Ultrasonic Field Modeling of Transient Wave Propagation in Homogenous and Non-Homogenous Fluid Media Using Distributed Point Source Method (DPSM)

Transcription

1 Ultrasonic Field Modeling o Transient Wave Propagation in Homogenous and Non-Homogenous Fluid Media Using Distributed Point Source Method (DPSM) A Thesis report Submitted in the partial ulillment o requirements or the award o the degree o Master o Engineering in CAD/CAM & ROBOTICS Submitted by RAGHU RAM TIRUKKAVALLURI Roll no Under the guidance o Dr. ABHIJIT MUKHERJEE Director Thapar University, Patiala Mr. SANDEEP SHARMA Sr. Lecturer, MED Thapar University, Patiala Department o Mechanical Engineering THAPAR UNIVERSITY PATIALA (PUNJAB) JUNE-008

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3 Acknowledgement I take this opportunity to express my sincere gratitude to Dr. Abhijit Mukherjee, Director, Thapar University or giving me the opportunity o doing my thesis work under his guidance. I am also thankul to him or his constant supervision and valuable suggestions. It is my proud privilege to express regards and sincere gratitude to Mr.Sandeep Sharma, Sr. Lecturer, Mechanical Engineering Department, Thapar University, Patiala, or his patient listening o my ideas and also suggesting new ways or implementing my ideas by his expert guidance through out my work. I am also thankul to Prashant Bhise, Reasearch Student, IIT Bombay, or sharing his conceptual ideas and guiding me by giving his helping hand in successul completion o my thesis work. I would like to extend my thanks to Mrs.Shruti Sharma, Sr. Lecturer, Civil Engineering Department, Thapar University, Patiala, or her guidance and providing necessary inormation and acilities or the successul completion o my thesis. I am also thankul to Dr.S.K.Mohapatra, Head Mechanical Engineering Department, Thapar University, Patiala, or the motivation and inspiration that triggered me or my thesis work. I also take this opportunity to thank to the entire aculty and sta o Mechanical Engineering Deparment, Thapar University, Patiala, or their help, inspiration and moral support, which went a long way in successully completion o this report. Tirukkavalluri.Raghu Ram (806806)

4 ABSTRACT In the ield o nondestructive evaluation (NDE), the newly developed distributed point source method (DPSM) is gradually gaining popularity. DPSM is a semi-analytical technique used to calculate the ultrasonic ields (pressure, velocity and displacement ields) generated by ultrasonic transducers o inite dimension immersed in homogeneous or non-homogeneous media. In this report the technique is extended to model the propagation o transient wave and its pressure tomo-grams generated both in homogeneous luid and non-homogeneous luid having single interace. Tone burst signal is used as input signal at one end o luid and relected as well as transmitted pulse is observed at various points. In the irst case DPSM model or homogeneous luids is developed and results are validated or closed orm solutions. In the second case, interace o two luids is kept perpendicular to wave propagation direction and wave propagation is studied at various points. In both cases, Fast Fourier transormation (FFT) is used convert time domain signal into requency domain and inverse FFT is used to again transorm results in time domain. Numerical results obtained rom DPSM model are compared with experimental results taken rom the experimental setup on dierent luids and the time-histories are ound to be matching with reasonable accuracy. Keywords: DPSM, FFT, Tomo-grams, Non-Homogeneous luid, Wave Propagations, Transducer, Interace

5 CERTIFICATE ACKNOWLEDGEMENT ABSTRACT LIST OF FIGURES CONTENTS Page i ii iii iv CHAPTER Introduction. NDT and its methods. Basic concepts o wave propagation.. Basic Theory.. Modes o Wave Propagation 5.. Mechanics o Wave Propagation 8. Ultrasonic Testing.. Basic Principle o Ultrasonic Testing.. Methods o Ultrasonic Testing.4 Excitation Signal analyses CHAPTER Literature Review 6 CHAPTER Numerical Techniques. Ray Tracing Method 0. Spectral Approach 5. Finite Element Approach.4 Distributed Point Source Method (DPSM) 6

6 CHAPTER 4 Ultrasonic Field Modeling in Homogenous Fluid Using DPSM Technique 4. Computation o Velocity, Pressure and Displacement Fields 44 in a luid generated by a group o point sources 4. Matrix representation 47 CHAPTER 5 Ultrasonic Field Modeling in Layered Fluids (or) Non- Homogeneous Media using DPSM Technique 5.. Introduction 5 5. Methods or Finding Source Strength vectors 5.. Computation o the source strength vectors when multiple 5 Relections between the transducer and the interace are ignored 5.. Computation o the source strength vectors considering the 55 Interaction eects between the transducer and the interace 5. Obtaining the Ultrasonic ields ater knowing the source strength vectors 56 CHAPTER 6 Numerical Results and Discussion 6. Method used or inding source point coordinates Ultrasonic Field in a Homogenous Fluid-DPSM technique 6.. Steady wave propagation in Homogenous luid Transient wave propagation in homogenous luid 6 6. Ultrasonic Field in a Non-Homogenous Fluid-DPSM technique 6.. Steady Wave Propagation in a Non-homogenous luid Transient wave ields in a Non-homogeneous luid Acoustic pressure Distribution with respect to time 85

7 CHAPTER 7 Experimental Validation 7. Experimental Setup Details Experimental procedure 9 7. Checking the Linearity o the transducer using DPSM-technique 9 and experimental results. 7.4 Comparison o DPSM Time history with Experimental Time history 94 CONCLUSIONS 00 SCOPE OF FUTURE WORK 0 REFERENCES 0

8 List o Figures Figures Page no Figure. Relection and Transmission o sound wave at normal incidence 4 Figure. Propagation o Longitudinal waves 5 Figure. propagation o Transverse or Shear waves 5 Figure.4 particle movement showing the propagation o Longitudinal and Shear 6 waves Figure.5: propagation o surace or rayleigh waves 6 Figure.6: lamb waves propagation (a) symmetrical (b) asymmetrical waves 7 Figure.7: Forces acting in the x direction on an elemental volume 8 Figure.8: General ultrasonic Inspection Principle (pulse echo method) Figure.9: Principle o pulse echo method o inspection Figure.0: Principle o through transmission o ultrasonic testing Figure.: Time domain representation o a periodic signal. 4 Figure.: The amplitude spectrum o the periodic signal 4 Figure.: igure showing a time record o N equally spaced samples o the input Figure..: -D stress wave propagation through discretely layered FGM 0 Figure..: Gradient Architecture o FGMs Figure..: -D stress wave propagation through discretely layered FGM Figure..: Flow diagram or wave reconstruction program 0 Figure...: Initial coniguration o a rod with a concentrated load, P, at the ree end. Figure...: Coniguration at the end o increment o a rod with a concentrated load, P, at the ree end. Figure...: Coniguration o the rod at the beginning o increment. 5 Figure..4: Coniguration o the rod at the beginning o increment Figure.4.:(a) Point source generating spherical waveront (b) Line source generating cylindrical waveront(c) Ininite plane source generating plane waveront 4 6 Figure.4.: Four point sources distributed over a inite surace 7 Figure.4.: Position o particles or (a) Point source (b) Distributed inite 8 number o points (c) Large number o point sources (d) components o motion o multiple point sources Figure. 4. (a) Position o an observation point (target point) and its distance 44

9 rom the nth point source on the transducer surace, (b) Side view o a transducer and actual positions o the point sources. Figure.4. Rotation o the transducer with respect to x -axis and velocity o the 47 n th observation point adjacent to the transducer ace Figure.5. Distribution o point sources in the layered luid system 5 Figure 5.: Point P can receive two rays, (direct ray) and (relected rom 5 interace) rom a single point source. Figure.6.: AUTOCAD drawing showing polar array o point sources 58 Figure.6.: Distribution o Point sources on the ace o lat circular 58 Transducer Figure.6.: AUTOCAD drawing showing polar array o 48 point sources 59 Figure.6.4: Distribution o 48 Point sources on the ace o lat circular 59 Transducer Figure.6.5: Acoustic Pressure Variation or point sources along Z axis 6 Perpendicular to Transducer Face. Figure. 6.6: Acoustic Pressure Variation or 48 point sources along Z axis 6 Perpendicular to Transducer Face. Figure 6.8: Flow chart or Wave Reconstruction program 66 Figure 6.7: Plot showing Acoustic pressure variation Vs requency 65 Figure 6.9: Hal sine wave as input pulse and its FFT 67 Figure 6.0: Ultrasonic ield response at transducer ace or Hal Sine Wave 68 Velocity Impulse Figure 6.: Full Sine wave as input pulse and its FFT 69 Figure 6.: Ultrasonic ield response at transducer ace or Full Sine Wave 70 Velocity Impulse Figure. 6.: Hal Triangular wave as input pulse and its FFT 7 Figure 6.4: Ultrasonic ield response at transducer ace or Hal Triangular 7 Wave Velocity Impulse Figure 6.5: Triangular wave as input pulse and its FFT 7 Figure 6.6: Ultrasonic ield response at transducer ace or Triangular Wave 74 Velocity Impulse Figure 6.7: Tone Burst wave as input pulse and its FFT 75 Figure 6.8: Ultrasonic ield response at transducer ace or Tone Burst Wave 76 Velocity Impulse Figure. 6.9: Acoustic pressure Response at various points on Normal to 77 Transducer Figure 6.0: Acoustic Pressure in XY Plane close to the transducer Face 78 Figure 6.: Acoustic Pressure in XY Plane at Interace (Z=0mm) o two luid 78 Figure 6.: Location o target point where the response o ultrasonic ield is 80 observed Figure 6.: Location o Target point where the response o ultrasonic ield is 80 observed Figure6.4: Tone Burst Signal -Pressure response at Target point (at Z=0mm) 8 Figure 6.5: Tone Burst Signal -Pressure response at Target point (atz=00mm) 8

10 Figure 6.6: Tone Burst Signal-Velocity response at Target point (at Z=0mm) 8 Figure 6.7: Tone Burst Signal-Velocity response at Target point (at 84 Z=00mm) Figure 6.8: Location o X-Z plane and the target point grid used to generate 85 tomo-grams Figure 6.9: Acoustic Pressure Tomograms or Homogenous luid 86 Figure 6.0: Acoustic Pressure Tomo-grams or homogenous luid (Rigid 87 interace) Figure 6.: Acoustic pressure Tomo-grams or Non-Homogenous Fluid 88 Figure 6.: Acoustic Pressure Tomo-grams or two luids 89 Figure 7.: Experimental setup used or testing 90 Figure 7.: Variation o acoustic pressure with respect to input velocity pulse 9 Figure 7.: Variation o out put with respect to input voltage 9 Figure 7.4: Variation o Output voltage Vs Time along with peak amplitudes 95 using pulse echo method in kerosene oil. (Homogenous luid) Figure7.5: Variation o Output voltage Vs Time along with peak amplitudes 96 using pulse echo method in water (homogenous luid) Figure7.6: Variation o Output voltage Vs Time along with peak amplitudes 97 using pulse echo method in kerosene and water (non-homogenous case). Figure7.7: Variation o Output voltage Vs Time along with peak amplitudes using pulse echo method in water but transducer at bottom. Figure7.8: Variation o Output voltage Vs Time along with peak amplitudes using through transmission method in water

11 CHAPTER INTRODUCTION. NDT AND ITS METHODS Non Destructive Testing: The ield o Nondestructive testing (NDT) is a very broad, interdisciplinary ield that plays a critical role in assuring that structural components and systems perorm their unction in a reliable and cost eective ashion. The term is generally applied to investigations o material integrity. These tests are perormed in a manner that does not aect the uture useulness o the object or material. Because it allows inspection without interering with a product's inal use, NDT provides an excellent balance between quality control and cost-eectiveness. Non Destructive Evaluation: Nondestructive evaluation (NDE) is a term that is oten used interchangeably with NDT. However, technically, NDE is used to describe measurements that are more quantitative in nature. For example, a NDE method would not only locate a deect, but it would also be used to measure something about that deect such as its size, shape, and orientation. NDE may be used to determine material properties such as racture toughness, ormability, and other physical characteristics. NDT or NDE methods: Although a number o dierent NDT methods have been developed, but ollowing methods are most commonly used. Visual and Optical Testing-Visual inspection involves using an inspector's eyes to look or deects. The inspector may also use special tools such as magniying glasses or mirrors gain access and more closely inspect the subject area. Penetrant Testing- Test specimens are coated with visible or luorescent dye solution. Excess dye is then wiped out rom the surace, and a developer is applied. The developer acts as blotter, drawing trapped penetrant out o

12 imperections open to the surace. With visible dyes, vivid colour contrasts between the penetrant and developer make bleedout" easy to see. Magnetic Particle Testing (MT) In this method a magnetic ield in a erromagnetic material is induced and then dusting the surace with iron particles (either dry or suspended in liquid) is done. Surace and near-surace imperections distort the magnetic ield and concentrate iron particles near imperections, previewing a visual indication o the law. Electromagnetic Testing (ET) or Eddy Current Testing- Eddy currents are generated in a conductive material by an induced alternating magnetic ield and they low in circles at just below the surace o the material. Interruptions in the low o eddy currents, caused by imperections, dimensional changes, or changes in the materials conductive and permeability properties, can be detected with the proper equipment. Radiography (RT) - Radiography involves the use o penetrating gamma or X- radiation to examine parts and products or imperections. An X-ray generator or radioactive isotope is used as a source o radiation. The resulting shadowgraph shows the dimensional eatures o the part. Possible imperections are indicated as density changes on the ilm in the same manner as medical X-ray shows broken bones. Ultrasonic Testing (UT) - ultrasonic testing use transmission o high-requency sound waves into a material to detect imperections or to locate changes in material properties. The most commonly used ultrasonic testing technique is pulse echo, wherein sound is introduced into a test object and relections (echoes) are returned to a receiver rom internal imperections or rom the part's geometrical suraces. Acoustic Emission Testing (AE) - when a solid material is stressed, imperections within the material emit short bursts o acoustic energy called "emissions." as in ultrasonic testing; acoustic emissions can be detected by special receivers. Emission sources can be evaluated through the study o their intensity, rate, and location.

