequency at the location of FBH center. The

corresponding vibration patterns for GR will be discussed in next section.

Figure 5.5 Scheme for aluminum plate Al 6061-T6 sized by 304 mm × 304 mm × 2.29 mm

with FBH D25 mm and depth 0.9 mm (grey dash line) underneath scanning area 40 mm × 40

mm (red box). A PZT is attached outside of scanning area for generating signals.

(a) (b)

144

Figure 5.6 Excitation and response at the center of FBH show the waveforms and their

spectrums. (a) Excitation waveform (b) excitation spectrum (c) response waveform and (d)

response spectrum shows LDR and geometric resonance (GR) at the location of FBH center.

5.2.3 LDR for FBH

To experimentally observe a local defect at resonance is measuring each individual response

in the scanning area and its spectrum in a broadband frequency range. For this purpose, PZT

and LDV with wideband excitation can be able to probe and identify all possible resonances

in the vibration spectrum for all scanning points. According to Eq. (5.1.5), each point in

scanning area is transformed to frequency domain as U(x, y, ω). The spectrum obtained

includes the resonances corresponding to natural frequencies of specimen as well as local

resonance. This task can be simpler to identify resonances of LDR or GR if a location of the

defect is known. In this case, an aluminum plate with FBH is examined and peaks in

spectrum are identified in previous section. The corresponding patterns are shown in the

Figure 5.7 for LDR at 24.7 kHz, GR at 14.1 kHz and 30.6 kHz.

(d)(c)

LDR

GRGR

145

In the Figure 5.7(b) and (c), the vibration patterns show higher amplitudes outside of

damage region (white circle), which are related to geometric resonance as natural frequency

of specimen. In the Figure 5.7(a), the area of vibration is within the damage region. The deep

blue contour area surrounding the higher amplitudes shows ellipse shape and matches the

damage region. However, the area of vibration pattern is smaller than damage region. The

reason to have such pattern is because the local resonance is affected by geometry of defect

and boundary conditions surrounding the damage. In studies [29, 30], the high ratio H/h and

a/h have closely consistence with theoretical evaluation. In contrast, low ratio H/h and a/h

have difficulties to identify defects in stiffer materials such as aluminum. In this study, the

ratio H/h is about 2 but a/h is relatively high about 10 so that geometric resonance takes

primary while observing vibration patterns and makes local resonance obscure.

(a) Vibration pattern of LDR in top view and 3-D at 24.7 kHz

146

(b) Vibration pattern of GR in top view and 3-D at 14.1 kHz

(c) Vibration pattern of GR in top view and 3-D at 30.6 kHz

Figure 5.7 Vibration patterns are shown according to the spectrum at the center of damage

with its region depicted in white circle.

147

5.3 Imaging Processing for Linear LDR

The key idea for cross-correlation imaging is based on the concept that damage exists at the

location where forward waves interact in phase with backward waves. The imaging condition

for reconstructing wavefields is formulated in time domain or in frequency domain as cross

correlation as zero-lag in phase. In this case, forward and backward waves are based on the

excitation source for certain bandwidth. ZLCC imaging condition is based on the certain

bandwidth to be accumulated. Therefore, ZLCC imaging condition is related to the input

signal such as five-peaked toneburst signal in narrowband [38]. However, it is time-

consuming to try different frequencies for unknown damage sizes. Pulse signal excited by

laser overcame the difficulty but it is uncontrollable in desired bandwidth. As wideband

signal, pulse signal showed the ability to detect damage by using CSWE technique which is

also on Fourier-based filtering. ZLCC imaging can cumulate all waves in bandwidth by using

pulse laser.

LDR shows the ability to enhance the images of damage at certain frequency. At the

frequency, damage is excited and vibrated for higher amplitude in contrast to other regions.

Therefore, LDR is used for enhancing the images of defect with ZLCC imaging condition in

frequency domain.

5.3.1 Zero-lag Cross-Correlation (ZLCC) Imaging Condition

ZLCC imaging is based on the frequency-wavenumber filtering to separate forward

wavefields and scattered wavefields in 3-D domains (x, y, t). Out-of-plane displacements or

velocities in scanning area by sensors are captured and stored to form a 3-D data matrix.

148

Each data point is individually transformed into U(kx, ky, ω) domain by applying 3-D

temporal and spatial Fourier transform which is expected as follows

(5.3.1)

Then windows functions WF and WB for propagating wavefileds (forward) UF(kx, ky, ω) and

scattered waves (backward) UB(kx, ky, ω) are introduced to the transformed total waves U(kx,

ky, ω) so that the waves can be decomposed in frequency-wavenumber domain

U F ( B) U (kx ,ky , )WF ( B) (5.3.2)

where

and

WF 0 ,kx( y ) 0

1 ,kx( y ) 0

WB

1 ,kx( y ) 0

0 ,kx( y ) 0

(5.3.3)

For ZLCC imaging, forward and backward wavefield coincide in space domain. Thus filtered

forward UF and backward UB are transformed inversely to space domain by 2-D Fourier

transform as follows

(5.3.4)

Transformed forward uF and backward uB are employed in frequency domain and visualized

by taking the value of cross-correlation at all scanned points as follows

I(x, y) uF (x, y, )uB* (x, y, )

(5.3.5)

149

where I(x, y) is the ZLCC value at the location (x, y) with the multiple operation for forward

and backward wavefield in frequency domain. The superscript “*” symbolizes the complex

conjugation. These two wavefields coincide at the discontinuities and the value of I(x, y) will

be large in contrast to pristine region where has small value.

5.3.2 ZLCC Imaging without Bandwidth Filters

Although approximation of damage size and location can be deduced from vibration patterns

and identified by interaction between wave propagation and defects, a visualization method

is needed to determine the geometry, orientation, and location of small damages. Instead of

using the wave propagation and reflection in time domain, the cross-correlation method is

operated in frequency domain with summation of interested frequency range so that imaging

of ZLCC shows contour plot based on the values of coincident between incident and

reflected wavefields. According to Eq. (5.3.5), the values of ZLCC are higher in damaged

location than pristine region through all frequencies. Therefore, aluminum plate with FBH

first is examined by PZT and LDV system as mentioned previously.

The wavefield can be formed by excited linear chirp for wave propagation and sensed

to be a 3-D matrix data. Using the 3-D data, incident wave and reflected wave can be

separated by decomposed in frequency-wavenumber domain. ZLCC method is employed to

plot contour and to show the imaging of damage region as Figure 5.8 in the scanning area 40

mm × 40 mm. The interested frequency range is summed up from 0 to 100 kHz which is input

signal frequency range. In the Figure 5.8, the damage area drawn by white dash line is

fulfilled within the imaging of ZLCC. The damage location centered at x = 20 mm and y = 18

150

mm and size about 25 mm as expected are estimated by ZLCC. The damage region is

highlighted with red for high values as high sensitivity. This figures is consistent with

damage size and location as expected.

Figure 5.8 ZLCC imaging for linear chirp as wideband signal and damage region depicted in

white circle.

5.3.3 ZLCC Imaging at LDR

Vibration pattern shows the local out-of-plane motion of defect for the damage region in the

plate due to the excitation with certain frequency. The local vibration motion demonstrates

the defect geometry, size and location in specific frequency as LDR so that wave propagation

increases interaction with damage region. While observing at LDR, the imaging of ZLCC

only shows the area of vibration pattern as standing wave according to Figure 5.9. At LDR,

forward wave makes local vibration as standing wave in the area shown in the Figure 5.9,

151

and so as reflected waves. Therefore, ZLCC imaging condition has larger values due to

vibration pattern as dominate for the interaction between incident and reflected waves. Eq.

(5.3.5) can be represented as Eq. (5.3.6) based on the LDR frequency.

ILDR (x, y) uF (x, y, LDR )uB* (x, y, LDR ) (5.3.6)

Because of certain frequency for LDR, Eq. (5.3.6) presents waves interaction in one

frequency as one pattern rather than summation of ZLCC values through all frequencies. At

LDR, ZLCC imaging shows similarity to the vibration pattern according to Figure 5.7(a).

The result of imaging at LDR only shows part of damaged since the structure has low

H/h. When surface thickness of damage is much thinner than thickness of plate h << H, the

local vibration is more obvious than relatively high H/h. In this study, low H/h due to

fabricated flat bottom hole has two over third thickness of plate so that the vibrations in

damage region only shows potion because of the constrain by surrounding structure.

Figure 5.9 ZLCC image with vibration pattern at LDR frequency.

