design, construction and characterization of highpower ultrasound sources

Design, construction and characterization of highpower ultrasound sources

Søren Hartmann Kristensen

December 21, 2009

Abstract

In this Master’s thesis several areas of designing, constructing and characterizinghigh power ultrasonic sources are investigated.

An ultrasonic source, consisting of a prefabricated Langevin transducer and amechanical horn designed and constructed for this project, is used for experimentalvalidation of design simulations. The horn has a mounting plate for easy mountingof the joined ultrasonic source. Using this horn several experiments is carried out,including a successful attempt to levitate polystyrene balls and water drops.

To be able to characterize the vibrational pattern of radiation plates, an opticalsetup, based on the theory of speckle interferometry, is used. Using this setup, it ispossible to visualize nodal lines on the vibrating surfaces of radiation plates.

The investigation of the design parameters, as well as investigation of the behav-ior and radiation or radiation plates, is done trough simulations in Comsol Multi-physics. To achieve accurate material parameters, Young’s modulus is measured forthe material used to design the mechanical horn for the source.

Abstrakt pa dansk

I dette speciale er flere felter indenfor design, konstruktion og karakterisering afhøjeffekt ultralydskilder undersøgt.

En ultralydskilde, sammensat af en præfabrikeret Langevin transducer og etmekanisk horn designet og konstrueret til dette projekt, er brugt i eksperimentellevalideringer af design simuleringer. Hornet har en monteringsplade for let monteringaf den samlede ultralydskilde. Ved brug af dette horn er flere eksperimenter udført,bl.a. et succesfuld forsøg pa at levitere polystyrenkugler og vanddraber.

For at kunne karakterisere vibrationsmønsteret for udstralingsplader, er en optiskopstilling, baseret pa teorien om speckle interferometri, brugt. Ved hjælp af denneopstilling, er det muligt at visualisere nodallinier pa den vibrerende overflade afudstralingspladerne.

Undersøgelsen af designparametre, sa vel som undersøgelsen af udstralingspladernesopførsel og udstraling, er udført vha. simuleringer i Comsol Multiphysics. For atopna præcise materialeparametre, er Young’s modul malt for det materiale som erbrugt i designet af det mekaniske horn til kilden.

Contents

1 Introduction 11.1 The structure of high power ultrasonic transducers . . . . . . . . . . 1

1.1.1 The Langevin transducer . . . . . . . . . . . . . . . . . . . . . 11.1.2 Mechanical horns . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.3 Radiation plates . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Objective of the project . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Theory 62.1 Vibrations in bars with circular cross section . . . . . . . . . . . . . . 6

2.1.1 Longitudinal vibrations . . . . . . . . . . . . . . . . . . . . . . 62.1.2 Transversal bending waves vibrations . . . . . . . . . . . . . . 72.1.3 Torsional vibrations . . . . . . . . . . . . . . . . . . . . . . . . 92.1.4 Radial vibrations . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Radiation from a circular piston . . . . . . . . . . . . . . . . . . . . . 102.3 Vibrating discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Simulation in Comsol Multiphysics . . . . . . . . . . . . . . . . . . . 13

2.4.1 Axisymmetric simulations . . . . . . . . . . . . . . . . . . . . 132.4.2 Simulations types . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.3 Simulation of coupled physics . . . . . . . . . . . . . . . . . . 152.4.4 Simulation conclusion . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Acoustic levitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Effect of mechanical horn shapes 183.1 Conical horns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Exponential horns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 Stepped horns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3.1 Nodal position in stepped horn . . . . . . . . . . . . . . . . . 243.4 Resonance frequencies for horn types . . . . . . . . . . . . . . . . . . 25

4 Design of stepped horn for Langevin transducer based ultrasonicsource 284.1 Work flow stepped horn design . . . . . . . . . . . . . . . . . . . . . . 284.2 Mounting plate and transition effects . . . . . . . . . . . . . . . . . . 30

4.2.1 Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2.2 Mounting plate . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3 Radiation from a stepped horn with no radiation plate . . . . . . . . 33

4.4 Stepped horn designed for practical experiments . . . . . . . . . . . . 34

5 Experimental validation 365.1 Test of assembled sources . . . . . . . . . . . . . . . . . . . . . . . . . 365.2 Frequency analysis of horn . . . . . . . . . . . . . . . . . . . . . . . . 385.3 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.3.1 Simulation model errors . . . . . . . . . . . . . . . . . . . . . 395.3.2 Error analysis conclusion . . . . . . . . . . . . . . . . . . . . . 42

5.4 Sensitivity analysis of sources . . . . . . . . . . . . . . . . . . . . . . 435.5 Acoustic levitation test . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6 Radiation plate assessments 486.1 Node detection using talcum powder . . . . . . . . . . . . . . . . . . 486.2 Visualization of radiation plate displacement pattern using speckle

interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.3 Radiation plate simulations . . . . . . . . . . . . . . . . . . . . . . . 50

6.3.1 Vibration pattern of brass disc . . . . . . . . . . . . . . . . . . 516.3.2 Sound radiation from radiation plates . . . . . . . . . . . . . . 51

6.4 Radiation plate conclusion . . . . . . . . . . . . . . . . . . . . . . . . 55

7 Conclusions and Future Work 577.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577.2 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Appendices 61

A Measuring Young’s Modulus of Aluminum bars 61

B Optical experiment for vibrational displacement patterns 68

C Data sheet for Langevin transducer 71

D Data sheet for aluminum EN AW 2011 73

E Work sketch of designed horn 75

F Equipment list 77

G CD containing the thesis and videos of levitation experiments 78

Bibliography 79

Preface

For this project several people has contributed with knowledge and practical help,for which thy deserve thanks. First I would like to thank Associate Professor ReneSkov at University of Southern Denmark (SDU) for his kind help with the opticalsetup used in the project. Thanks also goes to the mechanical workshop at SDUfor teaching me how to use a metalworking lathe and for their help with practicalmetalwork. I would also like to thank my advisor Arturo Santillan, for his help withthe acoustic levitation experiments and for sharing his knowledge of the subjectscovered in this thesis.

Table of symbols

Symbol Description Unit

ca Speed of sound in air m/scL Propagation speed of longitudinal vibration m/scT Propagation speed of torsional vibration m/sρ Density kg/m3

ρ0 Density of air kg/m3

ε StrainE Young’s modulus, modulus of elasticity PaG Shear modulus Paσ Poisson’s ratiof Frequency HzD Diameter mS Surface area m2

w Displacement mW Displacement amplitude mx Distance mL Length mA Amplification gainp Pressure Pav Surface/particle velocity m/sλ Wavelength mω Angular frequency ω = 2πf rad/sk Wavenumber k = ω/c m−1

U Gorkov potentialRobj Radius of levitated object mF Force NZ Mechanical impedance N

m/s

1 Subscript 1, naming for input- or wider part of mechan-ical horns

2 Subscript 2, naming for output- or narrower part of me-chanical horns

∆ Change in e.g. frequency ∆ = |f2 − f1|‖A‖ Euclidean norm of errorsai Individual error

1

Chapter 1

Introduction

In this thesis some of the aspects of designing a high power ultrasonic source isdescribed. These aspects include the structure of such sources, design aspects inconstructing an efficient source, characterization of the such sources and measure-ments of the performance of a source.

The term ultrasonic sources means a source that radiates sound with frequenciesabove the human hearing range, normally said to be 20kHz (though most peoplecannot hear that high frequencies). Ultrasound has many practical applications [1]ranging from chemical such as large scale treatment of gasses [2–4] over medicalsuch as wound healing and ultrasonic imaging, to industry applications like weldingand cleaning. Some of these applications demand a very high sound pressure to beproduced, thus needing a high power ultrasonic source.

In this chapter the structure of a typical hight power ultrasonic transducer is out-lined and the objective for the thesis is stated. Furthermore the naming conventionsused in the thesis are described.

1.1 The structure of high power ultrasonic trans-

ducers

This section contains a short review the elements that are normally combined tobuild high power ultrasonic sources.

High power ultrasonic transducers are widely build on basis of a Langevin trans-ducer, coupled to a mechanical horn, providing a velocity amplification, and with aradiation plate to increase the sound radiating surface of the source, and there byenabling the source to deliver a higher sound pressure.

The elements of a typical ultrasound transducer can be seen in figure 1.1.

1.1.1 The Langevin transducer

The Langevin transducer is a mechanical vibration source invented by the Frenchphysicist Paul Langevin at the end of world war one [5]. The transducer was firstbuild as part of an ultrasonic submarine detector and consisted of a piezo electricelement sandwiched between two metal parts. When electrical voltage is appliedto the piezo elements, they expand or contract depending on the polarization of

2

Figure 1.1 - The elements of a typical ultrasonic transducer

the voltage. This expansion (or contraction) is very strong and reliable, and cantherefore be used to drive mechanical vibrations in metal parts that are attached tothe piezo elements.

A Langevin transducer usually consist of an even number of piezo elements sand-wiched between two metal blocks. By designing the metal blocks attached to thepiezo elements to be λ/4 longitudinal resonators with the same resonance frequency,the combined transducer will be a λ/2 longitudinal resonator in balance, havingmirrored vibrations on the two sides of the piezo elements. Using this technique,the Langevin transducer can produce a very strong vibrations on the surface of themetal blocks.

A sketch of a typical Langevin transducer arrangement can be seen in figure 1.2.

Figure 1.2 - Setup of a Langevin type transducer

The metal blocks in a Langevin transducer are usually made from aluminumand/of steel. An asymmetric design with one steel and one aluminum block can beused, but in that case the blocks should not have the same physical length, but thesame “Acoustical length”, meaning the length with which the λ/4 resonance has thesame frequency.

Much research is done to optimize the characteristics of Langevin transducersthis is described in e.g. [6, 7].

For this project a Langevin transducer from the Swiss company M.P. Intercon-sulting has been used. The transducer has the type number MPI-3025F-30H, a

3

diameter of D = 30mm and a resonance frequency of f = 29.9kHz. The transducercan be seen in figure 1.2.

Figure 1.3 - Langevin transducer used in this project

1.1.2 Mechanical horns

Mechanical horns is a means of amplifying mechanical vibrations. In ultrasonicsources for a variety of applications, mechanical horns are used to maximize thevibrational amplitude the source can produce. This amplification is described ingreater detain in a later chapter, but can intuitively be understood by a conservationof momentum example: Consider a free-free bar vibrating at a λ/2 longitudinalresonance. The ends vibrate in the longitudinal direction while there at the middleof the bar is a node. If one end of the bar is lighter than the other, this end willvibrate more, since the conservation of momentum applies. Hereby there will be anamplification of the vibration from the wider heavier end of the horn to the narrowerlighter end.

In section 3 the shape and theoretical performance of different horn types aredescribed in greater detail and investigated though Comsol simulations.

With the Langevin transducer bought for this project came a mechanical hornmeant to be the coupling to additional tools which can be attached using a threadedhole in the front of the supplied horn. This horn will be referenced to as the Swisshorn due to the origin of the company producing the Transducer-horn combination.The Swiss horn can be seen in figure 1.4

1.1.3 Radiation plates

Radiation plates attached to the mechanical horn of a ultrasonic transducer is ameans of achieving different acoustic radiation properties, than the often small sur-face of the horn would facilitate. By using a plane radiation plate with radius largerthan the horn end, the surface area of the transducer can be increased causing agreater volume velocity and thereby greater sound pressure radiation, than the hornsurface alone would produce. This principle is used in e.g. [8]. Furthermore specialshapes of radiation plates can be designed to produce a sound field suitable for cer-

4

Figure 1.4 - Swiss horn provided by the manufacturer of the Langevin trans-ducer. Also seen is the screw used for connecting the horn andLangevin transducer into a ultrasonic source

tain applications of the source. Such special designs of radiation plates can be seenin [3, 4, 9].

An example of a radiation plate with a special design is described in [9] where aradiation plate with special groves improves the acoustic power of the source. Thesource in [9] is designed to process large volumes of gas.

A way of changing the sound field radiated by a source is to make a flexuralradiation plate that does not vibrate in a piston-like fashion. This type of vibra-tion in a radiation plate is investigated in chapter 6 where some ways of visualizingthe vibrational pattern is introduced. Furthermore the vibrational pattern is sim-ulated in Comsol Multiphysics, where the mechanical vibrations simulation in theradiation plate can be coupled to an acoustic simulation, which can be useful forcharacterization of a given radiation plate design.

1.2 Objective of the project

Designing a high power ultrasonic transducer involves a range of different steps.From designing or choosing the Langevin source, designing an optimal horn anddesigning a radiation plate that gives a sound field usable for the purpose at hand.Other things that could be taken into the design and optimization of such sources,is e.g. optimizing the electric circuits that delivers the excitation signal to thepiezo elements. These parts are all places where great improvements can be made.Some of these are described in [8], where it is seen that the optimization of thesource facilitates the use of a much smaller amplifier for a stable setup for acousticlevitation.

