Design, construction and characterization of highpower ultrasound sources by Søren Kristensen

Design, construction and characterization of high power ultrasound sources SĂ¸ren Hartmann Kristensen December 21, 2009

Abstract In this Master’s thesis several areas of designing, constructing and characterizing high power ultrasonic sources are investigated. An ultrasonic source, consisting of a prefabricated Langevin transducer and a mechanical horn designed and constructed for this project, is used for experimental validation of design simulations. The horn has a mounting plate for easy mounting of 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 optical setup, based on the theory of speckle interferometry, is used. Using this setup, it is possible to visualize nodal lines on the vibrating surfaces of radiation plates. The investigation of the design parameters, as well as investigation of the behavior and radiation or radiation plates, is done trough simulations in Comsol Multiphysics. To achieve accurate material parameters, Young’s modulus is measured for the material used to design the mechanical horn for the source.

Abstrakt p˚ a dansk I dette speciale er flere felter indenfor design, konstruktion og karakterisering af højeffekt ultralydskilder undersøgt. En ultralydskilde, sammensat af en præfabrikeret Langevin transducer og et mekanisk horn designet og konstrueret til dette projekt, er brugt i eksperimentelle valideringer af design simuleringer. Hornet har en monteringsplade for let montering af den samlede ultralydskilde. Ved brug af dette horn er flere eksperimenter udført, bl.a. et succesfuld forsøg p˚ a at levitere polystyrenkugler og vanddr˚ aber. For at kunne karakterisere vibrationsmønsteret for udstr˚ alingsplader, er en optisk opstilling, baseret p˚ a teorien om speckle interferometri, brugt. Ved hjælp af denne opstilling, er det muligt at visualisere nodallinier p˚ a den vibrerende overflade af udstr˚ alingspladerne. Undersøgelsen af designparametre, s˚ a vel som undersøgelsen af udstr˚ alingspladernes opførsel og udstr˚ aling, er udført vha. simuleringer i Comsol Multiphysics. For at opn˚ a præcise materialeparametre, er Young’s modul m˚ alt for det materiale som er brugt i designet af det mekaniske horn til kilden.

Contents 1 Introduction 1.1 The structure of high power ultrasonic transducers 1.1.1 The Langevin transducer . . . . . . . . . . . 1.1.2 Mechanical horns . . . . . . . . . . . . . . . 1.1.3 Radiation plates . . . . . . . . . . . . . . . 1.2 Objective of the project . . . . . . . . . . . . . . . 1.3 Conventions . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

1 1 1 3 3 4 5

2 Theory 2.1 Vibrations in bars with circular cross section 2.1.1 Longitudinal vibrations . . . . . . . . 2.1.2 Transversal bending waves vibrations 2.1.3 Torsional vibrations . . . . . . . . . . 2.1.4 Radial vibrations . . . . . . . . . . . 2.2 Radiation from a circular piston . . . . . . . 2.3 Vibrating discs . . . . . . . . . . . . . . . . 2.4 Simulation in Comsol Multiphysics . . . . . 2.4.1 Axisymmetric simulations . . . . . . 2.4.2 Simulations types . . . . . . . . . . . 2.4.3 Simulation of coupled physics . . . . 2.4.4 Simulation conclusion . . . . . . . . . 2.5 Acoustic levitation . . . . . . . . . . . . . .

. . . . . . . . . . . . .

6 6 6 7 9 10 10 12 13 13 14 15 15 17

3 Effect of mechanical horn shapes 3.1 Conical horns . . . . . . . . . . . . . . 3.2 Exponential horns . . . . . . . . . . . . 3.3 Stepped horns . . . . . . . . . . . . . . 3.3.1 Nodal position in stepped horn 3.4 Resonance frequencies for horn types .

. . . . .

18 18 20 24 24 25

. . . . .

4 Design of stepped horn for Langevin transducer based ultrasonic source 4.1 Work flow stepped horn design . . . . . . . . . . . . . . . . . . . . . . 4.2 Mounting plate and transition effects . . . . . . . . . . . . . . . . . . 4.2.1 Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Mounting plate . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Radiation from a stepped horn with no radiation plate . . . . . . . .

28 28 30 30 32 33

4.4

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

5 Experimental validation 5.1 Test of assembled sources . . . . 5.2 Frequency analysis of horn . . . 5.3 Error analysis . . . . . . . . . . 5.3.1 Simulation model errors 5.3.2 Error analysis conclusion 5.4 Sensitivity analysis of sources . 5.5 Acoustic levitation test . . . . .

. . . . . . .

6 Radiation plate assessments 6.1 Node detection using talcum powder . . . . . . . . . 6.2 Visualization of radiation plate displacement pattern interferometry . . . . . . . . . . . . . . . . . . . . . . 6.3 Radiation plate simulations . . . . . . . . . . . . . . 6.3.1 Vibration pattern of brass disc . . . . . . . . . 6.3.2 Sound radiation from radiation plates . . . . . 6.4 Radiation plate conclusion . . . . . . . . . . . . . . .

. . . . . . .

36 36 38 39 39 42 43 45

48 . . . . . . . . . 48 using speckle . . . . . . . . . 49 . . . . . . . . . 50 . . . . . . . . . 51 . . . . . . . . . 51 . . . . . . . . . 55

7 Conclusions and Future Work 57 7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7.2 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Appendices

A Measuring Youngâ&#x20AC;&#x2122;s Modulus of Aluminum bars

B Optical experiment for vibrational displacement patterns

C Data sheet for Langevin transducer

D Data sheet for aluminum EN AW 2011

E Work sketch of designed horn

F Equipment list

G CD containing the thesis and videos of levitation experiments

Bibliography

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 RenÂ´e Skov at University of Southern Denmark (SDU) for his kind help with the optical setup used in the project. Thanks also goes to the mechanical workshop at SDU for teaching me how to use a metalworking lathe and for their help with practical metalwork. I would also like to thank my advisor Arturo SantillÂ´an, for his help with the acoustic levitation experiments and for sharing his knowledge of the subjects covered in this thesis.

Table of symbols Symbol Description ca cL cT Ď Ď 0 E G Ď&#x192; f D S w W x L A p v Îť Ď&#x2030; k U Robj F Z 1

â&#x2C6;&#x2020; kAk ai

Unit

Speed of sound in air m/s Propagation speed of longitudinal vibration m/s Propagation speed of torsional vibration m/s Density kg/m3 Density of air kg/m3 Strain Youngâ&#x20AC;&#x2122;s modulus, modulus of elasticity Pa Shear modulus Pa Poissonâ&#x20AC;&#x2122;s ratio Frequency Hz Diameter m Surface area m2 Displacement m Displacement amplitude m Distance m Length m Amplification gain Pressure Pa Surface/particle velocity m/s Wavelength m Angular frequency Ď&#x2030; = 2Ď&#x20AC;f rad/s Wavenumber k = Ď&#x2030;/c mâ&#x2C6;&#x2019;1 Gorkov potential Radius of levitated object m Force N N Mechanical impedance m/s Subscript 1, naming for input- or wider part of mechanical horns Subscript 2, naming for output- or narrower part of mechanical horns Change in e.g. frequency â&#x2C6;&#x2020; = |f2 â&#x2C6;&#x2019; f1 | Euclidean norm of errors Individual error

Chapter 1 Introduction In this thesis some of the aspects of designing a high power ultrasonic source is described. These aspects include the structure of such sources, design aspects in constructing an efficient source, characterization of the such sources and measurements of the performance of a source. The term ultrasonic sources means a source that radiates sound with frequencies above the human hearing range, normally said to be 20kHz (though most people cannot hear that high frequencies). Ultrasound has many practical applications [1] ranging from chemical such as large scale treatment of gasses [2â&#x20AC;&#x201C;4] over medical such as wound healing and ultrasonic imaging, to industry applications like welding and cleaning. Some of these applications demand a very high sound pressure to be produced, thus needing a high power ultrasonic source. In this chapter the structure of a typical hight power ultrasonic transducer is outlined and the objective for the thesis is stated. Furthermore the naming conventions used in the thesis are described.