152

5.4 Chirp Signal on Composite Plate with Barely Visible Impact Damage (BVID)

In aerospace structures, low-velocity impact due to tool drops or runaway stones may result

in various forms of damages such as indentation, matrix cracking, or delamination to cause

severe reduction in strength and integrity of composite structures. Although structures are

designed for fail-safe principles to withstand partial system failures, impact damage is an

important issue in maintenance of structures. While visible damages can be easily detected

and repaired with naked eyes, a major concern is the growth of hidden damage caused by

low-velocity impact. The internal damage is known as barely visible impact damage (BVID).

Many techniques have been developed in last few decades for BVID detection for

composite structures. These include various non-destructive testing (NDT) methods based on

ultrasonic waves, acoustic emission or X-rays. Guided ultrasonic waves and nonlinear

acoustic techniques are particularly attractive because their abilities can inspect small

damages with a small number of transducers.

Staszewski et al.[85] used three-dimensional laser Doppler vibrometry to locate and

estimate delamination in a composite plate, demonstrating that the delamination can be

revealed by the amplitude profiles of Lamb waves. Michaels et al. [35] used guided

wavefield images and frequency–wavenumber domain analysis to show the wave interactions

with structural discontinuities in composites. Tian et al [86] simulated wave propagation with

delamination and used PZT and scanning laser Doppler vibrometer (SLDV) experimentally

to scan artificial Teflon inserted as delamination in composite plate. He et al [39] used a

enhanced zero-lag cross-correlation reverse-time migration (E-CCRTM) successfully to

obtain BVID imaging with PZT and one-dimensional LDV. Harb et al [87] scanned different

153

sizes of BVID with a fully non-contact system of air-coupled transducers and one-

dimensional vibrometer. These studies show the ability for analysis of wave propagation

interacting with BVID by linear phenomena and use mono-frequency excitation to purify the

mode of Lamb wave and interpret wave interaction due to damage. However, broadband

signal such as pulse signal generated by pulse laser can be used for detecting damage rapidly.

Dhitalet al [88] used a Q-switched Nd:YAG laser as excitation coupled with air-coupled

transducer or one-dimensional LDV to scan damage on the composite plate, having filtered

20-100 kHz signals for imaging. An et al [36] used laser to generate pulse as wideband signal

and one-dimensional LDV with galvanometers for scanning. In these studies, Lamb wave

propagation with broadband signal is generated and can be analyzed for imaging but

excitation is hard to be in control. Therefore, controllable signal such as linear chirp as

superposition of sinusoidal waves is considered in this study for analysis and wideband

purpose.

Delamination as small internal crack in the composite can be detected by nonlinear

ultrasonic methods. Aymerich et al. [61] demonstrated the application nonlinear acoustic for

impact damage detection in composite laminates. The plate was instrumented with an

electromagnetic shaker and two bonded low profile piezoceramic transducers, one as exciter

and another as receiver. Pieczonka et al. [32] used LDR and the second harmonic imaging

technique (SEHIT) implemented in the study with one piezoceramic actuator and LDV on a

laminated composite plate. The LDR method is that a structural defect has associated

resonant frequencies, being a function of damage size and geometry. The assumption is that

the amplitude level of higher harmonic in measured response spectra increases dramatically

154

near the location of damage. The second harmonic imaging technique (SEHIT) is an

alternative approach used for imaging. The method is based on the spatial mapping of higher

harmonic amplitudes generated by damage. The assumption is that the spatial distribution of

the ratio between the amplitude of a second harmonic over the amplitude at excitation

frequency reveals the location of damage. Kelpka et al. [45] used similar way to do an

arbitrary frequency and frequency corresponding to local defect resonance for examining the

composite panels. LDR enhanced the intensity of modulation.

In this study, linear guided ultrasonic waves and nonlinear ultrasonic based on LDR

use a toneburst linear chirp signal to examine a BVID in a carbon fiber-reinforced composite

plate with PZT and LDV system. Imaging method based on ZLCC mentioned previously is

implemented on damage region after analyzing spectrum for LDR. The size and location of

damage is imagined at the LDR frequency. Since LDR can evoke nonlinear phenomena,

delamination can be clearly observed after imaging.

5.4.1 Specimen Description and Experimental Setup

A composite plate with a BVID is examined in experiment with PZT and LDV system.

Experiment setup is shown in the Figure 5.10 as scheme in the Figure 5.4. PZT actuator is

attached on the composite plate with glue. Excitation signal is generated from waveform

function generator with a linear chirp signal from 0 to 100 kHz. The input signal goes into

amplifier and then connects to PZT actuator. Waveform with chirp signal propagates in the

panel and the signal point LDV captures the out-of-plane velocity. LDV mounted on the 2-

axis translation stage sensing on the reflective tape moves along scanning path depicted in

155

the figure and record data in the workstation. In order to enhance SNR, the records are

averaged for 25 times. The transient records have 16,384 samples on each point for 3.2 ms.

The scanning area is 40 mm by 40 mm with equally spacing Δx = Δy = 1 mm. The BVID is at

the center of scanning area.

The material of composite plate T800/3900-2 carbon/epoxy with properties listed in

the Table 5.1 is used with 16-ply layups [0/45/90/−45]2s and dimensions 304 mm × 304 mm ×

2.4 mm. The specimen is impacted by ball-shaped head with energy 14J from a height

dropping on the panel. The damage region of BVID is about 7 mm from C-scan shown in the

Figure 5.11. From C-scan, delamination regions are depicted with dash white lines by

different layer scans inside the damage region. The dark region in the Figure 5.11 is the dent

area seen on the top surface by C-scan. The results of C-scan will be compared to the

imaging of vibration patterns and ZLCC imaging condition later.

Figure 5.10 Experimental setup for damage imaging on BVID in a composite plate.

156

Figure 5.11 Damage region (dash red) and delamination (dash white) of different layer are

shown in the C-scan. The size of C-scan matches the scanning area.

Table 5.1 Material properties of T800/3900-2 composite lamina [89]

E1 (GPa)

E2 (GPa)

G12 (GPa)

F1t (MPa)

F1c (MPa)

F2t (MPa)

F2c(MPa)

F6(MPa) v12

ρ(kg/m3)

160 9 6.2 2840 1550 95 165 116 0.28 1645

5.4.2 Spectrum on Impact Area and LDR

The spectrum on the damage is analyzed for LDR before imaging. The detecting point is at

the center of damage region shown in the Figure 5.12 using yellow cross indicating the

location. At the detecting point, received response is transformed into frequency domain. The

spectrum shows multiple peaks in different frequencies. Examining the patterns of each

frequency indicates that the first peak frequency is local resonance for damage region. The

patterns will be discussed later in the next section.

The first peak at 29.4 kHz indicates the local defect resonance (LDR). According to

Eq. (5.1.1), the theoretical resonance is about 31.4 kHz for radius of damage region a = 3.5

157

mm, effective elastic modulus E = 46.34 GPa and h = 0.15 mm due to delamination happened

from second layer based on C-scan result. The second peak corresponds to GR of specimen.

The frequencies at 55.6 kHz and 69.4 kHz are related to LDR of delamination since the

detecting location is close to delamination region. These vibration patterns will be discussed

later.


area. (a) The detcting point at the center of damage region. (b) Peaks in spectrum at detecting

point.

5.4.3 Vibration Pattern on at LDR

According to the spectrum at the center, the vibration patterns are shown in the Figure 5.13

and Figure 5.14 in 3-D and top view. In the vibration patterns, the shape of damage is clear to

be seen. Based on spectrum analysis, the LDR is at 29.4 kHz on the detecting point. Figure

5.13(a) and Figure 5.14(a) at LDR show clear images of larger amplitude location which is

(a) (b)

158

corresponding to damage region in the C-scan. Therefore, vibration pattern at LDR provides

good estimation of damage.

In higher two frequencies, small areas are shown in the figure (c) and (d) of Figure

5.13 and Figure 5.14. The small areas are matched to the delamination region from C-scan.

Therefore, the spectrum in these two frequencies can be seen as the local resonance of

delamination since the detecting location is close to delamination. In order to estimate the

accurate frequency, the confirmation of spectrum needs to be examined at the location of

delamination. It will be discussed in further section later.

The vibration pattern of Figure 5.13(b) and Figure 5.13(b) at 41.9 kHz shows higher

amplitude because the GR vibrates at the detecting point. From images, the frequency

doesn’t cause any vibration related to damage.