13 Leak Testing (LT) - Several techniques are used to detect and locate leaks in pressure containment parts, pressure vessels, and structures. Leaks can be detected by using electronic listening devices, pressure gauge measurements, liquid and gas Penetrant techniques, and/or a simple soap-bubble test. Wave propagation provides an eicient means o characterizing deects in structures. For this purpose it is necessary to analyze scattering o waves by such deects. The sudden occurrence o small laws initiated rom damage sites in structural solids generates elastic waves that carry important inormation on the nature o damage.so next section describes about it.. BASIC CONCEPTS OF WAVE PROPAGATION.. Basic Theory: Sound Waves: - Sound waves are simply organized mechanical vibrations traveling through a medium, which may be a solid, a liquid, or a gas. These waves will travel through a given medium at a speciic speed or velocity, in a predictable direction, and when they encounter a boundary with a dierent medium they will be relected or transmitted according to simple rules. This is the principle o physics that underlies ultrasonic law detection. Frequency: - All sound waves oscillate at a speciic requency, or number o vibrations or cycles per second, which we experience as pitch in the amiliar range o audible sound. Human hearing extends to a maximum requency o about 0,000 cycles per second (0 KHz), while the majority o ultrasonic law detection applications utilize requencies between 500 KHz to 0 MHz. At requencies in the Megahertz range, sound energy does not travel eiciently through air or other gasses, but it travels reely through most liquids and common engineering materials. Wave Speed: - The speed o a sound wave varies depending on the medium through which it is traveling, aected by the medium's density and elastic properties. Dierent types o sound waves will travel at dierent velocities.

14 Wavelength o a Wave: - Wavelength is related to requency and velocity by the simple equation Wavelength ( λ ) =velocity (V)/requency (). In ultrasonic law detection, the generally accepted lower limit o detection or a small law is onehal wavelength, and anything smaller than that will be invisible. In ultrasonic thickness gauging, the theoretical minimum measurable thickness is one wavelength. Acoustic impedance: Sound travels through materials under the inluence o sound pressure. Because molecules or atoms o a solid are bound elastically to one another, the excess pressure results in a wave propagating through the solid. The Acoustic impedance (Z) o a material is deined as the product o its density ( ρ ) and acoustic velocity (V). Z = ρ V Figure.: Relection and Transmission o sound wave at normal incidence Z Z ρv ρv Relection coeicient, R = = Z + Z ρv + ρv Z ρv Transmission coeicient, T = = Z + Z ρ V + ρ V 4

15 .. Modes o Wave Propagation: The ultrasonic waves propagate in a number o ways in a medium. On the basis o the mode o particle displacement, these waves can be classiied as: a) Longitudinal or Compressional waves(l-waves) b) Transverse or Shear waves (S-waves) c) Surace or Rayleigh waves d) Lamb or Plate waves Longitudinal or Compressional waves: In longitudinal waves, the oscillations occur in the longitudinal direction or the direction o wave propagation. Since compressional and dilatational orces are active in these waves, they are also called pressure or compressional waves. Compression waves can be generated in liquids, as well as solids because the energy travels through the atomic structure by a series o comparison and expansion (rareaction) movements. Figure.: Propagation o Longitudinal waves Transverse or Shear waves: In the transverse or shear wave, the particles oscillate at a right angle or transverse to the direction o propagation. Figure.: propagation o Transverse or Shear waves 5

16 Shear waves require an acoustically solid material or eective propagation, and thereore, are not eectively propagated in materials such as liquids or gasses. Shear waves are relatively weak when compared to longitudinal waves. Figure.4: Particle Movement Showing the Propagation o Longitudinal and Shear Waves. Surace (or Rayleigh) waves: Surace (or Rayleigh) waves travel the surace o a relatively thick solid material penetrating to a depth o one wavelength. The particle movement has an elliptical orbit as shown in the image and animation below. Rayleigh waves are useul because they are very sensitive to surace deects and they ollow the surace around curves. Because o this, Rayleigh waves can be used to inspect areas that other waves might have diiculty reaching. Figure.5: Propagation o Surace or Rayleigh Waves 6

17 Lamb waves or plate waves: Plate waves can be propagated only in very thin metals. Lamb waves are the most commonly used plate waves in NDT. Lamb waves are complex vibrational waves that travel through the entire thickness o a material. Propagation o lamb waves depends on the density and the elastic material properties o a component. They are also inluenced a great deal by the test requency and material thickness. With lamb waves, a number o modes o particle vibration are possible, but the two most common are symmetrical and asymmetrical. The complex motion o the particles is similar to the elliptical orbits or surace waves. Symmetrical lamb waves move in a symmetrical ashion about the median plane o the plate. This is sometimes called the extensional mode because the wave is stretching and compressing the plate in the wave motion direction. Wave motion in the symmetrical mode is most eiciently produced when the exciting orce is parallel to the plate. The asymmetrical lamb wave mode is oten called the lexural mode because a large portion o the motion moves in a normal direction to the plate, and a little motion occurs in the direction parallel to the plate. In this mode, the body o the plate bends as the two suraces move in the same direction. Figure.6: Lamb Waves Propagation (a) Symmetrical (Dilatational) and (b) Asymmetrical (Bending) waves 7

18 .. Mechanics o Wave propagation Equilibrium Equations: Figure.7: Forces acting in the x direction on an elemental volume I a body is in equilibrium, then the resultant orce and moment on that body must be equal to zero. We have two equilibrium equations: σ ji Force Equilibrium Equation: + i = σ ji, j + i = 0 x j (.) Moment Equilibrium Equation: σ ij = σ ji (.) The stress strain relation or isotropic material is given by Green as: σ ij = λ δ ij ε kk + µ ε ij (.) λ = ν Where ( ν )( + µ ) E µ = E, ( + ν ) are Lame s irst and second constants ε ij = σ ii (.4) Putting the above value o stress strain relationship. in equilibrium equation. we λ + µ. u + µ u + = 0 (.5) get: - ( ) ( ) 8

19 This is the Navier s Equation. One dimensional problem can be easily solved by Navier s equation where only one component o the problem is nonzero, and this nonzero displacement component is a unction o only one variable. But the displacement ield in the hal space material has two components o displacement, u and u, and both o them will be unctions o x and x in general. Thus it is very diicult to solve two and three dimensional problems directly rom the Navier s equation. Thus Stokes- Helmholtz decomposition o the displacement ield transorms the Navier s governing equation o motion into simple wave equation below. Wave equations or a two dimensional case obtained by Stokes-Helmholtz decomposition are given by φ ρ ( λ + µ ) && φ = φ & φ = 0 c P (.6) ρ A A&& = A µ c s A&& = 0 (.7) These equations have solutions in the ollowing orm: φ A ( x, t) = φ( n. x c t) ( x, t) = A( n. x c t) s p (.8) (.9) These equations represent two waves propagating in the n direction with the velocity o c p and c s, respectively. Note that n is the unit vector in any direction. When A = 0 and φ = nonzero, then rom the above solutions one gets u = φ = nφ. ( n x c t) p (.0) When A is not equal to 0 and φ is 0, then rom the above solutions one gets u = A = A n. x ( c t) s (.) 9

20 Three components o displacement in the Cartesian coordinate system can be written rom Equation.: (. s ) (. s ) (. s ) (. s ) (. ) (. ) u = n A n x c t n A n x c t u = n A n x c t n A n x c t u = n A n x c t n A n x c t s s (.) Clearly the dot product between n and u (given in Equation.) is zero; hence, the direction o the displacement vector us is perpendicular to the wave propagation direction n. Displacement ields given in Equation.0 and Equation. correspond to P- and S-waves, respectively. P- and S-Waves Elastic waves in an ininite elastic solid can propagate in two dierent modes: P-wave mode and S-wave mode. When an elastic wave propagates as the P-wave, then only normal stresses (compressional or dilatational) are generated in the solid and the wave propagation speed is c p λ + µ =. When the elastic wave propagates as the S- ρ wave, then only shear stresses are generated in the solid and the propagation speed is c s µ =.Wave potentials or these two types o waves, propagating in a three- ρ dimensional space in direction n, are given by Equations.8 &.9. I the problem is simpliied to an in-plane problem where the waves propagate in one plane (say x x plane), then the wave potentials, φ andψ, or these two types o waves can be written in the ollowing orm: ( x, t ) = ( n. x c pt) = ( nx + nx c pt ) = ( x cos + x sin c pt) φ φ φ φ θ θ ( x, t) = ( n. x c t ) = ( n x + n x c t) = ( x cos + x sin c t) ψ ψ ψ ψ θ θ s s s (.) Equation. represents waves propagating in direction n in the x x - plane. Note that in any plane normal to the wave propagation direction n the displacement and stress components are identical. 0

21 . ULTRASONIC TESTING:.. Basic Principle o Ultrasonic Testing: Ultrasonic testing (UT) uses high requency sound energy to conduct examinations and make measurements. A typical UT inspection system consists o several unctional units, such as the pulser/receiver, transducer, and display devices. A pulser/receiver is an electronic device that can produce high voltage electrical pulses. Driven by the pulser, the transducer generates high requency ultrasonic energy. The sound energy is introduced and propagates through the materials in the orm o waves. When there is a discontinuity (such as a crack) in the wave path, part o the energy will be relected back rom the law surace. The relected wave signal is transormed into an electrical signal by the transducer and is displayed on a screen. Figure.8: General ultrasonic Inspection Principle (pulse echo method).. Methods o Ultrasonic Testing:. Pulse echo method. Through transmission method. Two transducer method Pulse Echo Method: In the pulse-echo method, a piezoelectric transducer with its longitudinal axis located perpendicular to and mounted on or near the surace o the test material is used to transmit and receive ultrasonic energy. The ultrasonic waves are relected by the opposite ace o the material or by discontinuities, layers, voids, or inclusions in the material, and received by the same transducer where the relected energy is converted into an electrical signal. The electrical signal is computer processed or display on a

22 video monitor or TV screen. The display can show the relative thickness o the material, depth into the material where laws are located, and (with proper scanning hardware and sotware), where the laws are located in the X-Y plane. Figure.9: Principle o pulse echo method o inspection Through Transmission Method: In the through-transmission method, an ultrasonic transmitter is used on one side o the material while a detector is placed on the opposite side. One unit acts as transmitter and the other unit as receiver. The beam rom the transmitter T travels through the material to its opposite surace where the receiving transducer R is placed. Scanning o the material using this method will result in the location o deects, laws, and inclusions in the X-Y plane. Figure.0: Principle o through transmission o ultrasonic testing

23 Two Transducer Method The pulse echo method can be used with either single or double crystal unit in single transducer unit the probe acts as both transmitter and receiver.in two transducer arrangement,one transmits and other receives the ultrasonic waves.these are placed on same side o specimen.pulse wave is send in to the specimen by the transducer T. And the echoes relected rom the back surace or any deect.are received by the transducer R and displayed on the law detector screen. For speciic applications like wall thickness measurement special type o transducers in which the transmitting and the receiving crystals are housed in a single unit are also used.these transducers are popularly known as twin or T-R probes..4 EXCITATION SIGNAL ANALYSIS Fourier analysis: Although the inputs and outputs (excitations and responses) are unctions o time, they can also be represented as unctions o requency, through Fourier transormation. The resulting Fourier spectrum o a signal can be interpreted as the set o requency components that the original signal contains. One immediate advantage o the Fourier transorm is that, through its use, dierential operations (dierentiation and integration) in the time domain are converted into simpler algebraic operations (multiplication and division). Transorm techniques are quite useul in mathematical applications. Discrete Fourier transorm Relation name Fourier integral transorm(fit) (DFT) Forward transorm iωt X x t e dt = ( ω) ( ) n N X = T x e n= 0 n = 0,,..., N π mn j N m Inverse transorm iωt x( t ) = X ( ω) e dω π m N x = F X e k= 0 m = 0,,..., N n π nm j N Three versions o Fourier transorm are important: the Fourier integral transorm can be applied to any general signal, the Fourier series expansion is applicable only

