Post on 22-Oct-2021
Practical Aspects of Assessing Nonlinear Ultrasonic Response of
Cyclically Load 7075-T6 Aluminum
Byungseok Yoo
Thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE
IN
ENGINEERING MECHANICS
J. C. Duke, Jr., Ph.D., Chair
M. R. Hajj, Ph.D.
R. D. Kriz, Ph.D.
December 12, 2006 Blacksburg, Virginia
Keywords: Ultrasonic, NDE, Nonlinearity Parameter, Power Spectrum, Bispectrum, Bicoherence Spectrum
Practical Aspects of Assessing Nonlinear Ultrasonic Response of
Cyclically Load 7075-T6 Aluminum
Byungseok Yoo
ABSTRACT
The ultrasonic NDE technique to characterize the ultrasonic nonlinear response of the
cyclically load 7075-T6 aluminum is described in this thesis. In order to estimate the
nonlinear relation of the ultrasonic waves due to material fatigue damage or degradation,
the spectral analysis techniques such as the power spectrum, bispectrum, and bicoherence
spectrum are applied. The ultrasonic nonlinearity parameters by Cantrell and Jhang are
introduced and presented as a function of the material fatigue growth, the number of fatigue
cycles. This thesis presents the effectiveness of the bispectral analysis for evaluating the
nonlinear aspects of the ultrasonic wave propagation. The results show that the nonlinearity
parameters by Cantrell and Jhang are responsive to the output amplitude of the received
signal and vary for the various materials, and independent of the input frequency and the
ultrasonic wave propagation distance. By using the bispectral analysis tools, particularly the
bicoherence spectrum, the increase of the coupling levels between the fundamental, its
harmonic, and subharmonic frequency components is presented as the number of fatigue
cycles is increased. This thesis suggests that the application of the bicoherence spectrum
based on the nonlinear wave coupling relations be more effective for estimating the level of
the material fatigue life.
ii
ACKNOWLEDGEMENTS
I would like to thank Dr. John C. Duke, Jr. for his guidance, support, and patience as my
committee chair throughout this study. I have gained a lot from his experience and
knowledge. I wish to acknowledge the support from the department of Engineering Science
and Mechanics.
I wish to thank Dr. Muhammad Hajj for his assistance and advice. I would like to
recognize and thank Robert A. Simonds for his hours behind the controls of MTS machines.
I am grateful to Dr. Ron Kriz for taking time to join my committee. I also thank friends and
colleagues who have listened to me for their support and encouragement.
I would like to dedicate this thesis to my wife, Soo Ja Kim, for her love and faith in me
to achieve my goals, to my parents who show their unlimited love and support throughout
my life, and to my little angel, Connie Somyoung Yoo, who makes me smile. Thank you all.
iii
TABLE OF CONTENTS
ABSTRACT ………………………………………………………………………………………………. ii
ACKNOWLEDGEMENTS ………………………………….………………………………………… iii
LIST OF CONTENTS ………………………………………………………………………………… iv
LIST OF TABLES ……………………………………………………………………………………. vii
LIST OF FIGURES …………………………………………………………………………………... viii
I INTRODUCTION AND LITERATURE REVIEW …………………………………………... 1
1.1 Introduction and Literature Review ……………………………………………………. 1
II THEORETICAL BACKGROUND …………………………………..……………………….. 5
2.1 Physical Theory of Wave Propagation in Solids …………………………..………….. 5
2.1.1 Constitutive Equations of Motion in Three Dimensions ………..………….. 5 2.1.2 General Linear Wave Equation ………………………………………..………. 7
2.1.3 General Nonlinear Wave Equation ………………………..…………………... 8
2.1.4 Simplified Nonlinear Wave Equation ……………………….………………... 8
2.1.5 Solution of Nonlinear Wave Equation ………….………………..…………... 9
2.2 Spectral Analysis for Nonlinear Ultrasonic Response …………………………….... 11
2.2.1 General Background of Fourier Transform ………………………...………... 11
2.2.2 Power Spectrum and Bispectrum ……………………………………..……... 12
2.2.3 Bicoherence Spectrum ……………………………………………………….… 16
2.3 Nonlinearity Parameters ………………………………………………….………………. 18
2.3.1 Nonlinearity Parameter by Cantrell …………………………….…………… 18
2.3.2 Nonlinearity Parameter by Jhang …………………………………..………... 19
III EXPERIMENTAL BACKGROUND …………………….………………………………… 21
3.1 Ultrasonic Data Acquisition Setup ……………………………………………………. 21
iv
3.1.1 Ultrasonic Transducer ……..………………………………….………………... 21
3.1.2 Ultrasonic Hardware and System Setup ……………………………………. 22
3.2 Ultrasonic Measurement Testing ……………………………………………....………... 24
3.3 System Stability Examination ………………………………………………...…….…… 25
IV EXPERIMENTAL STUDIES ………………………………………………………………... 27
4.1 Experimental Setup ……………………………………………………………………… 27
4.2 Characterization of Response of Data Acquisition System ……………………. 29
4.3 Signal Data Reproducibility – Measurement Variability Test …………..………… 32
4.4 Input Signal Characterization …………………………………………………….……. 34
4.4.1 Introduction …………………………………………………….………………… 34
4.4.2 Test Setup and Experimentation Result ………………………….………… 34
4.5 System Characterization using Fused Quartz Sample ………….……….………... 40
4.5.1 Introduction ………………………………………………….…………………… 40
4.5.2 Test Setup and Experimentation Result ..………………………...………... 40
4.5.3 Summary of System Characterization using Fused Quartz Sample …….. 50
4.6 Nonlinear Response of Various Specimens ……………………..……………...……... 51
4.6.1 Introduction ………………………………………………………..……………… 51
4.6.2 Test Setup and Experimentation Result …………………………...………... 51
4.6.3 Summary of Nonlinear Response of Various Specimens ………………... 60
4.7 Ultrasonic Response Test of Cyclic Loading 7075-T6 Aluminum ……………… 61
4.7.1 Specimen Preparation ………………………………………………….………. 61
4.7.1.1 Specimen Description ……………………………………..………….. 61
4.7.1.2 Solution Heat Treatment for Specimen …….……………………... 62
4.7.2 Test Setup and Experimentation …………………………….…………………. 63
v
4.7.2.1 Cyclic Loading System and Specimen Fatiguing ……….……… 63
4.7.2.2 Ultrasonic Data Acquisition …………………………………………. 65
4.7.3 Result and Discussion ……………………………………………………………. 66
4.7.4 Summary of Ultrasonic Response Test of Cyclic Loading 7075-T6 Aluminum ………………………………... 72
V CONCLUSIONS …………………………………………………………………………………. 73
5.1 Thesis Summary ……………………………………………………………………………. 73
5.2 Conclusions and Future Work Recommendations …………………………………. 75
REFERENCES …………………………………………………………………………………………... 77
VITA ………………………………………………………………………………………………………... 79
vi
LIST OF TABLES Table 3.1 Description of the HP 3314A Function Generator …………………………. 22 Table 3.2 Details of the GageScope CS12100 A/D Card …………………………….. 23 Table 3.3 Settings for the GageScope Software ……………………………………….. 23 Table 4.1 Testing Parameters Table Format ………………………………………….… 28 Table 4.2 Testing Parameters for Input Signal Characterization-Voltage ………….…... 34 Table 4.3 Spectral Analysis Worksheet(INPUT and OUTPUT Signals) ………….…… 34 Table 4.4 Testing Parameters for Input Signal Characterization-Frequency …….…….. 37 Table 4.5 Spectral Analysis Worksheet(5 Volts Input Signal) ………………….……… 38 Table 4.6 Testing Parameters for System Characterization-Frequency
(5MHz Receiver) …………………………………………………….……… 40 Table 4.7 Spectral Analysis Worksheet(5 MHz Receiver) …………………….……….. 41 Table 4.8 Testing Parameters for System Characterization-Frequency
(10MHz Receiver) …………………………………………………..………..43 Table 4.9 Spectral Analysis Worksheet(10 MHz Receiver) …………………………… 44 Table 4.10 Testing Parameters for System Characterization-Voltage
(10MHz Receiver) ......................................................................................... 48 Table 4.11 Testing Parameters for Nonlinear Response of Various Specimens .………. 51 Table 4.12 Acoustic Properties of Test Materials …………………………….………... 56 Table 4.13 Mechanical Properties for 7075-T6 Aluminum ………………….………… 62 Table 4.14 Acoustic Properties for 7075-T6 Aluminum …………………….………… 62 Table 4.15 Testing Parameters for Ultrasonic Response Test of
Cyclic Loading Aluminum ………………………………………………… 66 Table 4.16 Spectral Analysis Calculation Table using MATLAB codes ….…………… 67
vii
LIST OF FIGURES Fig. 2.1 Higher Order Spectra Classification Map by Nikias and Mendel …………….. 12 Fig. 2.2 Linear and Quadratic Parts in a Parallel Structure …………………………….. 15 Fig. 2.3 Region of the Bispectrum Computation by Kim and Power ………………... 17 Fig. 3.1 Diagram of Ultrasonic Measurement System …………………………………. 24 Fig. 3.2 Input Signal, Total Received and Initial Disturbance Signal ………………….. 25 Fig. 3.3 System Stability Test ………………………………………………………….. 26 Fig. 4.1 Diagram of Experimental Setup ………………………………………………. 27 Fig. 4.2 Three Types of Experimental Setups …………………………………………. 29 Fig. 4.3 HP Function Generator and Initial Disturbance Capturing by GageScope …… 29 Fig. 4.4 Output Amplitudes depending on Input Signal Amplitude Change …………. 30 Fig. 4.5 Fundamental vs. Second Harmonic Amplitude Ratio Plot, A2/A1 ……………. 30 Fig. 4.6 Nonlinearity Slope Plot, A2/A1
2 ………………………………………………. 31 Fig. 4.7 Nonlinearity Parameters by Jhang in Log Scale Plot …………………………. 32 Fig. 4.8 Nonlinearity Parameters by Cantrell in Log Scale Plot ………………………. 33 Fig. 4.9 Power Spectrum Plot for Input Signal with Input Voltage Decrease ……..…... 35 Fig. 4.10 Power Spectrum Plot for 10 MHz Receiver with Input Voltage Decrease …. 35 Fig. 4.11 Input Signal Amplitude vs. Output Signal Amplitude Plot,
Pi(f1)1/2 vs. Po(f1)1/2 and Pi(f2)1/2 vs. Po(f2)1/2 ………………………………... 36 Fig. 4.12 Total Power Spectrum Change with Input Frequency Increase
(5 Volts Input Signal) ……………………………………………………….. 37 Fig. 4.13 Power Spectrum Change with Input Frequency Increase
(5 Volts Input Signal) ……………………………………………………….. 38 Fig. 4.14 Nonlinearity Parameter Change depending on Input Frequency Increase
(5 Volts Input Signal) ……………………………………………………….. 39 Fig. 4.15 Total Power Spectrum Change with Input Frequency Increase
(5 MHz Receiver) ……………………………………………………………. 41 Fig. 4.16 Power Spectrum Change with Input Frequency Increase(5 MHz Receiver) … 42 Fig. 4.17 Total Power Spectrum Change with Input Frequency Increase
(10 MHz Receiver) …………………………………………………………... 44 Fig. 4.18 Power Spectrum Change with Input Frequency Increase(10 MHz Receiver) .. 45
viii
Fig. 4.19 Nonlinearity Parameter Change depending on Input Frequency Increase (5MHz Receiver and10 MHz Receiver) ……………………………………... 46
Fig. 4.20 Nonlinearity Parameter Change Comparison (10 MHz Receiver & 5 Volts Input Signal) ………………………………….. 47
Fig. 4.21 Total Power Spectrum Change with Input Voltage Decrease (10 MHz Receiver) …………………………………………………………... 48
Fig. 4.22 Nonlinearity Parameter Change with Output Amplitude Change (3 Sets of Fused Quartz) ……………………………………………………... 49
Fig. 4.23 Difference Between Input Signal Voltage Plot and Output Amplitude Plot …. 52 Fig. 4.24 Cantrell Nonlinearity Parameter Change with Output Amplitude Change ….. 53 Fig. 4.25 Cantrell Nonlinearity Parameter Change for 7075-T6 Al Specimens ……….. 54 Fig. 4.26 Cantrell Nonlinearity Parameter Change for Various Material Specimens ….. 55 Fig. 4.27 Jhang Nonlinearity Parameter Change with Output Amplitude Change …….. 57 Fig. 4.28 Jhang Nonlinearity Parameter Change for Various Material Specimens …….. 58 Fig. 4.29 Jhang Nonlinearity Parameter Change for Copper Alloy and Fused Quartz … 59 Fig. 4.30 Schematic Diagram of the Specimen ………………………………………… 62 Fig. 4.31 Specimen Gripping and Cyclic Loading Configuration ……………………... 64 Fig. 4.32 Transducer Positioning ………………………………………………………. 65 Fig. 4.33 Power Spectrum Plots for Data Collected Every 5000 ………………………. 68 Fig. 4.34 Bicoherence Spectrum Colormap and Contour Plots ………………………... 69 Fig. 4.35 Bicoherence Spectrum Contour Plot only for [0.4(blue) 0.6(green) 0.8(red)].. 70 Fig. 4.36 Nonlinearity Parameter Changes by Cantrell and Jhang …………………….. 71
ix
CHAPTER I
INTRODUCTION AND LITERATURE REVIEW
1.1 Introduction and Literature Review
Fatigue of materials is a very common problem in almost every aspect of industry.