1.1

The structure of high power ultrasonic transducers

This section contains a short review the elements that are normally combined to build high power ultrasonic sources. High power ultrasonic transducers are widely build on basis of a Langevin transducer, coupled to a mechanical horn, providing a velocity amplification, and with a radiation plate to increase the sound radiating surface of the source, and there by enabling 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 French physicist Paul Langevin at the end of world war one [5]. The transducer was first build as part of an ultrasonic submarine detector and consisted of a piezo electric element sandwiched between two metal parts. When electrical voltage is applied to the piezo elements, they expand or contract depending on the polarization of

Figure 1.1 - The elements of a typical ultrasonic transducer

the voltage. This expansion (or contraction) is very strong and reliable, and can therefore be used to drive mechanical vibrations in metal parts that are attached to the piezo elements. A Langevin transducer usually consist of an even number of piezo elements sandwiched between two metal blocks. By designing the metal blocks attached to the piezo elements to be λ/4 longitudinal resonators with the same resonance frequency, the combined transducer will be a λ/2 longitudinal resonator in balance, having mirrored 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 the metal 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 aluminum and/of steel. An asymmetric design with one steel and one aluminum block can be used, but in that case the blocks should not have the same physical length, but the same “Acoustical length”, meaning the length with which the λ/4 resonance has the same frequency. Much research is done to optimize the characteristics of Langevin transducers this is described in e.g. [6, 7]. For this project a Langevin transducer from the Swiss company M.P. Interconsulting has been used. The transducer has the type number MPI-3025F-30H, a

diameter of D = 30mm and a resonance frequency of f = 29.9kHz. The transducer can 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 ultrasonic sources for a variety of applications, mechanical horns are used to maximize the vibrational amplitude the source can produce. This amplification is described in greater detain in a later chapter, but can intuitively be understood by a conservation of momentum example: Consider a free-free bar vibrating at a 位/2 longitudinal resonance. The ends vibrate in the longitudinal direction while there at the middle of the bar is a node. If one end of the bar is lighter than the other, this end will vibrate more, since the conservation of momentum applies. Hereby there will be an amplification of the vibration from the wider heavier end of the horn to the narrower lighter end. In section 3 the shape and theoretical performance of different horn types are described in greater detail and investigated though Comsol simulations. With the Langevin transducer bought for this project came a mechanical horn meant to be the coupling to additional tools which can be attached using a threaded hole in the front of the supplied horn. This horn will be referenced to as the Swiss horn 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 a means of achieving different acoustic radiation properties, than the often small surface of the horn would facilitate. By using a plane radiation plate with radius larger than the horn end, the surface area of the transducer can be increased causing a greater volume velocity and thereby greater sound pressure radiation, than the horn surface alone would produce. This principle is used in e.g. [8]. Furthermore special shapes of radiation plates can be designed to produce a sound field suitable for cer-

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

tain applications of the source. Such special designs of radiation plates can be seen in [3, 4, 9]. An example of a radiation plate with a special design is described in [9] where a radiation plate with special groves improves the acoustic power of the source. The source 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 flexural radiation plate that does not vibrate in a piston-like fashion. This type of vibration in a radiation plate is investigated in chapter 6 where some ways of visualizing the vibrational pattern is introduced. Furthermore the vibrational pattern is simulated in Comsol Multiphysics, where the mechanical vibrations simulation in the radiation plate can be coupled to an acoustic simulation, which can be useful for characterization 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 and designing 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 the piezo 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 the source facilitates the use of a much smaller amplifier for a stable setup for acoustic levitation. In this thesis the main focus will be the design of the mechanical horn to attach to the purchased Langevin transducer, which will be joined to form an ultrasonic source. The design process followed in the work described in the present thesis is based on the use of Comsol Multiphysics, using material properties measured in the

lab. Further more the project also covers ways to characterize radiation plates. The characterization is done trough simulations of the radiated sound field and practical experiments, with which it is possible to visualize the displacement patterns of radiation plates vibrating at ultrasonic frequencies. The main motivation for this work is to produce a source strong enough to be used for acoustic levitation.

1.3

Conventions

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

CHAPTER 2. THEORY

Chapter 2 Theory In this chapter the theory used further in this thesis will be outlined. First a review of vibrational theory in bars needed for designing mechanical horns is presented, and the 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 plate does not always vibrate in a piston like motion. For this reason the chapter also contains a section describing the vibrations of flexible circular plates. Apart from the physics theory an introduction to the simulation tool Comsol Multiphysics is also made in this chapter. In the end of the chapter a short introduction to the Gorkov potential, and some acoustic levitation theory is introduced to better be able to evaluate the sources used 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 theory involves longitudinal, transversal, torsional and radial vibrations and resonances in bars

2.1.1

Longitudinal vibrations

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

â&#x2C6;&#x201A;w â&#x2C6;&#x201A;x

(2.1)

and stress given by F = â&#x2C6;&#x2019;E Âˇ S

(2.2)

where w is a measure of displacement along the bar, x is a measure of distance and S is the cross section of the given bar that is substitute to the force F . The value E is the elasticity coefficient of the material, also known as Youngâ&#x20AC;&#x2122;s modulus.

CHAPTER 2. THEORY

Given the mentioned equations for stress and strain, the longitudinal wave equation can be deduced to be [10, page 70] : ∂ 2w 1 ∂ 2w = ∂x2 c2L ∂t2

(2.3)

where cL is the propagation speed of the longitudinal vibration. cL can be calculated from the density of the given material, ρ, and Young’s modulus the following equation [10, page 70] s E (2.4) cL = ρ By assuming a general solution of the wave equation given in equation 2.3 and applying the boundary conditions of the geometry at hand, the mode pattern for the resonances can be calculated. This is described in greater detail in [10, Page 71-72]. For a bar that is free in both ends, meaning that the support for the bar does not apply any strain to the ends of the bar, the mode shapes can be calculated from the following equation [10, page 72]: w (x, t) = 2W ejωn t cos(kn x)

(2.5)

where W is the amplitude of a an initial vibration reflected at the bar end, kn is the wavenumber 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 =

ncL 2L

(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 interest when building a high power ultrasonic resonator source. The equations here mentioned is a formulation for pure longitudinal vibrations in a bar, and it is assumed that the displacement and thereby the strain is uniform over the cross section of the bar. This is not always the case, especially for bars that are relatively wide compared to their length. In that case the longitudinal vibrations are affected by the radial vibrations (see section 2.1.4) ultimately changing the resonance frequency 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 bending vibrations. This vibration can also be experienced, for instance, in a swaying flag pole or a vibrating spring.

CHAPTER 2. THEORY

Figure 2.1 - Example of a longitudinal resonance in a circular bar. The displacement is shown as deformation (exaggerated) and by the color scale indicating absolute values of displacement. The original bar is shown by the black wire frame. This is a resonance of a fixedfree 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. An example 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] Ď&#x20AC;acL f = 1.1942 , 2.9882 , 52 , 72 , ... (2.7) 12L2 where L is the length of the bar and a is the radius of the bar. The expression has been written as a function of cL , which is the propagation speed of longitudinal waves for the material. This is not because cL is the same as the propagation speed of the bending wave, but because it makes the the formulations for the longitudinal and transversal waves easier to compare. It is seen from equation 2.7 that the frequency of 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 the radius of the horn parts, given the rest of the horn design allows this change.

CHAPTER 2. THEORY

In the design of a Langevin transducer based ultrasonic source, transversal vibrations are not desired. If transversal vibrations have resonances in the same frequency range as the longitudinal vibrations it is possible that some of the vibrational energy would 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 axis of the bar. An example of the twisting movement is shown in figure 2.3, where a torsional resonance for a fixed-free bar is seen.

Figure 2.3 - Example of torsional resonance in a circular bar. The bar has been 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 the first longitudinal resonance. In this and other ways the torsional and longitudinal vibrations behave much alike the. The torsional wave equation is, like the longitudinal waves, a second order differential equation. The propagation speed of torsional waves is [11, page 46] s G cT = (2.8) ρ where G is the shear modulus, determined by the material composition. G has in the description of torsional vibrations the role that the Young’s modulus E has in the longitudinal vibration description. From this similarity it is noted, that if torsional and longitudinal resonances coincide, it is not possible to move the frequency of the torsional resonance to a different 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 most cases the sheer modulus will be a lot less. For some materials though, it will be possible to excite a higher torsional resonance when driving the first longitudinal resonance. In the case of a Langevin transducer driving a mechanical horn, it is not likely that the torsional resonances will be excited very much, since the piezo electric discs of the Langevin transducer vibrates primarily in the longitudinal direction.