24 to periodic signals, and the discrete Fourier transorm is used or discrete signals. As shall be seen, all three versions o transorm are interrelated. In particular, one must use the discrete Fourier transorm in digital computation o both Fourier integral transorm and Fourier series expansion Frequency Spectrum: An alternative graphical representation o the periodic signal shown in Figure. is given in Figure.. In this representation, the amplitude o each harmonic component o the signal is plotted against the corresponding requency. This is known as the amplitude or requency spectrum o the signal, and it orms the basis o the requency domain representation. Note that this representation is oten more compact and can be ar more useul than the time domain representation Figure.: Time domain representation o a periodic signal. Figure.: The amplitude spectrum o the periodic signal Sampling: Taking measurements at regular time intervals T is called sampling. Sampling requency: s = T 4

25 Figure.: Figure showing a time record o N equally spaced samples o the input Fast Fourier Transorm (FFT): FFT To perorm a N point DFT,N operations are required. To perorm an N point N *log( N ) operations required. FFT is just a smart and eicient algorithm or computing DFT. FFT Transorms these N equally spaced samples as shown in igure (.) to N/ equally spaced lines in the Frequency Domain Nyquist requency: Shannon s sampling theorem states that i a time signal x (t) is sampled at equal steps o T, no inormation regarding its requency spectrum X () is obtained or requencies higher than c = / ( T), and the limiting (cuto) requency (c) is called the Nyquist requency. In this chapter it is concluded that among all NDT techniques our area o interest is ultrasonic testing.principle and methods in the ultrasonic testing are also discussed in brie. In the last section it is also described how a signal is analyzed beore it is ed to a DPSM model. 5

26 CHAPTER LITERATURE REVIEW The problem o wave propagation can be solved by several methods such as ray tracing method, spectral approach, inite element method etc. Ray tracing method is limited to one dimension wave propagation and cannot be applied to or dimensions. Spectral approach is a requency based method and involves decomposition o the applied impulse into its many sinusoidal components (Fourier components). In this method, the governing partial dierential wave equation is reduced to a set o ordinary dierential equations. Their solution is easier than the original dierential equation. However, otenapproximate solutions are sought. The transormation is eected by Fast Fourier Transorm (FFT) Algorithm. But the disadvantage o this method is exact solutions or complex dierential equations are diicult to obtain and hence this method becomes ineicient in this case. Also non-linear problems are diicult to solve using this approach. The above two methods seem to be very useul to determine stress wave propagation in the -D models, however they prove to be utile when complex models are to be analyzed. In order to analyze linear and non-linear problems, conventional FEM proves to be more useul. This chapter presents a review o literature on propagation o elastic waves through solids and luids. This gives an idea o study carried out in this area up to this stage. This work can be classiied based on the theoretical/analytical studies and experimental studies. In analytical studies the prominent work done is listed here. Abrahams et al. (99) have examined the scattering o Rayleigh waves by an inclined two-dimensional plane surace breaking crack in an isotropic elastic hal-plane. Biwa et al. (00) presented a computational procedure or multiple wave scattering in unidirectional iber-reinorced composite materials. Bruck (000) proposed a simple, one-dimensional model to develop insight into stress wave management issues. This model is initially applied to FGMs with discrete layering, and then extended to continuously graded architectures. The beneit o 6

27 the FGM over the sharp interace is to introduce a time delay to the relected wave propagation when stresses approach peak levels which are highly dependent on the composition gradient and the dierences in base material properties. Gilchrist, (999) showed how horizontal symmetric crack-like deects can be detected rapidly in thin isotropic plates by using longitudinal ultrasonic waves. Joshi et al. (00) discussed the characterization o unctionally graded materials. Karagiozova (004) obtained the speeds o the stress waves that can propagate in an elastic plastic medium with isotropic linear strain hardening in a plane stress state, to analyze the inluence o the transient deormation process on the initiation o buckling in square tubes under axial impact. It is shown that the plastic wave speeds depend on the stress state and on the direction o wave propagation. The material hardening properties have a stronger eect on the speed o the slow plastic wave, while the shear stress aects both the speeds o the ast and slow plastic waves. Krawczuk et al. (004) presented the method o analysis o the wave propagation process in cracked plates. Elastic behavior o the plate at the crack location was considered as a line spring with a varying stiness along the crack length. Lima and Hamilton (00) investigated the propagation o inite-amplitude waves in a homogeneous, isotropic, stress-ree elastic plate theoretically. Mal (00) analyzed elastic waves generated by localized dynamic sources in structural composites. Sharma et al. (004) studied the thermoelastic interaction in an ininite viscoelastic, thermally conducting plate whose upper and lower suraces o the plate are subjected to stress-ree, thermally insulated or isothermal conditions. Voyiadjis and Baluch (98) developed a technical theory or the lexural motions o isotropic elastic plates, taking into account the inluence o transverse normal strain and transverse normal stress, together with rotary inertia and transverse shear. Wang (00) investigated shear horizontal (SH) wave propagation in a semi-ininite solid medium surace bonded by a layer o piezoelectric material abutting the vacuum. The dispersive characteristics and the mode shapes o the delection, the electric potential, and the electric displacements in the thickness direction o the piezoelectric layer are obtained theoretically. Yang et al. (966) deducted a twodimensional linear theory o motions o heterogeneous plates. Transverse shear deormations and rotatory inertia were also included. Zak et al. (006) present certain results o the analysis o wave propagation in an isotropic panel with damage in the orm 7

28 o a atigue crack. Mukherjee et al. (006, ) used ultrasonic piezoelectric transducers that are employed in various applications to produce a broadband requency spectrum. However, intererence rom the sympathetic pulses generated by the transducer limits the duration o the waveorm to a very short time. The paper discusses grading o the transducer as a means o alleviating the sympathetic pulses.a simple one dimensional model based on the spectral approach has been presented. The piezoelectric constant e is graded in various manners and their perormances are evaluated.the signal qualities are evaluated by their Euclidean distances rom the applied voltage pulse. Linearly graded transducers show the best results. The second part o the literature review is about the experimentation carried out in this area. Francesco et al. (004) dealt with the propagation o ultrasonic guided waves in adhesively bonded lap shear joints. Phillips et al (978) studied mechanical stress wave propagation in a long, thin, isotropic, elastic rod containing a single transverse edge crack theoretically and experimentally and ound that one-dimensional wave theories, coupled with an eective compliance o the cracked region, predict reasonably well the observed dynamic strains induced by a longitudinal impact. Giurgiutiu (005) explored the capability o embedded piezoelectric waer active sensors (PWAS) to excite and detect tuned Lamb waves or structural health monitoring. Mukherjee et al propagation through solids to locate the position and extent o crack with single actuator and several surace mounted sensors. Mukherjee et al (006, ) did the characterization o discretely graded materials using acoustic wave propagation. The third part o the literature review deals with the work done till date in the area o DPSM technique. This technique or ultrasonic ield modelling was irst developed by Placko and Kundu(00).they successully used this technique to model ultrasonic ields in a homogeneous luid, and in a non homogeneous ield with one interace (Lee et al. 00,Placko et al 00)and multiple interaces (Banerjee,Kundu and Placko,006).The interaction between two transducers,or dierent transducer arrangements and source strengths,placed in a homogenous luid has been studied by Ahmed et. al. (00).The scattered ultrasonic ield generated by a solid scatter o inite dimension placed in a 8

29 homogeneous luid has been modeled by the DPSM technique (Placko et al,00).the method has been extended to model the phased array transducers (Ahmad et. al. 005).All these works modeled the ultrasonic ield in a luid medium.this method is also applied to model the ultrasonic ield in plates immersed in luids (Banerjee and Kundu,007).DPSM technique is applied or modeling ultrasonic ield at luid -solid interace also(banerjee, Kundu and Alnuaimi,007).semi-analytical modeling o ultrasonic ields in solids with internal anomalies immersed in a luid(banerjee and Kundu,007). Recently distributed point source method is applied on Elastic wave scattering in a solid hal space with a circular cylindrical Hole (Das, Banerjee and Kundu, 008). 9

30 CHAPTER NUMERICAL TECHNIQUES TO MODEL WAVE PROPAGATION There are various methods or numerical modeling o wave propagation problems. These methods either use time based approach or requency based techniques. Some o them are hybrid or extended rom them using certain manipulations. The characteristics o wave propagation problems are that the requency content o the exciting orce is very high. As we know that, at very high requencies the system becomes mass dominated where inertial eects need to be very accurately modeled. Some popular methods o modeling wave propagation are:. Ray tracing method. Spectral Approach. Finite Element Method 4. Distributed Point Source Method (DPSM). RAY TRACING METHOD A simple elegant, one-dimensional model based on ray tracing the path o wave movement was proposed by Bruck (000) to develop insight into stress wave management issues. Ray Tracing is a unique method working independent o time or spectral approach. Figure..: -D stress wave propagation through discretely layered FGM 0

31 .. Explanation o ray tracing method using FGMs: Ray tracing method has wide applications in characterization o materials. So, it is used as a very promising means or characterization o unctionally graded materials (FGMs).The method adopted by Bruck (000), to characterize the discretely graded FGM, is a simple relection-transmission method. When the stress waves come across an interace, some part o it is relected and the remaining is transmitted in the next layer. The behavior o these waves, moving across the FGM is traced and hence it is called as the ray-tracing method. In this method, the path o waves emerging out o a point or a group o points is traced as rays through all interaces. Ray tracing is useul when only a ew trains o wave emerge and they traverse through simple interaces (Figure..). Figure..: Gradient Architecture o FGMs For designing the gradient architecture, a discrete graded layered structure is modeled as shown in Figure... In addition, it can be ininitely reined to get the continuous graded structure. The layers ollow power-law or complete polynomial variation in

32 composition (x/d) n. Here d is the thickness o the graded part. Stress wave propagation in discretely layered FGMs can now be visualized as demonstrated in Figure... I the graded interace consists o m discretely graded layers between base materials and, then the propagating stress wave will encounter (m+) sharp interaces. At each sharp interace, the stress wave will be partially relected and transmitted. For one-dimension wave propagation, the amount o relection and transmission rom a sharp interace can be determined rom σ i i ( ) t and σ σ = σ α r =.... (.) ( + α) ( + α) Where σ i is the amount o stress in the incident wave, σ t is the amount o stress in the transmitted wave; σ r is the amount o stress in the relected wave, and α is the ratio o the acoustic impedance o base material to the acoustic impedance o base material. Acoustic impedance as deined in chapter can be deined as the ratio o pressure across the material to the low through it. It is deined as ρc/a. For unit area, it becomes the characteristic impedance, which is the material property. The thickness o each layer is d/m, and the total time, t, it takes or the incident wave to travel through a layer and then d get relected back is: t = where c is the longitudinal wave speed o the layer cm Figure..: -D stress wave propagation through discretely layered FGM For developing time-history proile o the stress wave relected into base material by the graded interace, we need summation o the stresses o waves relected rom each discrete layer and the time it takes or that wave to be generated and reach base

33 material. Assuming that the period o the stress wavelength, λ, is much longer than the thickness o the graded interace (i.e., λ>> d), the normalized magnitude o the relected wave as shown in base material will be, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) σ α 0 α α 0 α 0 α α α r 0 = + + σ i + α0 + α 0 + α + α0 + α + α + α + α0 This can be written in summation orm or m layers as ( α ) ( ) m ( α j j ) α k HOTs j ( ). (.) k ( ) σ r 4 σ + α + α + α 0 = + + i 0 = j = k where σ r is the amount o stress in the wave relected into base material, α j is the ratio o acoustic impedance o layer j+ to j, and HOTs are higher order terms comprised o three or more relections and one or more transmissions. Also, layer 0 and layer m+ are base materials and respectively, and each intermediate layer is a composite o the two base materials. The normalized time it takes or the wave relected rom the j th layer to reach base material, t is given by, t t c d c m c ( / ) 0 = =.. (.) 0 Where t is the time it takes to reach base material, c 0 is the wave speed in base material, and c k is the wave speed in layer k. k = For analyzing the eects o gradient architecture on stress wave propagation in continuous FGMs, assume that the normalized physical properties o each layer can be described using a linear rule-o-mixtures (ROM) ormula as ollows: p p j 0 k ( p ) = + p (.4) Where p 0 is the property o base material, p is the ratio o the property or base material to the property or base material, and p j is the property o the j th layer. Thus α k + ( k ) v j j+ j+ j = = (.5) k j + ( k ) v j Arranging the above equation in the orm α j ( k )( v j v j+ ) = + α + ( k )( v + v ) Now in equation as m, α k j j j+ (.6)