Components of vehicles, large structures and factory machines are subjected to cyclic
loadings. This type of loading is tremendously dangerous to material as the smallest of
cracks or scratches can grow to cause physical damage in the whole material. A defect such
as a microcrack or simply the weakest region of the material causes localized deformation,
which leads to a microscopic crack if one is not initially present. After some additional time
and loading the microcrack grows in size, very slowly at first, and then at an increasing rate
as it develops into a macroscopic crack or damage that can be detected. Finally, the crack
leads to failure of the component. Moving vehicles incur cyclic loads, including the forces
necessary to operate the vehicle, vibrations, bouncing and wind turbulence, which make
fatigue a critical design aspect. Consequently, it is important to detect a microcrack or
degradation of strength of materials at the early stage in the fatigue life. Nondestructive
Evaluation(NDE) using ultrasonic waves has been commonly used to inspect the damage of
materials. The classical ultrasonic NDE techniques, however, are more sensitive to gross
forms of damage than microcracks or degradation of materials, which can be present in the
preservice materials. In order to avoid the sudden failure of structural components due to
1
fatigue, periodic inspections using ultrasonic NDE approaches have been conducted widely.
Christodoulou and Larsen[1] note that each aircraft components undergo fatigue cycles
to the components of the structure during flight. The major components which were retired
when they reach their book life (a very low probability of failure) would actually have
remaining material lifetime or capability. However, it is impractical to evaluate and predict
the remaining lifetime of materials. As the crack size increases, the material variability and
predictive uncertainty decreases. Unless a microcrack reaches a detectable size, the
conventional ultrasonic NDE approaches cannot help detect variations of material fatigue
process during the period prior to the formation of a microcrack. Christodoulou and
Larsen[1] also note that a safe-life approach to reduce the uncertainty in material life
prediction is conducted by discarding 1000 components to ensure the removal of one
component which has a small crack. Therefore, it is necessary to develop the reliable
damage prognosis techniques for the materials under fatigue to enable the management of
materials and reduce optional inspection costs. Consequently, a new application of the
ultrasonic NDE method, which would monitor the material fatigue stage to improve
material life prediction, is needed.
Hajj, Duke, and Yoo[2] note that the development and density of vein structures,
persistent slip bands, and microcrack nucleation characterizes the initial stage of the fatigue
life. These substructures would substantially distort ultrasonic waves propagating through
the fatigued material. Cantrell and Yost[3] and Cantrell[4] suggest that a material
nonlinearity parameter physically related to a quantitative measure of the wave distortion
can be used as a new ultrasonic NDE method to characterize the extent of the fatigue
process in materials. Cantrell and Yost[3] note that fatigue tests under uniaxial load were
2
conducted for 3 cycles, 10 kcycles, and 100 kcycles in three different specimens,
respectively, of an aluminum alloy 2024-T4. The material nonlinearity parameter was made
from the Fourier spectral amplitude measurements of the fundamental and second harmonic
signals and was plotted as a function of the number of fatigue cycles. The results showed
that the nonlinearity parameter increased as increasing fatigue cycles. On the other hand,
the results showed that the nonlinearity parameter of the specimen underwent 3 cycles
suddenly increases from the virgin state of the unfatigued specimen, but the difference in
the nonlinearity parameter between 10 kcycles and 100 kcycles is relatively small.
Jhang and Kim[5, 6] proposed that the bispectrum method can be used to estimate the
material nonlinearity parameter. Since the bispectrum approach can eliminate the Gaussian
noise effects to the signals and detect the coupling relation between the fundamental and
second harmonic frequency components, the bispectrum method can be considered as an
effective method to measure the material nonlinearity parameter. The nonlinearity
parameter was defined with the absolute bispectrum measurement, between the
fundamental and second harmonic components, divided by the squared power spectrum
measurement of the fundamental component. The results clearly showed that the quantities
of the second harmonic components and the material nonlinearity parameter increased as
the number of fatigue cycles increases. The results also showed that although the magnitude
of second harmonic component increased as the input signal voltage increases, the
nonlinearity parameter did not increase due to the increase of the magnitude of the
fundamental component. The increase trend of the nonlinearity parameter by Jhang agreed
with the nonlinearity parameter by Cantrell[3]. In addition, the difference of the
nonlinearity parameters between 1 kcycles and 100 kcycles is small. This aspect also
3
agreed with the aspect of Cantrell[3]. The previous experiments by Cantrell and Jhang
clearly show the limitation of both nonlinearity parameters. Although both nonlinearity
parameters can be used to characterize the early stage of the material fatigue life, those can
not distinguish the late stage of the material fatigue life.
Hajj, Duke, and Yoo[2, 7] note that the higher spectral analysis techniques were used to
estimate the overall nonlinear interaction of low and high frequency components. The
bicoherence, normalized bispectrum, approach which varies between zero and one, was
suggested to characterize the phase relationship among the frequency components.
Moreover, a single specimen rather than the use of multiple specimens was used to monitor
the level of the fatigue process[7, 8]. The use of a single specimen can reduce the
nonlinearity parameter variation due to the material variability. The results showed that the
power spectrum can be used to estimate the energy distribution of the received signal. The
results also showed that the level of the bicoherence between the fundamental and the
second harmonic components increased as more cycles were applied to the specimen. Hajj,
Duke, and Yoo[2, 7] suggested the use of the bicoherence contour plot to estimate the
nonlinear couplings among the frequency components.
4
CHAPTER II
THEORETICAL BACKGROUND
2.1 Physical Theory of Wave Propagation in Solids
2.1.1 Constitutive Equations of Motion in Three Dimensions
An arbitrary body with non-zero volume is considered,
ii admF ⋅=
iB
11σ ′
13σ
11σ
12σ
12σ ′
13σ ′
1x
2x
3x
Fig 2.1 Force and Stress Components at an Arbitrary Point
where is body force per unit mass and is an external force on the body, and iB iF ijσ is
the stress tensor components on the body. If and are not equal to zero,
should hold.
i i 1111' σσ ≠B F
5
For an equilibrium state, should hold and the equilibrium equation is defined
by,
0=∑ iF
( ) ( )( ) iiii
iiii
admdxdxdxBdxdxdxdx
dxdxdxdxdxdxdxdx
⋅=+−+
−+−
32121321'3
31231'232132
'1
σσ
σσσσ (2.1.1)
where 3,2,1=i
321 dxdxdxdm ⋅= ρ But , where ρ is the density of the material and the acceleration,
, where is displacement vector. iu22 / tua ii ∂∂=
With Taylor series expansion,
2133
321321
'3
3122
231231
'2
3211
132132
'1
..
..
..
dxdxOHdxx
dxdxdxdx
dxdxOHdxx
dxdxdxdx
dxdxOHdxx
dxdxdxdx
iii
iii
iii
⎟⎟⎠
⎞⎜⎜⎝
⎛+
∂∂
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛+
∂∂
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛+
∂∂
+=
σσσ
σσσ
σσσ
(2.1.2)
Neglecting higher order terms in Taylor series and substituting Eq.(2.1.2) into Eq.(2.1.1),
the equilibrium equations became,
3212
2
3213213
3
2
2
1
1 dxdxdxtu
dxdxdxBdxdxdxxxx
ii
iii
∂∂⋅=+⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
ρσσσ
(2.1.3)
Assuming there is no body force, we can finally obtain the three dimensional equations
of motion as follows,
2
2
3
3
2
2
1
1
tu
xxxiiii
∂∂
=∂∂
+∂∂
+∂∂
ρσσσ
, (2.1.4)
6
2.1.2 General Linear Wave Equation
The propagation of an elastic wave in a solid bar is considered. The three dimensional
equations of motion in Lagrangian coordinates are defined by
j
iji
xtu
∂
∂=
∂∂ σ
ρ 2
2
, (2.1.5)
with Einstein summation notation, where are the Lagrangian coordinates, are the
components of the wave displacement vector,
iujx
ρ is the mass density of the material, and
are the components of the stress tensor. ijσ
Assuming no initial stress and that the material is an isotropic solid, ijσ in terms of
displacement gradients was expanded as
(2.1.6) lkijklij uC ,=σ
where it the linear elastic coefficient of the material and ijklC )//(2/1, kllklk xuxuu +=
are the displacement gradients.
Substituting Eq.(2.1.5) into Eq.(2.1.6), we can obtain the general linear compressible wave
equations as follows
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
∂∂
=∂∂
k
l
l
kijkl
j
i
xu
xu
Cxt
u21
2
2
ρ (2.1.7)
Assuming that the material is homogeneous, which means is not a function of
position and could still vary with temperature or strain rate, the general linear wave
equations are defined by
ijklC
7
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂∂
+∂∂
∂=
∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
∂∂
=∂∂
kj
l
lj
kijkl
i
k
l
l
k
jijkl
i
xxu
xxu
Ctu
xu
xu
xC
tu
22
2
2
2
2
21
,21
ρ
ρ
(2.1.8)
2.1.3 General Nonlinear Wave Equation
Assuming no initial stress, nonlinear stress-strain relations is given by
L++= nmlkijklmnlkijklij uuAuA ,,, 21σ , (2.1.9)
where and are the propagation coefficients[9, 10]. ijklA ijklmnA
The relationship between the propagation coefficients and the elastic coefficients under
no initial stress condition was shown by Wallace[11] to be
(2.1.10) ijklmnimjknlkmijnlikjlmnijklmn
ijklijkl
CCCCA
CA
+++=
=
δδδ
,
where and are the second and third order elastic coefficients[9, 12].
Substituting Eq.(2.1.9) into Eq.(2.1.5), we can obtain the general nonlinear wave equations.
ijklC ijklmnC
2.1.4 Simplified Nonlinear Wave Equation
For the purpose of the present research, however, it is sufficient to consider the simple
one-dimensional case of the propagation of an elastic wave in the x direction. We assume
that the relation between the stress perturbation and elastic strain may be expanded in the
nonlinear Hooke’s law form
8
L3
32
21 31
21
xxxxxxxx eEeEeE ++=σ , (2.1.11)
where xxσ is the unidirectional stress component, is the displacement gradient
, are the coefficients of the higher order terms in
xxe
xu ∂∂ / xu ∂∂ /nE , which are called
Huang coefficients. Moreover the simple equation of motion can be given by
xttxu xx
∂∂
=∂
∂ σρ 2
2 ),( , (2.1.12)
xwhere ρ is the mass density of the material, is the propagation distance of ultrasonic
wave, xxσ is the stress, and is the wave displacement. Substituting Eq.(2.1.11)
into Eq.(2.1.12) and neglecting cubic and higher order terms of displacement gradient, and
rearranging them, we can obtain the nonlinear equation of motion for displacement
as follows
),( txu
),( txu
2
22
2
22
2
2
1
222
22
2
2
2
2
22
2
12
2
,
xu
xuC
xuC
xu
xu
EEC
xuC
tu
xu
xuE
xuE
tu
∂∂
∂∂
−∂∂
=∂∂
∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛−−
∂∂
=∂∂
∂∂
∂∂
+∂∂
=∂∂
β
ρ (2.1.13)
ρ/1EC =where is the longitudinal wave velocity of the material,C is always positive
and β is a nonlinear parameter terminology which is subjected to discuss later.
2.1.5 Solution of Nonlinear Wave Equation
The solution of the nonlinear wave equation was discussed with distortion and the
generation of higher harmonics[13].
In order to apply a perturbation method to solve for , Eq.(2.1.13) is recalled, ),( txu
2
2
22
2
12
2
xu
xuE
xuE
tu
∂∂
∂∂
+∂∂
=∂∂ρ (2.1.14)
9
and assuming that , and the solution of Eq.(2.1.14) can be regarded as the
combination
12 EE <<
10 uuu += (2.1.15)
where and is the solution of that part of Eq.(2.1.14) for which 01 uu << 0u
020
2
120
2
=∂∂
−∂∂
xuE
tuρ (2.1.16)
and is a perturbation arising from the remaining part of Eq.(2.1.14) indicated by , 1u p
pxuE
tu
=∂∂
−∂∂
2
2
12
2
ρ (2.1.17)
2
2
2 xu
xuEp∂∂
∂∂
= (2.1.18) where
Substituting Eq.(2.1.15) into Eq.(2.1.14), then Eq.(2.1.14) becomes
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
⋅⎟⎠⎞
⎜⎝⎛
∂∂
+∂∂
+∂∂
+∂∂
=∂∂
+∂∂
21
2
20
210
221
2
120
2
121
2
20
2
xu
xu
xu
xuE
xuE
xuE
tu
tu ρρ (2.1.19)
Due to Eq.(2.1.16), Eq.(2.1.19) becomes
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
+∂∂
∂∂
+∂∂
∂∂
+∂∂
∂∂
+∂∂
=∂∂
21
20
20
21
21
21
20
20
221
2
121
2
xu
xu
xu
xu
xu
xu
xu
xuE
xuE
tuρ (2.1.20)
20
21
xu
xu
∂∂
∂∂
21
21
xu
xu
∂∂
∂∂
21
20
xu
xu
∂∂
∂∂Since the terms of 01 uu << , , and may be neglected in
comparison with 20
20
xu
xu
∂∂
∂∂
. Eq.(2.1.20) can be reduced to
20
20
221
2
121
2
xu
xuE
xuE
tu
∂∂
∂∂
+∂∂
=∂∂ρ (2.1.21)
Since )sin(10 kxtAu −= ω , the perturbation can be defined by 1u
)(2cos81 2
12
1
21 kxtxAk
EEu −⎟⎟
⎠
⎞⎜⎜⎝
⎛−= ω (2.1.22)
10
Therefore the solution of nonlinear wave equation can be given by[13-15]
)(2cos81)sin(),(
),(2cos81)sin(),(
2
221
1
21
21
2
1
21
kxtC
xAEE
kxtAtxu
kxtxAkEEkxtAtxu
−⎟⎟⎠
⎞⎜⎜⎝
⎛−−=
−⎟⎟⎠
⎞⎜⎜⎝
⎛−−=
ωω
ω
ωω (2.1.23)
where is the phase velocity, C ω is the angular frequency, and is the wave vector. k
2.2 Spectral Analysis for Nonlinear Ultrasonic Response
Kim and Powers[16] and Hajj[17] note that the bispectrum is a very useful analysis
technique in experimental research of nonlinear wave couplings and the bicoherence
spectrum, which is normalized bispectrum, may be used to detect and measure between
nonlinearly coupled waves and fundamental waves.