CHAPTER 2. THEORY

2.1.4

Radial vibrations

Radial vibrations are mechanical vibrations that runs in a direction perpendicular to the axial direction of the bar. This means that the resonance of this type of vibrations will be a standing wave in this direction. Since the dimensions of a bar are usually greater in the axial direction than in the radial direction, the resonance of radial vibrations will have a higher frequency than the longitudinal resonance frequency. An example of a radial resonance can be seen in figure 2.4, where the dimensions of 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 â&#x20AC;&#x153;barâ&#x20AC;? that is very short relative to its diameter. The two images are of opposite phase

In this project the radial vibrations play a part since they have an effect on the longitudinal resonance frequencies of cylindric bars as described in [7]. Here it is stated that even away from the radial resonance frequency, the radial vibration affect the longitudinal resonance by lovering the resonance frequency. This results in the longitudinal wave equation not being sufficient to calculate the resonance frequency. Since Comsol simulates the geometry defined and do not simply assume longitudinal vibrations, the mix of longitudinal and radial vibrations will not be a problem in the simulations. In the theoretical calculations however it cannot be assumed that the longitudinal vibrations equations can be accurately used to calculate the resonance frequency 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 planar circular piston vibrating with a fixed velocity v can be described as the following

CHAPTER 2. THEORY

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

a 2 1

kr p (r, a) = 2ρ0 ca v sin 1+ −1 (2.9)

2 r 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 k · a2 1 p(r, a) = ρ0 ca v 2 r

(2.10)

It should be noticed that the far field the pressure at a given distance is proportional to the radius of the piston squared. pf ar−f ield ∝ a2

(2.11)

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

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

From the figure it can be seen that for some specific distances, a higher sound pressure 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 distance to radius ratios of 1.2 and 1.7. With the distance constant the jump from 1.2 to 1.7 could be done by decreasing the disc radius by 30% but vould lead to an increase in the radiated sound pressure level by approximately 18dB. This is only a local effect and it does not comply in the far field, but if a high sound pressure in the near field is the main interest in a certain source design, the source radius should not necessarily be as big as possible.

CHAPTER 2. THEORY

2.3

Vibrating discs

By vibrations in discs is in this thesis meant a bending vibration traveling in the radial direction of the disc. Examples of resonances of free vibrating discs can be seen in figure 2.6, where the first two axisymmetric resonances, as well as one of the non axisymmetric resonances of an arbitrary free vibrating disc can be seen.

(a) First axisymmetric resonance

(b) Second 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 arbitrary non-axisymmetric resonance of the disc

CHAPTER 2. THEORY

In the figure it is seen that the first axisymmetric resonance has a nodal circle around 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 characterization of 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 base such research on finite element method (FEM) simulations [6, 7, 13]. In this project the simulation tool Comsol Multiphysics has been used for analysis of mechanical vibrations, as well as sound propagation. For a thorough introduction to FEM for vibrational simulation see [14, Chapter 8]. In this section it is described how the geometries in the project have been simulated. Simulations of stress strain vibration as well as acoustical field radiation, has been carried out using the Comsol Multiphysics (from here referred to as â&#x20AC;&#x153;Comsolâ&#x20AC;?). This simulation software has a wide range of simulation facilities e.g. electrical field simulations, magnetic field simulations, mechanical stress-strain simulations and acoustic field simulations. The software also has the ability to connect multiple physics in one simulation, as it is done in a later chapter where both mechanical vibrations and acoustical radiation is simulated in one combined simulation. When simulating a given physical situation in Comsol, the model of the physical object can be made in several different geometry formulations. To reduce calculation time is useful to reduce the problem to as simple a geometry as possible. If a given situation can be simulated as a two dimensional problem, the calculation time is far less than if a full 3D simulation is run of the same geometry. The reason for using the more elaborate geometry formulations is, that they will reveal solutions, that the simpler formulations would not. It is for instance not possible to simulate transversal waves in the axisymmetric formulation, since the transversal waves are not axisymmetric.

2.4.1

Axisymmetric simulations

In the axisymmetric formulation of Comsol, the geometry defined is solved with the conditions that all physical properties are symmetric around a user defined symmetry axis. This means the geometry is defined in a 2D space and the user must imagine the geometry rotated around the symmetry axis. Therefore, a geometry solved for stress/strain vibrations in the axisymmetric formulation will only reveal the longitudinal and radial waves in the geometry, since they are the only axisymmetric waves that can occur in a bar. In figure 2.7 examples are seen to illustrate how the Axisymmetric geometry represents a 3D shape. The geometry in 2.7(a) is defined for Comsol simulations and is rotational symmetric around the straight side on the left. In 2.7(b) a 3D model of a similar geometry is seen. This model is made in Autodesk Inventor by

CHAPTER 2. THEORY

(a)

(b)

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

actually drawing a 2D axisymmetric sketch and revolving it around the same axis as Comsol does (virtually) The simplification of the geometries enables the definition of these from a Matlab script, 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 to reveal different properties of the geometry. For the stress-strain simulations in this project two simulation types have been used, namely eigenfrequency analysis and frequency response analysis. The eigenfrequency simulations In eigenfrequency simulations Comsol finds a user defined number of resonances for the defined geometry. This is done trough eigenfrequency equations predefined within the Comsol software. When solving for eigenfrequencies Comsol includes the stress-strain interaction and mass of the geometry, but not defined forces or displacement. If any displacement 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 response simulations must be used. 1

Using these automated simulations, it is possible to make geometry sweeps; a functionality not implemented 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

CHAPTER 2. THEORY

Frequency response simulations In the frequency response simulations the simulations tool solves a stress-strain wave equation numerically with the given boundary conditions. This type of simulation corresponds to doing a physical experiment, in the way that an interaction must be applied to the model or nothing will happen. The resulting behavior of the model geometry can, in a frequency analysis simulation like in a physical experiment, be complex, and the results of a frequency response simulation can complicated to analyze automatically using programmed scripts. For this reason frequency analysis simulations are mostly used to analyze geometries where the behavior of a single model geometry is investigated, rather than in e.g. a sweep of geometry dimensions. To ensure consistent simulations results all simulations have been done with fixed-force boundary conditions for the input excitation. This ensures that both eigenfrequency and frequency response simulations can be run revealing the same resonances of the same geometry. To prove that this is the correct way to define input excitations for the free bars, see figure 2.8. Here it is seen that when the input excitation is set to a fixed displacement the resonance calculated is the 位/4 rather than 位/2 as is expected for a free-free bar. Frequency response simulation have been preformed to find the optimal horn parameters 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 simulations of different physics (hence the name Multiphysics). Using this functionality it is possible e.g. to couple a mechanical vibration problem to an acoustical problem, and simulate the two together, also including the interaction between the acoustical field and the mechanical system. This functionality is used in section 6.3, where radiation plate vibrations simulation are coupled to acoustic field simulations enabling the calculation of e.g. directivity pattern of radiation plates at a given mechanical vibration frequency.

2.4.4

Simulation conclusion

When simulating a certain physical problem in Comsol, choosing a geometry formulation that only just satisfy the problem at hand, can reduce simulation time in comparison to doing an elaborate 3D simulation for all problems. For this reason the axisymmetric geometry formulation has been used throughout this project. While the axisymmetric simulations only reveal axisymmetric vibration properties of the geometry, the longitudinal vibration problems can be simulated in the axisymmetric formulation to increase simulations speed and to have a simpler simulation result to analyze. Furthermore the axisymmetric geometries are simpler to define from a Matlab script making the combination of the two programs a strong tool for e.g. sweeps in geometry which is not possible to do in Comsol alone.

CHAPTER 2. THEORY

(a) Fixed displacement

(b) Fixed force

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

CHAPTER 2. THEORY

2.5

Acoustic levitation

This section consists of a short introduction to acoustic levitation. The fact that a sound field produces forces that can be used for the purpose of levitating small objects, has been described in various literature based on both a theoretical explanations [15–17] and practical setups, that can be used for levitating small objects or even small animals like in [18]. The original proposition that such phenomenon exists was first deduced in [19], but practical validations of the theory was not conducted until the 1970’s where NASA conducted some experiments with micro gravity using acoustic levitation. To calculate the force on an object in a sound field, the Gorkov potential U is used. The Gorkov potential, which was first defined by L.P. Gor’kov in [20], can be calculated from the following equation: v¯2 p¯2 3 (2.12) − ρ0 · U = 2 · π · Robj 3 · ρ0 · c2 2 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 object for 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 3 volume, and thereby the weight of a the object, are both proportional to Robj 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 levitation are carried out in section 5.5.