34 Hence, 4α k lim = m ( + αk ) Also we can write that Vj Vj+ dv and Vj + V j+ dv Hence, rom Equation (.) and above conditions we can write that as m i.e. or continuous grading, v r = dv. (.7) ( k )( + ( k ) V ) i 0 Proceeding in the same manner as above we can write the normalized time as, tc d 0 x V c Vx dv.. (.8) 0 = ( + ( ) ) When there is a sharp interace, the peak magnitude predicted by Equation (.7) should be identical to the magnitude o a wave relected rom sharp interace. However, it is not so and peak magnitude exceeds by a actor ( + k ) ( k ) ln( k ) This discrepancy is a direct result o neglecting higher order terms. Simulation o stress waves in discretely and continuously graded FGMs was done to validate the same. But rom this ray tracing model, it is determined that the peak stress o waves relected rom the FGM interace is slightly greater than or materials with sharp interaces. The beneit o the FGM over the sharp interace is to introduce a time delay to the relected wave propagation when stresses approach peak levels. This time delay is highly dependent on the composition gradient and the dierences in base material properties, consequently the optimal choice o FGM architecture will depend signiicantly on the critical design conditions or speciic applications. Disadvantages in this method o This method is limited only up to -dimension wave propagation and cannot be extended to or dimensions. o It cannot be used under arbitrary orcing unctions. 4

35 . SPECTRAL APPROACH One o the shortcomings o Ray Tracing Method is its diiculty in extending it to twodimensional wave propagation. Also another problem aced in modeling o wave propagation was that the requency content o the exciting orce is very high. Thereore, a very ine mesh o inite elements is necessary to adequately model the problem. This problem can be alleviated i we use requency-based methods, such as the spectral method instead o the time-based techniques. It has many advantages over time-based approaches like it takes very less time or running simulation models. In addition, the inertial eects are exactly represented in it and hence oten-exact solutions are obtained or the transormed partial dierential equation. In this method, the governing partial dierential wave equation is reduced to a set o ordinary dierential equations. Their solution is easier than the original dierential equation. However, oten-approximate solutions are sought. The transormation occurs as a result o Fast Fourier transormation (FFT). These solutions to the governing equations are used as shape unctions or spectral element ormulation. In addition, oten the resulting element is super convergent and very ew elements are required to model the system. Spectral analysis method (Doyle, 989) is a means o solving wave propagation problems in structures. While it is possible to solve structural dynamics problems by starting with partial dierential equations o motion and integrating, the task is horrendously large even or the biggest computer available. It has been known long back that an arbitrary time signal can be thought o as the superposition o many sinusoidal components. This is the basis o Fourier (or spectral) analysis. In wave analysis, the time domain or the disturbance is rom minus ininity to plus ininity and thus the components have a continuous distribution (known as Continuous Fourier transorm). However, the numerical evaluation o the transorm requires discretizing the distribution in some manner, and the one chosen here is by the way o discrete Fourier transorm (DFT). This has two advantages. First, many o the ideas and methods o time series analysis can be borrowed and used or present purposes. Second, it allows the use o the very eicient Fast Fourier Transorm (FFT) computer algorithm. 5

36 Spectral analysis o dierential equations The key to the spectral description o waves is to be able to express the phase changes incurred as the wave propagates rom location to location. This is done conveniently through the use o the governing dierential equations or particular structural models (although other schemes are possible).the idea o representing the time variation o a unction by a summation o harmonic unctions is to represent arbitrary unctions o time and positions resulting rom solution to wave equation. The approach is to remove the time variation by using the spectral representation o the solution. This leaves a new dierential equation or the coeicients, which in many cases can be integrated directly. General unctions o space and time The solutions in wave propagation are general unctions o space and time. I the time variation o the solution is ocused on at a particular point in space, then it has the spectral representation i t u( x, y, t) = ( t) = C e ω.. (.9) n At another point, it behaves as a time unction (t) and is represented by the Fourier coeicients. That is, the coeicients are dierent at each spatial point. Thus, the solution at any arbitrary position has the ollowing spectral representation i nt u( x, y, t) = uˆ ( x, y, ω ) e ω.. (.0) where u are the spatially dependent Fourier coeicients. n n Derivatives The dierential equations have terms o both space and time derivatives. Now apply the spectral representation to each dierential term appearing in the dierential equation. Thus the spectral representation or the time derivatives is u iωnt iωnt = uˆ ˆ ne = iωnune t t In act, time derivatives o general order have the representation. (.) m u m m m m i ω uˆ or i ω uˆ m t n n n n. (.) 6

37 Herein lays the advantages o the spectral approach to solving dierential equations. The algebraic expressions in the Fourier coeicients replace the time derivatives. That is, there is a reduction in the number o derivatives occurring. Similarly, the spatial derivatives are represented by, u iωt uˆ n = uˆ ne = e x x x iωt (.) And in shorthand notation, it becomes u uˆ ˆ n un or x x x Dierential Equations Consider the ollowing general, linear, one-dimensional, homogenous dierential equation in the dependent variable u(x,t). u u u u u u + a + b + c + d + e +... = 0 (.4) x t x t x t The coeicients a, b, c are assumed not to depend on time, but could be unctions o positions, i now, the solution is given the spectral representation u( x, t) = uˆ ( x, ω ) e iωt.. (.5) then on substitution into the dierential equation we get uˆ n uˆ uˆ iωt u ( ) ˆ ( ) ˆ n + a + iω bun + c + iω du ( )... 0 n + iω e + e = n x x x. (.6) Since each term is independent, then this equation must be satisied or each n. That is, there are n simultaneous equations o the orm. duˆ d uˆ ( + ( iω ) ( )..) ˆ n b + iωn d + un + ( a + ( iωn ) e +...) + ( c +...) +... = 0. (.7) dx dx These equations become, duˆ d uˆ A ˆ ( x, ω) u + A ( x, ω) + A ( x, ω) +... = 0 (.8) dx dx on grouping terms, where, A, A depend on requency and position and are complex. It is seen that the original partial dierential equations becomes a set o ordinary linear dierential equations in the Fourier coeicients. n n n 7

38 Spectral relation Linear dierential equations with constant coeicients have solutions o the orm, where λ is obtained by solving algebraic characteristics equation A + A λ + A λ + =.. (.9)... 0 It is usual in wave analysis, however to assume that λ is complex to begin with, that is, that the solutions are o the orm. uˆ( x) = Ce For example, consider the dierential equation This gives k and the solution as ikx duˆ au + = a k C = dx 0 0 k = a and uˆ( x) = Ce ax where C is a constant o integration. Similarly, ollowing second-order dierential equation gives k and the solution as, d uˆ au + = a k C = dx 0 0 i ax i ax k = ± a and uˆ( x) = Ce + Ce There are two solutions (and constant o integration) because occurred to the power o two. Note that, even i the coeicients in the dierential equation are real, it is possible or to be complex. In general, then, the characteristics equation becomes A + ( ik) A + ( ik) A +... = 0 and this has many values o k that satisies it. That is, k k = ( A, A, A,..., ω ) mn m n This relation between the exponent k (called the wave number) and requency ω is called the spectral relation and is undamental to the spectral analysis o waves. The dierent values o m correspond to the dierent modes. The solution is given as the superposition o modes in the orm, u x C e C e C e ikx ikx ikmx ( ) = mn 8

39 There are as many modes (or solutions) are there are roots o the characteristics equation and these should be conused with the number o solutions at each requency. To reinorce this, the solution in total orm is written as, ik x { mn } ik x ik x m i t u( x) = C e + C e C e e ω.. (.0) n The exponential orm or each term is due to the coeicients o the dierential equation being constant; however, the solution or any problem can be always be expressed as u( x, t) = F G( K x) e iωt... (.) n where F n is the amplitude spectrum and G (which may be combination o modes) is the system transer unction. Analysis o the partial dierential equation combined with the boundary conditions determines the particular orms or G and in act, G determines the phase shits with respect to space. Further, it is noted that the wave number k acts as a scale actor on the position variable in the same way that the requency acts on the time. The analysis o the scaling done by k provides a good deal o insight into the solution beore the actual solution is obtained. mn Propagating and Reconstructing waves The signiicance o the spectral approach to waves coupled with the use o the dierential equations is that once the signal is characterized at one space position then it is known at all positions, thereore propagating it becomes a airly simple matter. This section illustrates the basic algorithm or doing this. Basic Algorithm In its simplest terms, the solutions to a waves problems is represented as, ikx ikx m { ik x i } ω t ˆ iωt mn n mn. (.) u( x) = C e + C e C e e = F G( K x) e n where G is the analytically known transer unction o the problems. It is a unction o position x and has dierent numerical values at each requency. ˆF is the amplitude spectrum; this is known rom the input conditions or rom some measurement. This F ˆnG is recognized as the Fourier transorm o the solution. O course it is dierent at each position but once it is evaluated at a particular position then its inverse immediately gives the time history o the solution at that point. Figure... is a low diagram or its 9

40 evaluation. Briely, the time input F(t) is converted to its spectrum F ˆn through a use o the orward FFT. The transorm solution is the obtained by evaluating the product, at each requency. uˆ = Fˆ G( k ).. (.) n n mn ( ) F t m ˆ ( ) F ω n uˆ = Gˆ ( ω ) Fˆ n n n ˆ( ) u ω n ( ) u t m Figure..: Flow diagram or wave reconstruction program This is inally reconstructed in the time domain by the use o the inverse FFT. In the process, it is necessary to realize (when using the FFT to perorm the inversion) that F ˆnG is evaluated only up to the Nyquist requency and the remainder is obtained by imposing 0

41 that it must be complex conjugate o the initial part, This ensures that reconstructed time history is real only. Disadvantages o Spectral element method o Exact solutions or complex dierential equations are diicult to obtain, hence this method becomes ineicient in this case o Non-linear problems are diicult to solve using this approach.. FINITE ELEMENT APPROACH As already discussed, Ray tracing method is limited to -D wave propagation and cannot be applied to or dimensions. Spectral approach is a requency based method and involves decomposition o the applied impulse into its many sinusoidal components (Fourier components). In this method, the governing partial dierential wave equation is reduced to a set o ordinary dierential equations whose solution is easier than the original dierential equation. The transormation is eected by Fast Fourier Transorm (FFT) Algorithm. But the disadvantage o this method is exact solutions or complex dierential equations are diicult to obtain and hence this method becomes ineicient in this case. Also non-linear problems are diicult to solve using this approach. The above two methods seem to be very useul to determine stress wave propagation in the -D models, however they prove to be utile when complex models are to be analyzed. In order to analyze linear and non-linear problems, conventional FEM proves to be more useul. A coordinated theoretical and experimental program was carried out by Deepti et al. (006) in an eort to develop the knowledge base required or the design o a damage monitoring system in structures consisting o distributed surace mounted sensors. The behavior o isotropic was studied numerically and experimentally or undamaged and damaged conditions. Study aimed at detection o the damage in beam/plate using wave propagation technique. The experimental and numerical investigations were being made to locate the position and extent o crack approximately with single actuator and several surace mounted sensors. Numerical modeling o wave propagation was done using

42 ABAQUS/EXPLICIT through isotropic and isotropic medium with damages. Now a days so many solvers or solving FEM problems are coming among them ABAQUS/EXPLICIT is superior in solving wave propagation problems so a brie introduction is given in this chapter about it... Finite element method or explicit dynamics This section contains a conceptual and an algorithmic description o the ABAQUS/Explicit analysis product as well as a discussion on the advantages o the method.... Stress wave propagation illustrated This section attempts to provide some conceptual understanding o how orces propagate through a model when using the explicit dynamics method. In this illustrative example we consider the propagation o a stress wave along a rod modeled with three elements, as shown in igure... We study the state o the rod as we increment through time. Figure..: Initial coniguration o a rod with a concentrated load, P, at the ree end. In the irst time increment node has acceleration, u&& as a result o the concentrated orce, P, applied to it. The acceleration causes node to have a velocity, u& which, in turn, causes a strain rate, ε& el in element. The increment o strain, d el in element is obtained by integrating the strain rate through the time o increment. The total strain, ε el, is the sum o the initial strain, ε 0, and the increment in strain. In this case the initial strain is zero. Once the element strain has been calculated, the element stressσ el is obtained by applying the material constitutive model. For a linear elastic material the

43 stress is simply the elastic modulus times the total strain. This process is shown in igure... Nodes and do not move in the irst increment since no orce is applied to them. Figure..: Coniguration at the end o increment o a rod with a concentrated load, P, at the ree end. P u&& u&& = u& = u&& dt & ε = dε = & ε dt ε = ε + dε σ = Eε (.0) el el el el 0 el el el M l In the second increment the stresses in element apply internal, element orces to the nodes associated with element, as shown in igure... These element stresses are then used to calculate dynamic equilibrium at nodes and. Figure..: Coniguration o the rod at the beginning o increment. P I u&& = el u& = u& + u&& dt (.) old M I = el M = u&& u& u dt. (.) u& u& & ε el = dεel = εeldt ε + dεel σ el = Eεel l &.. (.) The process continues so that at the start o the third increment there are stresses in both elements and, and there are orces at nodes,, and, as shown in igure..4. The process continues until the analysis reaches the desired total time.