159

Figure 5.13 The corresponding three-dimensional vibration pattern for the peaks of spectrum

at center of damage region. (a) 29.4 kHz (b) 41.9 kHz (c) 55.6 kHz (d) 69.4 kHz

(a) (b)

(c) (d)

160

Figure 5.14 The corresponding vibration pattern in top view for the peaks of spectrum at

center of damage region. (a) 29.4 kHz (b) 41.9 kHz (c) 55.6 kHz (d) 69.4 kHz

5.4.4 ZLCC Imaging without Filtering on Composite Plate

Before applying the vibration patterns as experiment in aluminum as previous section, ZLCC

imaging examines the damage in range of excited frequencies. Wave propagation through the

damage region cause interaction happens in the damage region and trapped energy around

damage is expected. Figure 5.15 shows the damage location and damage region with red dash

(a) (b)

(c) (d)

161

line. Two delamination in red take primary indications for the imaging. The image shows

clear and higher values related to delamination regions which are depicted by white dash

lines from C-scan. In addition, a good agreement for ZLCC imaging condition is that the

method works for the wideband signal like linear chirp. Therefore, BVID damage can be

detected by such excitation signal without determining the excitation frequency in

narrowband ahead.

To be noted, the imaging processing needs to be operated when incident waves

passes through the scanning area so that the processing would not be affected by residual

waves in certain time frames. Since the chirp signal needs to be recorded in longer duration

relative to five-peaked toneburst signal, the operating time frame should be longer than usual.

Figure 5.15 ZLCC imaging cumulating all frequencies on composite plate with BVID.

162

5.4.5 ZLCC Imaging at LDR on Composite Plate

Through confirming the resonance at the center of damage, imaging method ZLCC is

employed with the vibration pattern. Figure 5.16 shows the result of ZLCC at the LDR

frequency. The higher values with red shows good agreements with damage region depicted

from C-scan in red dash line and delamination in white dash lines. In the figure,

delaminations are not clear to be observed but local vibration pattern for damage region is

shown with depiction of C-scan.

Figure 5.16 ZLCC imaging at LDR for vibration pattern with dicption from C-scan. (damage

region for red dash line and delamination for white dash lines)

5.4.6 Nonlinearity at LDR and Imaging on Delamination

Through moving the detecting point at the location of delamination in the Figure 5.17(a), the

spectrum of it is shown in the Figure 5.17(b). In the spectrum, there are two high amplitudes

163

shown as LDR of delamination. At frequency 56.3 kHz, the vibration pattern in the Figure

5.18(a) shows higher amplitudes around the location of delamination, so as at frequency 68.4

kHz in the Figure 5.18(b) but in different shape. Two LDR of the same delamination indicate

that the delamination has cross-section between layers. From C-scan in different layer in the

Figure 5.19, different depths of layer 4 through layer 5 have delamination at the same

location but in different size. According to Eq. (5.1.1), the theoretical resonance is related to

thickness h above delamination. When delamination happens in more depth, the resonance is

higher. For isotropic material, there is only one thickness for FBH. However, the laminated

composite plate is impacted and delamination is happened in any possible layup so that the

local resonance will change. Therefore, two LDR are found due to cross-section through

layers.

Outside of excitation signal range, there are also two relatively higher frequencies

found. The frequencies are about twice of LDR of delamination as second harmonics as

nonlinearity. Due to delamination happened in different layers, there are two harmonics for

nonlinearity as expected. The vibration patterns for the nonlinearity will be discussed later.

The ZLCC imaging condition is also employed with vibration patterns at LDR of

delamination in the

Figure 5.20. The results of images are compared to depiction of C-scan for damage

region in red dash line and delamination regions in white dash lines. The higher values

indicate the location of delamination, consisting with the depiction of delamination from C-

scan.

164


area. (a) The detcting point at location of delamination. (b) Peaks in spectrum at detecting

point.

From spectrum in Figure 5.17(b), there are two frequencies found outside excitation

range over 100 kHz. According to the LDR of delamination, these frequencies are about

twice of them. Based on LDR theory described previously, they are second harmonics from

nonlinearity. At the frequency 115.6 kHz, higher amplitude shows in the vibration pattern

other than regions without damages in the Figure 5.21(a). It is clear to see the delamination

location and size. At frequency 137.8 kHz, higher amplitudes also show in the figure with

different size at the same location in the Figure 5.21(c). For both images, nonlinearity causes

higher harmonics (HH) to have local vibration of delamination. The locations having higher

amplitudes are only related to the delaminations. Comparing the images with depiction of C-

scan, the areas with higher values are all in the white dash lines. Therefore, nonlinear

ultrasonic can demonstrate the imaging and identify the location of delamination only where

nonlinearity happens due to LDR.

(a) (b)

165

Figure 5.18 Vibration pattern on the delamination at (a) 56.3 kHz (b) in 3-D view and at (c)

68.4 kHz (d) in 3-D view.

(a) (b)

(c) (d)

166

(a) layer 4 (b) layer 5

Figure 5.19 C-scan imaging in (a) layer 4 and (b) layer 5 shows different size delamination in

the same location as cross-section.

Figure 5.20 ZLCC imaging with vibration patterns at LDR frequency (a) 56.3 kHz (b) 68.4

kHz.

(a) (b)

(

a)

(

b)

167

Figure 5.21 Vibration patterns for nonlinearity happened at LDR for frequency (a) 115.6 kHz

(b) in 3-D view and (c) 137.8 kHz (d) in 3-D view.

5.5 Summary

The concept of local mechanism on the defect in solid materials is demonstrated. A local

defect resonance (LDR) based on the fact of a local drop of rigidity for a mass in the area can

manifest in characteristic frequency of the defect. A straightforward and analytical approach

is investigated in an aluminum plate with a FBH and a CFRP plate with a BVID to evaluate

(c) (d)

(a) (b)

168

the fundamental LDR frequencies and nonlinearity. A direct way to reveal vibration patterns

experimentally is to measure individual contribution of each point in the scanning are with

vibrometry system combining with piezoelectric transducer and laser Doppler vibrometer. In

addition, ZLCC imaging condition in frequency domain according to vibration patterns at

LDR is applied by cross-correlating the forward and backward wavefields separated from the

measured data. Imaging conditions are verified and enhanced for location and size of

damages by ZLCC at LDR in experiments. According to LDR concept, nonlinearity happens

at smaller defect such as delamination in laminated composite plate. Strong increases in

vibration amplitudes at harmonics generated by LDR enable to detect reliably and visualize

delamination.

For the spectrum analysis, individual peaks of amplitudes should be careful to be

examined. Although the analytical formula provides a rough estimation of suitable LDR

frequency range, geometric properties of defect affects the accuracy such as the ratio of a/h

and h/H. However, previous studies show simulation results which provide good prediction

of LDR frequency on aluminum plate. In the future, simulation for composite plate with

delamination can be done for good estimation and evidences of LDR. Automated detection

methodology of LDR on BVID can also be developed for visualizing defects.

In this study, a single point LDV as sensing tool is used for detecting out-of-plane

velocities in the scanning area. The data is stored and analyzed in post-processing for

spectrum analysis, LDR and ZLCC imaging condition. However, it is time-consuming to

collect all information in the scanning area with high resolution by spacing 1 mm and long

duration for transient response. In order to reduce the time, the conventional approach such

169

as scanning LDV can be used for increasing speed and integration of synchronization of

system.

170

CHAPTER 6

Conclusions and Future Works

6.1 Dispersion Relationships with Chirp Signal

Chirp signal is applied in this research as excitation and analyzed by MP method and chirplet

transform for dispersion relationships on aluminum and composite plates. MP method

approaches dispersion relationships from frequency-wavenumber domain to group velocity.

Chirplet transform provides another option directly to group velocity with less points. Chirp

signal as the union of these two methods offers the opportunity to reduce the laboring time

and shows accuracy analysis with these methods for dispersion relationships of wave

propagating in plates.

Here are some conclusions for Matrix Pencil method:

1) MP method is compared and confirmed on different thicknesses aluminum plates by

theoretical curves that are numerically computed by the dispersion equations of Lamb

waves.

2) More modes up to A2 and S2 on thickness 6.35 mm aluminum plate are obtained with

higher frequency 5-1000 kHz.

3) MP method analysis shows accuracy and dispersion curves from the results in x-ω

domain can be reconstructed comparing to theoretical ones with coefficients of

determination over 97%.

4) MP method shows the ability to examine unknown material for dispersion

relationships.