In this thesis the main focus will be the design of the mechanical horn to attachto the purchased Langevin transducer, which will be joined to form an ultrasonicsource. The design process followed in the work described in the present thesis isbased on the use of Comsol Multiphysics, using material properties measured in the

5

lab.Further more the project also covers ways to characterize radiation plates. The

characterization is done trough simulations of the radiated sound field and practi-cal experiments, with which it is possible to visualize the displacement patterns ofradiation plates vibrating at ultrasonic frequencies.

The main motivation for this work is to produce a source strong enough to beused for acoustic levitation.

1.3 Conventions

In this thesis the parts of the horn are named after their place in the combinedsource. The part of the horn that is closest to the Langevin transducer (the widerpart of the horn) is named with the subscript “1” and the part closest to the soundfield is named with the subscript “2”. For example the diameter of the second partof the horn is named D2 and the length of the first part is named L1. In relation tofigure 1.1 this convention is a top-down naming of the horn parts. This conventionis used throughout the thesis.

CHAPTER 2. THEORY 6

Chapter 2

Theory

In this chapter the theory used further in this thesis will be outlined. First a reviewof vibrational theory in bars needed for designing mechanical horns is presented, andthe chapter continues with some theory on radiation of sound from circular pistons.When dealing with radiation plates attached to the end of a horn the radiation platedoes not always vibrate in a piston like motion. For this reason the chapter alsocontains a section describing the vibrations of flexible circular plates.

Apart from the physics theory an introduction to the simulation tool ComsolMultiphysics is also made in this chapter.

In the end of the chapter a short introduction to the Gorkov potential, and someacoustic levitation theory is introduced to better be able to evaluate the sourcesused in this project.

2.1 Vibrations in bars with circular cross section

In this section the theory of vibrations in circular bars is outlined. This theoryinvolves longitudinal, transversal, torsional and radial vibrations and resonances inbars

2.1.1 Longitudinal vibrations

For calculation of longitudinal vibrations in bars, it is commonly used, to use thestress strain formulation [10, chapter 3] given but the expressions for strain, ε

ε =∂w

∂x(2.1)

and stress given by

F

S= −E · ε (2.2)

where w is a measure of displacement along the bar, x is a measure of distance andS is the cross section of the given bar that is substitute to the force F . The valueE is the elasticity coefficient of the material, also known as Young’s modulus.

CHAPTER 2. THEORY 7

Given the mentioned equations for stress and strain, the longitudinal wave equa-tion can be deduced to be [10, page 70] :

∂2w

∂x2=

1

c2L

∂2w

∂t2(2.3)

where cL is the propagation speed of the longitudinal vibration. cL can be calcu-lated from the density of the given material, ρ, and Young’s modulus the followingequation [10, page 70]

cL =

√E

ρ(2.4)

By assuming a general solution of the wave equation given in equation 2.3 andapplying the boundary conditions of the geometry at hand, the mode pattern forthe resonances can be calculated. This is described in greater detail in [10, Page71-72].

For a bar that is free in both ends, meaning that the support for the bar doesnot apply any strain to the ends of the bar, the mode shapes can be calculated fromthe following equation [10, page 72]:

w (x, t) = 2Wejωntcos(knx) (2.5)

where W is the amplitude of a an initial vibration reflected at the bar end, kn is thewavenumber and ωn is the angular frequency of the n’th resonance.

The resonance frequency of the n’th resonance can now be calculated as:

fn =ncL2L

(2.6)

where L is the length of the bar.It is seen that the first resonance in a free-free bar is a half-wavelength resonance

(λ/2 resonator). As we shall later see, this is the resonance that is of most interestwhen building a high power ultrasonic resonator source.

The equations here mentioned is a formulation for pure longitudinal vibrations ina bar, and it is assumed that the displacement and thereby the strain is uniform overthe cross section of the bar. This is not always the case, especially for bars that arerelatively wide compared to their length. In that case the longitudinal vibrations areaffected by the radial vibrations (see section 2.1.4) ultimately changing the resonancefrequency of the bar. This phenomenon is described in detail in [7].

In figure 2.1 an example of a longitudinal resonance in a circular bar can be seen.

2.1.2 Transversal bending waves vibrations

Another types of vibrations that can occur in circular bars is transversal bendingvibrations. This vibration can also be experienced, for instance, in a swaying flagpole or a vibrating spring.

CHAPTER 2. THEORY 8

Figure 2.1 - Example of a longitudinal resonance in a circular bar. The dis-placement is shown as deformation (exaggerated) and by the colorscale indicating absolute values of displacement. The original baris shown by the black wire frame. This is a resonance of a fixed-free bar

Figure 2.2 - Example of a transversal resonance in a fixed-free circular bar

The wave is a result of bending moments and shear forces acting in the bar. Anexample of a transversal wave can be seen in figure 2.2

In the case of a circular bar that is clamped at one end and free at the other,the resonance frequencies are given by [10, page 83]

f =(1.1942, 2.9882, 52, 72, ...

) πacL12L2

(2.7)

where L is the length of the bar and a is the radius of the bar. The expression hasbeen written as a function of cL, which is the propagation speed of longitudinal wavesfor the material. This is not because cL is the same as the propagation speed of thebending wave, but because it makes the the formulations for the longitudinal andtransversal waves easier to compare. It is seen from equation 2.7 that the frequencyof the transversal waves is dependent of the radius of the bar. For this reason,unwanted transversal waves in a mechanical horn can be avoided by changing theradius of the horn parts, given the rest of the horn design allows this change.

CHAPTER 2. THEORY 9

In the design of a Langevin transducer based ultrasonic source, transversal vibra-tions are not desired. If transversal vibrations have resonances in the same frequencyrange as the longitudinal vibrations it is possible that some of the vibrational energywould go to transversal vibrations instead of the wanted longitudinal vibrations.

2.1.3 Torsional vibrations

A torsional vibration occurs when a bar is excited by a twists torque around its axisof the bar. An example of the twisting movement is shown in figure 2.3, where atorsional resonance for a fixed-free bar is seen.

Figure 2.3 - Example of torsional resonance in a circular bar. The bar hasbeen sliced trough to in 5 slices better see the rotation

The first torsional resonance is a λ/4 resonance, which was also the case for thefirst longitudinal resonance. In this and other ways the torsional and longitudinalvibrations behave much alike the. The torsional wave equation is, like the longitudi-nal waves, a second order differential equation. The propagation speed of torsionalwaves is [11, page 46]

cT =

√G

ρ(2.8)

where G is the shear modulus, determined by the material composition. G has inthe description of torsional vibrations the role that the Young’s modulus E has inthe longitudinal vibration description.

From this similarity it is noted, that if torsional and longitudinal resonancescoincide, it is not possible to move the frequency of the torsional resonance to adifferent frequency without the longitudinal resonance also moving the same amount.On the other hand G and E are not likely to coincide for many materials; in mostcases the sheer modulus will be a lot less. For some materials though, it will bepossible to excite a higher torsional resonance when driving the first longitudinalresonance.

In the case of a Langevin transducer driving a mechanical horn, it is not likelythat the torsional resonances will be excited very much, since the piezo electric discsof the Langevin transducer vibrates primarily in the longitudinal direction.

CHAPTER 2. THEORY 10

2.1.4 Radial vibrations

Radial vibrations are mechanical vibrations that runs in a direction perpendicularto the axial direction of the bar. This means that the resonance of this type ofvibrations will be a standing wave in this direction. Since the dimensions of a barare usually greater in the axial direction than in the radial direction, the resonanceof radial vibrations will have a higher frequency than the longitudinal resonancefrequency.

An example of a radial resonance can be seen in figure 2.4, where the dimensionsof the example rod is changed, so that the rod is very short but rather wide.

(a) (b)

Figure 2.4 - Radial resonance in a cylindric “bar” that is very short relative toits diameter. The two images are of opposite phase

In this project the radial vibrations play a part since they have an effect on thelongitudinal resonance frequencies of cylindric bars as described in [7]. Here it isstated that even away from the radial resonance frequency, the radial vibration affectthe longitudinal resonance by lovering the resonance frequency. This results in thelongitudinal wave equation not being sufficient to calculate the resonance frequency.Since Comsol simulates the geometry defined and do not simply assume longitudinalvibrations, the mix of longitudinal and radial vibrations will not be a problem in thesimulations. In the theoretical calculations however it cannot be assumed that thelongitudinal vibrations equations can be accurately used to calculate the resonancefrequency of a bar if the length to width ratio is not very high.

2.2 Radiation from a circular piston

From a summation of point sources, the on axis pressure generated from a planarcircular piston vibrating with a fixed velocity v can be described as the following


equation [12, Chapter 7] for pressure as function of distance r and piston radius a

p (r, a) = 2ρ0cav

∣∣∣∣∣sin(

1

2kr

(√1 +

(ar

)2− 1

))∣∣∣∣∣ (2.9)

where ca is the speed of sound in air, ρ0 is the density of air and k is the wavenumber.In the far field the pressure amplitude is asymptotic to the curve given by

p(r, a) =1

2ρ0cav

k · a2

r(2.10)

It should be noticed that the far field the pressure at a given distance is propor-tional to the radius of the piston squared.

pfar−field ∝ a2 (2.11)

For distances closer to the source, the pressure amplitude has not yet meet theasymptotic decay in equation 2.10, and therefore the proportionality to the radiussquared does not comply. A graphical representation of the pressure radiated froma circular disc is shown in figure 2.5 for k = 20 for a range of distance to radiusratios.

Figure 2.5 - Sound pressure radiated from circular piston for different radiusto distance ratios. k = 20

From the figure it can be seen that for some specific distances, a higher soundpressure can be achieved by choosing a smaller surface area for the sound source.This can be seen by comparing the normalized radiation in figure 2.5 at a distanceto radius ratios of 1.2 and 1.7. With the distance constant the jump from 1.2 to 1.7could be done by decreasing the disc radius by 30% but vould lead to an increasein the radiated sound pressure level by approximately 18dB. This is only a localeffect and it does not comply in the far field, but if a high sound pressure in thenear field is the main interest in a certain source design, the source radius shouldnot necessarily be as big as possible.


2.3 Vibrating discs

By vibrations in discs is in this thesis meant a bending vibration traveling in theradial direction of the disc.

Examples of resonances of free vibrating discs can be seen in figure 2.6, where thefirst two axisymmetric resonances, as well as one of the non axisymmetric resonancesof an arbitrary free vibrating disc can be seen.

(a) First axisymmetric resonance (b) Second axisymmetric resonance

(c) Non axisymmetric resonance

Figure 2.6 - Resonances in a thin unsupported disc. The discs in (a) and (b)are the two first axisymmetric resonances and (c) is an arbitrarynon-axisymmetric resonance of the disc


In the figure it is seen that the first axisymmetric resonance has a nodal circlearound the center of the disc.

In this thesis the non axisymmetric resonances will not be investigated further,but the axisymmetric vibrations of discs is used in connection to the characterizationof radiation plates in chapter 6.

The images are from a 3D simulation in Comsol Multiphysics.

2.4 Simulation in Comsol Multiphysics

When investigating the behavior of ultrasonic transducers it is widely used to basesuch research on finite element method (FEM) simulations [6, 7, 13]. In this projectthe simulation tool Comsol Multiphysics has been used for analysis of mechanicalvibrations, as well as sound propagation. For a thorough introduction to FEM forvibrational simulation see [14, Chapter 8].

In this section it is described how the geometries in the project have been simu-lated. Simulations of stress strain vibration as well as acoustical field radiation, hasbeen carried out using the Comsol Multiphysics (from here referred to as “Comsol”).This simulation software has a wide range of simulation facilities e.g. electricalfield simulations, magnetic field simulations, mechanical stress-strain simulationsand acoustic field simulations. The software also has the ability to connect multiplephysics in one simulation, as it is done in a later chapter where both mechanicalvibrations and acoustical radiation is simulated in one combined simulation.

When simulating a given physical situation in Comsol, the model of the physicalobject can be made in several different geometry formulations. To reduce calculationtime is useful to reduce the problem to as simple a geometry as possible. If a givensituation can be simulated as a two dimensional problem, the calculation time isfar less than if a full 3D simulation is run of the same geometry. The reason forusing the more elaborate geometry formulations is, that they will reveal solutions,that the simpler formulations would not. It is for instance not possible to simulatetransversal waves in the axisymmetric formulation, since the transversal waves arenot axisymmetric.