This section describes the applications of spectral analysis methods in the field of
nonlinear ultrasonic wave propagation in solids.
2.2.1 General Background of Fourier Transform
We briefly review the concept of the Fourier transform and the discrete Fourier transform.
The Fourier transform of a signal may be considered as the signal in the frequency
domain. The Fourier transform of is defined by
)(tx
)(tx
(2.2.1) dtetxX tiωω −∞
∫= )()(∞−
Assuming that is discrete, stationary, real-valued, zero-mean, Eq. (2.2.1) can be
rewritten by
)(tx
11
∫−
∞→=
2/
2/
)( )(1lim)(T
T
ti
T
kT dtetx
TX ωω (2.2.2)
( )*where , is the duration of , and )()( )*()( ωω kk XX =− T )(txTT denotes the complex
conjugate,.
is the discrete Fourier transform(DFT) of kth)()( ωkTX ensemble of the time series
taken over a time
)(tx
T . Using a fast Fourier transform, which is an efficient technique to
compute DFT, we can obtain the Fourier amplitudes given by[16]
∑=
−=N
n
NfnikkT enx
NfX
1
/2)()( )(1)( π ∑=
−=N
n
NnikkT enx
NX
1
/)()( )(1)( ωω or (2.2.3)
where is the kth)()( nx k ensemble of the time series, and n ,and Mk ,,1L= N,,1L=
is sets of data records and is the length of each record. NM
2.2.2 Power Spectrum and Bispectrum
Fig.
The power spectrum, which is a linear spectral analysis, is one of the main tools used in
2.1 Higher Order Spectra Classification Map by Nikias and Mendel[18]
Auto-correlation
Computation [ ]1F
Third-order-statistical
Computation
[ ]2F
)(τxxR
),( 21τ τxxxR),( 21 ffBxxx
)( fPxx
Discrete time signal
Fourier Transform
12
digital signal processing. The power spectral density function of signal is given by )(tx
, (2.2.4) ∫∞
= ττω ωτ deRP jxx
)()()(∞−
where )(τxxR is the autocorrelation function of . )(tx
For a real valued signal, the energy of the signal is defined by[19]
, (2.2.5) ∫∫∞∞
= dffEdttx xx )()(2
∞−∞−
∞−
The energy spectrum is given by )( fExx
)()()()()( * fXfXfXfXfExx =−= (2.2.6)
For stationary process, the power of signal can be defined as
∫∞
= dffPtxE xx )()]([ 2 (2.2.7)
where is the statistical expectation operator and the power spectrum is given
by
[ ]E )( fPxx
)]()([1lim)( * fXfXET
fPTxx ∞→
= (2.2.8)
For a discrete, stationary, real valued, zero-mean process, the power spectrum is estimated
as[2, 7, 16]
∑∑==
==M
k
kT
M
k
kT
kTxx fX
MfXfX
MfP
1
2
1)(
11
)(*1
)(1 )(1)()(1)( (2.2.9)
The power spectrum has a remarkable disadvantage that the background noise on the
signal remains in the power spectrum domain and the power spectrum contains no phase
information of the signal. The components of the energy or power spectra could be affected
13
by the nonlinear interactions of the other modes. Hajj[2] note that the power spectrum
provides no accurate estimate of the level of nonlinear couplings between spectral
components.
On the other hand, the bispectrum, which is a quadratic spectral analysis and the next
higher order moment to the power spectrum, includes both phase and energy information of
signal. The bispectrum is the two-dimensional Fourier transform of the third order
correlation function and is generally complex valued. The bispectrum is given by,
21)(
2121),(),( ττττωω τωτω ddeRB nmj
xxxnmxxx+
∞
∫ ∫= (2.2.10) ∞−
where ),( 21 ττxxxR is the third order correlation function.
Rather than decomposing the energy of a signal, it is possible to conduct similar analysis
on the cubed signal,[19]
21213 ),()( dfdfffEdttx xxx∫∫
∞∞
= (2.2.11) ∞−∞−
∞− ∞−
where the bispectrum can be defined as,
(2.2.12) )()()(),( 212*
1*
21 ffXfXfXffExxx +=
For a stationary process, we can obtain:
21213 ),()]([ dfdfffBtxE xxx∫ ∫
∞ ∞
= (2.2.13)
where the bispectrum is given as,
)]()()([1lim),( 212*
1*
21 ffXfXfXET
ffBTxxx +=
∞→ (2.2.14)
For a discrete, stationary, real valued, zero-mean process, the bispectrum is estimated
14
as[16]
)()()(1),( 21)(
12
)(*1
)(*21 ffXfXfX
MffB k
T
M
k
kT
kTxxx += ∑
=
(2.2.15)
Since Gaussian noise becomes zero through the bispectral analysis, Gaussian noise can be
eliminated completely in the bispectrum domain. The bispectrum is significantly useful in
detecting small high-order harmonic components induced by the nonlinear effects. In
averaging over many ensembles, the magnitude of the bispectrum is determined by the
presence of a phase relationship. If there is a arbitrary phase relationship among , ,
and , the undetectable quantity will be obtained by the bispectrum. If there is a
strong phase relationship among these frequency components, the corresponding
bispectrum will provide a large value. The bispectrum can be used as a measure to detect
quadratic couplings or interactions among different frequency components of a signal since
a quadratic nonlinear interaction between two frequency components and yields a
phase relation between them and their sum component
1f 2f
21 ff +
1f 2f
21 ff + [7].
Fig. 2.2 Linear and Quadratic Parts in a Parallel Structure
Linear Part
Quadratic Part
)sin(1 kxtA −ω
)sin( tA ω
System
)(2cos2 kxtA −ω
)2,,( ωωtD : Detected Signal
where )(2cos)sin()2,,( 21 kxtAkxtAtD −+−= ωωωω .
15
2.2.3 Bicoherence Spectrum
The bicoherence spectrum, which is the normalized bispectrum, is commonly used for
signal analysis. The bicoherence spectrum is most often used to detect and measure
quadratic phase coupling. In general, the bicoherence spectrum is defined as
)()()(),(
),(2121
2121 ffPfPfP
ffBffb
xxxxxx
xxxxxx
+≅ or
)()()(),(
),(2121
221
212
ffPfPfPffB
ffbxxxxxx
xxxxxx +
≅ (2.2.16)
Eq. (2.2.16) can be also written by the statistical expectation operator, . [ ]E
])([])()([
),(),( 2
212
21
221
212
ffXEfXfXE
ffBffb xxx
xxx+
= (2.2.17)
For a discrete, stationary, real valued, zero-mean process, the bicoherence spectrum is
estimated as[16]
∑∑
∑
==
=
+
+= M
k
kT
M
k
kT
kT
M
k
kT
kT
kT
xxx
ffXM
fXfXM
ffXfXfXMffb
1
2
21)(
1
2
2)(
1)(
1
2
21)(
2)*(
1)*(
212
)(1)()(1
)()()(1
),(
By Schwarz inequality[16], the bicoherence spectrum has a range between zero and one.
or . (2.2.18) 1),(0 << ffb 21xxx 21 1),(0 2 << ffb xxx
As noted before, because the bicoherence spectrum is the normalized bispectrum, unless
there is a phase relationship among the frequency components at , , and their
sum , the bicoherence spectrum quantity will be close to zero. If there is a phase
relationship among the frequency components at , , and , then the
1f 2f
21 ff +
1f 2f 21 ff +
16
bicoherence spectrum quantity will be close to one as we can see in Eq.(2.2.18). Values of
the bicoherence spectrum between zero and one indicate that the mechanical waves are
partially coupled in their phases. It is natural to interpret the bicoherence spectrum as a
measure of the quadratic nonlinearity of the signal with a value of one being quadratically
coupled and zero being not coupled at all.
1fq
q
2f021 =+ ff 021 − ff =
A
B
Fig. 2.3 Region of the Bispectrum Computation by Kim and Power[16]
The bispectrum can be defined over a hexagon[16] ),( 21 ffB
, and 1fq ≤− qf ≤2 qffq ≤+≤− 21 (2.2.19)
where and , and Nyquist frequency ffq N Δ= / )2/(1 tf N Δ=Tf /1=Δ and the record
length . tNT Δ=
Due to the relation , the hexagon is reduced to the region of
“A” and “B” in Fig. 2.3, defined as
),(),(),( * lkBklBlkB −==
Region “A”: and 2/0 2 qf ≤≤ 212 fqff −≤≤ (2.2.20)
Region “B”: and 02 ≤≤− fq qff ≤≤ 12 . (2.2.21)
17
We are only interested in those two regions of the bispectrum and bicoherence spectrum
to understand the wave interaction or coupling among different modes.
2.3 Nonlinearity Parameters
2.3.1 Nonlinearity Parameter by Cantrell
Cantrell and Yost[3] note that the nonlinearity parameter is obtained as follows.
A solution of Eq.(2.1.14), assuming a purely sinusoidal input wave of the form
)sin(1 tA ω 0=xω of frequency applied at ,and applying the perturbation theory, we
can obtain a solution after the wave travels a distance in the material, it is l
L+−⎟⎟⎠
⎞⎜⎜⎝
⎛−−= )(2cos
81)sin( 2
221
1
21 klt
ClA
EE
kltAu ωω
ω (2.3.1)
where is the wave vector, k λπ /2=k , and fC /=λ is the wave length, and l is
wave propagation distance in the material, is the frequency, f fπω 2= is the angular
frequency, and is the fundamental amplitude of the ultrasonic wave signal. The second
harmonic amplitude is given by
1A
2A
.81
2
221
1
22 C
lAEE
Aω
⎟⎟⎠
⎞⎜⎜⎝
⎛−= (2.3.2)
The measurement of and is the basis of calculation of the nonlinearity
parameter
1A 2A
β ,
12 / EE−=β (2.3.3)
Substituting Eq.(2.3.3) into Eq.(2.3.2), we can obtain the ultrasonic nonlinearity
parameter β in terms of measured quantities:
18
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
lC
AA
2
2
21
2 8ω
β (2.3.4)
We can rewrite the solution of the Eq.(2.3.1) by using β and as follows k
L+−+−= )](2cos[81)sin( 22
11 kltlkAkltAu ωβω (2.3.5)
where . 2222 /)/2( Ck ωλπ ==
ωEq.(2.3.5) shows that in addition to the fundamental sinusoidal signal of frequency , a
harmonic signal of frequency ω2 is generated with a certain amplitude that is dependent
on the magnitude of the nonlinearity parameter.
In this study, we use the normalized nonlinearity parameter for Cantrell’s calculation,
because we can eliminate the constant term following the amplitude term on Eq.(2.3.4) if
we know the length of the specimen, l , the input signal frequency,ω , and the wave velocity
in the specimen,C . Now the normalized nonlinearity parameter for Cantrell is given by
⎟⎟⎠
⎞⎜⎜⎝
⎛≡
)()(
1
2
ωω
βPP
C (2.3.6)
( )P12 2ωω =where a harmonic frequency, and is the power spectrum at an certain
frequency.
2.3.2 Nonlinearity Parameter by Jhang
Jhang[20] note that the magnitude of a specific frequency component can be obtained
from the power spectrum and the higher spectra can be used to estimate the nonlinearity
parameter.
Recalling the amplitude of subharmonic frequency component from Eq.(2.3.5), we can
19
obtain the nonlinearity parameter,
21
22
2
21
22
2212
8881
AA
lC
AA
lklkAA ⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟
⎠⎞
⎜⎝⎛=⇔=
ωββ , (2.3.7)
In this paper we also use the normalized nonlinearity parameter for Jhang’s calculation
technique[5],
21
2
AA
J ≡β , (2.3.8)
12 2ωω =1ωThe power spectra at frequency and are briefly given by,
21
211 )()( AXP == ωω 2
22
22 )()( AXP == ωω, (2.3.9)
1ω 1ω, is obtained as, The magnitude of the bispectrum for the same signal at
22
111*
1111 )()()(),( AAXXXB =+= ωωωωωω (2.3.10)
Now Jhang’s nonlinearity parameter using the power spectrum and bispectrum quantities
can be provided as,
⎟⎟⎠
⎞⎜⎜⎝
⎛=
)(1
)(),(
11
112
1
2
ωωωω
PPB
AA
21
11
)(),(
ωωω
βP
BJ ≡, (2.3.11)
The nonlinearity parameter by Jhang is more accurate than the nonlinearity parameter by
Cantrell since Jhang uses the bispectrum values, These values will include consideration of
the phase coupling between ω ω2 and parts of the detected signal, and eliminates any
Gaussian noise detected. The nonlinearity parameter by Cantrell includes background noise
errors for his calculation, and does not discriminate energy at ω2 from sources other than
the excitation.