171

5) MP method on a layered composite plate in three different directions 0°, 45°, and 90°

show compromised results with theoretical curves by 3-D elasticity theory.

6) For MP method, there are two approaches from k-t domain or x-ω domain to

determine the relationship of dispersion curves. Both approaches show accurate

results but different points and they can complement each other to complete

dispersion curves.

7) For k-t domain, it has lower resolution due to much less detection points x relative to

transient time t. Resolution for each approach is based on the Nyquist theorem and

sampling theory according to APPENDICES.

8) MP method derives the relations of wavenumber and frequency and the relations are

used for phase velocity cp and group velocity cg.

9) Group velocity for isotropic plate is derived from wavenumber and frequency as

gradients of curves. But group velocity for composite plate is affected by the

propagation angle for dispersion behavior and has different expressions with different

numerical implementations.

10) MP method can extract the dispersion relationships between wavenumber and

frequency directly without identifying the peak magnitudes and then defining them.

MP method can provides relationship from frequency-wavenumber to group velocity of

dispersion relationships but group velocities are based on the calculation of theoretical

relation. The results are easy to be affected by the gradient between analysis points due to

sampling issue. Therefore, chirplet transform provides another option to obtain group

velocity directly for chirp signal.

172

This application of chirplet transform (CT) to the time-frequency analysis on group

velocity of wave propagation has shown good agreement with theoretical dispersion curves n

this research. A linear chirp signal with a broadband frequency provides robust excitation to

carry rich information for the relationship of dispersion. It was found that CT is an effective

tool for linear chirp signal excitation on experimental analysis of dispersion curves in solid

media. In addition, only two measurements for one chirp signal excitation were received and

analyzed. For isotropic (aluminum plate) or laminated materials (composite plate), the group

velocities of the flexural plate mode were measured in all directions and had good

agreements with theoretical curves.

6.2 Chirp signal for Nonlinear Ultrasonic Imaging

Chirp signal as excitation can be used for vibrating the local mechanism in the plate. The

concept of local mechanism on the defect in solid materials is demonstrated in the research.

Here are some conclusions for ultrasonic imaging by using chirp signal:

1) A local defect resonance (LDR) based on the fact of a local drop of rigidity for a mass

in the area can manifest in characteristic frequency of the defect.

2) At LDR frequency, vibration patterns of defect become more obvious region in

spatial-frequency domain.

3) An aluminum plate with a FBH for evaluating the fundamental LDR frequencies and

imaging is used and confirmed by the analysis.

4) Imaging conditions are verified and demonstrated for location and size of damages by

ZLCC at LDR in experiments.

173

5) According to LDR concept, nonlinearity happening at smaller defect such as

delamination in laminated composite plate can be observed from spectrum.

6) Strong increases in vibration amplitudes at harmonics generated by LDR enable to

detect reliably and visualize delamination.

LDR is one of efficient methods to observe the location and sizing of defect images. ZLCC

imaging condition is also an effective method to analyze the wave propagation. Combining

LDR and ZLCC as analysis as in frequency domain provides more understandings about

local vibration patterns and wave interaction, which also opens up an opportunity to enhance

damage imaging condition. In addition, LDR causes nonlinearity in complex structure so that

small defect such as delamination in barely visible impact damage can be visualized.

Nonlinear ultrasonic imaging derives evidences to show high magnitudes obviously on small

defect region rather than intact regions.

6.3 Future Works

MP method yields dominant modal characteristics through the coefficients of the waveform.

In this study, frequencies or wavenumbers in poles were used. Damping factors in poles can

be studied in the future, which are referred to evanescent waves. MP method may be

developed to approach higher mode accurately and efficiently with a broadband excitation on

more complicated materials of unknown mechanical properties that cannot compute

theoretical curves ahead.

In the future, simulation for composite plate with delamination can be done for good

estimation and evidences of LDR. Automated detection methodology of LDR on BVID can

174

also be developed for visualizing defects. Moreover, in this study, a single point LDV as

sensing tool is used for detecting out-of-plane velocities in the scanning area. The data is

stored and analyzed in post-processing for spectrum analysis, LDR and ZLCC imaging

condition. However, it is time-consuming to collect all information in the scanning area with

high resolution by spacing 1 mm and long duration for transient response. In order to reduce

the time, the conventional approach such as scanning LDV can be used for increasing speed

and integration of synchronization of system.

Disbond or delamination in sandwich structure arises interests in past few years.

Future work will focus on detecting damages due to low velocity impact with linear and

nonlinear phenomena generated by LDR in sandwich structures. Imaging processing and

excitation signal are also investigated for enhancing images and fully non-contact system

such as using air-coupled transducer as excitation source.

175

REFERENCES

1. W.J. Staszewski, C. Boller, and G.R. Tomlinson, Health Monitoring of Aerospace Structures: Smart Sensor Technologies and Signal Processing. John Wiley & Sons. 2004

2. A. Raghavan and C.E.S. Cesnik, Review of Guided-Wave Structural Health Monitoring. Shock and Vibration Digest, Vol.39, No.2: p. 91-116. 2007.

3. K.F. Graff, Wave Motion in Elastic Solids. Courier Corporation. 2012

4. Z. Su, L. Ye, and Y. Lu, Guided Lamb Waves for Identification of Damage in Composite Structures: A Review. Journal of Sound and Vibration, Vol.295, No.3–5: p. 753-780. 2006.

5. N. Gandhi, Determination of Dispersion Curves for Acoustoelastic Lamb Wave Propagation, in Electrical and Computer Engineering. 2010, Georgia Tech.

6. L. Lu and B. Zhang, Analysis of Dispersion Curves of Rayleigh Waves in the Frequency Wavenumber Domain. Canadian geotechnical journal, Vol.41, No.4: p. 583-598. 2004.

7. K.F. Graff, Wave Motion in Elastic Solids. Courier Corporation. 1975

8. F.A. Amirkulova, Dispersion Relations for Elastic Waves in Plates and Rods, in Mechanical and Aerospace Engineering. 2011, Rutgers University-Graduate School-New Brunswick.

9. B. Köhler, Dispersion Relations in Plate Structures Studied with a Scanning Laser Vibrometer, in 9th European NDT Conference, ECNDT. 2006: Berlin. p. 1-11.

10. D. Alleyne and P. Cawley, A Two-Dimensional Fourier Transform Method for the Measurement of Propagating Multimode Signals. The Journal of the Acoustical Society of America, Vol.89, No.3: p. 1159-1168. 1991.

11. R.D. Costley Jr and Y.H. Berthelot, Dispersion Curve Analysis of Laser-Generated Lamb Waves. Ultrasonics, Vol.32, No.4: p. 249-253. 1994.

12. F. Schöpfer, F. Binder, A. Wöstehoff, T. Schuster, S. von Ende, S. Föll, and R. Lammering, Accurate Determination of Dispersion Curves of Guided Waves in Plates by Applying the Matrix Pencil Method to Laser Vibrometer Measurement Data. CEAS Aeronautical Journal, Vol.1-8. 2013.

176

13. M. Meunier and F. Brouaye. Fourier Transform, Wavelets, Prony Analysis: Tools for Harmonics and Quality of Power. in Harmonics and Quality of Power Proceedings, 1998. Proceedings. 8th International Conference On. IEEE.1998.

14. T. Zieliński and K. Duda, Frequency and Damping Estimation Methods-an Overview. Metrology and Measurement Systems, Vol.18, No.4: p. 505-528. 2011.

15. C.I. Chen and Y.C. Chen, Comparative Study of Harmonic and Interharmonic Estimation Methods for Stationary and Time-Varying Signals. IEEE Transactions on Industrial Electronics, Vol.61, No.1: p. 397-404. 2014.

16. L.L. Grant and M.L. Crow. Comparison of Matrix Pencil and Prony Methods for Power System Modal Analysis of Noisy Signals. in North American Power Symposium (NAPS).2011.

17. C.Y. Chang and F.G. Yuan. Dispersion Curve Extraction of Lamb Waves in Metallic Plates by Matrix Pencil Method. in SPIE Smart Structures and Materials+ Nondestructive Evaluation and Health Monitoring. International Society for Optics and Photonics.2017.

18. K. Kishimoto, H. Inoue, M. Hamada, and T. Shibuya, Time Frequency Analysis of Dispersive Waves by Means of Wavelet Transform. Journal of Applied Mechanics, Vol.62, No.4: p. 841-846. 1995.