2.4.1 Axisymmetric simulations

In the axisymmetric formulation of Comsol, the geometry defined is solved with theconditions that all physical properties are symmetric around a user defined symmetryaxis. This means the geometry is defined in a 2D space and the user must imaginethe geometry rotated around the symmetry axis. Therefore, a geometry solvedfor stress/strain vibrations in the axisymmetric formulation will only reveal thelongitudinal and radial waves in the geometry, since they are the only axisymmetricwaves that can occur in a bar.

In figure 2.7 examples are seen to illustrate how the Axisymmetric geometryrepresents a 3D shape. The geometry in 2.7(a) is defined for Comsol simulationsand is rotational symmetric around the straight side on the left. In 2.7(b) a 3Dmodel of a similar geometry is seen. This model is made in Autodesk Inventor by


(a) (b)

Figure 2.7 - A similar geometry represented as (a) an axisymmetric geome-try defined for Comsol axisymmetric simulations and (b) in a 3Dmodel designed in Autodesk Inventor

actually drawing a 2D axisymmetric sketch and revolving it around the same axisas Comsol does (virtually)

The simplification of the geometries enables the definition of these from a Matlabscript, which enables automated simulations driven from Matlab 1.

2.4.2 Simulations types

When simulating a given geometry in Comsol, different simulations can be run toreveal different properties of the geometry. For the stress-strain simulations in thisproject two simulation types have been used, namely eigenfrequency analysis andfrequency response analysis.

The eigenfrequency simulations

In eigenfrequency simulations Comsol finds a user defined number of resonancesfor the defined geometry. This is done trough eigenfrequency equations predefinedwithin the Comsol software.

When solving for eigenfrequencies Comsol includes the stress-strain interactionand mass of the geometry, but not defined forces or displacement. If any displace-ment is set, these will be seen as a fixed boundary by the eigenfrequency solver.To include forces and displacement in the simulation model a frequency responsesimulations must be used.

1Using these automated simulations, it is possible to make geometry sweeps; a functionality notimplemented in Comsol as of version 3.5. A new version of Comsol, version 4, is under development.This version supposedly supports geometry sweeps from within the Comsol environment


Frequency response simulations

In the frequency response simulations the simulations tool solves a stress-strain waveequation numerically with the given boundary conditions. This type of simulationcorresponds to doing a physical experiment, in the way that an interaction must beapplied to the model or nothing will happen.

The resulting behavior of the model geometry can, in a frequency analysis sim-ulation like in a physical experiment, be complex, and the results of a frequencyresponse simulation can complicated to analyze automatically using programmedscripts. For this reason frequency analysis simulations are mostly used to analyzegeometries where the behavior of a single model geometry is investigated, ratherthan in e.g. a sweep of geometry dimensions.

To ensure consistent simulations results all simulations have been done withfixed-force boundary conditions for the input excitation. This ensures that botheigenfrequency and frequency response simulations can be run revealing the sameresonances of the same geometry. To prove that this is the correct way to defineinput excitations for the free bars, see figure 2.8. Here it is seen that when the inputexcitation is set to a fixed displacement the resonance calculated is the λ/4 ratherthan λ/2 as is expected for a free-free bar.

Frequency response simulation have been preformed to find the optimal hornparameters in section 4 and for simulations of radiation plates in chapter 6.

2.4.3 Simulation of coupled physics

An important strength of Comsol Multiphysics is the ability to couple simulationsof different physics (hence the name Multiphysics). Using this functionality it ispossible e.g. to couple a mechanical vibration problem to an acoustical problem, andsimulate the two together, also including the interaction between the acoustical fieldand the mechanical system. This functionality is used in section 6.3, where radiationplate vibrations simulation are coupled to acoustic field simulations enabling thecalculation of e.g. directivity pattern of radiation plates at a given mechanicalvibration frequency.

2.4.4 Simulation conclusion

When simulating a certain physical problem in Comsol, choosing a geometry for-mulation that only just satisfy the problem at hand, can reduce simulation time incomparison to doing an elaborate 3D simulation for all problems. For this reasonthe axisymmetric geometry formulation has been used throughout this project.

While the axisymmetric simulations only reveal axisymmetric vibration proper-ties of the geometry, the longitudinal vibration problems can be simulated in theaxisymmetric formulation to increase simulations speed and to have a simpler sim-ulation result to analyze. Furthermore the axisymmetric geometries are simpler todefine from a Matlab script making the combination of the two programs a strongtool for e.g. sweeps in geometry which is not possible to do in Comsol alone.


(a) Fixed displacement

(b) Fixed force

Figure 2.8 - Two simulations of the same geometry with either fixed displace-ment (a) or fixed force (b) on the top of the bar. Only with fixedforce is the λ/2 resonance revealed


2.5 Acoustic levitation

This section consists of a short introduction to acoustic levitation. The fact thata sound field produces forces that can be used for the purpose of levitating smallobjects, has been described in various literature based on both a theoretical expla-nations [15–17] and practical setups, that can be used for levitating small objects oreven small animals like in [18]. The original proposition that such phenomenon existswas first deduced in [19], but practical validations of the theory was not conducteduntil the 1970’s where NASA conducted some experiments with micro gravity usingacoustic levitation.

To calculate the force on an object in a sound field, the Gorkov potential U isused. The Gorkov potential, which was first defined by L.P. Gor’kov in [20], can becalculated from the following equation:

U = 2 · π ·R3obj

(p2

3 · ρ0 · c2− ρ0 ·

v2

2

)(2.12)

Where p and v is the mean sound pressure and particle velocity for a given point,ρ0 is the density of air, ca is the speed of sound and Robj is the radius of the objectfor which the potential is calculated.

The force on a levitated object can be calculated from the Gorkov potential using

F = −∇U (2.13)

where F is the force on the object.Since the Gorkov potential, and there by the force on a levitated object, and the

volume, and thereby the weight of a the object, are both proportional to R3obj the

strength of a given levitator should not be measured by the weight it can levitate,but by the density of the object it can levitate. Experiments with acoustic levitationare carried out in section 5.5.

CHAPTER 3. EFFECT OF MECHANICAL HORN SHAPES 18

Chapter 3

Effect of mechanical horn shapes

In this chapter the effect of the shape of mechanical horns is investigated. Byevaluating different kinds of mechanical horns, mainly with respect to their expectedamplification of the vibration amplitude from one end to the other, the optimal hornshape for use in an ultrasonic source is determined.

The idea of mechanical horns is that by making a horn with different diametersat each end the end where the diameter is smallest will vibrate with the largestamplitude.

In this chapter three horn types are described and their theoretical amplifica-tion, deduced in [21], is attempted verified by simulations in Comsols axisymmetricformulation.

The horns are all meant to be λ/2 resonators with pure longitudinal waves, anda resonance frequency of f = 25kHz is chosen, as it lies within the normal frequencyrange of ultrasonic sources. The length of the horns are calculated from the followingequation, assuming the material of the horn is aluminum with cL = 5150m/s [10, p526].

L =λ

2=cL2f

=5150

2 · 29900= 103mm (3.1)

This length is used for all the simulated horns as is the material properties foundin [10] E = 7.1 · 1010Pa, ρ = 2700kg/m3 and σ = 0.33.

3.1 Conical horns

The simplest way of varying the diameter from one end of a mechanical horn to theother is to change the diameter gradually forming a conical horn. Such a horn isshown in figure 3.1 which shows the displacement of a conical horn in an axisym-metric Comsol simulation.

From the longitudinal wave equation (equation 2.3), the amplification A of aconical horn can be deduced, as it is done in [21, Chapter 4, III E]. Note the anothersubscript convention is being used:

A =v2v1

= N cos

(ωL

c

)− c

ω

N − 1

Lsin

(ωL

c

)(3.2)


Figure 3.1 - First resonance in an axisymmetric simulation of a conical hornwith aspect ratio D1/D2 = 7

where N = D2

D1.

For a half wavelength resonator (λ/2 resonator) this equation can be simplifiedusing

L =λ

2=cL2f

=πcLω

(3.3)

ωL

cL= π (3.4)

Inserting equation 3.4 in 3.2 the simple equation

A = −N = −D2

D1

(3.5)

is achieved for the amplification of a conical horn.To validate this amplification a series of simulations has been conducted in Com-

sol Multiphysics axisymmetric formulation. The simulation result for amplificationis seen in figure 3.2.

From the simulation result it is clear, that the amplification does only follow thetheoretical amplification for small aspect ratios of the horn. The reason for the dropin amplification at larger aspect rations could point to, that the assumption of purelongitudinal waves in equation 3.3 is not valid for the larger aspect ratios.

Figure 3.3 shows the displacement of a conical horn with a large aspect ratio ofD1/D2 = 7. Here the vibrations are not pure longitudinal waves, which is seen bythe curves of the coloring of the displacement.


Figure 3.2 - Simulation of the absolute amplification in a conical horn. Onlyfor small diameter rations does the simulation amplification matchthe theoretical amplification

3.2 Exponential horns

When designing a horn to go from one diameter to another, another way to make thistransition is in a gradual exponentially changing curve. This transition is describedby equation 3.6 for the cross section as a function of the length variable x.

S (x) = S1e−kx (3.6)

where

k =− ln

(D2

D1

)L

(3.7)

From 3.6 an expression for the diameter at a certain point in the transition canbe found:

D (x) =√D2

1ek(x−L) (3.8)

An example of an exponential horn can be seen in figure 3.4.For this type of horn the amplification can be proven to be [21, Chapter 4 III C,

p 117]:

A =v2v1

= −(D2

D1

)(3.9)

which is the same as the theoretical amplification of the conical horn, but in thededuction the assumption of longitudinal resonance made in 3.3 is not taken in thecase of the exponential horn. Like with the conical horn the exponential horn has


Figure 3.3 - An example of the deformation of the conical horn with an aspectratio of 7

been simulated for a range of diameter aspect ratios. The results of the simulationcan be seen in figure 3.5.

It is seen from the simulation results in figure 3.5 that the amplification of theexponential horn follows the theoretical amplification throughout the aspect ratiosweep, to a good extend. This shows that the exponential horn will be able todeliver a higher order of amplification than the conical horn though the theory saidtheir amplification would be the same. This is thought to be on account of thelongitudinal wave theory being more accurate for the exponential horn since it isthinner a longer way up the horn. In figure 3.6 the deformation of a wide exponentialhorn at resonance can be seen.

By comparing the deformation pattern of the exponential horn with the conicalhorn, which is seen in figure 3.3, it can be seen that in the conical horn there is acircular displacement pattern around y = 0.07m. This points to the presence of nonlongitudinal vibrations which are not assumed in the deduction of the amplificationtheory. To determine the reason for the lack of amplification for large horn ratios,the phenomenon would have to be investigated further.


Figure 3.4 - Example of an exponential horn

Figure 3.5 - Amplification simulation for the exponential horn (Absolute val-ues). Here the simulated amplification matches the theoreticalvalue for a wide range of diameter ratios


Figure 3.6 - Resonance of exponential horn with an aspect ratio of 7


3.3 Stepped horns

A simple way of going from one diameter to another is in one quick step. This is thereasoning behind the stepped horn as it is described in [21, Chapter 4, III. B] andused in [8]. This type of horn consist of two cylindric steps with an abrupt transitionat the midpoint. By assuming pure longitudinal waves in the horn, the length of thehorn steps should be equal to one another, and together form a λ/2 resonator. Inthis way the transition should be located at the nodal point of the resonator. Thisis sketched in figure 3.7

Figure 3.7 - The stepped horn and the displacement as function of the positionin the horn. This assumes pure longitudinal waves giving l1 =l2 and a nodal plane located at the transition between the twodiameters

For the stepped horn the amplification can be shown to be [21, page 113]

A =v2v1≈ Z1

Z2

=

(D1

D2

)2

(3.10)

where Z1 and Z2 are the mechanical impedance of the two horn steps. This expres-sion for the amplification assumes identical material properties (such as propagationspeed) in the two horn steps, but as long as the waves are pure longitudinal this isa reasonable assumption. The equation shows a great amplification potential whichsurpasses the two previous horn types by the power of two.

Like with the other horn type a series of simulations has been carried out, toverify the horn amplification for the stepped horn. The result of the simulation canbe seen in figure 3.8

As the figure shows, the amplification of the stepped horn follows the theoreticalamplification to a good extend for the entire aspect ratio sweep. This points to thestepped horn being the optimal horn for amplifying vibrations.

3.3.1 Nodal position in stepped horn

When looking at the displacement throughout the horn it can be seen, that thetransition is not a clear nodal plane as expected from the theory in [21].This can be


Figure 3.8 - Simulated and theoretical amplification (absolute values) of astepped horn. The simulated results match the theoretical ampli-fication to an acceptable degree

seen in figure 3.9 where the height of the plot shows the amount of displacement ata given position in the horn.