CHAPTER 3. EFFECT OF MECHANICAL HORN SHAPES

Chapter 3 Effect of mechanical horn shapes In this chapter the effect of the shape of mechanical horns is investigated. By evaluating different kinds of mechanical horns, mainly with respect to their expected amplification of the vibration amplitude from one end to the other, the optimal horn shape for use in an ultrasonic source is determined. The idea of mechanical horns is that by making a horn with different diameters at each end the end where the diameter is smallest will vibrate with the largest amplitude. In this chapter three horn types are described and their theoretical amplification, deduced in [21], is attempted verified by simulations in Comsols axisymmetric formulation. The horns are all meant to be λ/2 resonators with pure longitudinal waves, and a resonance frequency of f = 25kHz is chosen, as it lies within the normal frequency range of ultrasonic sources. The length of the horns are calculated from the following equation, assuming the material of the horn is aluminum with cL = 5150m/s [10, p 526]. L=

cL 5150 λ = = = 103mm 2 2f 2 · 29900

(3.1)

This length is used for all the simulated horns as is the material properties found in [10] E = 7.1 · 1010 P a, 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 the other is to change the diameter gradually forming a conical horn. Such a horn is shown in figure 3.1 which shows the displacement of a conical horn in an axisymmetric Comsol simulation. From the longitudinal wave equation (equation 2.3), the amplification A of a conical horn can be deduced, as it is done in [21, Chapter 4, III E]. Note the another subscript convention is being used: v2 A= = N cos v1

ωL c

c N −1 − sin ω L

ωL c

(3.2)

CHAPTER 3. EFFECT OF MECHANICAL HORN SHAPES

Figure 3.1 - First resonance in an axisymmetric simulation of a conical horn with aspect ratio D1 /D2 = 7 2 . where N = D D1 For a half wavelength resonator (λ/2 resonator) this equation can be simplified using

λ cL πcL = = 2 2f ω ωL =π cL

(3.3) (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 Comsol Multiphysics axisymmetric formulation. The simulation result for amplification is seen in figure 3.2. From the simulation result it is clear, that the amplification does only follow the theoretical amplification for small aspect ratios of the horn. The reason for the drop in amplification at larger aspect rations could point to, that the assumption of pure longitudinal 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 of D1 /D2 = 7. Here the vibrations are not pure longitudinal waves, which is seen by the curves of the coloring of the displacement.

CHAPTER 3. EFFECT OF MECHANICAL HORN SHAPES

Figure 3.2 - Simulation of the absolute amplification in a conical horn. Only for small diameter rations does the simulation amplification match the theoretical amplification

3.2

Exponential horns

When designing a horn to go from one diameter to another, another way to make this transition is in a gradual exponentially changing curve. This transition is described by equation 3.6 for the cross section as a function of the length variable x. S (x) = S1 e−kx

(3.6)

where − ln k=

D2 D1

(3.7)

From 3.6 an expression for the diameter at a certain point in the transition can be found: q D (x) = D12 ek(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]: v2 D2 A= =− (3.9) v1 D1 which is the same as the theoretical amplification of the conical horn, but in the deduction the assumption of longitudinal resonance made in 3.3 is not taken in the case of the exponential horn. Like with the conical horn the exponential horn has

CHAPTER 3. EFFECT OF MECHANICAL HORN SHAPES

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

been simulated for a range of diameter aspect ratios. The results of the simulation can be seen in figure 3.5. It is seen from the simulation results in figure 3.5 that the amplification of the exponential horn follows the theoretical amplification throughout the aspect ratio sweep, to a good extend. This shows that the exponential horn will be able to deliver a higher order of amplification than the conical horn though the theory said their amplification would be the same. This is thought to be on account of the longitudinal wave theory being more accurate for the exponential horn since it is thinner a longer way up the horn. In figure 3.6 the deformation of a wide exponential horn at resonance can be seen. By comparing the deformation pattern of the exponential horn with the conical horn, which is seen in figure 3.3, it can be seen that in the conical horn there is a circular displacement pattern around y = 0.07m. This points to the presence of non longitudinal vibrations which are not assumed in the deduction of the amplification theory. To determine the reason for the lack of amplification for large horn ratios, the phenomenon would have to be investigated further.

CHAPTER 3. EFFECT OF MECHANICAL HORN SHAPES

Figure 3.4 - Example of an exponential horn

Figure 3.5 - Amplification simulation for the exponential horn (Absolute values). Here the simulated amplification matches the theoretical value for a wide range of diameter ratios

CHAPTER 3. EFFECT OF MECHANICAL HORN SHAPES

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

CHAPTER 3. EFFECT OF MECHANICAL HORN SHAPES

3.3

Stepped horns

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

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

For the stepped horn the amplification can be shown to be [21, page 113] v2 Z1 A= â&#x2030;&#x2C6; = v1 Z2

D1 D2

2 (3.10)

where Z1 and Z2 are the mechanical impedance of the two horn steps. This expression for the amplification assumes identical material properties (such as propagation speed) in the two horn steps, but as long as the waves are pure longitudinal this is a reasonable assumption. The equation shows a great amplification potential which surpasses the two previous horn types by the power of two. Like with the other horn type a series of simulations has been carried out, to verify the horn amplification for the stepped horn. The result of the simulation can be seen in figure 3.8 As the figure shows, the amplification of the stepped horn follows the theoretical amplification to a good extend for the entire aspect ratio sweep. This points to the stepped 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 the transition is not a clear nodal plane as expected from the theory in [21].This can be

CHAPTER 3. EFFECT OF MECHANICAL HORN SHAPES

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

seen in figure 3.9 where the height of the plot shows the amount of displacement at a given position in the horn. From figure 3.9 it can be seen, that the node is not a clean cut perpendicular to the axial direction trough the transition of the horn, but rather a curved line from some point on the line between the two diameters and a point inside the first cylinder of the horn. For this reason a specific design process is established which should ensure that the 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 next chapter.

3.4

Resonance frequencies for horn types

When doing the geometry sweep simulations in the previous sections, the resonance frequency 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 the resonance frequency will drop below the pure longitudinal resonance frequency if the bar does not have a large length to width ratio. This is investigated in [7]. The resonance frequency for the three horn types, as a function of aspect ratio of the horns, 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

CHAPTER 3. EFFECT OF MECHANICAL HORN SHAPES

(a)

(b)

Figure 3.9 - Displacement of the stepped horn. The height of the plot shows displacement in the axial direction of the horn. In (b) there is zoomed on the transition of the horn

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

CHAPTER 3. EFFECT OF MECHANICAL HORN SHAPES

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

CHAPTER 4. DESIGN OF STEPPED HORN FOR LANGEVIN TRANSDUCER BASED ULTRASONIC SOURCE

Chapter 4 Design of stepped horn for Langevin transducer based ultrasonic source In this chapter the steps in designing a stepped horn for a specific resonance frequency 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 would interfere as little as possible with the vibrations in the horn. Simulations of the symmetric stepped horn used in the previous section, where l1 = l2 showed, that the nodal plane of the resonance does not coincide with the transition step. Further more there is no nodal point on the outside of either of the horn 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 vibrates in a fashion where the transition is moving as little as possible. This method is used for designing stepped horns both with and without mounting plates, for holding the horn, 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, a design work flow is used where the lengths of each of the horn steps are found individually. This is done by doing a dimension sweep in a Matlab driven Comsol simulation while looking at the achieved resonance frequency. Since the horn is to be designed for a specific Langevin transducer, the diameter of the horn step close to the Langevin transducer should have the same size, as the transducer, which is D = 30mm. In [7] it is shown that the longitudinal resonances can be coupled to the radial vibrations in the bar. As a consequence the resonance frequency will be lower than the pure longitudinal resonance. For this reason, the resonance of the horn steps cannot be assumed to have the same frequency even if the lengths are the same. For this reason, the lengths of the horn steps are found individually one at the time.

CHAPTER 4. DESIGN OF STEPPED HORN FOR LANGEVIN TRANSDUCER BASED ULTRASONIC SOURCE

Before the design process is started the simulation tool should be loaded with the material properties of the material the horn is to me made from. In this project an aluminum alloy called EN AW-2011 has been used to fabricate the parts. The properties for this material can be seen in appendix D, though not all properties needed to preform a stress-strain simulation (E, ρ and σ) were available. For this reason the value of Young’s modulus has been measured to be E = 7.54 · 1010 P a using the method described in appendix A. For density the data sheet value of ρ = 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 horn step. The optimal length for the first horn step is calculated trough a series of simulations, where the two horn steps have the same diameter (forming a two part cylinder). 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 the length are changed symmetrically and the horn steps have the same diameters. In design step two only the second horn step is changed

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

CHAPTER 4. DESIGN OF STEPPED HORN FOR LANGEVIN TRANSDUCER BASED ULTRASONIC SOURCE

Design step two To find the optimal length of the second and narrower horn step, a similar set of simulations are conducted. With the length and diameter of the first horn step fixed at the value found in design step one, the length of the second horn step is swept, at each length doing a frequency analysis to see the resonance frequency of the horn step with this particular length. When the desired resonance frequency is meet the length of the second horn step is at its optimal length. This is done for a single diameter of the second 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 designed source will now have the resonance frequency defined as the target frequency, and the 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 place to hold the transducer, without interfering with the resonance and amplification of the horn. Since the horn steps are designed to be 位/4 resonators, they will together form a 位/2 resonator with a node at the transition. This makes the transition a good position to introduce a plate for holding the horn, as this will interfere as little as possible with the vibration of the two horn steps. This plate will be called a mounting plate in this thesis. Another important thing to address when designing a mechanical horn for velocity amplification, is the stress in the transition. It is widely used to smooth the transition to avoid the high stress that a right angle transition intuitively gives. The influence on the characteristics of the horn, by introduction of the mounting plate 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 in figure 4.2.