19. M. Niethammer, L.J. Jacobs, J. Qu, and J. Jarzynski, Time-Frequency Representation of Lamb Waves Using the Reassigned Spectrogram. 2000.

20. H. Jeong, Analysis of Plate Wave Propagation in Anisotropic Laminates Using a Wavelet Transform. NDT & E International, Vol.34, No.3: p. 185-190. 2001.

21. H. Kuttig, M. Niethammer, S. Hurlebaus, and L.J. Jacobs, Model-Based Analysis of Dispersion Curves Using Chirplets. The Journal of the Acoustical Society of America, Vol.119, 2122. 2006.

22. F. Kerber, H. Sprenger, M. Niethammer, K. Luangvilai, and L.J. Jacobs, Attenuation Analysis of Lamb Waves Using the Chirplet Transform. EURASIP Journal on Advances in Signal Processing, Vol.2010, 5. 2010.

23. J.E. Michaels, S.J. Lee, A.J. Croxford, and P.D. Wilcox, Chirp Excitation of Ultrasonic Guided Waves. Ultrasonics, Vol.53, No.1: p. 265-270. 2013.

24. C. Boller, F.-K. Chang, and Y. Fujino, Encyclopedia of Structural Health Monitoring. Vol. 2960. Wiley New York, NY, USA:. 2009

177

25. I. Solodov, N. Krohn, and G. Busse. Nonlinear Ultrasonic Ndt for Early Defect Recognition and Imaging. in European Conf. on NDT (ECNDT), Moscow. Citeseer.2010.

26. D. Broda, W.J. Staszewski, A. Martowicz, T. Uhl, and V.V. Silberschmidt, Modelling of Nonlinear Crack–Wave Interactions for Damage Detection Based on Ultrasound—a Review. Journal of Sound and Vibration, Vol.333, No.4: p. 1097-1118. 2014.

27. K.-A. Van Den Abeele, P.A. Johnson, and A. Sutin, Nonlinear Elastic Wave Spectroscopy (News) Techniques to Discern Material Damage, Part I: Nonlinear Wave Modulation Spectroscopy (Nwms). Research in nondestructive evaluation, Vol.12, No.1: p. 17-30. 2000.

28. I. Solodov, J. Bai, S. Bekgulyan, and G. Busse, A Local Defect Resonance to Enhance Wave-Defect Interaction in Nonlinear Spectroscopy and Ultrasonic Thermography. 2012.

29. J. Hettler, M. Tabatabaeipour, S. Delrue, and K. Van Den Abeele, Detection and Characterization of Local Defect Resonances Arising from Delaminations and Flat Bottom Holes. Journal of Nondestructive Evaluation, Vol.36, No.1: p. 2. 2017.

30. I. Solodov, J. Bai, and G. Busse, Resonant Ultrasound Spectroscopy of Defects: Case Study of Flat-Bottomed Holes. Journal of Applied Physics, Vol.113, No.22: p. 223512. 2013.

31. I. Solodov. Local Defect Resonance (Ldr): A Route to Highly Efficient Thermosonic and Nonlinear Ultrasonic Ndt. in 40TH ANNUAL REVIEW OF PROGRESS IN QUANTITATIVE NONDESTRUCTIVE EVALUATION: Incorporating the 10th International Conference on Barkhausen Noise and Micromagnetic Testing. AIP Publishing.2014.

32. L. Pieczonka, A. Klepka, W.J. Staszewski, and T. Uhl. Nonlinear Acoustic Imaging of Structural Damages in Laminated Composites. in EWSHM-7th European Workshop on Structural Health Monitoring.2014.

33. O. Mesnil, C.A.C. Leckey, and M. Ruzzene, Instantaneous and Local Wavenumber Estimations for Damage Quantification in Composites. Structural Health Monitoring, Vol.14, No.3: p. 193-204. 2015.

34. E.B. Flynn, S.Y. Chong, G.J. Jarmer, and J.-R. Lee, Structural Imaging through Local Wavenumber Estimation of Guided Waves. Ndt & E International, Vol.59, 1-10. 2013.

178

35. T.E. Michaels, M. Ruzzene, and J.E. Michaels, Frequency-Wavenumber Domain Methods for Analysis of Incident and Scattered Guided Wave Fields. 729513-729513. 2009.

36. Y.-K. An, B. Park, and H. Sohn, Complete Noncontact Laser Ultrasonic Imaging for Automated Crack Visualization in a Plate. Smart Materials and Structures, Vol.22, No.2: p. 025022. 2013.

37. J.-R. Lee, H.-J. Shin, C.C. Chia, D. Dhital, D.-J. Yoon, and Y.-H. Huh, Long Distance Laser Ultrasonic Propagation Imaging System for Damage Visualization. Optics and Lasers in Engineering, Vol.49, No.12: p. 1361-1371. 2011.

38. M.S. Harb and F.-G. Yuan, Damage Imaging Using Non-Contact Air-Coupled Transducer/Laser Doppler Vibrometer System. Structural Health Monitoring, Vol.15, No.2: p. 193-203. 2016.

39. J. He and F.-G. Yuan. An Enhanced Ccrtm (E-Ccrtm) Damage Imaging Technique Using a 2-D Areal Scan for Composite Plates. in SPIE Smart Structures and Materials+ Nondestructive Evaluation and Health Monitoring. International Society for Optics and Photonics.2016.

40. R. Zhu, G.L. Huang, and F.G. Yuan, Fast Damage Imaging Using the Time-Reversal Technique in the Frequency–Wavenumber Domain. Smart Materials and Structures, Vol.22, No.7: p. 075028. 2013.

41. M.S. Harb and F.G. Yuan, A Rapid, Fully Non-Contact, Hybrid System for Generating Lamb Wave Dispersion Curves. Ultrasonics, Vol.61, 62-70. 2015.

42. J. Lin, J. Hua, L. Zeng, and Z. Luo, Excitation Waveform Design for Lamb Wave Pulse Compression. IEEE transactions on ultrasonics, ferroelectrics, and frequency control, Vol.63, No.1: p. 165-177. 2016.

43. G. Betta, L. Ferrigno, M. Laracca, P. Burrascano, M. Ricci, and G. Silipigni, An Experimental Comparison of Multi-Frequency and Chirp Excitations for Eddy Current Testing on Thin Defects. Measurement, Vol.63, 207-220. 2015.

44. R.A. Guyer and P.A. Johnson, Nonlinear Mesoscopic Elasticity. 2009.

45. A. Klepka, L. Pieczonka, W.J. Staszewski, and F. Aymerich, Impact Damage Detection in Laminated Composites by Non-Linear Vibro-Acoustic Wave Modulations. Composites Part B: Engineering, Vol.65, No.0: p. 99-108. 2014.

46. K.R. McCall and R.A. Guyer, A New Theoretical Paradigm to Describe Hysteresis, Discrete Memory and Nonlinear Elastic Wave Propagation in Rock. Nonlinear processes in geophysics, Vol.3, No.2: p. 89-101. 1999.

179

47. L.D. Landau and E.M. Lifshitz, Theory of Elasticity, 1986. Course of Theoretical Physics. 1986

48. P.A. Johnson, B. Zinszner, P. Rasolofosaon, F. Cohen‐Tenoudji, and K. Van Den Abeele, Dynamic Measurements of the Nonlinear Elastic Parameter Α in Rock under Varying Conditions. Journal of Geophysical Research: Solid Earth (1978–2012), Vol.109, No.B2: p., 2004.

49. L.A. Ostrovsky and P.A. Johnson, Dynamic Nonlinear Elasticity in Geomaterials. Rivista del nuovo cimento, Vol.24, No.7: p. 1-46. 2001.

50. M. Meo and G. Zumpano, Nonlinear Elastic Wave Spectroscopy Identification of Impact Damage on a Sandwich Plate. Composite structures, Vol.71, No.3: p. 469-474. 2005.

51. M. Meo, U. Polimeno, and G. Zumpano, Detecting Damage in Composite Material Using Nonlinear Elastic Wave Spectroscopy Methods. Applied composite materials, Vol.15, No.3: p. 115-126. 2008.

52. Y.M. Cheong, Nonlinear Acoustic Effect Due to Evolution of a Crack and Detection of Micro-Scale Cracks in a Fatigue Process, in Nondestructive Testing of Materials and Structures, O. Güneş and Y. Akkaya, Editors., Springer Netherlands. p. 525-530. 2013

53. H.J. Lim, H. Sohn, M.P. DeSimio, and K. Brown, Reference-Free Fatigue Crack Detection Using Nonlinear Ultrasonic Modulation under Various Temperature and Loading Conditions. Mechanical Systems and Signal Processing, Vol.45, No.2: p. 468-478. 2014.