20
CHAPTER III
EXPERIMENTAL BACKGROUND
3.1 Ultrasonic Data Acquisition Setup
In order to determine the nonlinearity parameter it is necessary to measure , ,C , l
for ultrasonic wave of frequency
1A 2A
ω . This section will describe the data acquisition system
setup, which ultrasonic transducers and instrumentation were used, and which software was
used to control the system. It also details how the ultrasonic signal was transmitted and
received through a specimen, and how the measurement variability was assessed.
3.1.1 Ultrasonic Transducer
While acquiring ultrasonic signals, two main types of ultrasonic transducers were used.
All transducers used, however, generate mechanical waves using piezoelectric elements.
Acquisition of ultrasonic wave signals was performed by using a 10 MHz surface contact
transducer. It was a Videoscan longitudinal transducer by Panametrics, model V111-RM,
with a half inch element and the serial number is 51012. Ultrasonic waves were excited
using a 5MHz contact Videoscan transducer made by Panametrics, model V109-RB, with a
half inch diameter; the serial number is 103910. Videoscan transducers are untuned
transducers that provide heavily damped broadband performance.
Both transducers were coupled to the stationary ends of the specimen using a
21
SONOTRACE 30 couplant made by SONOTECH INC., which is an ambient temperature
and glycerine-free couplant.
3.1.2 Ultrasonic Hardware and System Setup
The general setup for generating and recording ultrasonic wave signals on the specimen
is diagramed in Fig. 3.1.
Electrical function signals were generated by using a function generator manufactured by
Hewlett Packard, 3314A model. The function generator was used as an external trigger. A
computer based acquisition system using a CS12100 A/D card (PCI slot) made by
GageScope was used for data acquisition.
The data acquisition is controlled with GagaScope software, standard version 3.10,
designed by GageScope. The software package performs as a digital oscilloscope, showing
the current digitalized signal wave form in real time.
Table 3.1 Description of the HP 3314A Function Generator
Frequency, Amplitude, N Cycle,
[MHz] [Volts] [Cycles]
0.001 Hz to 19.99 MHz 0.00 mV to 10.00 V 1 to 1999 cycle
22
Table 3.2 Details of the GageScope CS12100 A/D Card
Max. Sample Rate on 1 Channel 100 M/s (100MHz)
Vertical Resolution 12 Bits
Full Power Bandwidth 50 MHz
Voltage Ranges ± 100 mV to 5V
Input Impedance 1 MΩ to 50 Ω
Input Coupling 1 MΩ: AC or DC / 50 Ω: DC only
Table 3.3 Settings for the GageScope Software
Sampling Rate 50 M/s (50 MHz)
Trigger setting External
Time base 500 ㎲/div
CS Input Range ± 1V DC coupling
Impedance 1 MΩ
23
External Trigger Line
Couplant
Transducer, f
Transducer, 2f CS12100 A/D Board
GageScope Software
Digital
Oscilloscope
HP Function
Generator
Channel 2
Channel 1
: Reflected UT Wave : Transmitted UT Wave
Fig. 3.1 Diagram of Ultrasonic Measurement System
3.2 Ultrasonic Measurement Testing
A 5 MHz tone burst input signal of 15 cycle duration with 5 volt peak to peak voltage
was generated from a function generator. 5 MHz and 10 MHz ultrasonic transducers were
used as a transmitter and a receiver, respectively on each end of the prepared specimen.
Transducers were coupled by means of ethylene glycol based couplant to the stationary
specimen ends, allowing propagation of the sound wave along the major axis. The
ultrasonic wave signal was collected by using a 12 bit analog to digital computer
acquisition system sampling at a rate of 50 MHz.
The initial disturbance was isolated and captured. 16 ensemble signal segments were
recorded in order to analyze the correlation among those nonlinear ultrasonic waves.
24
0 1 2 3 4 5 6 7 8 9
x 10-6
-5
0
5
5V Input Signal,Total Received Signal and Initial Disturbance
Am
plitu
de, V
Time, Sec
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
x 10-4
-0.04-0.02
00.020.040.06
Am
plitu
de, V
Time, Sec
3.45 3.5 3.55 3.6 3.65 3.7 3.75 3.8 3.85 3.9
x 10-5
-0.04-0.02
00.020.040.06
Am
plitu
de, V
Time, Sec
Fig. 3.2 Input Signal, Total Received and Initial Disturbance Signal
3.3 System Stability Examination
Since the measurement requires a relatively long time, it is necessary to check the
stability of the system over an extended time period. The received voltage and average
frequency detected by the 10 MHz receiver were recorded every 10 minutes with the
measurement system set with the same input frequency, voltage, and number excitation
cycles.
25
0 200 400 600 800 1000 1200200
210
220
230
240
250
260
270
280
290
300A
mpl
itude
,mV
Time,min
System Stability Test depending on Time
0 200 400 600 800 1000 12004
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5
Rec
eive
d Si
ngal
Ave
rage
Fre
quen
cy,M
Hz
Fig. 3.3 System Stability Test
This small fluctuation is thought to be from the changes in the room temperature during
the test time in the laboratory and couplant layer thickness due to a constant weight used to
hold the receiving transducer in place.
26
CHAPTER IV
EXPERIMENTAL STUDIES
4.1 Experimental Setup
The experimental setup use to systematically study all aspects of the measurement
system and process is shown schematically in Fig. 4.1. It is divided into three main
portions labeled A, B, and C; the input, the sample of interest, and the detection system,
respectively.
10 MHz UT transducer
5 MHz UT transducer
Centering Collar
Sonotrace couplant
Given Mass
GageScope
CS12100
HP Function
Generator
Voltage(p-p): Varied
Frequency: Varied
No. of Cycle: 15 cycles
External Triggered Always
Sampling Frequency: 50 MHz
No. of Samples: Varied
No. of Data Sets: 16 ensembles
B
A
C
Fig. 4.1 Diagram of Experimental Setup
A Hewlett-Packard 3314A Function Generator was used as the input electrical signal
source. It allowed for the frequency to be varied, the peak-to-peak voltage, as well as the
27
number of cycles of the tone burst excitation.
The data were collected using a GageScope CS 12100 PC based analog to digital data
acquisition board. The acquisition was synchronized with the excitation signal by means of
a signal from the function generator for all testing. While the acquisition sampling rate was
selectable all signals were sampled at a rate of 50 MHz. The number of data points sampled
was varied to improve the analysis in different situations, but in each instance 16 separate
temporal data sets were collected for analysis.
The portion B was varied the most since in one instance the function generator was
connected directly to the detection system, in other cases different receiving transducers
were used, and in others cases different specimen lengths and materials were examined; in
some preliminary testing the specimen was held in a testing grip assembly used for cyclic
loading the specimen. The following table 4.1 format will be used to indicate the testing
parameters for the different studies that will be described in this chapter.
Table 4.1 Testing Parameters Table Format
HP Function Generator GageScope Transducers Specimen
Voltage p-p Sampling Frequency 50MHz Transmitter Material
Frequency No. of Data points Receiver Length
No. of cycles 15 No. of ensembles 16 Couplant Diameter
28
GageScope
CS12100
HP Function
Generator
External Trigger Cable(Always)
B3. Cyclic Loading Test Setup B2. System Characterization and
Nonlinear Response Test Setup
B1. Input Signal Chracterization Test Setup
Cyclic Loading
Fig. 4.2 Three Types of Experimental Setups
Fig. 4.3 HP Function Generator and Initial Disturbance Capturing by GageScope
4.2 Characterization of Response of Data Acquisition System
It is necessary to assess if the response of the measurement system itself is linear or
nonlinear because the ultrasonic response of the specimen is being measured and
nonlinearity parameters calculated. Without this knowledge, it is hard to tell whether the
nonlinearity response is due to the tested sample or the measurement system itself.
A sample experiment with 1 inch long fused quartz specimen has been performed at
29
room temperature by changing the input peak to peak voltage through the transmitting
transducer. Fused quartz is a type of glass containing primarily silica in amorphous form
and has nearly ideal properties. Input voltage was increasing from 2V to 10V peak to peak.
2 3 4 5 6 7 8 9 1010
-4
10-3
10-2
10-1
100
Out
put A
mpl
itude
, Po(f j)1/
2
Input Singal Voltage peak to peak
Fundamental Frequency Amplitude, A1
1st Harmonic Frequency Amplitdue, A2
Fig. 4.4 Output Amplitudes depending on Input Signal Amplitude Change
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.2
0.4
0.6
0.8
1
1.2
1.4x 10
-3
1st
harm
onic
Freq
uncy
Am
plitu
de, A
2=Po(f 2)1/
2
Fundamental Frequncy Amplitude, A1=P
o(f
1)1/2
Slope, A2/A
1
linear
Fig. 4.5 Fundamental vs. Second Harmonic Amplitude Ratio Plot, A /A2 1
30
31
When the input signal voltage was increased from 2V through 10V, the fundamental
and its harmonic frequency component amplitudes increased in a similar manner. The Fig.
4.5 shows that the ultrasonic data acquisition system is linearly correlated with the
amplitude.
0 0.005 0.01 0.015 0.02 0.025 0.03 0.0352
4
6
8
10
12
14x 10
-4
1st
harm
onic
Freq
uncy
Am
plitu
de, A
2=Po(f 2)1/
2
Fundamental Frequncy Amplitude Sqaure, A12=P
o(f
1)
Slope, A2/A
12
Fig. 4.6 Nonlinearity Slope Plot, A2/A12
One can notice that our data acquisition system has nonlinear property from Fig. 4.6. It is
obvious that nonlinearity presents everywhere. We, however, want to track the change of
nonlinear parameter as the following tests are conducted.
Even though some nonlinearity is present in the system itself, this study will be valuable
assets if one can track the change of the nonlinear parameter during experiments. This also
suggests that tests should be performed within a region that is straight.
4.3 Signal Data Reproducibility – Measurement Variability
With a 2 inch long 7075-T6 Al cylindrical specimen, three sets of signal data, 16 records
for each set were collected, to eliminate measurement variability and to improve signal data
reproducibility. Each signal data set was collected on different dates, but the set-up for
signal acquisition was identical. The input signal frequency was set at 5MHz, but the input
voltage was varied from 5 volts to 0.5 volts in 0.5 volts increments. The rationale for
varying the input voltages will be discussed later in the section on Nonlinear Response of
Various Specimens. The collected signals were analyzed and compared with the
nonlinearity parameter by Jhang and Cantrell.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.110
-2
10-1
100
101 Logarithm Scale(2in. Al7075-T6)
Output Amplitude, Po(f1)1/2
Non
linea
rity
Val
ue b
y Jh
ang
Set 1Set 2Set 3
Fig. 4.7 Nonlinearity Parameters by Jhang in Log Scale Plot
32
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.110
-2
10-1
100
101 Logarithm Scale(2in. Al7075-T6)
Output Amplitude, Po(f1)1/2
Non
linea
rity
Val
ue b
y C
antr
ell
Set 1Set 2Set 3
Fig. 4.8 Nonlinearity Parameters by Cantrell in Log Scale Plot
Although each signal data set was sampled at a different day, those two plots show that
the nonlinearity parameters depend on the received signal amplitudes but consistent.
33
4.4 Input Signal Characterization
4.4.1 Introduction
In order to understand the nature of the input signal a study was done where the HP
function generator was connected directly to the GageScope detection system. First the
voltage of the input signal was varied while other parameters were held constant, then the
frequency of the input signal was varied while the other parameters were held constant.