From figure 3.9 it can be seen, that the node is not a clean cut perpendicularto the axial direction trough the transition of the horn, but rather a curved linefrom some point on the line between the two diameters and a point inside the firstcylinder of the horn.

For this reason a specific design process is established which should ensure thatthe horn is vibrating around a node positioned at the transition. While doing this,the length of the horn steps are not necessarily equal, as we shall see in the nextchapter.

3.4 Resonance frequencies for horn types

When doing the geometry sweep simulations in the previous sections, the resonancefrequency has been stored for each geometry step of each of the different horn types.This is interesting to look at, because it is expected that for cylindrical bars theresonance frequency will drop below the pure longitudinal resonance frequency ifthe bar does not have a large length to width ratio. This is investigated in [7]. Theresonance frequency for the three horn types, as a function of aspect ratio of thehorns, can be seen in figure 3.10.

It is noted that none of the horns have a resonance frequency of exactly f =25kHz as they were designed for using the pure longitudinal wave formulation.

From figure 3.10 it is seen that for the stepped horn the resonance frequency


(a)

(b)

Figure 3.9 - Displacement of the stepped horn. The height of the plot showsdisplacement in the axial direction of the horn. In (b) there iszoomed on the transition of the horn

drops when the aspect ratio of the horn increases. This is expected and correspondswell to the results achieved by [7]. When looking at the frequency changes for theconical and exponential horns, it is seen that the resonance frequency here increaseswith increased aspect ratio of the horn.


Figure 3.10 - Resonance frequencies for the first eigenfrequency of the differenthorn types.

CHAPTER 4. DESIGN OF STEPPED HORN FOR LANGEVIN TRANSDUCER BASEDULTRASONIC SOURCE 28

Chapter 4

Design of stepped horn forLangevin transducer basedultrasonic source

In this chapter the steps in designing a stepped horn for a specific resonance fre-quency are described.

For practical use it is wanted to have a means of mounting the horn in a setup, holding it at a nodal point or plane. This would ensure that the holder wouldinterfere as little as possible with the vibrations in the horn.

Simulations of the symmetric stepped horn used in the previous section, wherel1 = l2 showed, that the nodal plane of the resonance does not coincide with thetransition step. Further more there is no nodal point on the outside of either of thehorn steps, which can be used for the mounting of the horn.

For these reasons a work flow has been developed to ensure, that the horn vibratesin a fashion where the transition is moving as little as possible. This method is usedfor designing stepped horns both with and without mounting plates, for holding thehorn, and with curved and angled transitions.

4.1 Work flow stepped horn design

To construct a horn with a well defined nodal point for mounting of the horn, adesign work flow is used where the lengths of each of the horn steps are foundindividually. This is done by doing a dimension sweep in a Matlab driven Comsolsimulation while looking at the achieved resonance frequency.

Since the horn is to be designed for a specific Langevin transducer, the diameterof the horn step close to the Langevin transducer should have the same size, as thetransducer, which is D = 30mm.

In [7] it is shown that the longitudinal resonances can be coupled to the radialvibrations in the bar. As a consequence the resonance frequency will be lower thanthe pure longitudinal resonance. For this reason, the resonance of the horn stepscannot be assumed to have the same frequency even if the lengths are the same. Forthis reason, the lengths of the horn steps are found individually one at the time.


Before the design process is started the simulation tool should be loaded withthe material properties of the material the horn is to me made from. In this projectan aluminum alloy called EN AW-2011 has been used to fabricate the parts. Theproperties for this material can be seen in appendix D, though not all propertiesneeded to preform a stress-strain simulation (E, ρ and σ) were available. For thisreason the value of Young’s modulus has been measured to be E = 7.54 · 1010Pausing the method described in appendix A. For density the data sheet value ofρ = 2840kg/m3 has been used and for Poisson’s ratio, which is less material specific,the value σ = 0.33 has been used [21, page 206].

Design step one

The first step in designing a stepped horn is to define the length of the first hornstep. The optimal length for the first horn step is calculated trough a series ofsimulations, where the two horn steps have the same diameter (forming a two partcylinder). This method is used to ensure that the first horn step is a λ/4 resonator.

An example of such a geometry is show in figure 4.1(a).

(a) Design step one (b) Design step two

Figure 4.1 - Sketch of the two design steps. In design step one both thelength are changed symmetrically and the horn steps have thesame diameters. In design step two only the second horn step ischanged

By varying the lengths of the both horn steps the optimal length of a step withthis diameter can be found, for the desired resonance frequency.


Design step two

To find the optimal length of the second and narrower horn step, a similar set ofsimulations are conducted.

With the length and diameter of the first horn step fixed at the value foundin design step one, the length of the second horn step is swept, at each lengthdoing a frequency analysis to see the resonance frequency of the horn step with thisparticular length. When the desired resonance frequency is meet the length of thesecond horn step is at its optimal length. This is done for a single diameter of thesecond horn step.

After the length of the second horn step has been found the horn design is done,the horn steps having the lengths found in the two simulation sweeps. The designedsource will now have the resonance frequency defined as the target frequency, andthe node will be positioned at the transition of the horn.

4.2 Mounting plate and transition effects

When a horn is used in an ultrasonic transducer it is desirable to have a stable placeto hold the transducer, without interfering with the resonance and amplification ofthe horn. Since the horn steps are designed to be λ/4 resonators, they will togetherform a λ/2 resonator with a node at the transition. This makes the transition agood position to introduce a plate for holding the horn, as this will interfere as littleas possible with the vibration of the two horn steps. This plate will be called amounting plate in this thesis.

Another important thing to address when designing a mechanical horn for ve-locity amplification, is the stress in the transition. It is widely used to smooth thetransition to avoid the high stress that a right angle transition intuitively gives.

The influence on the characteristics of the horn, by introduction of the mountingplate and the change in transition shapes, are investigated in two simulation sweeps,as described in the following sections.

Examples of horns with mounting plates and different transitions can be seen infigure 4.2.

4.2.1 Transition

In this section the influence on horn characteristics by the transition shape is in-vestigated for a stepped horn without a mounting plate. This is done trough twosimulation sweeps, where the stress and amplification is measured as a function ofthe diameter ratio. The first sweep is done with a stepped horn that has a rightangle transition, that is, the diameter of the horn goes directly from D1 to D2 at thetransition. In the second sweep the transition follows a curve to lessen the stress.

When constructing a stepped horn for ultrasonic applications, a rule of thumb isto make the transition of the horn in the shape of a quarter-circle, where the radiusis the size of the difference between the radii of the two part that are joined together[21, page 114]. Since this rule of thumb is general, and also used in applicationswhere the horn is used in contact with other materials (like in ultrasonic welding),


(a) Horn design with mounting plate and an-gled transition

(b) Horn design with mounting plate andcurved transition, the transition following aBezier curve

Figure 4.2 - Examples of horns with mounting plates positioned at the nodalline of the horn and different transition types

the radius of the transition curve is, in this project, set to half this rule of thumb.The transition curve in this project therefore has the radius Rtransition.

Rtransition =D1 −D2

4(4.1)

In this project Bezier-curve shaped transition has been used as opposed to thequarter-circle transition.

A Bezier curve is a type of curve widely used in vector graphics. It is basically aquadratic interpolation of three points giving the smoothest transition between thetwo end points. The use of the Bezier curved transition is chosen since it is easier todefine in a Matlab driven Comsol simulation, than the quarter-circle. Furthermorethe interpolated transition is even more smooth than the circle transition making itmore ideal for the transition of a mechanical horn.

Simulations of stepped horn transitions

To investigate the effect of a curved or right angled transition, steeped horns withthe different transition shapes, have been simulated for a range of diameter ratios.In the simulation the maximum stress has been measured. This value is found bylooking for the place in the horn with the maximum stress and storing this value.

The simulation results of amplification and maximum stress can be seen in figure4.3, where it is evident, that the maximum stress in the horn is decreased by thecurved transition. For large aspect ratios there is a visible difference between thestress for the horn with the right angled transition, and the one with the Beziercurved transition.

When looking at the amplification of the horn it is seen, that the amplificationfor the two transition types are close to the same, but since there is a decrease inthe maximum stress, the source will be able to deliver a higher sound pressure witha curved transition before meeting its maximum stress limit.


Figure 4.3 - Simulation results for the amplification and stress normalized tothe velocity v1 for a free (not fixed) horn

4.2.2 Mounting plate

In this section simulations have been made to investigate the effects of an addedplate at the transition in the stepped horn. The plate is designed with a thicknessof 2mm and an outer diameter of 45mm.

Using the design steps mentioned in section 4.1, several stepped horns withmounting plates at the transition has been designed and simulated using an auto-mated Matlab script. The simulation result for these horns, having different diam-eter ratios as well as both curved and right angled transitions, can be seen in figure4.4.

From figure 4.4 it is seen that the maximum stress in the horn is decreased byadding the mounting plate to the horn. The curved transition still leads to lessstress than the angled transition, though the difference is not as large as in the hornwithout a mounting plate.

An interesting result for the simulation is, that the curved transition does notdecrease the amplification of the horn, rather it increases it when the horn hasa mounting plate at the transition. This result is very interesting and should beinvestigated further, as it could mean that the amplification could be increased byusing an even bigger transition curve or a different shape of the curved transition.

In the plot in figure 4.4 there are some irregularities in the curve of stress forthe Bezier curve transition. This error is thought to occur on account of the small


Figure 4.4 - Simulation results for the amplification and stress normalized tothe velocity v1 for a horn with a fixed mounting plate

curve radius in this area causing numerical errors in Comsol. This has not beeninvestigated further.

4.3 Radiation from a stepped horn with no radi-

ation plate

In this section the near field sound pressure radiated from a stepped horn with noradiation plate is investigated.

Since acoustic levitation is not necessarily a far field process, it can be interestingto look at the near field radiation from a piston of a given size at a distance closer tothe source than the far field. This is done introducing an expression for the pressureradiated from a stepped horn of a certain radius.

The expression is constructed of the expression for on axis pressure, given by 2.9,and the amplification of the stepped horn given by equation 3.10.

pHorn (r, a2) = p (r, a2) ·(D1/2

a2

)2

(4.2)

The expression in equation 3.10 is illustrated in figure 4.5, where the soundpressure level is plotted relative to the sound pressure radiated by a horn of D2 =


5mm. It is assumed that D1 = 30mm for all values of D2 and that the frequency isf = 29900Hz.

Figure 4.5 - Sound radiation in the near field relative to the radiation from apiston with the diameter 5mm. The different lines represent therelative sound pressure level at a specific distance. The two blueareas both represent a range of distances

It is clear from the figure, that for wider pistons the far field is further away.From the figure it is seen, that in the near field it is desirable to have a D2

as small as possible, since the amplification of the stepped horn will make up forthe loss in radiation surface. There is no gain in choosing a wider D2, other thanstability of the source, in respect to possible transverse vibrations.

4.4 Stepped horn designed for practical experi-

ments

To evaluate the theory on the stepped horn, described in this and previous chapters,a horn is designed for practical tests. The horn is designed to be attached to theLangevin transducer, why the diameter of the first horn step should be D1 = 30mmto ensure good transfer of vibrations.

Since the near field radiation will decrease for larger values of D2, this is set tobe D2 = 10.4mm.

Using the design process described in section 4.1 in a Matlab script, the optimallengths of the two horn steps have been found to be L1 = 43.95mm and L2 = 43mm.In the design a mounting plate of thickness 2mm and radius 45mm has been used.


The transition is shaped in a Bezier curve with the radius 4.9mm according toequation 4.1.

A work sketch of the designed horn can be seen in appendix E, and a picture ofa horn constructed from this sketch, can be seen in figure 4.6.

Figure 4.6 - Horn constructed for experimental validation experiments

CHAPTER 5. EXPERIMENTAL VALIDATION 36

Chapter 5

Experimental validation

In this chapter a stepped horn described in th previous chapter has been tested invarious experimental setups. The horn is designed to be mounted on a Langevintransducer from M.P. Interconsulting, which has a resonance frequency of 29.9kHzand a surface diameter of 30mm, at the end where the horn is attached. As well asthe designed horn, measurements are carried out on the Swiss horn.

Finally the performance of the designed horn is discussed in comparison of toexpected performance, as well as in comparison to the Swiss horn. To comparethe horns performances, a test of the acoustic levitation forces they supply is alsoconducted.

An elaborate list of the equipment used for the experiments, can be seen inappendix F.

5.1 Test of assembled sources

The first test carried out with the constructed horn, is a frequency analysis of thehorn joined with the Langevin transducer to form an ultrasonic source. For themeasurement the following equipment was used:

• Pulse system

• Laptop with the pulse software

• Yamaha audio amplifier

• Agilent multimeter

• Oscilloscope

• 1/8” microphone

In the test the source has been mounted in a holder made for the purpose. Thesource in the holder can be seen in figure 5.1.