54. Z. Parsons and W.J. Staszewski, Nonlinear Acoustics with Low-Profile Piezoceramic Excitation for Crack Detection in Metallic Structures. Smart Materials and Structures, Vol.15, No.4: p. 1110. 2006.

55. M. Ryles, F.H. Ngau, I. McDonald, and W.J. Staszewski, Comparative Study of Nonlinear Acoustic and Lamb Wave Techniques for Fatigue Crack Detection in Metallic Structures. Fatigue & Fracture of Engineering Materials & Structures, Vol.31, No.8: p. 674-683. 2008.

56. H.F. Hu, W.J. Staszewski, N.Q. Hu, R.B. Jenal, and G.J. Qin, Crack Detection Using Nonlinear Acoustics and Piezoceramic Transducers—Instantaneous Amplitude and Frequency Analysis. Smart Materials and Structures, Vol.19, No.6: p. 065017. 2010.

57. A. Klepka, W.J. Staszewski, R.B. Jenal, M. Szwedo, J. Iwaniec, and T. Uhl, Nonlinear Acoustics for Fatigue Crack Detection–Experimental Investigations of

180

Vibro-Acoustic Wave Modulations. Structural Health Monitoring, Vol.11, No.2: p. 197-211. 2012.

58. R.A. Guyer and P.A. Johnson, Nonlinear Mesoscopic Elasticity: Evidence for a New Class of Materials. Physics today, Vol.52, No.4: p. 30-36. 2008.

59. V. Zaitsev, V. Gusev, and B. Castagnede, Thermoelastic Mechanism for Logarithmic Slow Dynamics and Memory in Elastic Wave Interactions with Individual Cracks. Physical review letters, Vol.90, No.7: p. 075501. 2003.

60. V.Y. Zaitsev, V. Gusev, and B. Castagnede, Observation of the “Luxemburg–Gorky Effect” for Elastic Waves. Ultrasonics, Vol.40, No.1: p. 627-631. 2002.

61. F. Aymerich and W.J. Staszewski, Impact Damage Detection in Composite Laminates Using Nonlinear Acoustics. Composites Part A: Applied Science and Manufacturing, Vol.41, No.9: p. 1084-1092. 2010.

62. P. Liu, H. Sohn, T. Kundu, and S. Yang, Noncontact Detection of Fatigue Cracks by Laser Nonlinear Wave Modulation Spectroscopy (Lnwms). NDT & E International, Vol.66, No.0: p. 106-116. 2014.

63. C.H. Chapman, A New Method for Computing Synthetic Seismograms. Geophysical Journal International, Vol.54, No.3: p. 481-518. 1978.

64. G.A. McMechan and M.J. Yedlin, Analysis of Dispersive Waves by Wave Field Transformation. Geophysics, Vol.46, No.6: p. 869-874. 1981.

65. T.K. Sarkar and O. Pereira, Using the Matrix Pencil Method to Estimate the Parameters of a Sum of Complex Exponentials. IEEE Transactions on Antennas and Propagation Magazine,, Vol.37, No.1: p. 48-55. 1995.

66. Y. Hua and T.K. Sarkar, Generalized Pencil-of-Function Method for Extracting Poles of an Em System from Its Transient Response. IEEE Transactions on Antennas and Propagation, Vol.37, No.2: p. 229-234. 1989.

67. T.L. Boullion and P.L. Odell, Generalized Inverse Matrices. John Wiley & Sons. p. 101. 1971

68. Y. Hua and T.K. Sarkar, On Svd for Estimating Generalized Eigenvalues of Singular Matrix Pencil in Noise. IEEE Transactions on Signal Processing, Vol.39, No.4: p. 892-900. 1991.

69. W. Sachse and Y.H. Pao, On the Determination of Phase and Group Velocities of Dispersive Waves in Solids. Journal of Applied Physics, Vol.49, No.8: p. 4320-4327. 1978.

181

70. J. Achenbach, Wave Propagation in Elastic Solids. Elsevier. 2012

71. L. Wang and F.G. Yuan, Group Velocity and Characteristic Wave Curves of Lamb Waves in Composites: Modeling and Experiments. Composites Science and Technology, Vol.67, No.7: p. 1370-1384. 2007.

72. G. Neau, Lamb Waves in Anisotropic Viscoelastic Plates. Study of the Wave Fronts and Attenuation. PhD, University of Bordeaux, Vol., 2003.

73. Y. Lu, R. Demirli, G. Cardoso, and J. Saniie. Chirplet Transform for Ultrasonic Signal Analysis and Nde Applications. in Ultrasonics Symposium, 2005 IEEE. IEEE.2005.

74. S. Mann and S. Haykin, The Chirplet Transform: Physical Considerations. Signal Processing, IEEE Transactions on, Vol.43, No.11: p. 2745-2761. 1995.

75. I. Solodov, J. Bai, S. Bekgulyan, and G. Busse, A Local Defect Resonance to Enhance Acoustic Wave-Defect Interaction in Ultrasonic Nondestructive Evaluation. Applied Physics Letters, Vol.99, No.21: p. 211911. 2011.

76. D.J. Inman, Engineering Vibration. 2014.

77. I.Y. Solodov, N. Krohn, and G. Busse, Can: An Example of Nonclassical Acoustic Nonlinearity in Solids. Ultrasonics, Vol.40, No.1: p. 621-625. 2002.

78. D. Donskoy, A. Sutin, and A. Ekimov, Nonlinear Acoustic Interaction on Contact Interfaces and Its Use for Nondestructive Testing. NDT & E International, Vol.34, No.4: p. 231-238. 2001.

79. F. Semperlotti, K.W. Wang, and E.C. Smith, Localization of a Breathing Crack Using Super-Harmonic Signals Due to System Nonlinearity. AIAA journal, Vol.47, No.9: p. 2076. 2009.

80. D. Dutta, H. Sohn, K.A. Harries, and P. Rizzo, A Nonlinear Acoustic Technique for Crack Detection in Metallic Structures. Structural Health Monitoring, Vol.8, No.3: p. 251-262. 2009.

81. F. Aymerich and W.J. Staszewski, Experimental Study of Impact-Damage Detection in Composite Laminates Using a Cross-Modulation Vibro-Acoustic Technique. Structural Health Monitoring, Vol.9, No.6: p. 541-553. 2010.

82. U. Polimeno and M. Meo, Detecting Barely Visible Impact Damage Detection on Aircraft Composites Structures. Composite structures, Vol.91, No.4: p. 398-402. 2009.

182

83. I. Solodov, J. Bai, S. Bekgulyan, and G. Busse. A Local Defect Resonance to Enhance Wave-Defect Interaction in Nonlinear Spectroscopy and Ultrasonic Thermography. in 18th World Conference on Nondestructive Testing.2012.

84. I. Solodov, Resonant Acoustic Nonlinearity of Defects for Highly-Efficient Nonlinear Nde. Journal of Nondestructive Evaluation, Vol.33, No.2: p. 252-262. 2014.

85. W.J. Staszewski, S. Mahzan, and R. Traynor, Health Monitoring of Aerospace Composite Structures – Active and Passive Approach. Composites Science and Technology, Vol.69, No.11–12: p. 1678-1685. 2009.

86. Z. Tian, L. Yu, and C. Leckey, Delamination Detection and Quantification on Laminated Composite Structures with Lamb Waves and Wavenumber Analysis. Journal of Intelligent Material Systems and Structures, Vol.26, No.13: p. 1723-1738. 2015.

87. M.S. Harb and F.G. Yuan, Barely Visible Impact Damage Imaging Using Non-Contact Air-Coupled Transducer/Laser Doppler Vibrometer System. Structural Health Monitoring, Vol.1475921716678921. 2016.

88. D. Dhital, J.R. Lee, C.Y. Park, and E. Flynn. Laser Excitation and Fully Non-Contact Sensing Ultrasonic Propagation Imaging System for Damage Evaluation. in SPIE Smart Structures and Materials+ Nondestructive Evaluation and Health Monitoring. International Society for Optics and Photonics.2012.

89. T. Tran, D. Simkins, S. Lim, D. Kelly, G. Pearce, B.G. Prusty, J. Gosse, and S. Christensen. Application of a Scalarstrain-Based Damage Onset Theory to the Failure of a Complex Composite Specimen. in 28th Congress of the International Council of the Aeronautical Sciences (ICAS2012).2012.