44 Figure..4: Coniguration o the rod at the beginning o increment... Time integration ABAQUS/Explicit uses a central dierence rule to integrate the equations o motion explicitly through time, using the kinematic conditions at one increment to calculate the kinematic conditions at the next increment. At the beginning o the increment the program solves or dynamic equilibrium, which states that the nodal mass matrix, M, times the nodal accelerations,u&&, equals the total nodal orces (the dierence between the external applied orces, P, and internal element orces, I): Mu&& = P I.. (.4) The accelerations at the beginning o the current increment (time) are calculated as u&& = ( M ).( P I). (.5) ( t) ( t) Since the explicit procedure always uses a diagonal, or lumped, mass matrix, solving or the accelerations is trivial; there are no simultaneous equations to solve. The acceleration o any node is determined completely by its mass and the net orce acting on it, making the nodal calculations very inexpensive. The accelerations are integrated through time using the central dierence rule, which calculates the change in velocity assuming that the acceleration is constant. This change in velocity is added to the velocity rom the middle o the previous increment to determine the velocities at the middle o the current increment: t t t+ t ( t ( t+ t) + t ( t) ) u& = u& + u&& ( t). (.6) 4

45 The velocities are integrated through time and added to the displacements at the beginning o the increment to determine the displacements at the end o the increment: u = u + t u&. (.7) ( t+ t) ( t) ( t+ t ) t t+ Here is a summary o the explicit dynamics algorithm:. Nodal calculations. a. Dynamic equilibrium. ( ) u&& = M.( P I ) (.8) ( t) ( t) ( t) b. Integrate explicitly through time. t t t+ t ( t ( t+ t) + t ( t) ) u& = u& + u&& ( t).. (.9) u = u + t u&.. (.40) ( t+ t) ( t) ( t+ t ) t t+. Element calculations. a. Compute element strain increments, dε, rom the strain rate,ε&. b. Compute stresses, σ, rom constitutive equations. σ + = ( σ, ε ) (.4) ( t t) ( t) d c. Assemble nodal internal orces, ( t t). Set t + t to t and return to Step. I + 5

46 .4 DISTRIBUTED POINT SOURCE METHOD.4. Introduction: This method is used speciically or modeling o ultrasonic ield.the main originality o DPSM method is that it is not necessary to mesh the totality o the computation volume, but only the surace o interest, in the contrary to a classical inite elements method. The implementation o the model simply requires discretization o the active surace o the transducer or the interaces to obtain an array o point sources, so that the initial complexity is changed into a superposition o elementary problems. The active suraces like transducers, emitters, or interaces relecting a part o an incident ield are discretized into a inite number o elementary suraces, a point source being placed at the centroid o every elemental surace. DPSM technique or ultrasonic ield modeling was irst developed by Placko and Kundu (00). They successully used this technique to model ultrasonic ields in a homogeneous luid, and in a non-homogeneous luid with one interace (Lee et al. 00, Placko et al. 00) and multiple interaces (Banerjee, Kundu and Placko, 005). The interaction between two transducers, or dierent transducer arrangements and source strengths, placed in a homogeneous luid has been studied by Ahmad et al.(00). F F S F S S (a) (b) (c) Figure.4.: (a) Point source generating spherical waveront (b) Line source generating cylindrical waveront (c) Ininite plane source generating plane waveront 6

47 The scattered ultrasonic ield generated by a solid scatterer o inite dimension placed in a homogeneous luid has also been modeled by the DPSM technique (Placko et al. 00). Recently the method has been extended to model the phased array transducers (Ahmad et. al. 005). All these works modeled the ultrasonic ield in a luid medium. The ultrasonic ield generated inside the solid hal-space or the leaky waves in the luid produced by guided waves propagating along the luid-solid interace have not been modeled yet. Figure (.4.)) shows spherical waves generated by a point source in an ininite medium, cylindrical waves generated by a line source and plane waves generated by an ininite plane. The pressure ield due to a inite plane source can be assumed to be the summation o pressure ields generated by a number o point sources distributed over the inite source as shown in Figure.4.. The inite source can be the ront ace o the transducer. A harmonic point source, which expands and contracts alternately, can be represented by a point and a sphere as shown in Figure.4.(a). The point represents the contracted position and the sphere represents the expanded position. When a large number o point sources are placed side by side on a plane surace, then the contacted and expanded positions are shown in Figure.4.(b). The combined eect o a large number o point sources placed side by side is shown in Figure.4.(c). From this igure it is clear that the combined eect o a large number o point sources distributed on a plane surace is the vibration o particles in the direction normal to the plane surace. Non-normal components o motion at a point on the surace generated by neighboring source points cancel each other as shown in Figure.4.(d). However, non normal components do not vanish along the edge o surace. The particles not only vibrate normal to the surace but also expand to a hemisphere and contract to a point on the edge as shown. Figure.4.: Four point sources distributed over a inite surace 7

48 (a) (b) (c) Source expanded - Thin line Source contracted - Dark point (d) Figure.4.: Position o particles or (a) Point source (b) Distributed inite number o points (c) Large number o point sources (d) components o motion o multiple point sources 8

49 .4. THEORY DPSM is briely described in Section.4.. then detail mathematical derivations are presented in the ollowing sections..4.. Distributed Point Source Method I the ront ace o a transducer is considered as the main source o an ultrasonic ield, then the ultrasonic ield generated by that source can be assumed to be the summation o the ultrasonic ields generated by a number o point sources distributed near that inite source. Any interace is responsible or generating relected and transmitted ultrasonic ields. Thereore, the interace can be replaced by two layers o sources - one layer generating the relected ield and the second layer generating the transmitted ield. Two layers o interace sources are distributed on two sides o the interace. Strengths o the point sources distributed near the transducer ace and the interace are obtained by satisying the boundary conditions and interace continuity conditions. For solving this problem we need to calculate the stress and displacement Green's unctions in the solid, and pressure and displacement Green's unctions in the luid..4.. Mathematical derivations: Calculation o Displacement Green s unctions in the solid A point source acting in a solid, can be modeled as a concentrated body orce, F( x, t) = P ( t) δ ( x) or F = P ( t) δ ( x ). (.4) i i j i t where P is the orce vector. For harmonic time dependence [ ( t) = e ω ], when the point source is at y then the displacement ield at x can be expressed in terms o the Green s unctions G ij (x; y), iωt iωt u = U e = G ( x; y) P e.. (.4) i i ij j The displacement Green s unction or isotropic solids can be written as (Mal and Singh, 99), 9

50 G ( x; y) = ij ik pr e ik p k p Ri R j + ( Ri Rj δij ) + r r r πρω r r r 4 iksr e iks ks ( δij Ri R j ) ( Ri Rj δij ) xi yi where Ri = (.44) r where, x i are the coordinates at the observation point, y i are the coordinates at the source point, r is the distance between the observation point and the source point, k p is the P- wave number and k s is the S-wave number o the solid. In matrix orm, [ ] T G( x; y) = G ( x; y) G ( x; y) G ( x; y) and u =G(x;y)P (.45) I the unit excitation orce at y acts in the x j direction, then the displacement at x in x i direction is given by G ij (x;y). Calculation o Stress Green's unction in the solid For isotropic homogeneous solids the expression or stresses can be written as σ = µε + λδ ε (.46) ij Where, λ, µ are the two Lamé constants and δ ij is the Kronecker Delta. We know that strain can be expressed as a unction o displacement, ij ij kk ε ij = ( u i, j + u j, i ). Substituting the expression or displacement in the expression or strains, ( G ik, j + G jk, i ) P k ε ij =.... (.47) For isotropic homogeneous linearly elastic material, we can write the expression or the stress Green s unction at x due to a concentrated harmonic orce at y by substituting the expression or strains in Eq.(.46)as ( ) σ ( x; y ) = µ G + G P + λδ G P (.48) ij ik, j jk, i k ij kq, k q (,,, ) σ x y = µ G + G δ + λδ G P (.49) Or ( ; ) ( ) ij ik j jk i kq ij kq k q 40

51 Calculation o pressure and displacement Green s unctions in the luid A perect luid can be considered as a homogeneous isotropic medium, it cannot have shear stress and the pressure at a point in all directions in the luid must be the same. Thereore, the stress or pressure at any point can be written as σ = λδ ε = p.(.50) ij ij Following the usual notation ε ij = ( u i, j + u j, i ), the constitutive relation can be expressed in terms o the displacement components as ( u + u + u ) =. u = p kk λ (.5),,, λ The governing equation or the equation o motion in the luid can be written as.. p, i + Fi = ρ u i. (.5).. p + F = ρ u (.5) Applying divergence on both sides and using equation (.5) we get (. u) ρ... p +. F = ρ = p (.54) t λ Assuming =. F and the wave velocity in luid c = λ, we can rewrite the above ρ equation, p c.. p =... (.55) I. Bulk wave in the luid Spherical Bulk wave in a luid can be generated by a point source in an ininite luid medium as shown in Figure.4.(a). I the point source is a harmonic source, then it will generate harmonic spherical waves. I a point source is generating the bulk wave in a luid, then the harmonic dirac-delta impulsive orce will contribute to the body orce. 4

52 For an ininite luid medium with point source acting on it, Equation (.55) can be expressed as.. G G = δ ( ) c iωt x y e (.56) Where, y. G is the pressure Green s unction in luid at x due to the point source acting at I G iωt ( r, t) = G ( r, ω) e then the above equation takes the ollowing orm, Where, ω G ( r ω) + G ( r, ω) = δ ( x y).. (.57), c G is now a unction o r andω. The Laplacian operator in spherical coordinates can be written as v r r r v r.. (.58) Where, v is any scalar unction o r and r = y ( x y ) + ( x y ) + ( x ) x =. y Hence, the particular solution o Equation (.57) is G ik r e ( r, ω) = (.59) 4πr where k ω =. c The pressure in the luid can also be represented by a potential unction (Placko and Kundu 00). The pressure-potential and the displacement-potential relations can be written as p = ρω φ (.60) 4

53 u i = φ (.6) x i where φ is the scalar potential. Thereore, or the pressure Green s unction, the potential unction can be expressed as ik r e φ ( r, ω) =.. (.6) 4πρω r Taking derivatives o φ with respect to x i, the displacement components in the three directions can be obtained, u u u ik r ik e r = ik R R e 4πρω r r ik r ik e r = ik R R e 4πρω r r ik r ik e r = ik R R e 4πρω r r. (.6). (.64).. (.65) where xi R = i y r i 4

54 CHAPTER 4 Ultrasonic Field Modeling in Homogenous Fluid Using - DPSM Technique 4. Computation o Velocity, Pressure and Displacement Fields in a luid generated by a group o point sources: Homogenous luid is luid which is having same properties and they does not change rom point to point.for a lat surace transducer, where all point sources are excited at the same time, the pressure ield generated at a point x (see Figure. 4.(a)) by the transducer can be obtained by integrating the spherical waves, as done in the conventional surace integral technique. The combined eect o a large number o point sources distributed over a plane surace such as the transducer ace is the vibration o particles in a direction normal to the plane surace (a) (b) Figure 4. (a) Position o an observation point (target point) and its distance rom the nth point source on the transducer surace, (b) Side view o a transducer and actual positions o the point sources. 44

55 From the surace integral technique, the pressure ield at point x in ront o a group o point sources at y can be written as (Schmerr 998), ik r e p( x ) = B. G ds( y ) = B ds( ) r y. (4.) S where B represents the source strength o the point sources. This integral can also be written in the summation orm S N N B exp( ik rm ) exp( ik rm ) p( x ) = S m = Am.. (4.) m= 4π r m= r m In the DPSM technique it has been assumed that each point source distributed over the surace has dierent source strengths as speciied by th point source and r m is the distance o the target point x rom the m A m, where m designates the th m m point source. Hence, th pressure at any point at a distance r m rom the can be written as m point source with source strength A m ik r m e pm ( r) = Am (4.) r For N number o point sources distributed on a surace, the pressure at the target point is given by m ik r N N m e p( x ) = p ( r ) = A. (4.4) r m m m m= m= m From the pressure velocity relation, it is also possible to obtain the velocity in all three directions at x due to M number o point sources placed at y. p n v t n = ρ = ± iωρvn. (4.5) Where, p n is the derivative o pressure along the direction n. The velocity can be written as v n p = i ωρ n. (4.6) 45