4.2.1

Transition

In this section the influence on horn characteristics by the transition shape is investigated for a stepped horn without a mounting plate. This is done trough two simulation sweeps, where the stress and amplification is measured as a function of the diameter ratio. The first sweep is done with a stepped horn that has a right angle transition, that is, the diameter of the horn goes directly from D1 to D2 at the transition. 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 is to make the transition of the horn in the shape of a quarter-circle, where the radius is 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 applications where the horn is used in contact with other materials (like in ultrasonic welding),

CHAPTER 4. DESIGN OF STEPPED HORN FOR LANGEVIN TRANSDUCER BASED ULTRASONIC SOURCE

(a) Horn design with mounting plate and an- (b) Horn design with mounting plate and gled transition curved transition, the transition following a B´ezier curve

Figure 4.2 - Examples of horns with mounting plates positioned at the nodal line 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 Bézier-curve shaped transition has been used as opposed to the quarter-circle transition. A Bézier curve is a type of curve widely used in vector graphics. It is basically a quadratic interpolation of three points giving the smoothest transition between the two end points. The use of the Bézier curved transition is chosen since it is easier to define in a Matlab driven Comsol simulation, than the quarter-circle. Furthermore the interpolated transition is even more smooth than the circle transition making it more 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 with the 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 by looking 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 figure 4.3, where it is evident, that the maximum stress in the horn is decreased by the curved transition. For large aspect ratios there is a visible difference between the stress for the horn with the right angled transition, and the one with the Bézier curved transition. When looking at the amplification of the horn it is seen, that the amplification for the two transition types are close to the same, but since there is a decrease in the maximum stress, the source will be able to deliver a higher sound pressure with a curved transition before meeting its maximum stress limit.

CHAPTER 4. DESIGN OF STEPPED HORN FOR LANGEVIN TRANSDUCER BASED ULTRASONIC SOURCE

Figure 4.3 - Simulation results for the amplification and stress normalized to the 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 added plate at the transition in the stepped horn. The plate is designed with a thickness of 2mm and an outer diameter of 45mm. Using the design steps mentioned in section 4.1, several stepped horns with mounting plates at the transition has been designed and simulated using an automated Matlab script. The simulation result for these horns, having different diameter ratios as well as both curved and right angled transitions, can be seen in figure 4.4. From figure 4.4 it is seen that the maximum stress in the horn is decreased by adding the mounting plate to the horn. The curved transition still leads to less stress than the angled transition, though the difference is not as large as in the horn without a mounting plate. An interesting result for the simulation is, that the curved transition does not decrease the amplification of the horn, rather it increases it when the horn has a mounting plate at the transition. This result is very interesting and should be investigated further, as it could mean that the amplification could be increased by using 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 for the BÂ´ezier curve transition. This error is thought to occur on account of the small

CHAPTER 4. DESIGN OF STEPPED HORN FOR LANGEVIN TRANSDUCER BASED ULTRASONIC SOURCE

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

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

4.3

Radiation from a stepped horn with no radiation plate

In this section the near field sound pressure radiated from a stepped horn with no radiation plate is investigated. Since acoustic levitation is not necessarily a far field process, it can be interesting to look at the near field radiation from a piston of a given size at a distance closer to the source than the far field. This is done introducing an expression for the pressure radiated 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. 2 D1 /2 pHorn (r, a2 ) = p (r, a2 ) 路 (4.2) a2 The expression in equation 3.10 is illustrated in figure 4.5, where the sound pressure level is plotted relative to the sound pressure radiated by a horn of D2 =

CHAPTER 4. DESIGN OF STEPPED HORN FOR LANGEVIN TRANSDUCER BASED ULTRASONIC SOURCE

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

Figure 4.5 - Sound radiation in the near field relative to the radiation from a piston with the diameter 5mm. The different lines represent the relative sound pressure level at a specific distance. The two blue areas 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 for the loss in radiation surface. There is no gain in choosing a wider D2 , other than stability of the source, in respect to possible transverse vibrations.

4.4

Stepped horn designed for practical experiments

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 the Langevin transducer, why the diameter of the first horn step should be D1 = 30mm to ensure good transfer of vibrations. Since the near field radiation will decrease for larger values of D2 , this is set to be D2 = 10.4mm. Using the design process described in section 4.1 in a Matlab script, the optimal lengths 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.

CHAPTER 4. DESIGN OF STEPPED HORN FOR LANGEVIN TRANSDUCER BASED ULTRASONIC SOURCE

The transition is shaped in a BÂ´ezier curve with the radius 4.9mm according to equation 4.1. A work sketch of the designed horn can be seen in appendix E, and a picture of a 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

Chapter 5 Experimental validation In this chapter a stepped horn described in th previous chapter has been tested in various experimental setups. The horn is designed to be mounted on a Langevin transducer from M.P. Interconsulting, which has a resonance frequency of 29.9kHz and a surface diameter of 30mm, at the end where the horn is attached. As well as the designed horn, measurements are carried out on the Swiss horn. Finally the performance of the designed horn is discussed in comparison of to expected performance, as well as in comparison to the Swiss horn. To compare the horns performances, a test of the acoustic levitation forces they supply is also conducted. An elaborate list of the equipment used for the experiments, can be seen in appendix F.

5.1

Test of assembled sources

The first test carried out with the constructed horn, is a frequency analysis of the horn joined with the Langevin transducer to form an ultrasonic source. For the measurement 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. The source in the holder can be seen in figure 5.1. To find the resonance frequency of the combined source, an automated frequency sweep was run from the Pulse software, measuring the response an 1/8” microphone in a distance of 50mm. From this sweep the resonance frequency was found.

CHAPTER 5. EXPERIMENTAL VALIDATION

Figure 5.1 - Source mounted in the fabricated holder

The result of the frequency test was that the resonance of the designed horn connected to the Langevin transducer was f = 28, 594kHz. It is noted that the resonance 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 reason this could occur is, that it was not possible to fabricate the horn from the exact same bar of the aluminum alloy, as was used to fabricate the test bars used to measure Young’s modulus. For this reason there could be production deviations in the aluminum bars, that causes the measured value of Young’s modulus not to be sufficiently accurate. Other sources of errors are described in section 5.3. Another experience from this setup is, that when driving the source at its resonance frequency, vibrations can be detected in the source holder. This detection is done by lightly touching the surface of the holder with a set of tweezers. This produces an audible sound, that shows that the holder is vibrating.

CHAPTER 5. EXPERIMENTAL VALIDATION

This points to the resonance of the source not being entirely balanced, meaning that the horn has a different resonance frequency than the Langevin transducer. If this is the case, the horn would not be excited in its own resonance, why the mounting plate would not be located at a nodal plane of the vibration. To investigate this further, 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 resonance frequency of the aluminum bars is found, but in the case of the frequency analysis of the horn, no transversal resonances occurred within the frequency range of 25 â&#x2C6;&#x2019; 35kHz, which has been the focus for the analysis preformed. Since the desired resonance frequency of the horn is 29.9kHz, it is unlikely that any transversal resonance 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 the Pulse system, a force hammer and a 1/800 microphone. In this experiment the horn have been detached from the Langevin transducer an placed in holder like seen in figure 5.2.

Figure 5.2 - Horn in holder for frequency analysis. Here the excitation is given in 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, the microphone 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 this way shows a frequency of f = 28.72kHz. To ensure accurate measurements the test

CHAPTER 5. EXPERIMENTAL VALIDATION

is repeated with the excitation in the wider end of the horn and the microphone in the thinner end. This test confirms the resonance frequency from the first test of f = 28.72kHz. Since the horn was designed without the screw that connects the horn to the Langevin transducer, it was tested what influence this had on the horn 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 of f = 29.2kHz while the inverse test shows f = 29.09kHz. The reason for the small deviation is thought to be on account of unbalanced resonances where the two end have resonances of slightly different frequencies. From this experiment it is seen that the screw changes the resonance of the whole horn, not just the end where it is attached. The difference between the target frequency of 29.9kHz and the measured frequencies 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 same bar of aluminum as the test pieces for the Young’s modulus measurement was made from. This could mean that the value of Young’s modulus used in the design process is not the exact same as the value for the aluminum used to fabricate the horn, even though the material was said to be the same. This is thought to be a possible source of errors that has caused the calculated resonance frequencies to be incorrect. Apart from this error it is possible that the simulation model is not exact enough to simulate the correct behavior of the designed horns. In the following sections a number of changes has been made to the simulation model to investigate the influence of some of the different parameters in the simulation.