90. J.G. Proakis and D.G. Manolakis, Digital Signal Processing. 4th ed. Upper Saddle River, N.J.: Pearson Prentice Hall. xix, 1084 p. 2007

91. A.V. Oppenheim and A.S. Willsky, Signals and Systems. Prentice-Hall. 1997

183

APPENDICES

184

Appendix A

Nyquist Theorem and Sampling Theory

A.1. Introduction

This appendix explains the Nyquist’s criterion for signal sampling in time and space to set up

the parameters of experiment. Physically continuous signal for varying time (time) or particle

movement (space) in a moment is called “analog signal”. In contrast to continuous signal,

data stored discretely for sampling or digitizing in different time or different location with

evenly interval is called “digital signal”. Comparing to analog, digital signal on computer is

more convenient to do filters or edit, but analog signal is the true signal and full of

information. Pursuing to reconstruct digital signal to analog perfectly, signal sampling is very

important to be investigated. In the other words, most information in signal is lost with poor

sampling, which is call “aliasing”. In order not to cause signal aliasing, Nyquist criterion is

applied for the reconstruction of received signals. In addition, Nyquist criterion describes the

limitation of the measuring time and space intervals. In this appendix, sampling theorem is

introduced first and Nyquist theorem in time and space is conducted later.

A.2. Sampling Theorem

In order to describe the sampling theorem, it is a convenient way to represent the continuous

signal s(t) at regular intervals. A useful way is through a periodic impulse train p(t) as

sampling function as Dirac delta function and the regular period Δt as the sampling period. In

other term, the fundamental frequency of p(t), ωs = 2π/Δt is the angular sampling frequency.

185

p(t) (t nt)n

(A.1)

Therefore, a sampled function with the aid of sampling function is expressed as

s(n) s(t) p(t) s(t) (t nt)n

(A.2)

Because of the sampling property of the unit impulse, multiplying s(t) by a unit impulse

samples the value of the signal at the point which the impulse is located gets

s(t) (t nt) s(nt) (t nt) (A.3)

This is the sampled and discrete signal. For a periodic function, the sampling function p(t)

can be represented as

p(t) 1t

ein st

n

(A.4)

so the sampled signal becomes

s(n) s(t) 1t

ein st

n

(A.5)

From the multiplication property of Fourier transform, it is known as convolution so

that

S( ) 1

2S( ) 1

t 2 ( n s )

1t

S( n s )n

n

(A.6)

That Ŝ(ω) is a periodic function of ω consisting of a superposition of shifted S(ω) and scaled

by 1/Δt or fs.

This complex function represents Fourier transformation. This function distinguishes

amplitude and phase of every spectrum component. Absolute value of amplitude and n are

186

considered a real variable. The absolute values also represent the spectrum envelope.

Spectrum components frequencies are discrete values given as a multiplication of the

frequency and integer number n. That means the distance between two neighboring spectral

component is repeating by frequency ωs.

For the reconstruction, the figures in Figure A.1 are illustrated and can be clearly

pointed out how the signals form in time and frequency domain. The figure shows the

reconstruction of signal should be satisfied with ωs ≥ 2ωM. The basic result is lead to be as

Nyquist theorem and be stated later.

A.3. Signal Reconstruction in Time

The Nyquist theorem ensures analog signal is mapped into corresponding discrete signal with

all frequencies in the fundamental interval. Thus all the frequencies components of analog

signal are represented in sampled without ambiguity and distortion by using an ideal

interpolation method which is a way for analog-to-digital (A/D) conversion (quantization and

coding) and digital-to-analog (D/A) conversion (signal reconstruction).

From beginning, it is usually done by sampling the analog signal s(t) periodically

with a equally interval Δt to produce a discrete signal s(n), defined by

s(n) s(nt) (A.7)

The relationship describes the sampling process in time domain. The sampling frequency

must be selected large enough so that the sampling does not cause any loss of spectral

information. Consequently, finding the relationship between the spectra of analog signal s(t)

and discrete signal s(n) is investigated.

187

Figure A.1 Discrete-time s(n) and continuous-time (analog) s(t) signals are sampled in

scheme. (a) analog signal in general form s(t) is present in time domain and (b) frequency

domain. B presents the half of bandwidth and ωM is the largest frequency in signal. (c)

sampling function p(t) as Dirac delta function in time domain and (d) frequency domain. (e)

Discrete-time s(n) is transformed from analog signal in time domain and (f) frequency

domain. Ŝ(ω) is a periodic function and repeating with ωs.

188

In order to determine the relationship between analog signal s(t) and discrete signal

s(n), the periodic sampling imposes a relationship between independent variables t and n as

t nt nfs

(A.8)

where fs is sampling frequency (ωs = 2πfs). The relationship in time implies the

corresponding relationship between frequency variables. For distinguishing them, variables

ω and ϖ are presented in s(t) and s(n) respectively.

The discrete-time Fourier transform of a discrete-time sequence s(n) can represent the

sequence in term of the complex exponential einϖ with n points value. The spectrum of a

discrete-time signal s(n) is given by Fourier transform relation as

S( ) s(n)ein

n

(A.9)

and the sequence s(n) can be derived from its spectrum Ŝ(ϖ).

s(n) 1

2S( )e in d

(A.10)

which is called the inverse discrete-time Fourier transform. The equations constitute a

discrete-time Fourier transform pair for the sequence s(n).

Analog signals are continuous-time but digital signals are sampling by equal small

divided time. When processing a continuous-time signal using digital signal processing

techniques, it is necessary to convert signals into a sequence of numbers. Analog signal s(t)

can be expressed from its spectrum S(ω) by inverse Fourier transform

189

s(t) 1

2S( )e i t d

(A.11)

The correlation in discrete and analog time signal corresponds to frequency variables ϖ and

ω in Ŝ(ϖ) and S(ω). Based on the relationship between the independent variables n and t,

analog signal can be derived as

s(t) s(nt) 1

2S( )e in / fs d

(A.12)

For the relationship between discrete-time and analog signals as s(n) = s(nΔt) from

Eq. (A.7), the result is obtained

S( )e in d

S( )ein / fs d

(A.13)

The periodic sampling imposes a relationship between the discrete frequency variable ϖ and

the analog frequency variable ω and indicates

fs

(A.14)

It can be changed with the variables in above relationship

1fs

S( )ein / fs d fs

fs

S( )e in / fs d

(A.15)

The integration range of this integral can present an infinite number of intervals from width

−fsπ to +fsπ because the S(ω) is periodic. It becomes

190

S( )e in / fs d

S( )e in / fs d( L1/2)2 fs

( L1/2)2 fs

L

S( Lfs )e in / fs d

L

fs

fs(A.16)

In consequence, the result indicates

1fs

S( ) S( Lfs )L

(A.17)

or equivalently,

1fs

S( ) S[( L) fs]L

(A.18)

This equation means that the transformed analog signal consists of a periodic

repetition of spectrum fsS(ω) with period fs. When the sampling frequency fs is greater than

twice bandwidth of signal as Nyquist theorem, there is no aliasing and the discrete-time

signal is identical to the analog signal.

S( ) 1fs

S( ), s / 2 (A.19)

Sampling of an analog bandwidth limitation is explained by Figure A.2. By the Fourier

transform relationship from Eq. (A.9) with replacing ϖ to ω, that is

S( ) s(n)ein / fs

n

(A.20)

Respectively, the analog signal s(t) can be reconstructed by inverse Fourier transform of

S(ω). From Eq. (A.5) the analog signal s(t) can be reconstructed by inverse Fourier transform

of S(ω). Taking Eq. (A.19) and Eq. (A.20) into Eq. (A.11)

191

s(t) 1

2S( )e i t d

fs

fs

1

2[ 1

fs

S( )]e it d fs

fs

1

2[ 1

fs

s(n)ein / fs

n

]e i t d fs

fs

1

2 fs

s(n)n

e i (tnn/ fs ) d fs

fs

1

2 fs

s(n) e i (tn/ fs )

i(t n / fs ) fsfs

n

s(n) sin[( / t)(t nt)]( / t)(t nt)n

(A.21)

The reconstruction formula for the analog signal s(t) is called the ideal interpolation

formula which forms the basis for sampling theorem. If the Δt is going to be infinitesimal,

discrete signal values can be summed and recovered to analog signal. The reconstruction

formula involves the function

g(t) sin[( / t)t]

( / t)t (A.22)

appropriately shifted by nΔt and weighted by the corresponding sampling s(n). This is called

ideal interpolation formula for reconstructing s(t) from its samples s(n) and g(t). Substituting

Eq. (A.22) into the Eq. (A.21), the reconstructed signal s(t) becomes

s(t) s(n)g(t nt)n

(A.23)

Therefore, this Eq. (A.23) is that the analog signal is constructed from digitalized signal and

compares to Eq. (A.2) which is digitalized from analog signal.