56 Thereore, the velocity in the radial direction, at a distance r rom the m-th point source, is given by A exp( ) exp( ) exp( ) m ik r A ik m ik r ik r vm ( r) = = iωρ r r iωρ r r A exp( ik r) m = ik iωρ r r. (4.7) and the three components o velocity are v v Am r) = iωρ x exp( ik r r) A x m = iωρ exp( ik m ( r Am r) = iωρ x exp( ik r r) A x m = iωρ exp( ik m ( r r) ik r) ik r r. (4.8) (4.9) v Am r) = iωρ x exp( ik r r) A x m = iωρ exp( ik m ( r r) ik r. (4.0) When the contributions o all M sources are added, the total velocity in x, x and x directions at point x can be written as v v v N N A x m m exp( ik rm ) x ik. (4.) m= m= iωρ rm rm ( ) = vm ( rm ) = N N A x m m exp( ik rm ) x ik. (4.) m= m= iωρ rm rm ( ) = vm ( rm ) = N N A x m m exp( ik rm ) x ik. (4.) m= m= iωρ rm rm ( ) = vm ( rm ) = Where, x im is the shortest distance along x i direction between the th m point source and the target point, as shown in Figure 4.(a). I the transducer surace is parallel to the xx - plane and its velocity in the x direction is given by v 0 then or all x values on the transducer surace the velocity should be equal to v 0. Thereore, 46

57 v N A x m m exp( ik rm ) x ik = v0. (4.4) m= iωρ rm rm ( ) = Special Case: Transducer Face is Inclined at an Angle oθ : I the transducer ace is inclined at an angle oθ, measured rom x -axis when rotated about the x axis (Figure 4.), the velocity o the transducer ace can be expressed as ( x) θ ( x) v ( x) = v Sin + v Cosθ N A x m exp( ik rm ) xm exp( ik rm ) m = ik Sinθ + Cosθ v = m= iωρ rm rm rm 0. (4.5) Figure.4. Rotation o the transducer with respect to x -axis and velocity o the n th 4. Matrix representation: observation point adjacent to the transducer ace Velocity o the N target points placed on the transducer ace due to point sources distributed just below the transducer surace at a distance r s, can be written in matrix orm as V = M A. (4.6) S SS S where, V S is the (Nx) vector o the velocity components, perpendicular to the transducer surace. I the velocity o the transducer ace is given by v 0, then V S can be written as: 47

58 where, { } strengths, then T N v v v v N T 0 v 0 N V S =. (4.7) n v 0 is the velocity o the n-th target point. I A S is the (Nx) vector o the source T { } = [ A A A A A A A A ] T A. (4.8) S A N N From the earlier discussion, we know that each point source is placed inside a sphere and hence, the number o apex points o the spheres touching the transducer surace will be the same as the number o point sources. When the target points are placed at the apex o the spheres o the point sources, then M is equal to N. Thereore, or the target points at the apex o the spheres o the point sources, the square matrix M SS can be written as: ( xt, r ) ( xt, r ) ( xt, r ) ( xt 4, r4 ) ( xtn, rn ) ( xtn, rn ) ( xt, r ) ( xt, r ) ( xt, r ) ( xt 4, r4 ) ( xtn, rn ) ( xtn, rn ) ( xt, r ) ( xt, r ) ( xt, r ) ( xt 4, r4 ) ( xtn, rn ) ( xtn, rn ) M = ( xt, r ) ( xt, r ) ( xt, r ) ( xt 4, r4 ) ( xtn, rn ) ( xtn, rn ) SS where, N N N N N N N N N N N N ( xt, r ) ( xt, r ) ( xt, r ) ( xt 4, r4 ) ( xtn, rn ) ( xtn, rn ) (4.9) N N N n n n xtm exp( ik r ) exp( ) n n m ik rm n n ( xtm, rm ) = ik = ik x n mcos + x msin iωρ r i r rm n n ( ) r n m m ωρ ( m ) ( θ θ ) (4.0) and n r m is the distance between the m-th point source and the n-th target point. Special Case: For large number o point sources In Eq.(4.0) n r m appears in the denominator.thereore or small values o can be simpliied in the ollowing manner. n r m Eq.(4.0) 48

59 n n n n n n xtm exp( ik r ) exp( ) exp( ) n n m xtm ik rm xtm ik rm ( xtm, rm ) = ik n = n = iωρ r iωρ r iωρ r n n n ( m ) rm ( m ) rm ( m ) (4.) Note that all spheres have the same radius r m = r s = r, and thereore n xtm = r.substituting in the above expression and expanding the exponential term in the series expansion we get n n xtm exp( ik r ) n n m r n r ( xtm, rm ) = = + ik... rm + = (4.) iωρ r iωρ r iωρ r ( ) ( ) ( ) n n n m m ( m ) n m For m = n, r = r = r, substituting it in above equation, we get m m n n r r ( xtm, rm ). (4.) iωρ r ωρ ωρ Substitution o above two equations yields n ( ) i r i r m M SS iωρ r r r r... r r rn r r r... r r r N r r r... r r rn r r r... N N N r r r = N N. (4.4) For a general set o target points located on any surace, the velocity due to the transducer sources can be written as: V = M A. (4.5) T TS S where V T, the velocity vector (Nx) contains the normal velocity components o the target points distributed on the surace. The matrix M TS has elements that are similar to those o M SS, with dierent n x tm values and the size o the matrix is (MxN), where N is the number o target points and M is the number o source points. Following the same 49

60 concept, the pressure at any N number o target points due to M number o source points can be written as: PR = Q A. (4.6) T Where, PR T is the (Nx) vector o pressure values at N target points, and matrix given below TS S Q TS is a (NxM) Q TS ( ik r ) ( ik r ) ( ik r ) ( ik rm ) exp exp exp exp r r r rm exp( ik r ) exp ( ik r ) exp( ik r ) exp( ik rm ) r r r r M exp( ik r ) exp ( ik r ) exp ( ik r ) exp( ik rm ) = r r r rm N N N N exp( ik r ) exp( ik r ) exp ( ik r ) exp( ik rm ) N N N N r r r r M M N (4.7) When the target points are located at the apex o the spheres o the point sources, Equation (4.6) takes the orm, where, Q SS is a (NxN) matrix. The deinition o PR = Q A. (4.8) S SS S n r m is identical to that given in Equation (4.0). It is the distance between the m-th point source and the n-th target point. In the same manner, the matrix expression or displacements at general target points in the luid can be written as: U = DF A. (4.9) T T TS TS S U = DF A. (4.0) T TS S U = DF A. (4.) S 50

61 where g( R, r ) g( R, r ) g( R, r )... g( R, r ) g( R, r ) (, ) (, ) (, )... (, ) (, ) (, ) (, ) (, )... (, ) (, ) ) (, ) (, )... (, ) (, ) N N N N N N N N N N g( R, r ) g( R, r ) g( R, r )... g( R, r ) g( R, r ) i i i im M i M M g Ri r g Ri r g Ri r g RiM rm g Ri M rm g Ri r g Ri r g Ri r g RiM rm g Ri M rm DFiTS = g( Ri, r g Ri r g Ri r g RiM rm g Ri M rm i i i im M im M ( NxM ) (4.) where g( R, r ) ik R e R n ik rm n n n n ik r e m n im m = n im im ρω rm n ( r ) m. (4.) R x y = and i =,, n n n im im im n rm In this chapter through mathematical expressions it is shown how the ultrasonic ields such a pressure, displacement ields and velocity ields are generated in Homogenous luids using DPSM method. 5

62 CHAPTER 5 Ultrasonic Field Modeling in Layered Fluids (or) Non-Homogeneous Media using DPSM Technique 5. Introduction: We are interested in computing the ultrasonic ield in multilayered luid systems. In the multilayered problem geometry several interaces may be present. When luids with dierent densities and acoustic properties orm a multilayered system, the luid density should monotonically vary rom top to bottom. I we have n number o luids in the system, we should have (n-) number o interaces. Each interace acts as a transmitter as well as a relector o elastic wave energy generated by the ultrasonic transducers. When the entire system is considered, several continuity conditions across the interaces and boundary conditions at the transducer surace are to be satisied. x x Figure.5. Distribution o point sources in the layered luid system Let the transducer be immersed in a luid medium consisting o two luids, with a plane interace between two luids located in ront o the transducer, as shown in 5

63 Figure.5.. We can introduce three layers o point sources A S, AI and * I A, as shown in the Figure.5. to model the incident ield, relected ield and transmitted ield, respectively. The sources with source strength luid below it and the sources with source strength A I generate the ultrasonic ield in the A I * generate the ultrasonic ield in the luid above it. Observation points or target points are shown by open small circles in the Figure.5..The total ultrasonic ield in each medium is obtained by superimposing the ields generated by two sets o sources as listed below: Fluid : Summation o ields generated by A S and A I. P = Q A + Q A (5.) T T TS TS S S TI TI I V = M A + M A (5.) I Fluid : Fields generated by A I * T = QTI AI P (5.) T = M TI AI V (5.4) 5. Methods or Finding Source Strength vectors: 5.. Computation o the source strength vectors when multiple relections between the transducer and the interace are ignored: x x Figure 5.: Point P can receive two rays, (direct ray) and (relected rom interace) rom a single point source. 5

64 As shown in Figure 5., two rays can reach the point P that is located close to the interace, irst one is the direct ray denoted by ray and the second one is ray, which irst propagates rom the transducer to the interace, is relected by the interace and then reaches point P. The pressure ields due to rays and are modeled by Q A and TS S Q TI AI expressions, respectively. Note that pressure ield generated by ray can be alternately obtained by the irst multiplying the elements o Q TS terms by appropriate relection coeicient, thus orming matrix given in Eq. (5.5) and then multiplying this matrix with A S vector. R Q TS Q R TS exp R( θ ) r exp R( θ ) r = exp R( θ ) r exp N R( θ ) r ( ik r ) exp( ik r ) exp( ik r ) exp( ik r ) M ( ik r ) exp( ik r ) exp( ik r ) exp( ik r ) M ( ik r ) exp( ik r ) exp( ik r ) exp( ik r ) N N N N ( ik r ) exp( ik r ) exp( ik r ) exp( ik r ) N R( θ ) R( θ ) R( θ ) R( θ ) M r r r r N R( θ ) R( θ ) R( θ ) R( θ ) N r r r r N R( θ ) M R( θ ) M R( θ ) M R( θ ) N M r r r M r N M M M M M M N (5.5) where the n θ m denotes the angle o incidence at the interace o the ray traveling rom th m point source on the transducer surace to the th n target point close to the interace.the relection coeicient R( θ n m ) or the angle o incidence obtained rom the ollowing equation. n θ m is R( θ ) = c c c cos( ) c + cos ( ) c c ρ θ ρ θ c c c cos( ) + c + cos ( ) c c ρ θ ρ θ (5.6) R Because the pressure ield generated by Q TI AI and Q A TS S should be same Q A = Q A (5.7) TI I R TS S From the above equation A = [ Q ] Q A (5.8) I TI A I can be obtained ater knowing A S. TS S 54

65 In the same manner generated to rays traveling rom and rom A I can be computed by equating the pressure ields A I sources directly to the target points in luid A S sources ater being transmitted at the interace. Q A = Q * TI * I T TS A S (5.9) A = [ Q ] Q A (5.0) * I * TS T TS S Q T TS exp T ( θ ) r exp T ( θ ) r exp = T ( θ ) r exp N T ( θ ) r ( ik r ) exp( ik r ) exp( ik r ) exp( ik r ) M ( ik r ) exp( ik r ) exp( ik r ) exp( ik r ) M ( ik r ) exp( ik r ) exp( ik r ) exp( ik r ) N N N N ( ik r ) exp( ik r ) exp( ik r ) exp( ik r ) N T( θ ) T ( θ ) T ( θ ) T ( θ ) M r r r r N T ( θ ) T ( θ ) T ( θ ) T( θ ) N r r r r N T ( θ ) M T ( θ ) M T ( θ ) M T ( θ ) N M r r r M r N M M M M M M N (5.) n The transmission coeicient T ( θ ) or the angle o incidence rom the ollowing equation. m n θ m is obtained T ( θ ) = ρ c cosθ (5.) c c ρ c cosθ + ρc + cos ( θ ) c c 5.. Computation o the source strength vectors considering the interaction eects between the transducer and the interace: In this case A, A are to be computed simultaneously by satisying the given boundary conditions on the transducer surace and continuity conditions across the interace. I the transducer surace velocity is deined asv SO, then the velocity at the transducer S I and A I surace computed rom sources A S and AI should be equal tov SO ; thereore, On the transducer surace M A + M A = V (5.) SS S SI I S 0 On the interace, rom the continuity o the normal stress, 55