To find the resonance frequency of the combined source, an automated frequencysweep was run from the Pulse software, measuring the response an 1/8” microphonein a distance of 50mm. From this sweep the resonance frequency was found.


Figure 5.1 - Source mounted in the fabricated holder

The result of the frequency test was that the resonance of the designed hornconnected to the Langevin transducer was f = 28, 594kHz. It is noted that theresonance does not match the desired resonance frequency of f = 29.9kHz.

The reason for this difference between the simulated and the measured model,could be introduced from differences in the value of Young’s modulus. The reasonthis could occur is, that it was not possible to fabricate the horn from the exactsame bar of the aluminum alloy, as was used to fabricate the test bars used tomeasure Young’s modulus. For this reason there could be production deviations inthe aluminum bars, that causes the measured value of Young’s modulus not to besufficiently accurate.

Other sources of errors are described in section 5.3.

Another experience from this setup is, that when driving the source at its res-onance frequency, vibrations can be detected in the source holder. This detectionis done by lightly touching the surface of the holder with a set of tweezers. Thisproduces an audible sound, that shows that the holder is vibrating.


This points to the resonance of the source not being entirely balanced, meaningthat the horn has a different resonance frequency than the Langevin transducer. Ifthis is the case, the horn would not be excited in its own resonance, why the mountingplate would not be located at a nodal plane of the vibration. To investigate thisfurther, a frequency analysis of the horn alone is carried out in the next section.

5.2 Frequency analysis of horn

In this section a resonance frequency analysis is preformed with the designed horn.The analysis is made similar to the procedure in appendix A, where the resonancefrequency of the aluminum bars is found, but in the case of the frequency analy-sis of the horn, no transversal resonances occurred within the frequency range of25 − 35kHz, which has been the focus for the analysis preformed. Since the de-sired resonance frequency of the horn is 29.9kHz, it is unlikely that any transversalresonance outside this area should affect the longitudinal resonances of the horn.

For the experiment the same equipment as in appendix A is used. That is thePulse system, a force hammer and a 1/8′′ microphone.

In this experiment the horn have been detached from the Langevin transduceran placed in holder like seen in figure 5.2.

Figure 5.2 - Horn in holder for frequency analysis. Here the excitation is givenin the D1 end of the horn and the response measured in the D2

end.

With the horn mounted in the holder, without the Langevin source attached, themicrophone is placed near the wider end of the horn, and, using the force hammer,the horn is excited in the thinner end. Measuring the resonance frequency in thisway shows a frequency of f = 28.72kHz. To ensure accurate measurements the test


is repeated with the excitation in the wider end of the horn and the microphonein the thinner end. This test confirms the resonance frequency from the first testof f = 28.72kHz. Since the horn was designed without the screw that connectsthe horn to the Langevin transducer, it was tested what influence this had on thehorn resonance, by repeating the test with the screw inserted in the horn threading.With the excitation in the narrower end, this test gives a resonance frequency off = 29.2kHz while the inverse test shows f = 29.09kHz. The reason for the smalldeviation is thought to be on account of unbalanced resonances where the two endhave resonances of slightly different frequencies. From this experiment it is seenthat the screw changes the resonance of the whole horn, not just the end where itis attached.

The difference between the target frequency of 29.9kHz and the measured fre-quencies is discussed further next section.

5.3 Error analysis

In this section the errors found in the frequency test are discussed.When designing the horn used for the experimental tests Yong’s modulus has

been measured. Unfortunately it was not possible to construct the horn of the samebar of aluminum as the test pieces for the Young’s modulus measurement was madefrom. This could mean that the value of Young’s modulus used in the design processis not the exact same as the value for the aluminum used to fabricate the horn, eventhough the material was said to be the same.

This is thought to be a possible source of errors that has caused the calculatedresonance frequencies to be incorrect. Apart from this error it is possible thatthe simulation model is not exact enough to simulate the correct behavior of thedesigned horns. In the following sections a number of changes has been made to thesimulation model to investigate the influence of some of the different parameters inthe simulation.

5.3.1 Simulation model errors

The first test done is a simulation of the horn with the dimensions measured fromthe actual produced horn. These measurements revealed some small errors in thelengths and radii of the horn, which has the dimensions:

• L1 = 43, 95mm

• L2 = 43, 15mm

• D1 = 30, 1mm

• D2 = 10, 4mm

Apart from these fabrication errors, the mounting plate was measured to have athickness of 2, 1mm and a diameter 45mm. Since the shape of the transition israther complex, it has not been possible to measure the exact dimensions of this.


The curved transition is a very complex shape to manufacture, why this shape ispossibly not perfectly accurate.

Using these dimensions the resonance frequency did not change away from the29, 9kHz also achieved in the design simulations.

Simulation mesh density

Now that a geometry with more exact dimensions is established, the next step inoptimizing the simulation model is to refine the simulation mesh. By increasing themesh density in the simulation, it can be investigated weather the simulation meshis sufficiently dense, to calculate the true behavior of the horn. When changing themesh density to a finer mesh the resonance frequency of the mechanical system willasymptotically approach the most correct value for the simulated parameters. InComsol the mesh refinement is done by clicking the“refine mesh”button and runningthe simulation over. For testing the mesh refinement influence on the resonancefrequency, simulations have been preformed with both one and two mesh refinements,in both cases the resonance frequency was calculated to be 29.8kHz rather than thefrequency f = 29.9kHz calculated with the standard mesh. This shows that for thegeometry of the horn with a Bezier curve transition and a fixed mounting plate, themesh should be refined one time more than the standard mesh for better results.Since the second refinement of the mesh did not change the result is is not necessaryto refine the mesh more than once.

Mounting plate support

The next simulation alteration made is a change in the support conditions of themounting plate. When looking at the source mounted in the constructed holder(shown in figure 5.1), it is seen that rather than applying a plane pressure on themounting plate all the way to the wider part of the horn, the holder only supportsthe outer part of the mounting plate, and there in a squeezing fashion, applyingmore pressure to the outer part than to the parts further in the mounting plate.This is illustrated in figure 5.3, showing the optimal, the real and the simplifiedreal support of the mounting plate. The simplified real support is made, to tryto simulate a support that is more similar to the real situation than the optimalsupport used in design simulations. It would be very complicated to simulate thereal support situation, since it does not apply a plane pressure to the mountingplate. Furthermore the holder for the horn has a gap where it does not support themounting plate. This has been introduced to be able to mount the source withouttaking it apart. The gap has not been simulated, but could influence the hornbehavior.

When changing the support of the mounting plate to the simplified real situation,instead of the situation used in the design of the horn, the resonance frequencychanges to f = 29.5kHz. This drop in resonance frequency shows that the supportis an important part of the design of the horn. Since the simplified real support isnot the same as the real support it is very possible that the real support accounts foran even larger part of the difference between resonance frequency of the simulatedhorn and the measured horn. It has not been investigated, if this frequency change


(a) Mounting plate supportused in the design process

(b) Sketch of real mounting plate sup-port

(c) “Simplified real” support ofthe mounting plate used for er-ror analysis simulations

Figure 5.3 - Three different supports for the mounting plate of the horn. Thegray area marks the area that is fixed in the simulations


would also occur in the ideal lengthed horn, or if it is a consequence of an unbalancedhorn vibration.

Transition curve size

Since the horn is designed using on a computer, the tolerance of the dimensionsis ultimately zero. This is not the case with the constructed horn. A part of thedesign that is extraordinary difficult to produce with the desired low tolerance, isthe transition which is designed to follow a Bezier curve. Even though much efforthas been put into fabricating the transition as accurate as possible, it is very likelythat a great deal of error is introduced here. For this reason it has been investigatedwhat influence the transition size has on the resonance frequency. This has beendone by changing the radius of the Bezier curve transition from 4, 9mm to 4mm andsimulating the new geometry. In the simulation with the smaller curve radius theresonance frequency drops 200Hz to a value of 29.3kHz.

5.3.2 Error analysis conclusion

In this section some of the possible errors of the construction of the horn has beeninvestigated. Some errors in the design of the horn could be due to the fact, thatwhen fabricating the test bars for the measurement of Young’s modulus the materialused does not origin from the same bar, as the material used to fabricate the horn.This is thought to be a likely error resulting in the design process calculating designdimensions which gives another resonance frequency than the desired.

In the error analysis several changes to the simulation geometry has been madein succession. The changes all contributed to lowering the resonance frequency of thehorn, but not enough to account for the whole resonance frequency error. After thesimulations, it is thought that the most likely reason for the difference in resonancefrequency between the designed simulated horn and the actual constructed measuredhorn, is a combination construction inaccuracies, of poor mesh density in the simu-lations, inaccurate mounting plate support in the simulation, insufficient mountingplate support in the experiments with the constructed horn, and inaccuracy in thematerial properties, due to production variations in the supplied aluminum. Themost important things to investigate further is thought to be the material prop-erties, the support of the horn and the construction accuracy. These three thingscan be investigated by conducting more material experiments, using the proceduredescried in appendix A, constructing a new holder for the horn, and by having thehorn fabricated using a computer controlled turning lathe.

Furthermore it would be very useful to construct a way of directly measuring thevibrations of the horn in experimental setups, rather than the sound radiation. Thiscould perhaps be done using laser vibrometry, given a vibrometer that can measureultrasonic frequency vibration is available.


5.4 Sensitivity analysis of sources

In this section the sensitivity of the sources, consisting of the Langevin transducerand each of the two horns (the designed horn and the Swiss horn), has been mea-sured. For the test the same equipment, as in the test of the resonance frequency ofthe source, has been used.

From the Pulse software, the signal generator was set to deliver a single frequencyequivalent to the resonance of each of the the sources, as measured in section 5.1.These are for the source with the designed horn f = 28.594kHz and for the sourcewith the Swiss horn f = 29.9. Using this excitation the next step in setting upthe test is to adjust the gain of the amplifier, to as high level as the amplifierallows without clipping. This is done by carefully adjusting the amplification whilemeasuring the voltage over the source with the oscilloscope. In this way clippingcan be detected visually. When the amplification is fixed, the voltage over thesource terminals is measured using the multimeter, measuring RMS voltage. In theexperiments carried out with the available amplifier the maximum voltage that couldbe delivered to the sources was 25VRMS. When the amplification is set, the measuringsequence can be run from the Pulse software. The measuring sequence consists of thegenerator playing the fixed frequency at a fixed level, and a microphone recordingthe signal which then is displayed in the pulse system, as an auto spectrum. Thissequence is run with the microphone placed at different distanced to the horn surface.The microphone is placed on axis at distances of 10 − 100mm. The measurementresults for the two sources can be seen in figure 5.4

Figure 5.4 - Sound pressure radiated from the two horns


In figure 5.4 it is seen that the difference between the two sound pressure levellines is not the same for different distances. In figure 5.5 the difference between thesound pressure levels for the two horns is plotted as well as a line calculated from atheoretical estimation.

Figure 5.5 - Difference between the sound pressure radiated from the swissand the produced

In the figure the blue line shows the expected difference for the two sizes of pistonat the different distances. This theoretic line is calculated from equation 5.1 whichis deduced from equation 4.2, and is an expression of the change in sound pressurelevel, when changing from one horn size to another.

∆SPL (r) |D=10.4mm→25mm = 20 log

(p(r,(10.4mm

2

))·(25mm

2

)2p(r,(25mm

2

))·(10.4mm

2

)2)

(5.1)

This calculation assumes a flat piston, but since the Swiss horn does not have aplane surface but rather a big hole in the middle of the surface it must me expectedthat i radiates less than the equivalent piston. By making a simple estimation basedon the reduction in surface area from the equivalent piston SSwiss to the piston witha hole in the middle, S = SSwiss−SHole it must be expected that the radiated soundpressure level should be smaller by

20 log

(SSwiss − SHole

Sswiss

)= 20 log

((25mm

2

)2 − (10mm2

)2(25mm

2

)2)

= −1.5dB (5.2)


where the diameter of the hole is 10mm. This estimate, of the difference betweenthe radiation from a piston and a piston with a hole in the middle, is only a roughestimate, and the fact that the calculated correction only accounts for some of thedifference between the theoretical and the measured sound pressure levels, is thoughtto on account of the assumption, that the horn with the hole in the middle has thesame sound radiation pattern as a vibrating piston. This is also thought to be thereason for the differences larger than 1.5dB measured at some distances.

5.5 Acoustic levitation test

In this section some attempts to levitate different objects has been carried out, usingdifferent horns attached to the Langevin source. The horns are the designed horn,the Swiss horn and the Swiss horn with a brass disc to cover the hole in the surfaceof the horn.