4.4.2 Test Setup and Experimentation Result
Table 4.2 Testing Parameters for Input Signal Characterization-Voltage
HP Function Generator GageScope Transducers Specimen
none Voltage p-p 0.5-5.0 V Sampling Frequency 50MHz Transmitter none Material
none Frequency 5 MHz No. of Data points 180 Receiver none Length
none No. of cycles 15 No. of ensembles 16 (3sets) Couplant none Diameter
Table 4.3 Spectral Analysis Worksheet(INPUT and OUTPUT Signals)
Bxxx(f1,f1) Pi(f1) √Pi(f1) Pi(f2) Bxxx(f1,f1) Po(f1) √Po(f1) Po(f2)
5 0.21220539 2.64642687 1.62678421 0.00080631 1.6183E-05 0.00950612 0.09749934 3.6824E-074.5 0.15393794 2.13799123 1.46218714 0.00064955 1.17E-05 0.00771684 0.08784552 2.9164E-074 0.10582069 1.68421054 1.29777137 0.00049444 8.2404E-06 0.00610728 0.07814911 2.3119E-07
3.5 0.06928535 1.2870526 1.13448341 0.00036278 6.1895E-06 0.00468327 0.06843441 2.2413E-073 0.04474297 0.94602296 0.97263712 0.00028012 3.9347E-06 0.00344255 0.05867326 1.687E-07
2.5 0.02539847 0.65576745 0.8097947 0.00018778 2.4707E-06 0.00240054 0.04899529 1.3642E-072 0.01265652 0.41938128 0.64759654 0.00011418 1.2965E-06 0.00153392 0.0391653 9.5217E-08
1.5 0.00531641 0.23518038 0.484954 6.4195E-05 5.8845E-07 0.00086657 0.02943755 6.6026E-081 0.00157562 0.10441956 0.32314015 2.8807E-05 2.3659E-07 0.00038257 0.01955941 5.2248E-08
0.5 0.00020988 0.02821737 0.16798027 7.0723E-06 4.5163E-08 0.00010494 0.01024419 2.7915E-08
OUTPUT(from 10MHz Receiver_Fused Quartz)Input Voltage[V]
(peak to peak)
INPUT(from Function Generator)
34
0 5 10 15 2010
-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
Pow
er
Frequency (MHz)
Spectrum Change depending on Input Signal Voltage Change (Input Setting)
5V4.5V4V3.5V3V2.5V2V1.5V1V0.5V
Fig. 4.9 Power Spectrum Plot for Input Signal with Input Voltage Decrease
0 5 10 15 20 25
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
Pow
er
Frequency (MHz)
Spectrum Change Depending on Input Signal Voltage Change(10MHz Receiver-Fused Quartz)
5 V4.5 V4 V3.5 V3 V2.5 V2 V1.5 V1 V0.5V
Fig. 4.10 Power Spectrum Plot for 10 MHz Receiver with Input Voltage Decrease
35
Both power spectra, which are the energy distributions, show that the energy of the
detected signal is decreased when the voltage of the input signal is decreased. The power
spectrum in Fig. 4.9 suggests that the 5MHz input electrical signal generated from the HP
function generator contains not only 5 MHz frequency component, but also its harmonic
frequency components. The power spectrum in Fig. 4.10 shows that the detected signal
using 10 MHz receiver includes the obvious energy distribution of the fundamental
frequency and the second harmonic frequency component which may be related to the
nonlinear response of the material.
0 0.5 1 1.50
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Out
put P
o(f 1)1/2
Input Pi(f1)1/2
Output Po(f1)1/2/Input Pi(f1)
1/2
0 0.01 0.02 0.031
2
3
4
5
6
7x 10
-4
Out
put P
o(f 2)1/2
Input Pi(f2)1/2
Output Po(f2)1/2/Input Pi(f2)
1/2
Linear Fit Slope = 0.059884Linear Fit Slope = 0.059884Linear Fit Slope = 0.016502
Fig. 4.11 Input Signal Amplitude vs. Output Signal Amplitude Plot,
1/2Pi(f1) vs. Po(f1)1/2 1/2 and Pi(f2) vs. Po(f )1/2 2
36
This suggests that the 10 MHz input is probably attenuated and that the response
detected is due to the much larger amplitude 5MHz input. The offset from zero for the
detected response for a nominal zero input suggests that the detector noise level is increased
when the 10 MHz transducer is attached.
Table 4.4 Testing Parameters for Input Signal Characterization-Frequency
HP Function Generator GageScope Transducers Specimen
none Voltage p-p 5 V Sampling Frequency 50MHz Transmitter none Material
none Frequency 3.5-7.5 MHz No. of Data points 400 Receiver none Length
none No. of cycles 15 No. of ensembles 16 Couplant none Diameter
0 3.544.555.566.577.58 9 10 11 12 13 14 15 2010
-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
Pow
er
Frequency (MHz)
Spectrum Change depending on Input Signal Frequency Change (Input Signal)
3.5MHz4MHz4.5MHz5MHz5.5MHz6MHz6.5MHz7MHz7.5MHz
Fig. 4.12 Total Power Spectrum Change with Input Frequency Increase
(5 Volts Input Signal)
37
Table 4.5 Spectral Analysis Worksheet(5 Volts Input Signal)
Input Freq[MHz]
P(finput) |Χ(finput)| P(2finput) B(finput,finput)Beta byJhang
Beta byCantrell
3.5 0.975664 0.987757 0.000465 0.0595095 0.062515 0.0221054 0.579679 0.761367 0.000321 0.0293499 0.087344 0.030887
4.5 0.350417 0.591961 0.000201 0.0140492 0.114414 0.0404615 0.215812 0.464556 0.000125 0.0068237 0.14651 0.05186
5.5 0.136217 0.369075 8.01E-05 0.003445 0.185667 0.0657026 0.088345 0.297229 5.13E-05 0.0017865 0.228896 0.081097
6.5 0.059198 0.243306 3.02E-05 0.0009183 0.262039 0.0928677 0.039895 0.199737 1.87E-05 0.0004856 0.305099 0.108269
7.5 0.027793 0.166714 1.32E-05 0.0002835 0.366958 0.130642
HP Function Generator System Check(ndim=400)
0 3.5 7 10.5 14 17.520
100
Pow
er
Frequency (MHz)
3.5MHz
0 4 8 12 16 20
100
Pow
er
Frequency (MHz)
4MHz
0 4.5 9 13.5 1820
100
Pow
er
Frequency (MHz)
4.5MHz
0 5 10 15 20
100
Pow
er
Frequency (MHz)
5MHz
0 5.5 11 16.5 20
100
Pow
er
Frequency (MHz)
5.5MHz
0 6 12 1820
100
Pow
er
Frequency (MHz)
6MHz
0 6.5 13 20
100
Pow
er
Frequency (MHz)
6.5MHz
0 7 14 20
100
Pow
er
Frequency (MHz)
7MHz
0 7.5 15 20
100
Pow
er
Frequency (MHz)
7.5MHz
Fig. 4.13 Power Spectrum Change with Input Frequency Increase(5 Volts Input Signal)
This spectral analysis result shows the significant aspects of the response of the data
acquisition system. The power spectrum plot in Fig. 4.12 shows that the directly detected
38
input signal energy is decreased when the frequency of the input signal is increased. This
decrease is due to the increase of noise portion in the signal data points when the wave
length of the input signal is decreased.
The power spectra in Fig. 4.13 show that the input signal includes the energy distribution
of the fundamental frequency and its harmonic frequency components in all the stages. This
result agrees with the result of the previous input signal voltage change test.
3.5 4 4.5 5 5.5 6 6.5 7 7.50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Bet
a Pa
ram
eter
Input Frequency (MHz)
Beta Change depending on Input Signal Frequency Change (Input Signal)
JhangCantrell
Fig. 4.14 Nonlinearity Parameter Change depending on Input Frequency Increase
(5 Volts Input Signal)
This suggests that the nonlinearity parameter of the detected input signal is increased as
increasing the frequency of the input signal. The nonlinearity parameters by Jhang are
higher than those by Cantrell and the nonlinearity parameters are increased almost linearly.
39
4.5 System Characterization using Fused Quartz Sample
4.5.1 Introduction
This test was conducted to characterize the response of the ultrasonic data acquisition
system with respect to the change of the input signal frequency and voltage. The input
signal frequency was varied from 3.5 MHz to 7.5 MHz and the input signal voltage was
varied from 0.5 Volts to 5.0 Volts. Two main ultrasonic transducers, which have different
resonant frequencies 5 MHz and 10 MHz, were used to compare the response results. A 1
inch cylindrical fused quartz sample, which has low attenuation influence, was used as the
test specimen. The previous test showed that the detected signal using the ultrasonic
transducer contained two apparent frequency responses, at 5 MHz and 10 MHz, on the
power spectrum plots. The objective of this test is to investigate how response changed, and
what factors caused these resonant frequency response results.
4.5.2 Test Setup and Experimentation Result
Table 4.6 Testing Parameters for System Characterization-Frequency(5MHz Receiver)
HP Function Generator GageScope Transducers Specimen
Fused Quartz Voltage p-p 5 V Sampling Frequency 50MHz Transmitter 5 MHz Material
1 inch Frequency 3.5-7.5 MHz No. of Data points 400 Receiver 5 MHz Length
1 inch No. of cycles 15 No. of ensembles 16 Couplant Sonotrace30 Diameter
40
0 3.544.555.566.577.58 9 10 11 15 2010
-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Pow
er
Frequency (MHz)
Spectrum Change depending on Input Signal Frequency Change(5 MHz Receiver)
3.5MHz4MHz4.5MHz5MHz5.5MHz6MHz6.5MHz7MHz7.5MHz
Fig. 4.15 Total Power Spectrum Change with Input Frequency Increase
(5 MHz Receiver)
Table 4.7 Spectral Analysis Worksheet(5 MHz Receiver)
Input Freq[MHz]
P(finput) P(f1) P(f2) |Χ(finput)| |Χ(f1)| |Χ(f2)| P(2finput) B(finput,finput)Beta byJhang
Beta byCantrell
3.5 0.0013632 1.067E-06 2.681E-09 0.036921 0.001033 5.18E-05 1.45E-07 1.4571E-06 0.7841142 0.279011764 0.00101 3.503E-06 2.352E-09 0.031781 0.001872 4.85E-05 3.52E-08 5.171E-07 0.50688578 0.18573314
4.5 0.000685 1.827E-05 2.759E-09 0.026172 0.004274 5.25E-05 1.31E-08 1.9871E-07 0.4235186 0.167289185 0.0004313 0.0004313 3.931E-09 0.020769 0.020769 6.27E-05 3.93E-09 6.5106E-08 0.34993147 0.14534951
5.5 0.0002266 2.689E-05 2.334E-09 0.015054 0.005185 4.83E-05 2.71E-09 1.7982E-08 0.35011774 0.229916986 9.532E-05 6.175E-06 2.809E-09 0.009763 0.002485 5.3E-05 2.29E-09 2.7944E-09 0.30753517 0.50236514
6.5 3.475E-05 2.585E-06 3.25E-09 0.005895 0.001608 5.7E-05 2.8E-09 2.7974E-09 2.31589999 1.5216127 1.294E-05 1.311E-06 3.51E-09 0.003598 0.001145 5.92E-05 2.57E-09 8.2756E-10 4.94019905 3.91946798
7.5 5.061E-06 7.941E-07 3.239E-09 0.00225 0.000891 5.69E-05 2.64E-09 6.9789E-11 2.72416628 10.1545122
5 MHz Receiver (ndim=400)
41
0 5 10 15 20
10-5
Pow
er
Frequency (MHz)
3.5MHz
0 5 10 15 20
10-5
Pow
er
Frequency (MHz)
4MHz
0 5 10 15 20
10-5
Pow
er
Frequency (MHz)
4.5MHz
0 5 10 15 20
10-5
Pow
er
Frequency (MHz)
5MHz
0 5 10 15 20
10-5
Pow
er
Frequency (MHz)
5.5MHz
0 5 10 15 20
10-5
Pow
er
Frequency (MHz)
6MHz
0 5 10 15 20
10-5
Pow
er
Frequency (MHz)
6.5MHz
0 5 10 15 20
10-5
Pow
er
Frequency (MHz)
7MHz
0 5 10 15 20
10-5
Pow
erFrequency (MHz)
7.5MHz
Fig. 4.16 Power Spectrum Change with Input Frequency Increase(5 MHz Receiver)
The power spectrum results show several remarkable aspects. First of all, one can argue
that the power spectrum response result is strange since the 5MHz transducer, which has
relatively the highest energy distribution at 5MHz frequency, was used as a transmitter for
this test. Considering the total power spectrum plot in Fig. 4.15, this shows that the detected
signal when the HP function generator excites 3.5 MHz as the input signal frequency
displays the highest energy distribution even using a 5 MHz transmitter. Table 4.7 also
shows that the energy of the input signal, power spectrum quantity, decreases as the input
signal frequency increases. This result, however, is due to the relationship between the
input signal frequency change and the power spectrum calculation. For all stages, 400 data
points were used to analyze the received ultrasonic signal. If the input signal frequency is
42
increased, the received signal frequency will be increased and then more noise portions will
be recorded into the 400 data points. This leads to the decrease of the power spectrum as
the input frequency increases. If the 400 data points for the real signal can be constantly
recorded, the energy distribution of the received ultrasonic signal will show the highest
quantities when the input frequency is 5 MHz, since the 5 MHz ultrasonic transducer is
used as a transmitter.
The power spectra in Fig. 4.16 show that the second harmonic frequency response is
present when the frequency of the input signal is varied from 3.5 MHz to 4.5 MHz. The
second harmonic responses of the rest of the input signal frequency can hardly be
distinguished due to the noise. The change of the nonlinearity parameters by Cantrell and
Jhang will be presented in the later part of this section.
Table 4.8 Testing Parameters for System Characterization-Frequency(10MHz Receiver)
HP Function Generator GageScope Transducers Specimen
Fused Quartz Voltage p-p 5 V Sampling Frequency 50MHz Transmitter 5 MHz Material
1 inch Frequency 3.5-7.5 MHz No. of Data points 400 Receiver 10 MHz Length
1 inch No. of cycles 15 No. of ensembles 16 Couplant Sonotrace30 Diameter
At this time, in order to understand the detected signal response difference due to the
characterization of the receiver, the 10 MHz transducer which was mentioned before was
used as an ultrasonic signal detector.