66 IS S + Q II A I = QII A I Q A (5.4) M A (5.5) IS S + M II AI = M II AI Equations (5.) to (5.5) can be written in matrix orm M Q M SS IS IS M Q M SI II II 0 Q M II II M M A A A S I * I M V = 0 0 S 0 M (5.6) or, [ ]{ Λ} { V} ΜAT = (5.7) The vector o source strengths o the total system can be calculated by taking inverse MAT and multiplying it with the vector{ V }, o [ ] { Λ} [ MAT] { V} = (5.8) Among these two methods ray tracing method is easy to use and most promising results are coming rom this method compared to the matrix inversion method. So entire non-homogenous ields are modeled using ray tracing method. 5. Obtaining the Ultrasonic ields ater knowing the source strength vectors: Ater calculating the source strengths, the pressure, velocity, displacement values at any point in luid can be obtained by the ollowing equations: Pressure Field: PR = Q ) A (5.9) ( F ) Q( F ) S AS + ( F I I Velocity Field: V ( F ) M ( F ) S AS + M ( F ) S = A (5.0) I Displacement Field: U ( F ) DF( F ) S AS + DF( F ) I = A (5.) I For example, the pressure ield at the target point P in the luid as shown in the igure 5. can be written as: PR = Q A + Q A (5.) P PS S PI I In this chapter it shown how DPSM method is used to develop ultrasonic ields generated in Non-Homogenous luids. 56

67 57

68 CHAPTER 6 Numerical Results and Discussions MATHCAD programs have been developed to model the ultrasonic ield based on the DPSM ormulation presented above. In the simplest case, the transducer is immersed in homogenous luid. More complex problem geometries involve two luids with a plane interace. The numerical results clearly show how the ultrasonic ield decays as the distance rom the transducer increases and the ield becomes more collimated as the size o the transducer increase. Following two separate cases have been studied Case : Wave Propagation in Homogenous Fluid: In this case single luid is used to study generated Ultrasonic ield.here both steady and Transient Wave propagation is studied. Name o Fluid Density (gm/cc) P-Wave Speed ( C ) (Km/sec) p Water ( 0 C ).49 Case : Wave Propagation in Non-Homogenous Fluid: In this case two layers o luids are arranged such that their density monotonically increases rom bottom to top and Transient wave propagation is studied in this. Name o Fluid Density (gm/cc) P-Wave Speed ( C ) (Km/sec) p Fluid.49 Fluid.5.00 For convenience rom now onwards the X axis is called the X axis and X axis is called the Z axis. 57

69 6. Method used or Finding Source point Coordinates: Beore writing the MathCAD programs Center Point Coordinates o the Points sources must be obtained. For getting them AutoCAD drawings are made according to the given dimensions o the Transducer and number o point sources.the center point coordinates o all the point sources are then exported rom AutoCAD in to an excel ile and this data is ed to MathCAD programs. For example igure (6.) shows a Transducer having points sources and shows plot o igure (6.) extracted data o point sources distributed on lat circular Transducer. Similarly igure (6.) and igure (6.4) shows the same or 48 point sources or the transducer o same diameter. Dia.o Transducer =0.inch No. o point sources= Dia o source point sphere=0.06cm Figure.6.: AUTOCAD drawing showing polar array o point sources Point sources Y Axis (cm) X Axis(cm) Figure.6.: Distribution o Point sources on the ace o lat circular Transducer 58

70 Dia.o Transducer =0.inch No. o point sources=48 Dia o source point sphere=0.0095cm Figure.6.: AUTOCAD drawing showing polar array o 48 point sources Point Sources Y Axis (cm) X Axis (cm) Figure.6.4: Distribution o 48 Point sources on the ace o lat circular Transducer 59

71 6. Ultrasonic Field in a Homogenous Fluid-DPSM technique: 6.. Steady wave propagation in Homogenous luid: Following the DPSM technique described in previous chapters, the ultrasonic ield generated in a homogeneous luid (water) by a lat circular transducer is computed. For such simple problem geometry the expression o the near ield zone length and the divergence angle o the emitted beam can be calculated in closed orm (Kundu, 000). Numerically computed values o these two parameters are compared with the closed orm analytical values to check the accuracy o the numerical results. From Kundu (000), D NF = or λ D (6.) 4λ where N F is near ield length. The analytical expression o the pressure in a homogeneous luid along the central axis o a circular transducer is (Placko and Kundu, 00) ρ 0 p( Z ) =. c v exp( ik Z ) exp( ik Z + a (6.) Figure 6.5 and Figure 6.6 shows acoustic pressure variations or the DPSM modeling o point sources ( MHz and 5 MHz signal requencies) and 48 point sources ( MHz and 5 MHz signal requencies) along the central axis (Z axis) o a circular transducer computed by the DPSM technique described above (dashed curve) and the analytical expression [Eq. (6.), solid curve]. Coordinate Z is measured rom the transducer ace. The transducer area is 5.0mm, its diameter (D) =.58 mm. For example DPSM modeling or 48 sources is made by locating all o them at Z= - r s, while the transducer ace is at Z = 0. For this problem geometry π D D.58 rs =. = = = mm (6.) 4 π N 8N 8 48 Figure 6.5 shows that or point sources DPSM curve is closely moving above the EXACT curve but there is no well match with it. Figure 6.6 shows that with 48 point sources DPSM result match well with exact solution. Hence it can be concluded that as the number o point sources increases accuracy o DPSM increases. As the waves speed in the surrounding luid is.49 km/sec, wave length λ is equal to 0.98 mm or 5 MHz signal. From Eq.(6.) N F is ound to be 5.mm that matches with last peak obtained numerically as shown in igure (6.6). 60

72 For MHz Frequency For 5 MHz Frequency Figure.6.5: Acoustic Pressure Variation or point sources along Z axis Perpendicular to Transducer Face. 6

73 For MHz Frequency N F = 0.5cms For 5 MHz Frequency Figure 6.6: Acoustic Pressure Variation or 48 point sources along Z axis Perpendicular to Transducer Face. 6

74 6.. Transient Wave Propagation in homogenous Fluid: Same steady state model is used to predict transient wave propagation in homogeneous luid. Fast Fourier transorm (FFT) is used to convert time domain impulse into requency domain pulse. Then or each requency, DPSM method is applied and obtained response is again transormed to time domain using inverse Fourier transorms (IFFT). Frequencies up to Nyquist requencies are used or analysis and eect o remaining requencies are taken care by superimposing antisymmetry behavior o imaginary part about Nyquist requency. Figure 6.8 shows the lowchart or reconstruction o transient wave program in homogenous luid using DPSM techniques. For non-homogeneous luid case same lowchart is applicable only changes to be made are matrix Mss is replaced by MAT as per equation (5.8) and appropriate velocity vector V depending upon the location o transducer with respect to interaces. Figure 6.7 shows variation o acoustic pressure at transducer ace verses input signal requency. In this graph variation o acoustic pressure at three dierent locations (i.e. center, hal radius and periphery o circular transducer) is plotted and it is clear that variation o Acoustic pressure at center o transducer seems to be sensitive with change in input signal requency compared to that o acoustic pressure near the periphery o transducer surace. Due to this reason rest o the analysis is done about center o transducer. Same model is used to study the response o various input impulses and results are presented rom Figure. 6.9 To Figure 6.8. They are Hal sine wave pulse as shown in Figure.6.9 is applied as velocity input at transducer ace and ultrasonic ield response in terms o acoustic pressure, displacement and velocity at transducer centre are presented in Figure 6.0.Displacement response does not attenuate completely in given time window and hence the plot show typical leakage problem. This problem can be eliminated by proper selection o window instead o rectangular window. Figure 6. shows the input ull sine wave velocity pulse and Figure.6. shows its acoustic ield response. As the wave completes ull cycle, all responses (velocity, pressure and displacement) get attenuated within given time window. Figure.6. and Figure.6.4 shows the input positive spike pulse and ultrasonic wave response or hal triangular input pulse (positive spike). 6

75 Figure.6.5 and Figure.6.6 shows input triangular pulse and ultrasonic wave response or triangular input pulse. Figure.6.8 shows ultrasonic wave response or tone burst signal. It can be seen rom Figures 6.0, 6., 6.4, 6.6, 6.8 that or various input impulses, tone burst input pulse seems to be better than others as response peaks can be distinctly identiied in this case. The reason may be that the energy entered and get removed rom system smoothly. The same input pulse is used or urther study o transient wave propagation in homogeneous and non-homogeneous luids. Figure 6.7 show the tone burst signal in time domain and it s FFT. Figure 6.9((a), (b), (c)) shows the acoustic pressure at three dierent target points lying along Z-axis o transducer ace. Transducer is immersed in homogeneous luid having density ( ρ ) = gm/cc and wave speed is ( C ) =.49 km/sec. For point close to transducer ace (Z= 0) acoustic wave pressure peak lies around 95 µ-sec as shown in igure 6.9(a) with markers, same as the input velocity pulse peak time as shown in igure 6.7 (a). Acoustic pressure pulse response at target point at a distance o 50 mm rom Transducer ace shows that a peak arrives at 8 µ-sec as shown in the igure 6.9(b)(having time lag R = µ-sec). Similarly or third point at a distance 50 mm rom transducer ace, pressure peak reaches at time 95 µ-sec as shown in the igure 6.9(c) ( having time lag R = 00 µ-sec). These time lags are matching with the actual time required or wave to travel above two distances (50mm and 50 mm) as shown in the table o time validation given below. Also the pressure amplitudes are matching with the close orm solution as shown in the table o amplitude validation shown below. These two checks validate the transient wave propagation model result. 64

76 Time Validation: Dis tan ce Travelled by the wave = Velocity o wave( C ) Time taken to travel Method Z=50mm Z=50mm Analytical(ormula).55 µ-secs µ-secs DPSM (graph) µ-secs 00 µ-secs Pressure amplitude validation: Closed orm (Exact) solution: p( x ) = ρ. c v 0 [ exp( ik x ) exp( ik x + a ] Method Z=50mm Z=50mm Exact solution 0.0 KPa 0.04 KPa DPSM (graph) KPa 0.00 KPa Acoustic Pressure on Transducer Face (KPa) Center Periphery / Radius Frequency (MHz) Figure 6.7: Plot showing Acoustic pressure variation Vs requency 65

77 Figure 6.8: Flow chart or Wave Reconstruction program 66

78 Hal sine wave velocity Impulse:.5 Input Velocity Pulse Amplitude(m/s) Time (Micro-sec) FFT Amplitude FFT Frequency (khz) Figure 6.9: Hal sine wave as input pulse and its FFT 67

79 0.8 Pressure at Center o Transducer Acoustic Pressure( kpa) TIME(Micro-sec) 0.5 Displacement at Center o Transducer Displacement (Micro-m) Time (Micro-sec) Velocity Amplitude (m/s) Velocity at center o Transducer Time (Micro-sec) Figure 6.0: Ultrasonic ield response at transducer ace or Hal Sine Wave Velocity Impulse 68

80 Full Sine Wave Velocity Impulse: Input Velocity Pulse Amplitude(m/s) Time (Micro-sec) FFT Amplitude 4 FFT Frequency (khz) Figure 6.: Full Sine wave as input pulse and its FFT 69

81 0.8 Pressure at Center o Transducer Acoustic Pressure( kpa) TIME(Micro-sec) 0. Displacement at Center o Transducer Displacement (Micro-m) Time (Micro-sec) Velocity Amplitude (m/s) 0 Velocity at center o Transducer Time (Micro-sec) Figure 6.: Ultrasonic ield response at transducer ace or Full Sine Wave Velocity Impulse 70

82 Hal Triangular Wave Velocity Impulse: Input Velocity Pulse Amplitude(m/s) Time (Micro-sec) 0.6 FFT FFT Amplitude Frequency (khz) Figure 6.: Hal Triangular wave as input pulse and its FFT 7

83 .6 Pressure at Center o Transducer Acoustic Pressure( kpa) TIME(Micro-sec) 0. Displacement at Center o Transducer Displacement (Micro-m) Time (Micro-sec) Velocity Amplitude (m/s) Velocity at center o Transducer Time (Micro-sec) Figure 6.4: Ultrasonic ield response at transducer ace or Hal Triangular Wave Velocity Impulse 7

84 Triangular Wave Velocity Impulse: Input Velocity Pulse Amplitude(m/s) Time (Micro-sec) FFT Amplitude FFT Frequency (khz) Figure 6.5: Triangular wave as input pulse and its FFT 7

