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.