43
0 3.544.555.566.577.58 9 10 11 15 2010
-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Pow
er
Frequency (MHz)
Spectrum Change depending on Input Signal Frequency Change (10 MHz Receiver)
3.5MHz4MHz4.5MHz5MHz5.5MHz6MHz6.5MHz7MHz7.5MHz
Fig. 4.17 Total Power Spectrum Change with Input Frequency Increase
(10 MHz Receiver)
Table 4.9 Spectral Analysis Worksheet(10 MHz Receiver)
Input Freq[MHz]
P(finput) P(f1) P(f2) |Χ(finput)| |Χ(f1)| |Χ(f2)| P(2finput) B(finput,finput)Beta byJhang
Beta byCantrell
3.5 0.0031729 2.703E-06 8.754E-09 0.056329 0.001644 9.36E-05 9.06E-07 8.5281E-06 0.8470867 0.29990814 0.0022428 9.276E-06 5.502E-09 0.047358 0.003046 7.42E-05 4.25E-07 4.1243E-06 0.8199456 0.29084803
4.5 0.0015657 4.746E-05 1.32E-08 0.039568 0.006889 0.000115 2.11E-07 2.0226E-06 0.82513255 0.293205175 0.0010391 0.0010391 8.474E-08 0.032234 0.032234 0.000291 8.47E-08 8.3983E-07 0.77788747 0.28016551
5.5 0.0006608 6.671E-05 9.832E-09 0.025706 0.008168 9.92E-05 1.93E-08 2.4946E-07 0.57133719 0.210477586 0.0005233 1.728E-05 1.781E-08 0.022876 0.004157 0.000133 4.78E-09 8.4182E-08 0.30737905 0.13205565
6.5 0.0002079 6.319E-06 1.827E-08 0.01442 0.002514 0.000135 3.69E-09 2.2119E-08 0.51157148 0.292159747 0.0001136 3.237E-06 3.955E-08 0.01066 0.001799 0.000199 1.89E-09 2.8727E-09 0.22247848 0.38308605
7.5 6.719E-05 1.971E-06 4.799E-08 0.008197 0.001404 0.000219 2.77E-09 1.6766E-09 0.37137445 0.78398103
10 MHz Receiver (ndim=400)
44
0 5 10 15 20
10-5
Pow
er
Frequency (MHz)
3.5MHz
0 5 10 15 20
10-5
Pow
er
Frequency (MHz)
4MHz
0 5 10 15 20
10-5
Pow
er
Frequency (MHz)
4.5MHz
0 5 10 15 20
10-5
Pow
er
Frequency (MHz)
5MHz
0 5 10 15 20
10-5
Pow
er
Frequency (MHz)
5.5MHz
0 5 10 15 20
10-5
Pow
er
Frequency (MHz)
6MHz
0 5 10 15 20
10-5
Pow
er
Frequency (MHz)
6.5MHz
0 5 10 15 20
10-5
Pow
er
Frequency (MHz)
7MHz
0 5 10 15 20
10-5
Pow
erFrequency (MHz)
7.5MHz
Fig. 4.18 Power Spectrum Change with Input Frequency Increase(10 MHz Receiver)
The power spectrum results are similar to the response results when using the 5 MHz
receiver. Considering the total power spectrum plot in Fig. 4.17, one can find that the power
spectrum quantity is decreased when the frequency of the input signal is increased.
Figure 4.18 suggests that the receiver, a 10 MHz transducer, does not influence the
ultrasonic harmonic response. If the second harmonic frequency response is the effect of
the 10 MHz resonant receiver, the energy distribution at 10 MHz component of the detected
signal should be present in all stages. However the plots in Fig. 4.18 indicate that there is
no noticeable peak at 10 MHz frequency component unless the frequency of the input
signal is excited at 5MHz. This suggests that the energy distribution at 10 MHz is not
because of the 10 MHz receiver. In addition, the power spectra in Fig. 4.18 show that the
45
second harmonic frequency response is present when the frequency of the input signal is
varied from 3.5 MHz to 5.5 MHz. The second harmonic responses of the rest of the input
signal frequency can hardly be distinguished due to the noise.
3.5 4 4.5 5 5.5 6 6.5 7 7.510
-1
100
101
Non
linea
rity
Par
amet
er
Input Frequency (MHz)
when using 10 MHz Receiver
JhangCantrell
3.5 4 4.5 5 5.5 6 6.5 7 7.510
-1
100
101
Non
linea
rity
Par
amet
er
Input Frequency (MHz)
when using 5 MHz Receiver
JhangCantrell
Steady Region
Fig. 4.19 Nonlinearity Parameter Change depending on Input Frequency Increase
(5MHz Receiver and10 MHz Receiver)
The interesting and important features are shown in the change of the nonlinearity
parameters with respect to the change of the input signal frequency. The one of the
significant aspects is that the nonlinearity parameters are displayed in the steady state as
increasing the frequency of the input signal, from 3.5 MHz to 5 MHz. Although, in Fig.
4.18, one can find the small peak at the second harmonic frequency component when the
46
input signal was 5.5 MHz, the nonlinearity parameter plots in Fig. 4.19 show decrease at
that frequency component.
3.5 4 4.5 5 5.5 6 6.5 7 7.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Bet
a Pa
ram
eter
Input Frequency (MHz)
Beta Change depending on Input Signal Frequency Change (Input Signal & 10MHz Receiver)
Jhang-Input SignalCantrell-Input SignalJhang-10MHz ReceiverCantrell-10MHz Receiver
Fig. 4.20 Nonlinearity Parameter Change Comparison (10 MHz Receiver & 5 Volts Input Signal)
Fig 4.20 shows several conclusive aspects. The nonlinearity parameters by Jhang and
Cantrell are steady as changing the input signal frequency from 3.5 MHz to 5 MHz.
However, both nonlinearity parameters calculated with the input signal directly from the HP
function generator are increased as changing the input signal frequency from 3.5 MHz to 5
MHz. It seems advisable to operate the future test in the 3.5 MHz ∼ 5 MHz range.
47
Table 4.10 Testing Parameters for System Characterization-Voltage(10MHz Receiver)
HP Function Generator GageScope Transducers Specimen
Voltage p-p 0.5-5.0 V Sampling Frequency 50MHz Transmitter 5 MHz Material Fused Quartz
1 inch Frequency 5 MHz No. of Data points 180 Receiver 10 MHz Length
1 inch No. of cycles 15 No. of ensembles 16(3sets) Couplant Sonotrace30 Diameter
In order to understand the difference of the detected signal response, the voltage of the
input signal was varied. The 10 MHz transducer and the fused quartz sample were used as
an ultrasonic signal detector and a test specimen, respectively.
0 5 10 15 20 25
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
Pow
er
Frequency (MHz)
Fused Quartz - Input Signal Voltage Change(10MHz Receiver)
5 V4.5 V4 V3.5 V3 V2.5 V2 V1.5 V1 V0.5V
Fig. 4.21 Total Power Spectrum Change with Input Voltage Decrease
(10 MHz Receiver)
48
The power spectra in Fig. 4.21 show that the energy distribution is decreased as the
voltage of the input signal decreases. Moreover, one can clearly find that the energy
distributions around 5 MHz frequency component were dramatically decreased. This
suggests that one can reduce the low frequency couplings decreasing the input signal
voltage. Considering the power spectrum when the input signal voltage is 0.5 V, one can
easily discover that there are almost only two peaks at the fundamental and the second
harmonic frequency components considering the rest as noise portions.
0 0.02 0.04 0.06 0.08 0.110
-2
10-1
100
101
102 Logarithm Scale(Fused Quartz)
Output Amplitude, Po(f1)1/2
Non
linea
rity
Val
ue b
y C
antr
ell
0 0.02 0.04 0.06 0.08 0.110
-2
10-1
100
101
102 Logarithm Scale(Fused Quartz)
Output Amplitude, Po(f1)1/2
Non
linea
rity
Val
ue b
y Jh
ang
Cantrell Jhang
Fig. 4.22 Nonlinearity Parameter Change with Output Amplitude Change
(3 Sets of Fused Quartz)
This shows that the nonlinearity parameters are varied with the change of the output
amplitude as seen in Fig. 4.22, which is related to the change of the input signal voltage.
The result also shows that there are small fluctuations and the nonlinearity parameter by
Jhang is higher than that of Cantrell.
49
4.5.3 Summary of System Characterization using Fused Quartz Sample
The primary object of this test was to investigate how the ultrasonic data acquisition
system responds to the change of the input signal frequency and the input signal voltage.
The changes of the power spectrum and the nonlinearity parameter were shown. To
characterize the ultrasonic response, the ultrasonic wave signals from the 10 MHz and 5
MHz receivers were analyzed and the sinusoidal function signals from the HP function
generator were also analyzed.
This study showed that the frequency resonance of the ultrasonic transducers, the
receiver, was not related to influence of the ultrasonic second harmonic response. The
nonlinearity parameters by Jhang and Cantrell, recording and analyzing the ultrasonic
signals from the ultrasonic receiver, were reasonably constant as the input signal frequency
was changed from 3.5 MHz to 5MHz. This showed that both nonlinearity parameters are
less responsive to the low frequency of the input signal. The nonlinearity parameter,
however, varied after 5 MHz input signal frequency. Moreover by recording and analyzing
the sinusoidal signals received from the HP function generator, it was also shown that the
fundamental frequency response and its harmonic frequency responses present as
increasing the input signal frequency from 3.5MHz to 7.5MHz. This study also showed that
the nonlinearity parameters by Jhang and Cantrell were increasing linearly as the input
signal frequency increased from 3.5 MHz to 5 MHz.
50
4.6 Nonlinear Response of Various Specimens
4.6.1 Introduction
The objective of this experimentation was to show the relationship between the
ultrasonic nonlinear response and the output amplitude, the power spectrum amplitude at
the fundamental frequency of the received signal. Although the same peak to peak voltage
is used to excite the transmitting ultrasonic transducer, the received voltage of the receiving
ultrasonic transducer may vary due to the measurement variability as well as the response
of various parts of the system. In this study, therefore, the output amplitude instead of the
input signal voltage was focused. The study also shows the calculated nonlinearity
parameter, using power spectrum and bispectrum quantities, is independent of the
ultrasonic wave path length. To conduct this study, three identical 7075-T6 aluminum
samples which have three different lengths, 3 inch, 2 inch, and less than 1 inch, were
prepared. In addition, it is shown that the nonlinearity parameter difference is related to the
material diversity. Several samples such as 2 inch long steel, 1.5 inch long copper alloy, and
1 inch long fused quartz were examined as well.
4.6.2 Test Setup and Experimentation Result
Table 4.11 Testing Parameters for Nonlinear Response of Various Specimens
HP Function Generator GageScope Transducers Specimen
Various Voltage p-p 0.5-5.0 V Sampling Frequency 50MHz Transmitter 5 MHz Material
Frequency 5 MHz No. of Data points 180 Receiver 10 MHz Length Various
Various No. of cycles 15 No. of ensembles 16(3sets) Couplant Sonotrace30 Diameter
51
0 1 2 3 4 510
-2
10-1
100
101 (3in. Al7075-T6)
Input Signal Voltage, Vp-p
Non
linea
rity
Val
ue b
y C
antr
ell
0 1 2 3 4 510
-2
10-1
100
101 (2in. Al7075-T6)
Input Signal Voltage, Vp-p
0
Cantrell Cantrell
102
102
0.02 0.04 0.06 0.08 0.110
-2
10-1
100
101
Output Amplitude, Po(f1)1/2
Non
linea
rity
Val
ue b
y C
antr
ell
0 0.02 0.04 0.06 0.08 0.110
-2
10-1
100
Cantrell Cantrell 101
Output Amplitude, Po(f1)1/2
0 1 2 3 4 510
-2
10-1
100
101 (3in. Al7075-T6)
Input Signal Voltage, Vp-p
Non
linea
rity
Val
ue b
y Jh
ang
0 1 2 3 4 510
-2
10-1
100
101 (2in. Al7075-T6)
Input Signal Voltage, Vp-p
0 0.02 0.04 0.06 0.08 0.110
-2
10-1
100
101
Output Amplitude, Po(f1)1/2
Non
linea
rity
Val
ue b
y Jh
ang
0 0.02 0.04 0.06 0.08 0.110
-2
10-1
100
101
Output Amplitude, Po(f1)1/2
Jhang Jhang
Jhang Jhang
Fig. 4.23 Difference Between Input Signal Voltage Plot and Output Amplitude Plot
52
The nonlinearity parameter difference of the signal data with respect to the input voltage
is exhibited in Fig 4.23. However, one can find that the nonlinearity parameter difference of
the signal data set is reduced when the output signal amplitude is used. This suggests that
the output amplitude of the detected signal rather than the input signal voltage should be
used to estimate the nonlinearity parameters. This approach can reduce the measurement
variability with respect to the received signal voltage.
0 0.02 0.04 0.06 0.08 0.110
-2
10-1
100
101
102 Logarithm Scale
Output Amplitude, Po(f1)1/2
Non
linea
rity
Val
ue b
y C
antr
ell
0 0.02 0.04 0.06 0.08 0.110
-2
10-1
100
101
102Logarithm Scale(Al7075-T6 Cantrell Calculation)
Output Amplitude, Po(f1)1/2
Non
linea
rity
Val
ue b
y C
antr
ell
Total Various Length 7075-T6 Al
Fig. 4.24 Cantrell Nonlinearity Parameter Change with Output Amplitude Change
The result shows that the nonlinearity parameter is increased when the output signal
amplitude is decreased.