5.3.1

Simulation model errors

The first test done is a simulation of the horn with the dimensions measured from the actual produced horn. These measurements revealed some small errors in the lengths 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 a thickness of 2, 1mm and a diameter 45mm. Since the shape of the transition is rather complex, it has not been possible to measure the exact dimensions of this.

CHAPTER 5. EXPERIMENTAL VALIDATION

The curved transition is a very complex shape to manufacture, why this shape is possibly not perfectly accurate. Using these dimensions the resonance frequency did not change away from the 29, 9kHz also achieved in the design simulations. Simulation mesh density Now that a geometry with more exact dimensions is established, the next step in optimizing the simulation model is to refine the simulation mesh. By increasing the mesh density in the simulation, it can be investigated weather the simulation mesh is sufficiently dense, to calculate the true behavior of the horn. When changing the mesh density to a finer mesh the resonance frequency of the mechanical system will asymptotically approach the most correct value for the simulated parameters. In Comsol the mesh refinement is done by clicking the â&#x20AC;&#x153;refine meshâ&#x20AC;? button and running the simulation over. For testing the mesh refinement influence on the resonance frequency, 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 the frequency f = 29.9kHz calculated with the standard mesh. This shows that for the geometry of the horn with a BÂ´ezier curve transition and a fixed mounting plate, the mesh 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 necessary to refine the mesh more than once. Mounting plate support The next simulation alteration made is a change in the support conditions of the mounting 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 the mounting plate all the way to the wider part of the horn, the holder only supports the outer part of the mounting plate, and there in a squeezing fashion, applying more 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 simplified real support of the mounting plate. The simplified real support is made, to try to simulate a support that is more similar to the real situation than the optimal support used in design simulations. It would be very complicated to simulate the real support situation, since it does not apply a plane pressure to the mounting plate. Furthermore the holder for the horn has a gap where it does not support the mounting plate. This has been introduced to be able to mount the source without taking it apart. The gap has not been simulated, but could influence the horn behavior. 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 frequency changes to f = 29.5kHz. This drop in resonance frequency shows that the support is an important part of the design of the horn. Since the simplified real support is not the same as the real support it is very possible that the real support accounts for an even larger part of the difference between resonance frequency of the simulated horn and the measured horn. It has not been investigated, if this frequency change

CHAPTER 5. EXPERIMENTAL VALIDATION

(a) Mounting plate support (b) Sketch of real mounting plate supused in the design process port

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

CHAPTER 5. EXPERIMENTAL VALIDATION

would also occur in the ideal lengthed horn, or if it is a consequence of an unbalanced horn vibration. Transition curve size Since the horn is designed using on a computer, the tolerance of the dimensions is ultimately zero. This is not the case with the constructed horn. A part of the design that is extraordinary difficult to produce with the desired low tolerance, is the transition which is designed to follow a BÂ´ezier curve. Even though much effort has been put into fabricating the transition as accurate as possible, it is very likely that a great deal of error is introduced here. For this reason it has been investigated what influence the transition size has on the resonance frequency. This has been done by changing the radius of the BÂ´ezier curve transition from 4, 9mm to 4mm and simulating the new geometry. In the simulation with the smaller curve radius the resonance 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 been investigated. Some errors in the design of the horn could be due to the fact, that when fabricating the test bars for the measurement of Youngâ&#x20AC;&#x2122;s modulus the material used 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 design dimensions which gives another resonance frequency than the desired. In the error analysis several changes to the simulation geometry has been made in succession. The changes all contributed to lowering the resonance frequency of the horn, but not enough to account for the whole resonance frequency error. After the simulations, it is thought that the most likely reason for the difference in resonance frequency between the designed simulated horn and the actual constructed measured horn, is a combination construction inaccuracies, of poor mesh density in the simulations, inaccurate mounting plate support in the simulation, insufficient mounting plate support in the experiments with the constructed horn, and inaccuracy in the material properties, due to production variations in the supplied aluminum. The most important things to investigate further is thought to be the material properties, the support of the horn and the construction accuracy. These three things can be investigated by conducting more material experiments, using the procedure descried in appendix A, constructing a new holder for the horn, and by having the horn fabricated using a computer controlled turning lathe. Furthermore it would be very useful to construct a way of directly measuring the vibrations of the horn in experimental setups, rather than the sound radiation. This could perhaps be done using laser vibrometry, given a vibrometer that can measure ultrasonic frequency vibration is available.

CHAPTER 5. EXPERIMENTAL VALIDATION

5.4

Sensitivity analysis of sources

In this section the sensitivity of the sources, consisting of the Langevin transducer and each of the two horns (the designed horn and the Swiss horn), has been measured. For the test the same equipment, as in the test of the resonance frequency of the source, has been used. From the Pulse software, the signal generator was set to deliver a single frequency equivalent 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 source with the Swiss horn f = 29.9. Using this excitation the next step in setting up the test is to adjust the gain of the amplifier, to as high level as the amplifier allows without clipping. This is done by carefully adjusting the amplification while measuring the voltage over the source with the oscilloscope. In this way clipping can be detected visually. When the amplification is fixed, the voltage over the source terminals is measured using the multimeter, measuring RMS voltage. In the experiments carried out with the available amplifier the maximum voltage that could be delivered to the sources was 25VRM S . When the amplification is set, the measuring sequence can be run from the Pulse software. The measuring sequence consists of the generator playing the fixed frequency at a fixed level, and a microphone recording the signal which then is displayed in the pulse system, as an auto spectrum. This sequence is run with the microphone placed at different distanced to the horn surface. The microphone is placed on axis at distances of 10 â&#x2C6;&#x2019; 100mm. The measurement results for the two sources can be seen in figure 5.4

Figure 5.4 - Sound pressure radiated from the two horns

CHAPTER 5. EXPERIMENTAL VALIDATION

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

Figure 5.5 - Difference between the sound pressure radiated from the swiss and the produced

In the figure the blue line shows the expected difference for the two sizes of piston at the different distances. This theoretic line is calculated from equation 5.1 which is deduced from equation 4.2, and is an expression of the change in sound pressure level, when changing from one horn size to another. 25mm 2 ! · p r, 10.4mm 2 2 ∆SP L (r) |D=10.4mm→25mm = 20 log (5.1) 10.4mm 2 25mm p r, · 2 2 This calculation assumes a flat piston, but since the Swiss horn does not have a plane surface but rather a big hole in the middle of the surface it must me expected that i radiates less than the equivalent piston. By making a simple estimation based on the reduction in surface area from the equivalent piston SSwiss to the piston with a hole in the middle, S = SSwiss − SHole it must be expected that the radiated sound pressure level should be smaller by ! 25mm 2 10mm 2 − SSwiss − SHole 2 20 log = 20 log = −1.5dB (5.2) 2 25mm 2 Sswiss 2

CHAPTER 5. EXPERIMENTAL VALIDATION

where the diameter of the hole is 10mm. This estimate, of the difference between the radiation from a piston and a piston with a hole in the middle, is only a rough estimate, and the fact that the calculated correction only accounts for some of the difference between the theoretical and the measured sound pressure levels, is thought to on account of the assumption, that the horn with the hole in the middle has the same sound radiation pattern as a vibrating piston. This is also thought to be the reason 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, using different 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 surface of the horn. The first test of acoustic levitation has been done using the Langevin transducer with the Swiss horn attached. This horn has a hole with threading for attaching different tools, which was thought to be impractical for levitation experiments. To cover this hole a brass disc was fabricated and attached to the horn. Using this setup it was possible to levitate up to two polystyrene balls when adjusting the reflector carefully. This can be seen in figure 5.6 and in the video Levitation.wmv on the CD in appendix G.

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

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

CHAPTER 5. EXPERIMENTAL VALIDATION

careful adjusting of the reflector was needed. Other objects than the polystyrene balls has not been tried in this setup, but the levitation force did not feel strong enough to levitate denser materials such as water. Out of curiosity levitation of talcum powder was tried, unsuccessfully. The experiment did however reveal another interesting observation. As the powder was not levitated, it fell on the surface of the brass disc, and was immediately arranged in a circle about half way out the radius of the brass disc. This lead to the idea of investigating the vibrational pattern of the disc described in section 6.2. The vibrational pattern was found to form because the disc was not in contact with the horn 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 in chapter 4. Using this horn a very strong levitation was achieved. The levitation was strong enough to levitate polystyrene balls without adjusting the distance between the 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 between the source and reflector (up to four has been tried successfully). Furthermore levitation of water was achieved with up to two drops levitated at the same time. This can be seen in figure 5.7 and in several videos on the CD in appendix G with names starting 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, and the levitation was less stable than when levitating the polystyrene balls. This is a natural consequence of the higher density of water, than polystyrene balls.