192

Figure A.2 Signal is reconstructed with ideal interpolation using sinc function. (a) analog

signal s(t); (b) pulse train of samples s(n); (c) superposition of ideal interpolation using sinc

function to replace pulse train of samples.

A.4. Nyquist Theorem in time

The Nyquist theorem [90, 91] states that a time-varying signal is periodically sampled at a

rate twice the highest frequency sinusoidal signal, and then the original time-varying signal

can be exactly recovered from the periodic samples. A discrete-time sinusoidal signal are

characterized by one property which describes that its frequency can be separated by an

integer multiple of 2π are identical. Any sequence from a sinusoid has a range for frequency.

193

(A.24)

From the relationship between the frequencies in analog and discrete-time signal ω

and ϖ, the range can be expressed as

fs

(A.25)

The result means that any frequency in this range is identical. In order to avoid the

ambiguities result from aliasing, a sufficiently high sampling frequency should be selected.

The analog signal contains a variety of frequencies. To avoid the problem of aliasing for each

frequency ωs must be selected as

s 2 M (A.26)

or equivalently

fs 2 fmax (A.27)

where fmax is the largest frequency component in an analog signal and ωM is the largest

angular frequency.

Aliasing occurs when the sampling frequency is lower than twice of maximum

frequency. The spectrum of sampled signal avoids the aliasing when the sampling rate is

greater than or equal to two times the highest frequency in the signal ωs ≥ 2ωM. It causes the

distortion or artifact when the sampled signal is different from the original signal. There is an

easy way of illustrating the aliasing effect when higher frequency is sampled by less points as

lower frequency as Figure A.3. From Figure A.3, it also can clearly seen that aliasing

happens when ωs < 2ωM. The transformed signals are overlapped each other. Then the

inverse signals in time domain are not the same.

194

Figure A.3 Signal reconstruction while aliasing happens. The bandwidth B is related to the

largest frequency in the signal. When sampling ωs < 2ωM, it causes transformed spectrum

overlapping and the reconstructed signal is distorted.

Figure A.4 Example for insufficient samples: A higher frequency signal ω = 20π (blue) is

sampled at too low rate (circle) and it looks like a lower frequency signal ω = 4π (red).

195

A.5. Nyquist Theorem in Space

A data sampling parameter is used with the same meaning as Nyquist Frequency because of

the conceptual analogy between Frequency as a time-domain characteristic

and wavelength as a space-domain characteristic. For Nyquist criteria, the distance for

receiver spacing corresponds to half of the cycle length of the highest spatial frequency

passing through the detection spacing. It is called the Nyquist wavelength. Alternatively,

spatial sampling interval is defined as equal or less than half of wavelength as space-domain

characteristic as

d min / 2 (A.28)

where d is defined as sampling spatial distance. While the detection spacing is greater than

this threshold, waveform is indistinguishable and aliased. Aliasing can be reduced if

sampling distance is made smaller.

A.6. Spatial Signal Reconstruction for Measurements

Based on the Nyquist theorem in space, spatial signal as wave movements along propagating

direction on the material can be reconstructed with appropriate measuring. From signal

reconstruction in time and above, Δx can be seen as sampling spatial distance between

measurements equally. m refers to numbers of measurements in space. After transforming,

wavenumber k means the spatial frequency and ks as sampling wavenumber. Therefore, the

relationship between wave shape s(x) and discrete points s(m) imposes a relation between

variables x and m as

196

x mx mks

(A.29)

The reconstruction equation as ideal interpolation formula in space is defined as

s(x) s(m) sin[( / x)(x mx)]( / x)(x mx)m0

(A.30)

Reconstruction signal in space shows in scheme as Figure A.5. The measurements start from

the first position as x = 0 as m = 0.

According to Nyquist theorem in space, spatial sampling interval should be equal or

less than half wavelength. In terms of Δx, the criterion is defined with phase velocity cp and

excited frequency fe while wave propagating as

x cp

2 fe

(A.31)

This implies that the measurements are constrained to the excited frequency for perfectly

reconstructing the wave shape on the material.

A.7. Example

Assume that a non-dispersive wave propagates in sinusoidal form in a plate with consistent

frequency f (Hz). A linear array consisting of n sensors with equal spacing is mounted on the

plate Δx (mm). The sinusoidal wave travels in velocity C (m/s).

In order to completely detect the behavior and reconstruct the signal from the

excitation source, the velocity C = 3000 (m/s) and the excitation frequency f = 100 kHz. The

197

wavelength can be obtained as 30 mm and the interval space should be less than x < 15 mm.

An example is plotted in the Figure A.6.

Figure A.5 Measurements s(m) and wave movements s(x) are sampled in scheme. (a) wave

shape in general form s(x) is present and (b) wavenumber domain. Bk presents the half of

bandwidth and kM is the largest wavenumber. (c) Measurements s(m) is transformed from

analog signal in space and (d) wavenumber domain. Ŝ(k) is a periodic function and repeating

with ks.

198

Figure A.6 Discrete-time signals reconstruct the analog signals. Line with dots indicates the

spectrum of discrete time signals. Red line represents the Nyquist’s theorem to show the least

sampling rate constructing the original signal.

199

Appendix B

Operations for Chirplet Transform

For wavelet, a generalized three-dimensional time-frequency representation is written by

introducing the time and frequency operators Tt1 and Fω1 and the scale operator Ss as:

Time shifting

t1h(t) h(t t1) (B.1)

Tt1H ( ) e t1 H ( ) (B.2)

Frequency shifting

F1h(t) ei1th(t) (B.3)

F1H ( ) H ( 1) (B.4)

Scaling

Ssh(t) 1s

h( ts) (B.5)

SsH ( ) sH (s ) (B.6)

where h(t) is defined on smoothing kernels (window function) resulting in smoothed versions

of the exact energy distribution. For Morlet or Gabor function in wavelet transform as mother

wavelet function, the Gaussian distribution is used as window function:

h(t) 14

exp(12

t2 i0t) (B.7)

For Gabor function, it can be simplified as,

200

g (t) 14

0

ei0te

(0 / )2 t2

2 h( t / 0

) Ssh(t) (B.8)

Gabor function has another scale to adjust the signal from the center frequency.

Now Mann and Haykin [74] and Kuttig et al. [21, 22] uses his framework and

expend the idea by using two additional operators to formulate the chirp let transform,

frequency and time shear.

1) Shearing in frequency: The time-frequency kernel is multiplied by a harmonic

function of linearly changing frequency (like a “chirp”) Qq. This modulation has a

shear in the frequency direction with a slope 1/q.

Qqh(t) h(t)exp(i q2

t2 ) (B.9)

Qq H ( ) (iq)1/2 exp(i 2q

2 ) H ( ) (B.10)

It corresponds to the time-frequency domain as tilted as expressed

inst (t) qt (B.11)

2) Shearing in time: Since the corresponding multiplication of a chirp function in the

frequency domain constitutes the dual operation to the frequency shear, the effect to

the signal is that a semi axis of the time-frequency atom has the slope p.

Pph(t) (ip)1/2 exp[i( 12 p

t2 )] h(t) (B.12)

Pp H ( ) exp[i( p2

2 )]H ( ) F1h(t) ei1th(t) (B.13)

201

Pp has rotating effect to the time by an angle α = tan-1(p). The time shift influences its

group delay as expressed as tilting in time-frequency domain

g ( ) p (B.14)

Using the five operators, time-frequency kernel is translated, scaled, and sheared in the time-

frequency domain. The function is presented with five operators as Eq. (4.4.2) and the kernel

is represented as Eq. (4.4.3) shown in the following figures.

h(t)

(a)

(b)

202

(c)

(d)

(e)

203

(f)

(g)

Figure B.1 Visualization for the operators of mother chirplet. (a) original window function

(b) shifting in time (c) shifting in frequency (d) scaling (e) chirp rate in time domain (f) chirp

rate in frequency domain (g) all operators applied.

Documents

ABSTRACT Ultrasonic Damage Imaging. (Under the direction