By considering the second plot in Fig. 4.24, a significant feature can be observed. The
53
result notes that the nonlinearity calculation results of the 9 different sets of the ultrasonic
signal data have a similar trend with respect to the output amplitudes. The three tested
specimens, 7075-T6 aluminum, have the identical material properties and the same 1 inch
diameter, but they have various lengths; 3 inch, 2 inch, and less than 1 inch. This result
shows that the nonlinearity parameters are independent of the ultrasonic wave traveling
distance through the specimen. This is an interesting result since the amplitude term, , of
the second harmonic frequency component which includes the length information,
ultrasonic wave propagation distance, of the specimen was used to estimate the nonlinearity
parameters of both Cantrell and Jhang. In addition, considering that this plot is the semi-
logarithm scale, the nonlinearity parameters for the output amplitude greater than the 0.05
volts are constant.
2A
0 0.02 0.04 0.06 0.08 0.110
-2
10-1
100
101
102 (3in. Al7075-T6)
Output Amplitude, Po(f1)1/2
Non
linea
rity
Val
ue b
y C
antr
ell
0 0.02 0.04 0.06 0.08 0.110
-2
10-1
100
101
102 (2in. Al7075-T6)
Output Amplitude, Po(f1)1/2
Non
linea
rity
Val
ue b
y C
antr
ell
Fig. 4.25 Cantrell Nonlinearity Parameter Change for 7075-T6 Al Specimens
54
0 0.02 0.04 0.06 0.08 0.110
-2
10-1
100
101
102 (Less 1in. Al7075-T6)
Output Amplitude, Po(f1)1/2
Non
linea
rity
Val
ue b
y C
antr
ell
0 0.02 0.04 0.06 0.08 0.110
-2
10-1
100
101
102 (Steel)
Output Amplitude, Po(f1)1/2
Non
linea
rity
Val
ue b
y C
antr
ell
Steel ~1 inch 7075-T6
0 0.02 0.04 0.06 0.08 0.110
-2
10-1
100
101
102 (Copper Alloy)
Output Amplitude, Po(f1)1/2
Non
linea
rity
Val
ue b
y C
antr
ell
0 0.02 0.04 0.06 0.08 0.110
-2
10-1
100
101
102 (Fused Quartz)
Output Amplitude, Po(f1)1/2
Non
linea
rity
Val
ue b
y C
antr
ell
Copper Alloy Fused Quartz
Fig. 4.26 Cantrell Nonlinearity Parameter Change for Various Material Specimens
55
The relationship between the nonlinearity parameter and the material difference is shown
on Fig. 4.25 and Fig. 4.26. Since all the nonlinearity plots have the same axis range, one
can easily find the considerable dissimilarity among the different samples. Notice that on
Fig. 4.24 through Fig. 4.26, the nonlinearity parameters can be easily distinguished below
0.06 volts of the output amplitude. In addition, the nonlinearity parameter plot of the fused
quartz sample seems to be similar to the plot of the 7075-T6 Al samples. Simply looking at
the acoustical property table 4.12, one can find those two materials have similar acoustic
properties such as density and acoustic impedance. This acoustical property similarity may
lead to the similar nonlinearity parameter plot. In addition, the nonlinearity parameter plots
for the fused quartz and the steel samples show small fluctuation compared with the other
nonlinearity parameters.
Table 4.12 Acoustic Properties of Test Materials
Material Wave Velocity, cm/㎲ Density, g/㎤ Acoustic Impedance, g/㎤-sec 510×
0.632 2.70 17.10 Aluminum
0.428 8.56 36.70 Copper Alloy
0.589 7.71 45.41 Steel 1020
0.557 2.60 14.5 Fused Quartz
56
0 0.02 0.04 0.06 0.08 0.110
-2
10-1
100
101
102 Logarithm Scale
Output Amplitude, Po(f1)1/2
Non
linea
rity
Val
ue b
y Jh
ang
0 0.02 0.04 0.06 0.08 0.110
-2
10-1
100
101
102Logarithm Scale(Al7075-T6 Jhang Calculation)
Output Amplitude, Po(f1)1/2
Non
linea
rity
Val
ue b
y Jh
ang
Total Various Length 7075-T6 Al
Fig. 4.27 Jhang Nonlinearity Parameter Change with Output Amplitude Change
This result shows the similar outcomes of the nonlinearity parameter as in Fig. 4.24.
This notes that the nonlinearity parameter is increased when the output signal amplitude is
decreased. The nonlinearity parameter independence of the ultrasonic wave propagation
distance through the specimen is shown on the second plot in Fig 4.27. Moreover, this
shows that the nonlinearity parameters for the output amplitude greater than the 0.05 volts
are nearly constant. On the other hand, the first plot in Fig. 4.27 shows that there is
relatively large fluctuation of the nonlinearity parameter plot of fused quartz sample.
57
0 0.02 0.04 0.06 0.08 0.110
-2
10-1
100
101
102 Logarithm Scale(3in. Al7075-T6)
Output Amplitude, Po(f
1)1/2
Non
linea
rity
Val
ue b
y Jh
ang
0 0.02 0.04 0.06 0.08 0.110
-2
10-1
100
101
102 Logarithm Scale(2in. Al7075-T6)
Output Amplitude, Po(f
1)1/2
Non
linea
rity
Val
ue b
y Jh
ang
3 inch 7075-T6 Al 2 inch 7075-T6 Al
0 0.02 0.04 0.06 0.08 0.110
-2
10-1
100
101
102 Logarithm Scale(Less 1in. Al7075-T6)
Output Amplitude, Po(f
1)1/2
Non
linea
rity
Val
ue b
y Jh
ang
0 0.02 0.04 0.06 0.08 0.110
-2
10-1
100
101
102 Logarithm Scale(Steel)
Output Amplitude, Po(f
1)1/2
Non
linea
rity
Val
ue b
y Jh
ang
Steel ~1 inch 7075-T6
Fig. 4.28 Jhang Nonlinearity Parameter Change for Various Material Specimens
58
0 0.02 0.04 0.06 0.08 0.110
-2
10-1
100
101
102 Logarithm Scale(Copper Alloy)
Output Amplitude, Po(f1)1/2
Non
linea
rity
Val
ue b
y Jh
ang
0 0.02 0.04 0.06 0.08 0.110
-2
10-1
100
101
102 Logarithm Scale(Fused Quartz)
Output Amplitude, Po(f1)1/2
Non
linea
rity
Val
ue b
y Jh
ang
Copper Alloy Fused Quartz
Fig. 4.29 Jhang Nonlinearity Parameter Change for Copper Alloy and Fused Quartz
This shows the similar results of the nonlinearity parameter as in Fig. 4.25 and Fig. 4.26.
The result shows that the considerable dissimilarity among the different samples is present.
This also shows that the nonlinearity parameters can be easily distinguished below 0.06
volts of the output amplitude. In addition, the nonlinearity parameter plot of the fused
quartz sample seems to be similar to the plot of the 7075-T6 Al samples. On the other hand,
the nonlinearity parameter plots for the fused quartz and the steel samples show relatively
large fluctuation compared with those for the related plots in Fig. 4.28 and Fig. 4.29.
59
4.6.3 Summary of Nonlinear Response of Various Specimens
The primary object of this test was to inspect the change of the nonlinearity parameter
using the various samples and the change of nonlinearity parameter based on the magnitude
of the output signal amplitude when the input signal voltage was varied. The change plot of
the nonlinearity parameter was shown.
This study showed that the nonlinearity parameter whether computed according Cantrell
or Jhang exhibit some variation with the output amplitude rather than the input signal
voltage. It was suggested that the nonlinearity parameter is independent of the length, the
ultrasonic wave propagation distance, of the samples and is related to the material
properties of the various specimens. Moreover this study suggests the meaningful output
amplitude range, which should be considered for the future study.
60
4.7 Ultrasonic Response Test of Cyclic Loading 7075-T6 Aluminum
The primary purpose of this experimentation is to simulate fatigue and to understand and
observe the effect of cyclic loading on a 7075-T6 aluminum specimen and distinguish the
developed techniques for measuring the material degradation with respect to fatigue
damage. By researching fatigue behavior with spectral analysis of ultrasonic wave, we can
better understand and predict the remaining life of the specimen.
In this experiment, the aluminum alloy specimen was tested under the designated cyclic
loading. This test was conducted by recording ultrasonic signal data every 5000 cycles and
then analyzing the recorded data with statistical methods such as the power spectrum,
bispectrum, and bicoherence spectrum.
It is sufficient to measure changes in nonlinear response as long as the changes are large
enough to be reliably measured.
4.7.1 Specimen Preparation
4.7.1.1 Specimen Description
7075-T6 aluminum alloy “dog-bone” style specimens were prepared for this study. The
specimens were fabricated from 1 inch diameter rod stock. The specimens were tapered
from both ends toward a middle section of constant cross sectional area to reduce the stress
concentration near the end regions. Dimensions of a specimen were detailed in Fig. 4.30.
The mechanical and acoustic properties for general 7075-T6 aluminum alloy are found in
Table 4.13 and 4.14, respectively.
61
Table 4.13 Mechanical Properties for 7075-T6 Aluminum
7075-T6 Aluminum
Elastic Modulus Poisson’s Ratio Yield Strength Crystal Structure Ultimate Strength
ν0σ yσE [GPa( ksi)] 310 [MPa(ksi)] [MPa(ksi)]
71(10.3) 469(68) 578(84) 0.345 FCC
Table 4.14 Acoustic Properties for 7075-T6 Aluminum
Aluminum
Longitudinal Velocity Shear Velocity Acoustic Impedance
[m/s (in/㎲)] [m/s (in/㎲)] [g/cm2-sec ] 510×
6320 (0.2488) 3130 (0.1232) 17.10
1 in.
R=0.125 in. 0.5 in.
0.5 in.
0.65 in.
3 in. 1.05 in. 3 in.
Fig. 4.30 Schematic Diagram of the Specimen
4.7.1.2 Solution Heat Treatment for Specimen
7075-T6 aluminum alloys are Al-Zn-Mg(-Cu) alloys, which are heat treatable materials
containing Zn as the major alloying element. These alloys are fabricated by a precipitation
aging process. Relatively high strength, poor corrosion behavior and moderate fatigue
performance are the general characteristics of 7075-T6 aluminum alloys.
62
Solution heat treatment may be used to redisolve the Zn precipitate particles and make
dislocation movement easier. Solution heat treatment was performed by heating the
specimens in an oven at 900℉, for 4 hours and then quenching it in the agitated water.
4.7.2 Test Setup and Experimentation
4.7.2.1 Cyclic Loading System and Specimen Fatiguing
The specimen was subjected to uniaxial cyclic loading using a MTS closed loop
servohydraulic testing machine, which was used to apply sinusoidal loading varying
between 800 lb and 8000 lb. In this test, a load cell was employed to measure and control
the load. The frequency at which the load was applied was 10 Hz.
A specially fabricated set of gripping cages which provided access to the end of the test
specimen were used during the designated cyclical loadings. This set of cages was used to
reach the end of the sample to transmit and receive the ultrasonic signal using ultrasonic
transducers[21].
A universal coupler was used to connect the grip assembly, Fig. 4.31, to the MTS
machine. The major design criteria, for this assembly, were that easy to access be provided
at either end of the specimen for transducer placement, uniform uniaxial application of the
load and reasonable acoustic isolation from the cyclic loading machine. A cage-like design
of two steel plates, one of which was slotted to accept the specimen, connected by four rods
of steel was chosen. A split steel collar was used to accurately align the specimen as well as
uniformly distribute the load.
63
Anchor
Cyclic Loading
7075-T6
Specimen
Collar Specially Designed
Cage style Grip
1 inch
0.75 in.
0.75 in.
Anchor Part
Cyclic Loading Part
Fig. 4.31 Specimen Gripping and Cyclic Loading Configuration[21]
In this experiment, the specimen was cyclically loaded. After 5000 cycles we stopped the
MTS machine and unloaded until there is completely no load on the specimen, and then we
put the collars and positioned ultrasonic transducers as a receiver and a transmitter,
respectively on the top and bottom ends of the test specimen.
64
4.7.2.2 Ultrasonic Data Acquisition
The HP function generator was used to generate a 15 cycle 5 MHz sinusoidal tone burst
signal with 5 volt peak to peak voltage. As shown in Fig.4.3, 5 MHz and 10 MHz ultrasonic
transducers were used as a transmitter and a receiver respectively. Both ultrasonic
transducers were coupled to the unloaded specimen ends, allowing propagation of the
sound wave along the major axis. A small weight was used to press the receiver on to the
top end of the test specimen and a spring loading device was used to apply pressure to the
transmitter.
10MHz Receiver
Fig. 4.32 Transducer Positioning
A 12 bit analog to digital computer acquisition system was used to collect ultrasonic
wave signal, at a sampling rate of 50 MHz. The initial disturbance which separated on the
digital oscilloscope window was captured. 16 ensemble signal records were collected to
analyze the nonlinear ultrasonic wave responses.
Traveling UT Signal
5MHz Transmitter
65
4.7.3 Result and Discussion
The specimen was tested until failure. It failed in the middle section of the specimen at
20669 cycles, so 5 sets of ultrasonic signal data every 5000 cycles from 0 (base) to 20000
cycles were collected.