85 Pressure at Center o Transducer Acoustic Pressure( kpa) TIME(Micro-sec) 0. Displacement at Center o Transducer Displacement (Micro-m) Time (Micro-sec) Velocity Amplitude (m/s) Velocity at center o Transducer Time (Micro-sec) Figure 6.6: Ultrasonic ield response at transducer ace or Triangular Wave Velocity Impulse 74

86 Tone Burst Velocity Impulse: Amplitude(m/s) Input Velocity Pulse Time (Micro-sec) (a) FFT Amplitude FFT Frequency (KHz) (b) Figure 6.7: Tone Burst wave as input pulse and its FFT 75

87 0 8 Pressure at Center o Transducer Acoustic Pressure( kpa) TIME(Micro-sec) 0.0 Displacement at Center o Transducer Displacement (Micro-m) Time (Micro-sec) Velocity Amplitude (m/s) Velocity at center o Transducer Time (Micro-sec) Figure 6.8: Ultrasonic ield response at transducer ace or Tone Burst Wave Velocity Impulse 76

88 Pressure at Center o Transducer Acoustic Pressure( kpa) TIME(Micro-sec) (a) Acoustic Pressure at distance Z=0 From Transducer Face Acoustic Pressure (kpa) R Pressure at Center o Transducer Time(micro sec) (b) Acoustic Pressure at distance Z=50mm rom transducer Face Acoustic Pressure (kpa) R Pressure at Center o Transducer Time(Micro-sec) (c) Acoustic Pressure at distance Z=50mm rom transducer Face Figure 6.9: Acoustic pressure Response at various points along transducer axis (Zdirection) 77

89 6. Ultrasonic Field in a Non-Homogenous Fluid-DPSM technique: 6.. Steady Wave Propagation in a Non-homogenous luid: DPSM technique or homogenous luid can be extended to Non-homogeneous luid having interace using concepts mentioned in chapter 5, and MathCAD code is developed using the method mentioned in section 5... Developed code is irst validated or steady wave propagation in non homogeneous luid. Figure 6.0 and Figure 6. shows pressure ield developed at transducer ace and at interace. The distance between the transducer ace and the interace o two luids is kept as 0 mm. The transducer is kept in luid (having P-wave speed.49 km/sec and density = gm/cc) and the P-wave speed and density o luid is km/sec and.5 gm/cc respectively. Transducer diameter is 0. inch and transducer requency is MHz. The results are matching with kundu 5. Figure 6.0: Acoustic Pressure in XY Plane close to the transducer Face Figure 6.: Acoustic Pressure in XY Plane at Interace (Z=0mm) o two luids 78

90 6... Transient wave ields in a Non-homogeneous luid Same steady state model is used to study the transient wave using FFT engine. Tone burst signal is used as input signal and the acoustic pressure & velocity are recorded at two points lying along transducer axis at Z=0 as shown in igure 6. as target point and at Z=00 as shown in igure 6. as target point. To get clear distinct pulse peak, interace is kept at 50 mm rom the transducer ace and also the FFT sampling points are taken as 5. Figure 6.4 shows the acoustic pressure response at Target point close to transducer ace. First pulse is input pulse where as second pulse is relected pulse rom interace. The time lag between the peaks is 5 µ-secs, which is the matching with the theoretical time that the wave shall take to reach interace and rebound back to transducer ace. Figure.6.5 shows the acoustic pressure response at Target point. This plot show the time lag between input pulse peak and relected pulse peak is less as compared to earlier case. This is obvious as input pulse will take more time to reach the target point and also the relected pulse will reach early at point as it is nearer to interace. It is important to note that or clarity the relected pulse is magniied by 0 5 times in igure 6.4 (c) and by 0 times in igure 6.5(c). Similar behavior can be observed or velocity responses at two points rom transducer ace reer igure 6.6 (c) and igure 6.7(c). 79

91 Figure 6.: Location o Target point where the response o ultrasonic ield is observed Figure 6.: Location o Target point where the response o ultrasonic ield is observed 80

92 Input pulse Acoustic Pressure (kpa) Time (Micro-sec) Acoustic Pressure(kPa) Relected Pulse Time (Micro-sec) 0 Acoustic Pressure (Kpa) µ-sec I n p u t P u ls e R e le c t e d P u ls e T im e ( M ic r o - s e c ) Figure 6.4: Tone Burst Signal -Pressure response at Target point (at Z=0mm) (c) 8

93 Input pulse Acoustic Pressure (kpa) Time (M icro-sec) Acoustic Pressure (kpa) Relected Pulse Time (Micro-sec) I n p u t P u l s e R e l e c t e d P u l s e Acoustic pressure (KPa) µ-sec T im e ( M ic r o - s e c ) (c) Figure 6.5: Tone Burst Signal -Pressure response at Target point (at Z=00mm) 8

94 Input Pulse velocity (m/sec) Time (Micro-sec) Velocity (m/s) Relected pulse Time (Micro-sec) 5µ-sec I n p u t p u l s e R e l e c t e d p u l s e Velocity (m/sec) T i m e ( M i c r o - s e c ) (c) Figure 6.6: Tone Burst Signal-Velocity response at Target point (at Z=0mm) 8

95 Velocity (m/s) Input Pulse Time (Micro-sec) Velocity (m/s) Relected Pulse Time (Micro-sec) Velocity (m/sec) µ-sec In p u t p u ls e R e le d te d p u ls e T im e ( M ic r o - S e c ) (c) Figure 6.7: Tone Burst Signal-Velocity response at Target point (at Z=00mm) 84

96 6.. Acoustic Pressure Distribution with respect to Time Acoustic pressure contours (tomograms) are plotted at various times interval on X-Z plane as shown in the igure 6.8 or three types o interaces. Input velocity pulse at transducer ace is again tone burst pulse and having same requency spread as per igure Transducer having diameter 0. inch is immersed in water and interace is also kept o same size as 0. inch. In the ollowing igures dierent colors show intensity o pressure obviously red color indicates maximum pressure. Figure 6.9 shows the acoustics pressure tomogram or homogenous at dierent time instances o ininite boundary. Figure 6.0 shows the acoustic pressure developed in homogenous luid having a rigid interace. Relected waves can be seen ater µ sec. Figure 6. shows the acoustic pressure developed in case o non-homogenous luid and in this igure behavior o only one luid is shown. Figure 6. shows acoustic pressure developed due to non-homogenous luid showing behavior o two luids. Rigid interace or ree boundary interace are modeled by making second luid properties (ρ = 0) nearly zero or ree boundary and (ρ = 0 5 ) or rigid boundary. Figure 6.8: Location o X-Z plane and the target point grid used to generate tomo-grams 85

97 Z X Figure 6.9: Acoustic Pressure Tomograms or Homogenous luid (ininite boundary) 86

98 Figure 6.0: Acoustic Pressure Tomo-grams or homogenous luid (Rigid interace) 87

99 Figure 6.: Acoustic pressure Tomo-grams or Non-Homogenous Fluid 88

100 CHAPTER 7 EXPERIMENTAL VALIDATION 7. Experimental Setup: In the previous chapter numerical results or ultrasonic ields generated in homogenous and non-homogenous media and also their simulation have been presented.in this chapter to validate the numerical results with the experimental results the ollowing setup have been used. Testing Fluid R Transducer acting receiver S / R Transducer acting source & receiver Figure 7.: Experimental setup used or testing 90

101 Following are the salient eatures o the Pulser/ receiver system used: PRF Oscillator & Pulser Trigger control: The internal PRF oscillator generates repetitive trigger pulses or the pulser subsystem under the control o the PRF control. Pulser Trigger control selects between the internal PRF oscillator or an external source applied to the Trig/Sync connector as trigger sources or the DPR 00 Pulser. Pulser (Impedance/Energy/Damping): The pulser generates an excitation pulse upon receiving a trigger event rom a selected source. There are our energy and two impedance values, and the single Energy and impedance control adjusts the pulse energy and the pulser impedance. Receiver ampliier: It controls the ampliication or attenuation o signals processed by the DPR00 receiver. The receiver gain can be varied rom -db to 66 db. Low Pass and High Pass ilters: Low ilters are available or reducing the bandwidth o the DPR00 receiver. High Pass ilters are available or eliminating undesirable low requency energy rom the DPR00 receiver signal. High pass iltering can be used as a means o providing aster receiver recovery rom strong signals such as the excitation pulse or strong interace echoes. Digitizer Card: It converts the analog signal which is coming rom DPR 00 in to Digital signal and ed to computer. 7. Experimental Procedure: Pulser/receiver system sends a negative spike signal (00 V to 475 V) to the transducer as an input pulse.depending upon the method (pulse echo or through) the output pulse will be sensed by the respective transducer. The received output pulses are then ampliied and iltered by DPR 00 and sends to digitizer card where it converts analog signal to digital orm and on computer screen the waveorm can be seen with the help o sotware (Acquiris) and also graphical data can be captured. Thus dierent waveorms or dierent types o luids and interaces are obtained rom experimental setup.. 9

102 7. Veriying the Linearity o the Transducer using DPSM Technique and Experimental results: 7.. DPSM technique results: In the developed MATHCAD code or homogenous luid dierent input velocity impulses are inputted and the Acoustic pressures developed at dierent levels o luid (00,50,00mm) is observed. Observations: Velocity impulse (in m/s) Pressure Developed at Z=00mm (in dynes/cm ) Pressure Developed at Z=50mm (in dynes/cm ) Pressure Developed at Z=00mm ((in dynes/cm ) Acoustic pressures vary linearly with respect to input velocity impulses as shown in igure (7.).. Acoustic pressure values are decreasing as the level o luid increases this shows that as the level o luid is increasing the dissipation o pressure to the surrounding luid is also increasing. Pressure developed at Z=50(dynes/cm) Input velocity impuse (m/s) Figure 7.: Variation o acoustic pressure with respect to input velocity pulse 9

103 7.. Experimental Results: Dierent input voltages are inputted in steps o 5V (5 to 475 V) rom DPR00 and the corresponding output voltages are observed at dierent levels o water. Output voltage Output voltage Output voltage Input voltage or or or 00 mm o water 50mm o water 00mm o water (in volts) (in volts) (in volts) (in volts) Observations:. Output voltages vary linearly with respect to the input voltage as shown in the igure (7.).. Output voltage is decreasing as the level o water is increasing this shows that signal is attenuating as the level o water is increasing. (reer to columns,and 4 in the above table) output voltage (volts) Input voltage(volts) Figure 7.: Variation o output voltage with respect to input voltage 9

104 7.4 Time history Records-DPSM technique Vs Experimental Results: Experimental setup shown in igure 7. is used or generating experimental results. Dierent luids o dierent levels are tested in the tank using pulse echo method and through transmission methods which are described in chapter.depending upon the method DPR 00 activates respective transducer as shown in table below. Pulse Echo Through transmission Sender and receiver(s/r) transducer is same, signal is send rom T/R port o DPR00. Signal sender is (S) rom T/R port and signal receiver (R) is connected to THROUGH port o DPR00 Figure 7.4 and Figure 7.5 shows the waveorms o relected pulses which are coming when the wave hits the bottom surace o the glass beaker and touch the transducer ace back traveling two times same path as it is a pulse echo method. The tabular orm shows how well the experimental times are matching with DPSM time values and also the peak magnitude. Figure 7.6 shows time domain signal o relected pulses which are coming both rom interace o kerosene oil and water and also rom bottom o the glass beaker. In this also the times are matching with DPSM results. Figure 7.7 shows the relected pulses which are coming rom the stress ree boundary by placing the transducer at the bottom o the tank here also pulse echo method is used and the results are well matched with DPSM. Figure 7.8 shows the incident pulse which is received by the transducer placed at the other end as it is through transmission technique. It takes less time as the target point is near to the interace. Here also experimental results are showing well match with DPSM results. 94

105 .0.884E-4.65E E-4.5 Amplitude (Volts) E E E E E-4.665E E Time (secs) Time(µs) Ampl(v) Remarks DPSM/theoritical( µs) st rom bottom o beaker (0x )mm nd rom bottom o beaker (0x4)mm rd rom bottom o beaker (0x6) mm 54 Figure 7.4: Variation o Output voltage Vs Time along with peak amplitudes using pulse echo method in kerosene oil. (Homogenous luid) 95

106 E E-4.5 Amplitude (Volts) E E E E Time (secs) Time(µs) Ampl(v) Remarks DPSM (µs) 60.0 st From bottom o beaker(95x )mm nd rom bottom o beaker (95x4)mm rd rom bottom o beaker (95x6)mm 785 Figure7.5: Variation o Output voltage Vs Time along with peak amplitudes using pulse echo method in water (homogenous luid) 96