The first test of acoustic levitation has been done using the Langevin transducerwith the Swiss horn attached. This horn has a hole with threading for attachingdifferent tools, which was thought to be impractical for levitation experiments. Tocover this hole a brass disc was fabricated and attached to the horn. Using this setupit was possible to levitate up to two polystyrene balls when adjusting the reflectorcarefully. This can be seen in figure 5.6 and in the video Levitation.wmv on the CDin appendix G.

Figure 5.6 - Levitation using the Langevin transducer with the Swiss hornattached. To cover the hole in the horn, a brass disc has beenattached

As it is seen in figure 5.6 the setup with the Swiss horn and the brass discattached, levitation of some polystyrene balls was achieved. To achieve the levitation


careful adjusting of the reflector was needed. Other objects than the polystyreneballs has not been tried in this setup, but the levitation force did not feel strongenough to levitate denser materials such as water.

Out of curiosity levitation of talcum powder was tried, unsuccessfully. The ex-periment did however reveal another interesting observation. As the powder wasnot levitated, it fell on the surface of the brass disc, and was immediately arrangedin a circle about half way out the radius of the brass disc. This lead to the ideaof investigating the vibrational pattern of the disc described in section 6.2. Thevibrational pattern was found to form because the disc was not in contact with thehorn surface, though the gap between the two was not visible (see figure 5.6).

The next levitation test preformed, is a similar test with the horn designed inchapter 4. Using this horn a very strong levitation was achieved. The levitation wasstrong enough to levitate polystyrene balls without adjusting the distance betweenthe source and the reflector very carefully. By adjusting the distance more carefully,it was possible to levitate polystyrene balls at several of the pressure nodes betweenthe source and reflector (up to four has been tried successfully). Furthermore levi-tation of water was achieved with up to two drops levitated at the same time. Thiscan be seen in figure 5.7 and in several videos on the CD in appendix G with namesstarting with Levitation II.

Figure 5.7 - Levitation of two water drops at the same time

To levitate the water drops the reflector had to be adjusted very carefully, andthe levitation was less stable than when levitating the polystyrene balls. This is anatural consequence of the higher density of water, than polystyrene balls.


Levitation experiments has also been conducted using the Swiss horn withoutthe brass disc attached. Here it was also possible to levitate polystyrene balls, butnot water. In the fine tuning of the reflector the hole in the horn surface complicatedthe work.

The levitation experiments here described, has been conducted using the YamahaAudio amplifier. This amplifier cannot deliver a very high voltage (approximately25VRMS) without clipping. If an amplifier more suitable for driving a piezoelectrictransducer, such as the Langevin transducer was available, stronger levitation forceswould be achievable.

CHAPTER 6. RADIATION PLATE ASSESSMENTS 48

Chapter 6

Radiation plate assessments

In this chapter some methods for characterizing radiation plates are discussed. Thesemethods involve both simulations of mechanical and acoustic coupled setups in Com-sol, and a low tech and an optical method for visualization radiation plate displace-ment.

For the experiments with radiation plate displacement visualization, the Langevintransducer has been used, together with the Swiss horn with a brass disc attached.The disc can be seen in figure 6.1

Figure 6.1 - Brass disc used in experiments with radiation plate displacementpattern visualization

6.1 Node detection using talcum powder

The low tech method for visualization of displacement, is to apply talcum powder tothe excited radiation plate. This method delivers a fast and easy way of getting anidea about the displacement of the radiation plate. Examples of excited radiationplates with talcum power on the surface, can be seen in figure 6.2, where both aradiation plate with a nodal-circle, and one where the radiation plate is excited ina piston-like fashion, is seen.

When the radiation plate has a nodal circle, the powder will be pushed towardsthis nodal circle. In the case where the piston is behaving like a piston, the powder


(a) (b)

Figure 6.2 - Powder on the surface of a radiation plate forms a pattern visu-alizing the displacement. In (a) the plate does not vibrate in aplanar piston-like motion, because the backside of the radiationplate does not touch the horn. In (b) the radiation plate touchesthe horn, and a clear vibrational pattern is not visible

will not form any distinctive pattern, but move around on the vibrating surface ina random pattern.

A good thing about this approach for displacement visualization is, that it canbe carried out in situ, given there is no significant forces acting on the on the powdersuch as levitation-like forces. If such a force is present and is strong enough to movethe powder around it could happen that these forces, which normally are rotationalsymmetric around the radiation plate center, would move the powder to a circleshaped pattern looking like a nodal line, but in fact being a visualization of theacoustic force field on the surface.

6.2 Visualization of radiation plate displacement

pattern using speckle interferometry

The previous method for radiation plate displacement visualization has the disad-vantage, that it cannot be determined weather the displacement is measured, orthere is an acoustic force field on the surface of the radiation plate. For this reasonan optical setup has been used to visualize the radiation plate displacement. Thissetup has the advantage that the measurement is not contact dependent, since it isdriven by light.

The optical method for visualization of radiation plate displacement, is basedon the theory of speckles [22, Chapter 7.5] and the interference of these. A setuphas been made that can visualize the displacement pattern of radiation plates, giventhey have a surface with a fairly good reflection. The setup has been used withbrass plates with good results, but with a radiation plate made of acrylic plastic, itwas not possible to use the method, possibly because the reflection of the materialis not high enough.


The experimental setup for visualizing the displacement pattern of the radiationplate is described in greater detail in appendix B.

Figure 6.3 - (a) Setup for displacement visualization experiments seen fromthe USB camera. (b) Displacement visualization recorded withfrom the same camera position. The lighter the pixel the largerthe displacement amplitude at the corresponding position. It canbe seen that the mounting plate of the horn is also moving a little.

The image in figure 6.3(b) shows the displacement pattern of the brass radiationdisc. The figure should be interpreted the way, the illuminated areas are vibrating,while the black areas are not. From the figure it is clear that this experiment showsthe same circle of no vibration, approximately half way between the center and therim of the plate, as the powder experiment did.

Furthermore the experiment shows, that the mounting plate of the horn alsomoves. This movement has also been observed by placing a set of tweezers lightlyon the mounting plate, which gives an audible sound, that shows that the mountingplate moving. This ability to locate vibrations visually, could be useful in the furtheroptimization of ultrasonic horns and sources.

In a comparison between the powder and the speckle based experiments, it canseem a bit elaborate to do a complicated optical setup when you can just applysome powder to the surface of the disc, but it is here an important feature of theoptical setup, that it is independent of the physical forces on the plate; it only showsdisplacement.

Experiments has been conducted, with the speckle setup, to try and visualizethe displacement throughout the length of the horn, but with no success.

6.3 Radiation plate simulations

In this section a series of simulations of the behavior of radiation plates has beencarried out in Comsol Multiphysics. The simulations consist of simulations repro-ducing the vibration pattern of the brass radiation disc used in the visualizationexperiment in the previous section, and some simulations where an acoustic fieldsimulation has been coupled to the mechanical system simulation.


6.3.1 Vibration pattern of brass disc

In this section a simulation of the vibrations of a brass plate has been preformed onthe axisymmetric formulation in Comsol.

The brass disc is modeled using Comsol’s draw functionality as a rectangle ofsize 12.5x3.6mm with one of the shorter sides assigned as symmetry axis. A sectionof the top part of the rectangle, representing the radiation disc screw, is assigned afixed displacement.

The material of the disc is chosen to be brass with the material properties from[10, Page 526]:

E = 10.4 · 1010Pa (6.1)

σ = 0.37 (6.2)

ρ = 8500kg/m3 (6.3)

The first simulation run with the brass disc geometry is an eigenfrequency simu-lation. This reveals the two first resonance frequencies of the disc to be f1 = 19kHzand f2 = 87kHz. It is clear that the drive frequency used in section (f = 29.9kHz)6.2 is not a resonance of the disc.

This other simulation is a frequency response simulation. In this simulation thedisc is excited with the frequencies f = 19kHz and f = 29.9kHz respectively, andthe responses are calculated. From this simulation the results shown in figure 6.4are achieved.

It is seen from figure 6.4, that the disc simulations show a nodal line about halfway between the center and the rim of the disc, at f = 29.9kHz. This is the lineshown in section 6.2.

The final simulation run with the disc geometry alone, is a frequency sweep fromf = 10kHz to f = 35kHz. From this simulation a plot of the frequency responseof the disc can be produced. This is seen in figure 6.5, showing the displacement ofthe center and rim of the disc, at different frequencies.

From the plot is is noted, that the surface displacement of the center and rim ofthe disc are in phase until a little before the resonance frequency of 19kHz. Afterthis the displacement of the center is in counter phase to the rim displacement. Thisshows that until this frequency, the disc behaves more like a piston, but since thedisplacement at the two points are not of same magnitude, not entirely like a piston.The piston like movement can only be achieved well under 10kHz, this is seen atthe two lines approaching one another for lower frequencies. For frequencies higherthan the resonance, the piston will have a nodal circle somewhere. This is seen bythe counter phase behavior of the displacement of the center and rim of the disc.

6.3.2 Sound radiation from radiation plates

In this section it is presented how Comsol can be used to characterize the soundfield from a complex source, such as the vibrating radiation disc. This is doneby coupling the mechanical simulation to an acoustic simulation also conducted inComsol. When doing this the behavior of the mechanical system will affect theacoustic simulation as will the acoustic field affect the mechanical system.


(a)

(b)

Figure 6.4 - Simulation result of vibrational pattern of(a) brass disc at res-onance and (b) at a frequency higher than the first resonance.Note the different color scales; the displacement is much larger atresonance. The vibrational pattern in (b) is to be compared tothe experimental pattern visualized in section 6.2


Figure 6.5 - Frequency sweep of vibration in brass disc. f = 10 − 35kHz. Thelines show axial displacement of the center of the disc (green) andrim of the disc (blue)

The geometry used is the same as in the previous simulations of the disc, butmore geometries must be added to represent the sound field. The sound field is inComsol defined as a quarter circle attached to the surface of the disc.

For acoustic simulations the material properties (of the air) are set to Comsolsdefault values of

ρ0 = 1.25kg/m3 (6.4)

c = 343m/s (6.5)

When the geometry is setup and the material properties are defined and theinteraction of physics is setup the coupled simulations can be run. In coupled me-chanical and acoustical simulations in Comsol, eigenfrequency analysis can no longerbe simulated, so the introduction of a sound field limits the further simulations tofrequency response simulations.

In figure 6.6 an example of a simulation of the sound pressure level radiated fromthe brass plate is shown.

From the sound field simulation various data can be pulled out. An example ofthis i seen in figure 6.7, showing the directivity patterns of a variety of sources is seen.The sources simulated are the brass disc, driven at resonance and at f = 29.9kHz,and a planar moving piston at the same two frequencies. It is seen that the directivityof the brass disc changes when it is driven at the two frequencies, but that thedirectivity of the piston remains fairly uniform for these two frequencies.


Figure 6.6 - Sound field simulation of brass disc driven at f = 29, 9kHz. Thebrass disc is represented by the rectangle at the top of the plot.The color scale shows absolute sound pressure in Pa

In figure 6.7 it is seen that for the piston motion the radiated sound pressurelevel has a different in of ∆SPL = 8dB for all angles. This is a effect of assigninga fixed displacement amplitude for all frequencies. The reason for this is that thesurface velocity of the is dependent of the frequency in the following way

v =∂w

∂t(6.6)

Given a sinusoidal displacement of amplitude W the following is deduced

v =∂W sin (2πft)

∂t= 2πfW cos (2πft) (6.7)

Since the sound pressure is proportional to the surface velocity multiplied by thefrequency [12, Equation 2.7, page 9]

p ∝ v · f (6.8)

p ∝ W · f 2 (6.9)

must apply. This leads to a gain in sound pressure for going from f1 to f2 of

∆SPL|f1→f2 = 20 log

(f2f1

)2

(6.10)

For the case in figure 6.7 the rise in frequency from f = 19− 29.9kHz must leadto a rise of

∆SPL|19kHz→29.9kHz = 20 log

(29.9kHz

19kHz

)2

= 7.9dB (6.11)


Figure 6.7 - Comparison of the radiated sound pressure level for piston typevibrations (dashed line) and the brass discs flexural vibration (fullline) at different frequencies. All simulations have the same dis-placement amplitude of 1m

which accounts for the difference in the piston radiation for the two frequencies.With this knowledge of the increase in SPL for higher frequencies, it is even

more extraordinary that the flexural motion of the brass disc at resonance radiatesas high a sound pressure level as it does. This effect could be exploited in the sameway the horn amplification does to produce a highly efficient ultrasonic source.

6.4 Radiation plate conclusion

In this chapter some experiments and simulations regarding the behavior of radiationplates has been conducted. For the purpose of visualizing the displacement patternof a source mounted radiation plate, two method has been used, which had theirindividual strengths.