To analyze the nonlinear ultrasonic signal response a MATLAB code developed by
Hajj[7] based on the spectral analysis method, such as power spectrum, bispectrum, and
bicoherence spectrum was used. Table 4.16 lists the spectral analysis quantities calculated
by the MATLAB code. For the nonlinearity parameters by Cantrell and Jhang, and cβ
, respectively, we recalled Eq.(2.3.6) and Eq.(2.3.11). Jβ
Table 4.15 Testing Parameters for Ultrasonic Response Test of Cyclic Loading Aluminum
HP Function Generator GageScope Transducers Specimen
7075-T6 Al Voltage p-p 5 V Sampling Frequency 50MHz Transmitter 5 MHz Material
8.55 inch Frequency 5 MHz No. of Data points 180 Receiver 10 MHz Length
1 inch No. of cycles 15 No. of ensembles 16(3sets) Couplant Sonotrace30 Diameter
66
Table 4.16 Spectral Analysis Calculation Table using MATLAB codes
Set P(f1)=A12 P(f2) |B(f1,f1)| A2=√P(f2) βc βJ
1 0.002007091 3.39139E-08 1.03078E-06 0.000184157 0.091753266 0.2558785152 0.001592146 2.59823E-08 7.00806E-07 0.00016119 0.101240874 0.2764598613 0.001415216 2.21395E-08 5.76826E-07 0.000148794 0.10513835 0.288004535
Set P(f1)=A12 P(f2) |B(f1,f1)| A2=√P(f2) βc βJ
1 0.002622512 1.50299E-07 2.8623E-06 0.000387684 0.147829351 0.4161787812 0.002528257 1.43351E-07 2.69874E-06 0.000378617 0.149754307 0.4221997183 0.002142387 1.30123E-07 2.16639E-06 0.000360725 0.168375477 0.471998382
Set P(f1)=A12 P(f2) |B(f1,f1)| A2=√P(f2) βc βJ
1 0.003099352 1.37111E-07 3.22634E-06 0.000370285 0.119471667 0.335867252 0.0027522 1.28991E-07 2.77172E-06 0.000359153 0.130496667 0.3659223423 0.002463467 1.30158E-07 2.49178E-06 0.000360774 0.146449826 0.410597021
Set P(f1)=A12 P(f2) |B(f1,f1)| A2=√P(f2) βc βJ
1 0.00328659 2.09445E-07 4.23583E-06 0.000457651 0.139248008 0.3921454182 0.003052593 2.11009E-07 3.95542E-06 0.000459357 0.150480777 0.4244770133 0.002589382 1.7885E-07 3.09122E-06 0.000422907 0.163323521 0.461038313
Set P(f1)=A12 P(f2) |B(f1,f1)| A2=√P(f2) βc βJ
1 0.002765444 1.56414E-07 3.07355E-06 0.000395492 0.143012 0.4018935722 0.002483052 1.48307E-07 2.68515E-06 0.000385106 0.155093872 0.4355095123 0.002322883 1.33568E-07 2.37615E-06 0.000365469 0.157334205 0.440371226
Spectral analysis with Solution Heat Treated Specimen(20000 cycles)
Spectral analysis with Solution Heat Treated Specimen(base, 0 cycle)
Spectral analysis with Solution Heat Treated Specimen(5000 cycles)
Spectral analysis with Solution Heat Treated Specimen(10000 cycles)
Spectral analysis with Solution Heat Treated Specimen(15000 cycles)
Although three sets of ultrasonic signal data were recorded and used for calculation, the
highlighted set for each point in the cycle count from Table 4.16 is plotted in Fig. 4.33
through Fig. 4.36. The reason why we used only one set for each cycle is subjected to
discuss briefly in the conclusions of this section. From Table 4.16 one could easily find that
both nonlinearity parameters are increasing as the power spectrum values decrease.
67
0 5 10 15 2010
-10
10-5
Pow
er
Frequency (MHz)
Spectrum
0 5 10 15 2010
-10
10-5
Pow
er
Frequency (MHz)
Spectrum
base
0 5 10 15 2010
-10
10-5
Pow
er
Frequency (MHz)
5000
0 5 10 15 2010
-10
10-5
Pow
er
Frequency (MHz)
10000
0 5 10 15 2010
-10
10-5
Pow
er
Frequency (MHz)
15000
0 5 10 15 2010
-10
10-5
Pow
er
Frequency (MHz)
20000
Total
Fig. 4.33 Power Spectrum Plots for Data Collected Every 5000
The power spectra in Fig. 4.33 show the energy distribution of frequency components
in the received nonlinear ultrasonic signals. Since 5 MHz ultrasonic signal was generated,
the most significant energy at the 5 MHz frequency component in all the power spectra was
present. The results on the other hand indicate that the second harmonic frequency
component at 10 MHz can be seen in all the plots. In addition, the noise in the power
spectra is increasing as the specimen is subjected to increasing cyclic loading.
68
0 5 10 15 20 25
-20
-10
0
10Auto-Bicoherence at Base
Freq
uenc
y, M
Hz
Frequency, MHz
0 5 10 15 20 25
-20
-10
0
10Auto-Bicoherence at 5000 cycles
Freq
uenc
y, M
Hz
Frequency, MHz
0 5 10 15 20 25
-20
-10
0
10Auto-Bicoherence at 10000 cycles
Freq
uenc
y, M
Hz
Frequency, MHz
0 5 10 15 20 25
-20
-10
0
10Auto-Bicoherence at 15000 cycles
Freq
uenc
y, M
Hz
Frequency, MHz
0 5 10 15 20 25
-20
-10
0
10Auto-Bicoherence at 20000 cycles
Freq
uenc
y, M
Hz
Frequency, MHz
0.20.4
0.6
0.8
Fig. 4.34 Bicoherence Spectrum Colormap and Contour Plots
69
Auto-Bicoherence at Base
Freq
uenc
y, M
Hz
Frequency, MHz0 2.5 5 7.5 10
0
2.5
5
7.5
Auto-Bicoherence at 5000 cycles
Freq
uenc
y, M
Hz
Frequency, MHz0 2.5 5 7.5 10
0
2.5
5
7.5
Auto-Bicoherence at 10000 cycles
Freq
uenc
y, M
Hz
Frequency, MHz0 2.5 5 7.5 10
0
2.5
5
7.5
Auto-Bicoherence at 15000 cycles
Freq
uenc
y, M
Hz
Frequency, MHz0 2.5 5 7.5 10
0
2.5
5
7.5
Auto-Bicoherence at 20000 cycles
Freq
uenc
y, M
Hz
Frequency, MHz0 2.5 5 7.5 10
0
2.5
5
7.5
Fig. 4.35 Bicoherence Spectrum Contour Plot only for [0.4(blue) 0.6(green) 0.8(red)]
Since the power spectrum is affected by Gaussian noise, the bispectral analysis such as
bispectrum and bicoherence spectrum was used to detect and measure the nonlinear
relations of the ultrasonic response signals[2, 20].
These plots, Fig. 4.34 and Fig. 4.35, show several features. The bicoherence spectrum
plot shows that high levels of coupling among low frequency components present before
applying cyclic load to the test specimen. This coupling could be found in the power
spectra plots. Low frequency components between 1 MHz and 8 MHz have lots of energy
in Fig. 4.33 and this unwanted ultrasonic signal energy distribution results in the
undesirable coupling. Moreover it is hardly to see the change of the nonlinear interactions
70
between the fundamental and the second harmonic frequency components after 5000 cycles.
To obtain more valuable bicoherence spectrum plots, it was decided to decrease the input
signal in order to investigate the ultrasonic signal energy distribution at low frequencies.
0 5 10 15 20
10-1
Beta Change by Cantrell
Kcycles
Non
linea
rity
Par
amet
er
0 5 10 15 20
10-1
Beta Change by Jhnag
Kcycles
Non
linea
rity
Par
amet
er
Fig. 4.36 Nonlinearity Parameter Changes by Cantrell and Jhang
These two plots of change in nonlinearity parameters are similar, with relatively small
amounts of nonlinearity exhibited before the specimen was cyclically loaded. Both
nonlinearity parameters by Cantrell and Jhang increase at 5000 cycles and then were steady
through 20000 cycles of loading. This trend can be found in the power spectra and
bicoherence spectra plots, Fig. 4.33 through Fig. 4.35. Cantrell noticed this trend in his
previous work[3].
71
The change in the Jhang nonlinearity parameter is preferred because he used bispectrum
value for his calculation, which is insensitive to background noise and incorporates the
phase information of the signal.
4.7.4 Summary of Ultrasonic Response Test of Cyclic Loading 7075-T6 Aluminum
In this work, the nonlinear response of the ultrasonic signal traveling through the 7075-
T6 Al specimen that was tested under cyclic loading was investigated. The primary goal of
this study was to measure and analyze the material fatigue degradation using spectral
analysis techniques. One solution treated 7075-T6 aluminum specimen was tested in the
cyclic loading setting. Ultrasonic signals were transmitted and received using the ultrasonic
equipment. The recorded ultrasonic signal data were analyzed with the higher order spectral
analysis techniques.
The results reported suggest that the spectral analysis such as the power spectrum and the
bispectrum is the reliable method to monitor the change of nonlinear relation between the
excited frequency and the second harmonic frequency components. In addition, the results
show that the spectral analysis technique is closely related to the received signal energy, i.e.
the output amplitude of the received signal.
72
CHAPTER V
CONCLUSIONS
5.1 Thesis Summary
By using the magnitudes of the power spectrum and the bispectrum, the nonlinearity
parameters by Cantrell and Jhang were provided in Chapter 2. The bicoherence spectrum,
normalized bispectrum, was also defined for the characterization of the wave interactions.
The experimental setup which includes the HP function generator, the ultrasonic
transducers and the data acquisition system was described in Chapter 3. This chapter
detailed how the ultrasonic signal was transmitted and received through a specimen, and
how the measurement variability was assessed.
In section 4.4 of Chapter 4, the input signal with respect to the voltage change and the
frequency change was characterized where the HP function generator was connected
directly to the GageScope detection system.
In section 4.5, using a fused quartz specimen, the response of the ultrasonic data
acquisition system with respect to the change of the input signal frequency and voltage was
described. To characterize the ultrasonic data acquisition system response, the ultrasonic
wave signals received from the 10 MHz and 5 MHz receivers were analyzed and the
sinusoidal signals directly received from the HP function generator were also analyzed.
In section 4.6, the nonlinearity parameter change related to the material difference was
73
investigated. The relationship between the nonlinearity parameter and the output signal
amplitude was described. The result also showed that the nonlinearity parameter is
independent of the ultrasonic wave propagation distance.
The nonlinear response of the ultrasonic signal traveling through the solution treated
7075-T6 aluminum specimen that was tested under cyclic loading was described in section
4.7. The material fatigue degradation using spectral analysis techniques was measured and
analyzed. The result showed that the nonlinearity parameter and the bicoherence spectrum
vary as the number of fatigue cycles is increased.
74
5.2 Conclusions and Future Work Recommendations
This thesis discusses the practical aspects related to assessing nonlinear ultrasonic
response. The research is focused on the ultrasonic NDE technique to characterize the
ultrasonic nonlinear response of the cyclically load 7075-T6 aluminum. Studies conducted
in this thesis make the nonlinear response estimation using spectral analysis techniques
more useful. In order to estimate the level of the fatigue life, the examination of the
nonlinearity parameters by Cantrell and Jhang, and the bicoherence spectrum was
performed.
The results show that the nonlinearity parameters by Cantrell and Jhang are responsive
to the output amplitude, the power spectrum magnitude at the fundamental frequency
component, of the received signal. Both nonlinearity parameters vary for the various
materials, are less responsive to the low frequency (from 3.5 MHz to 5 MHz) of the input
signal, and are independent of the ultrasonic wave propagation distance. In addition, the
results presented show that the nonlinearity parameter by Jhang, using the power spectrum
and the bispectrum quantities of the fundamental and the second harmonic frequency
components, is in agreement with the nonlinearity parameter by Cantrell, using only power
spectrum quantities. Moreover, the results of the solution treated 7075-T6 aluminum
specimen test show that the material nonlinearity parameters by Cantrell and Jhang are
increased and the coupling levels between the fundamental, its harmonic, and subharmonic
frequency components increase as the number of fatigue cycles is increased. This suggests
that the application of the bicoherence spectrum is more effective for evaluating the level of
the material fatigue damage or degradation due to the growth of the dislocation
substructures. Basically, the bicoherence spectrum is based on the nonlinear wave coupling
75
relations between the fundamental and its harmonic frequency components rather than on
the spectral magnitudes of the fundamental and the second harmonic frequency components.
Future work should be conducted to reduce experimental error and create more
meaningful data to estimate the ultrasonic signal nonlinear response as a function of the
material fatigue life. It is recommended that the output amplitude from the receiving
transducer, which may be related to the ultrasonic nonlinear response, should be carefully
considered when the ultrasonic measurements are performed and the voltage of the input
signal should be decreased until the level of the bicoherence spectrum, the low frequency
wave coupling, is small. In order to eliminate the unwanted frequency components from the
ultrasonic transducer excitation, frequency filters are suggested.
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REFERENCES
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VITA
Byungseok Yoo
Byungseok Yoo was born on November 18, 1977 to Mansik Yoo and Nohsook Kim in
Nonsan, South Korea. Byungseok was raised in Changwon, South Korea and graduated
from Masan High School. He started pursuing his undergraduate degree in Inje University,
South Korea. Byungseok graduated top student with honors from Inje University with a
Bachelor of Science degree in Mechanical and Automotive Engineering in February of
2003. He decided to continue the engineering education and started graduate studies at
Virginia Polytechnic Institute and State University in August of 2004, finishing his M.S. in
Engineering Science and Mechanics in December, 2006. His research focused on studying
ultrasonic NDE method and spectral analysis applications to estimate fatigue damage in
aluminums.