CHAPTER 5. EXPERIMENTAL VALIDATION

Levitation experiments has also been conducted using the Swiss horn without the brass disc attached. Here it was also possible to levitate polystyrene balls, but not water. In the fine tuning of the reflector the hole in the horn surface complicated the work. The levitation experiments here described, has been conducted using the Yamaha Audio amplifier. This amplifier cannot deliver a very high voltage (approximately 25VRM S ) without clipping. If an amplifier more suitable for driving a piezoelectric transducer, such as the Langevin transducer was available, stronger levitation forces would be achievable.

CHAPTER 6. RADIATION PLATE ASSESSMENTS

Chapter 6 Radiation plate assessments In this chapter some methods for characterizing radiation plates are discussed. These methods involve both simulations of mechanical and acoustic coupled setups in Comsol, and a low tech and an optical method for visualization radiation plate displacement. For the experiments with radiation plate displacement visualization, the Langevin transducer 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 displacement pattern visualization

6.1

Node detection using talcum powder

The low tech method for visualization of displacement, is to apply talcum powder to the excited radiation plate. This method delivers a fast and easy way of getting an idea about the displacement of the radiation plate. Examples of excited radiation plates with talcum power on the surface, can be seen in figure 6.2, where both a radiation plate with a nodal-circle, and one where the radiation plate is excited in a piston-like fashion, is seen. When the radiation plate has a nodal circle, the powder will be pushed towards this nodal circle. In the case where the piston is behaving like a piston, the powder

CHAPTER 6. RADIATION PLATE ASSESSMENTS

(a)

(b)

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

will not form any distinctive pattern, but move around on the vibrating surface in a random pattern. A good thing about this approach for displacement visualization is, that it can be carried out in situ, given there is no significant forces acting on the on the powder such as levitation-like forces. If such a force is present and is strong enough to move the powder around it could happen that these forces, which normally are rotational symmetric around the radiation plate center, would move the powder to a circle shaped pattern looking like a nodal line, but in fact being a visualization of the acoustic 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 disadvantage, that it cannot be determined weather the displacement is measured, or there is an acoustic force field on the surface of the radiation plate. For this reason an optical setup has been used to visualize the radiation plate displacement. This setup has the advantage that the measurement is not contact dependent, since it is driven by light. The optical method for visualization of radiation plate displacement, is based on the theory of speckles [22, Chapter 7.5] and the interference of these. A setup has been made that can visualize the displacement pattern of radiation plates, given they have a surface with a fairly good reflection. The setup has been used with brass plates with good results, but with a radiation plate made of acrylic plastic, it was not possible to use the method, possibly because the reflection of the material is not high enough.

CHAPTER 6. RADIATION PLATE ASSESSMENTS

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

Figure 6.3 - (a) Setup for displacement visualization experiments seen from the USB camera. (b) Displacement visualization recorded with from the same camera position. The lighter the pixel the larger the displacement amplitude at the corresponding position. It can be 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 radiation disc. 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 shows the same circle of no vibration, approximately half way between the center and the rim of the plate, as the powder experiment did. Furthermore the experiment shows, that the mounting plate of the horn also moves. This movement has also been observed by placing a set of tweezers lightly on the mounting plate, which gives an audible sound, that shows that the mounting plate moving. This ability to locate vibrations visually, could be useful in the further optimization of ultrasonic horns and sources. In a comparison between the powder and the speckle based experiments, it can seem a bit elaborate to do a complicated optical setup when you can just apply some powder to the surface of the disc, but it is here an important feature of the optical setup, that it is independent of the physical forces on the plate; it only shows displacement. Experiments has been conducted, with the speckle setup, to try and visualize the 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 been carried out in Comsol Multiphysics. The simulations consist of simulations reproducing the vibration pattern of the brass radiation disc used in the visualization experiment in the previous section, and some simulations where an acoustic field simulation has been coupled to the mechanical system simulation.

CHAPTER 6. RADIATION PLATE ASSESSMENTS

6.3.1

Vibration pattern of brass disc

In this section a simulation of the vibrations of a brass plate has been preformed on the axisymmetric formulation in Comsol. The brass disc is modeled using Comsol’s draw functionality as a rectangle of size 12.5x3.6mm with one of the shorter sides assigned as symmetry axis. A section of the top part of the rectangle, representing the radiation disc screw, is assigned a fixed displacement. The material of the disc is chosen to be brass with the material properties from [10, Page 526]: E = 10.4 · 1010 P a σ = 0.37 ρ = 8500kg/m3

(6.1) (6.2) (6.3)

The first simulation run with the brass disc geometry is an eigenfrequency simulation. This reveals the two first resonance frequencies of the disc to be f1 = 19kHz and 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 the disc is excited with the frequencies f = 19kHz and f = 29.9kHz respectively, and the responses are calculated. From this simulation the results shown in figure 6.4 are achieved. It is seen from figure 6.4, that the disc simulations show a nodal line about half way between the center and the rim of the disc, at f = 29.9kHz. This is the line shown in section 6.2. The final simulation run with the disc geometry alone, is a frequency sweep from f = 10kHz to f = 35kHz. From this simulation a plot of the frequency response of the disc can be produced. This is seen in figure 6.5, showing the displacement of the center and rim of the disc, at different frequencies. From the plot is is noted, that the surface displacement of the center and rim of the disc are in phase until a little before the resonance frequency of 19kHz. After this the displacement of the center is in counter phase to the rim displacement. This shows that until this frequency, the disc behaves more like a piston, but since the displacement 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 at the two lines approaching one another for lower frequencies. For frequencies higher than the resonance, the piston will have a nodal circle somewhere. This is seen by the 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 sound field from a complex source, such as the vibrating radiation disc. This is done by coupling the mechanical simulation to an acoustic simulation also conducted in Comsol. When doing this the behavior of the mechanical system will affect the acoustic simulation as will the acoustic field affect the mechanical system.

CHAPTER 6. RADIATION PLATE ASSESSMENTS

(a)

(b)

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

CHAPTER 6. RADIATION PLATE ASSESSMENTS

Figure 6.5 - Frequency sweep of vibration in brass disc. f = 10 â&#x2C6;&#x2019; 35kHz. The lines show axial displacement of the center of the disc (green) and rim of the disc (blue)

The geometry used is the same as in the previous simulations of the disc, but more geometries must be added to represent the sound field. The sound field is in Comsol defined as a quarter circle attached to the surface of the disc. For acoustic simulations the material properties (of the air) are set to Comsols default values of Ď 0 = 1.25kg/m3 c = 343m/s

(6.4) (6.5)

When the geometry is setup and the material properties are defined and the interaction of physics is setup the coupled simulations can be run. In coupled mechanical and acoustical simulations in Comsol, eigenfrequency analysis can no longer be simulated, so the introduction of a sound field limits the further simulations to frequency response simulations. In figure 6.6 an example of a simulation of the sound pressure level radiated from the brass plate is shown. From the sound field simulation various data can be pulled out. An example of this 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 directivity of the brass disc changes when it is driven at the two frequencies, but that the directivity of the piston remains fairly uniform for these two frequencies.

CHAPTER 6. RADIATION PLATE ASSESSMENTS

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

In figure 6.7 it is seen that for the piston motion the radiated sound pressure level has a different in of ∆SP L = 8dB for all angles. This is a effect of assigning a fixed displacement amplitude for all frequencies. The reason for this is that the surface velocity of the is dependent of the frequency in the following way ∂w (6.6) ∂t Given a sinusoidal displacement of amplitude W the following is deduced v=

∂W sin (2πf t) = 2πf W cos (2πf t) (6.7) ∂t Since the sound pressure is proportional to the surface velocity multiplied by the frequency [12, Equation 2.7, page 9] v=

p∝v·f p ∝ W · f2

(6.8) (6.9)

must apply. This leads to a gain in sound pressure for going from f1 to f2 of 2 f2 ∆SP L|f1 →f2 = 20 log (6.10) f1 For the case in figure 6.7 the rise in frequency from f = 19 − 29.9kHz must lead to a rise of 2 29.9kHz ∆SP L|19kHz→29.9kHz = 20 log = 7.9dB (6.11) 19kHz