The first and simplest method was to apply some talcum powder to the surfaceof the excited radiation plate. This method revealed a circular nodal line on thebrass plated used to cover the hole in the swiss horn. While this method is quickto use, and do not need the source to be removed from its place of use, the methodhas the disadvantage that it is not known weather the powder forms a given patternon account of the vibrational pattern of the radiation plate, or because of acousticforces in the field in front of the radiation plate.

The second method for visualizing radiation plate displacement, was an opticalsetup using speckle interferometry to detect the vibrational pattern. This methodis not dependent of physical contact to the radiation plate as long as there is anoptical contact. Using this method the acoustic forces in the sound field in front of


the source does not affect the measurements. For this reason this is a more accurateway of visualizing the radiation plate displacement patterns. Using this method it ishowever needed that the source is removed from the setup it is used in, and placedin the optical setup. For this reason this method is more elaborate to conduct, butsince the results from the method are more reliable this is acceptable in some cases.

The optical setup used can be expanded to a more exact setup which enablesmeasurement of absolute displacement as well as phase information [23]. This setupcould be very useful when working with radiation plates and radiation plate design,since the displacement pattern of the radiation plate is crucial to understanding thedirectivity of the given radiation plate.

Apart from the visualization experiments, this chapter also covered some sim-ulations of radiation plates and radiation plate radiation. In the simulations itwas possible to reproduce the displacement pattern seen in the visualization experi-ments. Furthermore mechanical and acoustic coupled simulations proved very usefulfor characterizing the sound radiation from different radiation plates.

Coupled mechanical and acoustical simulations in Comsol could be used as apowerful design tool when designing radiation plates for a given application, since alot of information such as directivity and sound field characteristics can be calculatedusing this tool. Apart from the information drawn from the simulations in this thesisit would also be possible, directly in Comsol, to implement the calculation of theGorkov potential and from this the acoustic levitation force on a given object in thesound field. These calculations can be done from the sound field calculation such asthe one seen in figure 6.6.

CHAPTER 7. CONCLUSIONS AND FUTURE WORK 57

Chapter 7

Conclusions and Future Work

7.1 Conclusion

In this thesis several subjects in designing, constructing and characterizing highpower ultrasonic sources has been investigated.

In the theoretical investigation of different mechanical horn types, it was foundthat the stepped horn delivers the largest vibrational amplification, which was theratio of the horn diameters squared. This was, using the Comsol simulation tool,proven to be true for even large ratios of horn diameters. While the theory saidthat the conical and exponential horn types would have similar amplification, it wasshown, that the expression for the conical amplification is only valid for small ratiosof diameters, while it showed to be usable for larger ratios in the exponential horn.

In simulations of the influence of the shape of the transition in a stepped horn, itwas shown that by making the transition follow a curved line, the stress in the hornwas decreased more than the amplification was decreased. For this reason the hornwith a curved transition would be able to vibrate at a larger amplitude, withoutmeeting its stress limit, where it would deform or break.

To facilitate a convenient way of holding the horn in a practical setup, simu-lations has been conducted where a mounting plate is introduced in the transitionof the stepped horn. The presence of a mounting plate showed, that in connectionwith a mounting plate, the curved transition did not decrease the amplification ofthe horn, but increased it at large aspect ratios, while still decreasing the maximumstress in the horn. From these results it could be interesting to see what wouldhappen if the radius of the transition curve was increased.

From simulations of the stepped horn it was shown, that the width of the hornsteps has an effect on the resonance frequency of this horn step. For that reason, amethod for designing a stepped horn was used, where each step length was foundindividually, ensuring that each step had the optimal length for the desired resonancefrequency.

Using this method a horn was designed to be attached to a Langevin from theSwiss company M.P. Interconsulting.

In the experiments conducted with the designed horn, it was found that the horn


had a resonance frequency of 28.72kHz rather than the design frequency of 29.9kHzused in the simulations.

To evaluate this error, changes were made to the simulations model to try toget a more accurate result of the horn simulations. These changes showed that thesimulation would be more accurate if the mesh density of the simulation is increased.It was found that the resonance frequency of the horn changed when meshing onestep finer than Comsol’s default mesh density. With more mesh refinements, thefrequency did not change.

In simulations of the support for the horns mounting plate, it was found that thesupport plays a role in the resonance frequency of the horn, even though the hornwas designed to have a node at the mounting plate position. This simulation is onlyconducted with the measured dimensions of the constructed horn, not with a hornwith design dimensions.

Other reasons for the inaccurate resonance frequency of the constructed horncould be due to construction imprecision, or from deviation between the value ofYoung’s modulus in the bar used to construct the test bars, an bar used for con-structing the horn.

For comparison between the designed horn and a horn provided by the Langevintransducer company (the Swiss horn), some experiments with acoustic levitationwas conducted. These experiments showed that when using the horn designed inthis project, a stronger levitation force was produced. In this setup it was possibleto levitate water drops, using a standard audio amplifier, and an excitation signalto the Langevin transducer of 25V . If used with an amplifier more suitable for driv-ing Langevin typed transducers the levitation force could be improved considerably.Using the Swiss horn levitation was possible but not as strong as with the hornconstructed in this project. Levitation was also achieved with the Swiss horn witha brass disc attached to the front of the horn.

To characterize the vibrational pattern of radiation plates several experimentshas been conducted. A low tech method, that consist of applying talcum powder tothe surface of the radiation plate, proved a fast and easy means of detecting nodallines on radiation plates. For more precise measurements an optical setup, basedon the theory of speckle interferometry, has been used. This setup showed a greatpotential, as a way of determining nodal lines on the vibrating surfaces of a radiationplate.

In acoustic simulations of the sound radiation of radiation plates, it was showedthat a radiation plate at resonance will radiate a far greater sound pressure thana radiation plate driven off resonance. It was also shown that the directivity ofthe radiation plate sound radiation changes when the radiation plate is driven atresonance.

7.2 Further work

Since the horn produced in this project did not have the desired resonance frequency,a natural next step in the work, could be to optimize the horn for the wanted


resonance frequency. This could be done by altering the horn already produced.For the characterization of the horn vibrations it could be very useful to constructa method (optical or other), to investigate the local vibration of the horn. It wasin this project tried by use of the speckle interferometry setup used in 6.2, butunsuccessful.

The use of optical setups to measure mechanical and acoustical behavior couldalso be expanded with a setup using Schlieren Imaging [24] to visualize the standingwave field in the acoustic levitator. Using this technique it be possible to visualizethe forcefield inside the levitator, which would be most useful in designing an efficientlevitator.

To produce a strong ultrasonic source it could be interesting, to construct asource using a resonant radiation plate, since it proved to radiate a far larger soundpressure than the radiation plate driven off resonance. For the characterization ofthese sources an optical setup for measuring absolute displacement amplitude wouldbe very useful. This can be obtained by expanding the optical setup already used[25].

Furthermore the source constructed in this project could be the basis of furtherexperiments with acoustic levitation.

60

Appendices

APPENDIX A. MEASURING YOUNG’S MODULUS OF ALUMINUM BARS 61

Appendix A

Measuring Young’s Modulus ofAluminum bars

In this appendix an experiment for measuring Young’s Modulus is described. Thismethod has been used to measure Young’s modulus for the aluminum alloy used forfabricating the mechanical horn parts in this project.

The aluminum available is not pure aluminum, but an alloy called EN AW-2011.This alloy is purchased for metalworking and is therefore optimized for that purpose.The alloy consists of mostly aluminum and copper, but it also has componentslike iron and others (see data sheet in appendix D). The material has a density ofρ = 2840kg/m3.

To measure Young’s modulus for the material, a test is advised, where the res-onance frequencies of well defined rods made from the material, is measured. In[21, p. 181] this method is known as the “Resonance Method”, but here the firstresonance is measured instead of the distance between resonance frequencies. Sincethe pure longitudinal resonances are harmonic this is a reasonable thing to do.

cL = 2L∆f (A.1)

where cL is the propagation speed of longitudinal vibrations, L is the length of thebar, and f is the recorded resonance frequency of the first longitudinal resonance.

By knowing the resonance frequency for the first longitudinal resonance, Young’smodulus can be deduced from:

cL =

√E

ρ(A.2)

where E is Young’s modulus and ρ is the density of the material. Finally Young’smodulus can be calculated from this equation

E = ρ (f2L)2 (A.3)

This calculation assumes a pure longitudinal resonance. If this is not the case, thecombined resonance frequency will be lower than that of the pure longitudinal mode,as described in [7]. This fact is not taken into account when directly calculating


Young’s modulus from the resonance frequency of the bar, but when the value ofYoung’s modulus is optimized for simulations, this fact is naturally included, sincethe simulations do assume any specific kind of vibrations.

The main part of the experiment for measuring Young’s modulus for the alu-minum bars, is the bars mounted in a fabricated holder that supports the bar onrubber bands (see figure A.1). For excitation of the bar a force hammer (Bruel &Kjær type 8203) is used. For measurement of the bars response to the excitation a1/8” microphone (Bruel & Kjær type 4138) is placed close to the end of the rod. Byavoiding mounting an accelerometer on the bar, it is avoided, that the weight of theaccelerometer changes the resonance frequencies of the bars.

By recording the frequency response both at the front and to the side of the testbar, the longitudinal resonances, which are only present at the front of the bar, canbe distinguished from the transversal resonances which are present at both positions.

Equipment

Using the setup described the frequency spectra shown in figures A.2 and A.3 wasrecorded.

The figure shows the spectra recorded to the side of the bars, overlaid with thespectra recorded in front of the bar end. Displayed in this way it is clear that somefrequencies are not represented in the side spectrum; these are the longitudinalresonance frequencies. In figure A.4 a good argument for measuring the spectrain both the front and side position can be seen. Here the frequencies of the tworesonances are very close, and it would be difficult to distinguish which was thelongitudinal resonance, if the side spectrum was not recorded. Having both spectrait is clear, that the higher frequency is not represented in the spectrum recorded tothe side of the bar, why it must be the longitudinal resonance frequency.

The measured longitudinal resonance frequencies are shown on table A.1. HereYoung’s modulus, calculated using equation A.3, is also displayed. The measurementhas been repeated two times (with several weeks in between) for better accuracy.Each time the same resonance frequencies were measured.

L [mm] D [mm] f1 [Hz] E [Pa]

100.10 9.80 25781 7.57 · 1010

99.75 30.01 25656 7.44 · 1010

49.85 10.00 51578 7.51 · 1010

50.00 30.00 50094 7.13 · 1010

Table A.1 - Frequencies for longitudinal resonances in a selection of aluminumalloy bars

From table A.1 it is seen that Young’s modulus is not calculated to be the exactsame for the different aspect ratios of the bars. The calculated Young’s modulusis higher for the bars with higher aspect radio, than for bars with lower aspectratio. This points to the same conclusion as in [7], where it is shown that for widerbars, the resonance frequency is lowered, which would in this case lead to a lower


calculated Young’s modulus. For this reason, and to produce a simulation modelthat has good use in design purposes, the optimal Young’s modulus has been foundusing simulations of the bars, with a range of values for Young’s modulus. This isdescribed in the next section.

Optimizing Young’s modulus for simulations

In this section an optimal value of Young’s modulus is found by means of the Comsolsimulation tool. This is done to have a better starting point of the simulations usedto design the mechanical horn. The Young’s modulus found by this method maynot be the “real” value for the material, but it will be the value that gives the mostaccurate simulation result. Thereby it is assumed that the simulations of the horn,will be the most accurate possible from the measured data.

To find the optimal value for Young’s modulus, the aluminum bars are simulatedwith a sweep of Young’s modulus in Comsol. For each value of Young’s modulusthe resonance frequency of the four bars is calculated and stored. After the sweep iscomplete the optimal value will be the one that gives the least error on the resonancefrequencies. Since there are four measurements, the combined error of the resonancefrequencies is calculated as the norm of the individual errors, the errors being thedifference between the measured and the calculated resonance frequency. The errornorm is calculated as the Euclidean norm given by

‖A (E)‖ =

√∑i

|ai (E)|2 (A.4)

where a ai (E) is the difference between the measured and the simulated resonance,for a given value of E, for the i’th bar.

From this simulation, the optimal value found for Young’s modulus for simula-tions, is E = 7.54·1010. For this value the error norm has a value of ‖A (7.54 · 1010)‖ =76.6Hz.


(a)

(b)

Figure A.1 - Setups used for measuring the resonance frequencies of aluminumbars. Response recorded in (a) is dominated by the longitudinalresonances while it in (b) is dominated by the transversal res-onances. The aluminum bars are mounted in rubber support.The force hammer used is seen to the right and the microphoneis visible to the left


(a) Bar with diameter 10mm

(b) Bar with diameter 30mm

Figure A.2 - Frequency response for the bar with length 50mm


(a) Bar with diameter 10mm

(b) Bar with diameter 30mm

Figure A.3 - Frequency response for the bar with length 100mm


Figure A.4 - Closer look at the frequency response for the 100x10mm bar. Theclose resonances for the transversal and longitudinal vibrationscan be distinguished because the response is recorded both infront of and to the side of the aluminum bar.

APPENDIX B. OPTICAL EXPERIMENT FOR VIBRATIONAL DISPLACEMENTPATTERNS 68

Appendix B

Optical experiment for vibrationaldisplacement patterns

In this appendix an experiment is described, that is used for visualizing small dis-placements of vibrating surfaces such as the surface of a radiating plate. The setupis based on optical speckles and the interference of these [22, 26]. The setup is basedon the theory and practical experiments in [25], but a simpler setup has been used.In the setup made for this project, it is not possible to do absolute measurements,but vibrational displacement patterns can be visualized.

The setup for visualizing surface displacement, is based on a few pieces of opticalequipment and a USB digital camera. The optical equipment consist of the following:

• Laser - Uniphase 1135/P - 632.8nm Helium-Neon laser

• Mirror with micrometer screws

• Microscope objective

• Semi matte glass plate

• Plain glass plate mounted in a sturdy frame

• USB digital camera

Other than the optical equipment, a sturdy mounting for the test object is alsoneeded. It is important that the object under test is very stable mounted, since asmall displacement of the mounting will ruin the visualization of the displacement,adding another displacement which will be smearing the image.

The setup of the equipment is shown in figure B.1The setup functions in the following way: The light leaving the laser is reflected

on the mirror, to get a longer optical distance from the laser to the test object. Afterthe mirror the light passes trough a microscope objective, which expands the lightbeam. The expanded beam hits the plane side of the semi matte glass plate, and isspread from the matte side of the plate. Statistical optics say that the light spreadfrom a matte glass plate, will make speckles which is a pattern popularly describedas “light cigars”. An example of speckle a pattern is shown in figure B.2.


Figure B.1 - Setup for visualizing the vibrational displacement

Figure B.2 - The speckle pattern formed by a laser beam diffracted by a semimatte glass plate

The speckles emitted from the semi matte glass plate are reflected partially by theplane glass plate, and from the test object as well. The interference between thesetwo reflections is what enables one to visualize the displacement of the test object.By recording a reference pattern of the resulting speckle interference pattern, withthe USB camera without excitation of the test object, and subtracting this imagefrom the patterns recorded with excitation, it is possible to visualize the displacementof the test object.

When excitation is applied the areas where displacement occurs will have anintensity difference between the reference pattern and the excited pattern. Since anUSB camera can only record absolute values of intensity, the areas with displacementis shown in the difference image as an illuminated area. An example of this is shownin figure B.3 which shows the vibration pattern of a brass radiation plate of radius25mm. The figure also shows the reference image B.3(a) where the test object canstill be seen.


Figure B.3 - Image recorded with the USB camera (a) reference image and(b) image recorded with excitation with the reference image sub-tracted.

APPENDIX C. DATA SHEET FOR LANGEVIN TRANSDUCER 71

Appendix C

Data sheet for Langevintransducer

On the next page the data sheet for the Langevin transducer used in this projectcan be seen. The transducer is made by a Swiss company called M.P. Intercon-sulting. The specific transducer used in this project has a resonance frequency off = 29.9kHz.

5

Typ

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(g

)

Len

gth

mm

Maxim

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(m

m)

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(m

m)

Freq

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(Fs±

0.5

kH

z)

Im

ped

an

ce

min

.

(!

)

Cap

acit

an

ce

(C

±1

0%

PF)

In

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t

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)

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Meta

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MP

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02

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75

APPENDIX D. DATA SHEET FOR ALUMINUM EN AW 2011 73

Appendix D

Data sheet for aluminum EN AW2011

On the next page the data sheet for the aluminum alloy used for the fabricatedparts can be seen. The data sheet does not contain a value of Young’s modulus, andfor that reason it has been necessary to determine this value experimentally. Theexperiment for determining Young’s modulus for the aluminum alloy, is describedin appendix A

2011 - ALUMINIUM ALLOY

Cu Fe Pb Bi Si Zn Al Typical Analysis (Ave. values %) 5.5 0.7 0.4 0.4 0.4 0.3 92.0

BS ISO NEAREST STANDARD EN AW-2011 Al Cu6 BiPb

DESCRIPTION Heat treatable alloy with high strength, good machinability and fatigue strength. Standard condition of supply - Temper T4.

APPLICATIONS Applications where good machinability and high strength are required, such as machine parts, bolts, nuts, screws etc.

Temper Tensile strength

MPa

0.2% Proof

stress MPa

Elong.A5 %

Elong.A50 %

Hardness Vickers

T3 310-365 260-290 10-15 10-15 100 T4 320-350 270-270 10-18 18 95 T6 310-395 230-300 8-12 10-12 115

MECHANICAL PROPERTIES

T8 370-420 275-315 10-13 10-12 120

Machinability Excellent

Deep drawing Poor

Spinning Poor

FABRICATION PROPERTIES

Extruding Good

Density 2.84 (kg/dm3)

Melting point (Liquidus) 645oC

Melting point (Solidus, Eutectic) 540oC

Coefficient of thermal expansion 23-µmm-1K-1

Thermal conductivity 177-W m-1K-1

Specific heat capacity 863 JKg-1 K-1

Electrical resistivity 38 nΩ m

PHYSICAL PROPERTIES

Electrical conductivity 45% IACS

APPENDIX E. WORK SKETCH OF DESIGNED HORN 75

Appendix E

Work sketch of designed horn

In this appendix a work sketch of the horn designed for experimental validation canbe seen.

APPENDIX E. WORK SKETCH OF DESIGNED HORN 76

Figure E.1 - Horn designed for experimental validation

APPENDIX F. EQUIPMENT LIST 77

Appendix F

Equipment list

This appendix consists of a list of equipment used throughout this project.The used Pulse system consist of the following

• Bruel & Kjær Pulse Front end in frame 3560-D

Controller module - type 7536

Power module - type 2826

Input/Output module - type 3110

• Pulse software version 14.0.0.124

• Pulse USB dongle with FFT Analysis license

The Bruel & Kjær 1/8” microphone used is used in the following setup

• Bruel & Kjær Type 4138 - 1/8” Pressure-field Microphone

Bruel & Kjær Type 2670 - preamplifier for 1/4” microphone

Bruel & Kjær Type UA-0160 - 1/8” to 1/4” Adapter

The Force hammer used is used in a combination of the following equipment

• Bruel & Kjær Type 8203 - Force hammer

Bruel & Kjær Type 2647 A - Charge to DeltaTron Converter

Also used are:

• Agilent 34405 A 5¡ Digit Multimeter

• Kikusui 65MHz Oscilloscope cos 5060

• Yamaha Natural Sound Amplifier AX-396

• Uniphase 1135/P 632.8nm Helium-Neon laser

• Bruel & Kjær Type 4231 - Calibrator for condenser microphone

APPENDIX G. CD CONTAINING THE THESIS AND VIDEOS OF LEVITATIONEXPERIMENTS 78

Appendix G

CD containing the thesis andvideos of levitation experiments

This CD contains the following:

• The thesis: High Power Ultrasound Sources.pdf

• Video of acoustic levitation experiments using a source with Swiss horn witha brass disc attached to the surface.

Levitation.wmv - Fine tuning of the reflector and levitation of polystyreneballs

• Videos of acoustic levitation using the source constructed in this project.

Levitation II.wmv - Powder experiment and levitation of polystyrene ball

Levitation II - Bare hands.wmv - Levitation of polystyrene balls, withcrude instruments: The fingers

Levitation II - Water drops.wmv - Levitation of water

The videos can be played in Windows Media player or seen on http://www.

youtube.com/shkristensen

http://www.youtube.com/shkristensen

http://www.youtube.com/shkristensen

BIBLIOGRAPHY 79

Bibliography

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[2] J.A. Gallego-Juarez, G. Rodriguez-Corral, and L. Gaete-Garreton. An ultra-sonic transducer for high power applications in gases. Ultrasonics, 16(6):267 –271, 1978.

[3] J. A. Gallego-Juarez, G. Rodrıguez-Corral, E. Riera-Franco de Sarabia,C. Campos-Pozuelo, F. Vazquez-Martınez, and V. M. Acosta-Aparicio. Amacrosonic system for industrial processing. Ultrasonics, 38(1-8):331 – 336,2000.

[4] J. A. Gallego-Juarez, G. Rodrıguez-Corral, E. Riera-Franco de Sarabia,F. Vazquez-Martınez, C. Campos-Pozuelo, and V. M. Acosta-Aparicio. Re-cent developments in vibrating-plate macrosonic transducers. Ultrasonics, 40(1-8):889 – 893, 2002.

[5] Michel Barran. Biography of paul langevin. http://scienceworld.wolfram.

com/biography/Langevin.html, confirmed in December 2009.

[6] D. Chacon, G. Rodrıguez-Corral, L. Gaete-Garreton, E. Riera-Franco de Sara-bia, and J.A. Gallego-Juarez. A procedure for the efficient selection of piezo-electric ceramics constituting high-power ultrasonic transducers. Ultrasonics,44(Supplement 1):e517 – e521, 2006. Proceedings of Ultrasonics International(UI’05) and World Congress on Ultrasonics (WCU).

[7] Antonio Iula, Riccardo Carotenuto, Massimo Pappalardo, and Nicola Lamberti.An approximated 3-d model of the langevin transducer and its experimentalvalidation. The Journal of the Acoustical Society of America, 111(6):2675–2680,2002.

[8] Christopher R. Field and Alexander Scheeline. Design and implementation of anefficient acoustically levitated drop reactor for in stillo measurements. Reviewof Scientific Instruments, 78(12):125102, 2007.

[9] C. Campos-Pozuelo, A. Lavie, B. Dubus, G.Rodrıguez-Corral, and J.A. Gallego-Juarez. Numerical study of air-borne acoustic field of stepped-plate high-powerultrasonic transducers. Acta Acustica united with Acustica, 84(6):1042–1046,November/December 1998. Torrough deduction of plane, torsional, transver-sal(called bending) and other waves.

http://scienceworld.wolfram.com/biography/Langevin.html

http://scienceworld.wolfram.com/biography/Langevin.html

BIBLIOGRAPHY 80

[10] Lawrence E. Kinsler, Austin R. Frey, Alan B. Coppens, and James V. Sanders.Fundamentals of Acoustics. John Wiley & Sons, inc., fourth edition, 2000.

[11] L. Cremer; M. Heckl; B.A.T. Petersson. Structure-Borne Sound. Springer BerlinHeidelberg, December 2005.

[12] Finn Jacobsen and Peter Juhl. Radiation of sound. DTU website:http://server.elektro.dtu.dk/ftp/fja/Radiation.pdf, August 2006.

[13] Jan Kocbach. Finite Element Modeling of Ultrasonic Piezoelectric Transducers.PhD thesis, University of Bergen Department of Physics, September 2000.

[14] Frank Fahy and Paolo Gardonio. Sound and Structural vibration. Elsevier, 2.ndedition, 2007.

[15] T. G. Wang and C. P. Lee. Nonlinear Acoustics, chapter 6. Radiation pressureand acoustic levitation. Academic Press, Incorporated, 1997.

[16] M. Barmatz and P. Collas. Acoustic radiation potential on a sphere in plane,cylindrical and spherical standing wave fields. J. Acoust. Soc. Am., 77:928–945,1985.

[17] W. J. Xie and B. Wei. Dependence of acoustic levitation capabilities on geo-metric parameters. Phys. Rev. E, 66(2):026605, Aug 2002.

[18] W. J. Xie, C. D. Cao, Y. J. Lu, Z. Y. Hong, and B. Wei. Acoustic method forlevitation of small living animals. Applied Physics Letters, 89(21), November2006.

[19] Louis V. King. On the acoustic radiation pressure on spheres. Proceedings ofthe Royal Society of London, 147(861):212–240, November 1934.

[20] L.P. Gor’kov. On the forces acting on a small particle in an acoustical field inan ideal fluid. Soviet Physics Doklady, 6:773–+, March 1962.

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[22] S.J. Frank L. Pedrotti and Leno S. Predotti. Introduction to optics. Prenticehall, 2.nd edition, 1993.

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BIBLIOGRAPHY 81

[26] Joseph W. Goodman. Statistical Optics. Wiley Interscience, 2000.

design, construction and characterization of highpower ultrasound sources

Documents

amechanical horn

designet af det mekaniske

ved brug af dette horn

design parameters

mechanical horns

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en ultralydskilde

ved hjlp af denneopstilling