CHAPTER 6. RADIATION PLATE ASSESSMENTS

Figure 6.7 - Comparison of the radiated sound pressure level for piston type vibrations (dashed line) and the brass discs flexural vibration (full line) at different frequencies. All simulations have the same displacement 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 radiates as high a sound pressure level as it does. This effect could be exploited in the same way 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 radiation plates has been conducted. For the purpose of visualizing the displacement pattern of a source mounted radiation plate, two method has been used, which had their individual strengths. The first and simplest method was to apply some talcum powder to the surface of the excited radiation plate. This method revealed a circular nodal line on the brass plated used to cover the hole in the swiss horn. While this method is quick to use, and do not need the source to be removed from its place of use, the method has the disadvantage that it is not known weather the powder forms a given pattern on account of the vibrational pattern of the radiation plate, or because of acoustic forces in the field in front of the radiation plate. The second method for visualizing radiation plate displacement, was an optical setup using speckle interferometry to detect the vibrational pattern. This method is not dependent of physical contact to the radiation plate as long as there is an optical contact. Using this method the acoustic forces in the sound field in front of

CHAPTER 6. RADIATION PLATE ASSESSMENTS

the source does not affect the measurements. For this reason this is a more accurate way of visualizing the radiation plate displacement patterns. Using this method it is however needed that the source is removed from the setup it is used in, and placed in the optical setup. For this reason this method is more elaborate to conduct, but since 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 enables measurement of absolute displacement as well as phase information [23]. This setup could be very useful when working with radiation plates and radiation plate design, since the displacement pattern of the radiation plate is crucial to understanding the directivity of the given radiation plate. Apart from the visualization experiments, this chapter also covered some simulations of radiation plates and radiation plate radiation. In the simulations it was possible to reproduce the displacement pattern seen in the visualization experiments. Furthermore mechanical and acoustic coupled simulations proved very useful for characterizing the sound radiation from different radiation plates. Coupled mechanical and acoustical simulations in Comsol could be used as a powerful design tool when designing radiation plates for a given application, since a lot of information such as directivity and sound field characteristics can be calculated using this tool. Apart from the information drawn from the simulations in this thesis it would also be possible, directly in Comsol, to implement the calculation of the Gorkov potential and from this the acoustic levitation force on a given object in the sound field. These calculations can be done from the sound field calculation such as the one seen in figure 6.6.

CHAPTER 7. CONCLUSIONS AND FUTURE WORK

Chapter 7 Conclusions and Future Work 7.1

Conclusion

In this thesis several subjects in designing, constructing and characterizing high power ultrasonic sources has been investigated. In the theoretical investigation of different mechanical horn types, it was found that the stepped horn delivers the largest vibrational amplification, which was the ratio 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 said that the conical and exponential horn types would have similar amplification, it was shown, that the expression for the conical amplification is only valid for small ratios of 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, it was shown that by making the transition follow a curved line, the stress in the horn was decreased more than the amplification was decreased. For this reason the horn with a curved transition would be able to vibrate at a larger amplitude, without meeting its stress limit, where it would deform or break. To facilitate a convenient way of holding the horn in a practical setup, simulations has been conducted where a mounting plate is introduced in the transition of the stepped horn. The presence of a mounting plate showed, that in connection with a mounting plate, the curved transition did not decrease the amplification of the horn, but increased it at large aspect ratios, while still decreasing the maximum stress in the horn. From these results it could be interesting to see what would happen if the radius of the transition curve was increased. From simulations of the stepped horn it was shown, that the width of the horn steps has an effect on the resonance frequency of this horn step. For that reason, a method for designing a stepped horn was used, where each step length was found individually, ensuring that each step had the optimal length for the desired resonance frequency. Using this method a horn was designed to be attached to a Langevin from the Swiss company M.P. Interconsulting. In the experiments conducted with the designed horn, it was found that the horn

CHAPTER 7. CONCLUSIONS AND FUTURE WORK

had a resonance frequency of 28.72kHz rather than the design frequency of 29.9kHz used in the simulations. To evaluate this error, changes were made to the simulations model to try to get a more accurate result of the horn simulations. These changes showed that the simulation 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 one step finer than Comsolâ&#x20AC;&#x2122;s default mesh density. With more mesh refinements, the frequency did not change. In simulations of the support for the horns mounting plate, it was found that the support plays a role in the resonance frequency of the horn, even though the horn was designed to have a node at the mounting plate position. This simulation is only conducted with the measured dimensions of the constructed horn, not with a horn with design dimensions. Other reasons for the inaccurate resonance frequency of the constructed horn could be due to construction imprecision, or from deviation between the value of Youngâ&#x20AC;&#x2122;s modulus in the bar used to construct the test bars, an bar used for constructing the horn. For comparison between the designed horn and a horn provided by the Langevin transducer company (the Swiss horn), some experiments with acoustic levitation was conducted. These experiments showed that when using the horn designed in this project, a stronger levitation force was produced. In this setup it was possible to levitate water drops, using a standard audio amplifier, and an excitation signal to the Langevin transducer of 25V . If used with an amplifier more suitable for driving Langevin typed transducers the levitation force could be improved considerably. Using the Swiss horn levitation was possible but not as strong as with the horn constructed in this project. Levitation was also achieved with the Swiss horn with a brass disc attached to the front of the horn. To characterize the vibrational pattern of radiation plates several experiments has been conducted. A low tech method, that consist of applying talcum powder to the surface of the radiation plate, proved a fast and easy means of detecting nodal lines on radiation plates. For more precise measurements an optical setup, based on the theory of speckle interferometry, has been used. This setup showed a great potential, as a way of determining nodal lines on the vibrating surfaces of a radiation plate. In acoustic simulations of the sound radiation of radiation plates, it was showed that a radiation plate at resonance will radiate a far greater sound pressure than a radiation plate driven off resonance. It was also shown that the directivity of the radiation plate sound radiation changes when the radiation plate is driven at resonance.

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

CHAPTER 7. CONCLUSIONS AND FUTURE WORK

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 construct a method (optical or other), to investigate the local vibration of the horn. It was in this project tried by use of the speckle interferometry setup used in 6.2, but unsuccessful. The use of optical setups to measure mechanical and acoustical behavior could also be expanded with a setup using Schlieren Imaging [24] to visualize the standing wave field in the acoustic levitator. Using this technique it be possible to visualize the forcefield inside the levitator, which would be most useful in designing an efficient levitator. To produce a strong ultrasonic source it could be interesting, to construct a source using a resonant radiation plate, since it proved to radiate a far larger sound pressure than the radiation plate driven off resonance. For the characterization of these sources an optical setup for measuring absolute displacement amplitude would be 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 further experiments with acoustic levitation.

Appendices

APPENDIX A. MEASURING YOUNG’S MODULUS OF ALUMINUM BARS

Appendix A Measuring Young’s Modulus of Aluminum bars In this appendix an experiment for measuring Young’s Modulus is described. This method has been used to measure Young’s modulus for the aluminum alloy used for fabricating 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 components like iron and others (see data sheet in appendix D). The material has a density of ρ = 2840kg/m3 . To measure Young’s modulus for the material, a test is advised, where the resonance 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 first resonance is measured instead of the distance between resonance frequencies. Since the 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 the bar, and f is the recorded resonance frequency of the first longitudinal resonance. By knowing the resonance frequency for the first longitudinal resonance, Young’s modulus can be deduced from: s E (A.2) cL = ρ where E is Young’s modulus and ρ is the density of the material. Finally Young’s modulus can be calculated from this equation E = ρ (f 2L)2

(A.3)

This calculation assumes a pure longitudinal resonance. If this is not the case, the combined 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

APPENDIX A. MEASURING YOUNG’S MODULUS OF ALUMINUM BARS

Young’s modulus from the resonance frequency of the bar, but when the value of Young’s modulus is optimized for simulations, this fact is naturally included, since the simulations do assume any specific kind of vibrations. The main part of the experiment for measuring Young’s modulus for the aluminum bars, is the bars mounted in a fabricated holder that supports the bar on rubber bands (see figure A.1). For excitation of the bar a force hammer (Br¨ uel & Kjær type 8203) is used. For measurement of the bars response to the excitation a 1/8” microphone (Br¨ uel & Kjær type 4138) is placed close to the end of the rod. By avoiding mounting an accelerometer on the bar, it is avoided, that the weight of the accelerometer changes the resonance frequencies of the bars. By recording the frequency response both at the front and to the side of the test bar, the longitudinal resonances, which are only present at the front of the bar, can be 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 was recorded. The figure shows the spectra recorded to the side of the bars, overlaid with the spectra recorded in front of the bar end. Displayed in this way it is clear that some frequencies are not represented in the side spectrum; these are the longitudinal resonance frequencies. In figure A.4 a good argument for measuring the spectra in both the front and side position can be seen. Here the frequencies of the two resonances are very close, and it would be difficult to distinguish which was the longitudinal resonance, if the side spectrum was not recorded. Having both spectra it is clear, that the higher frequency is not represented in the spectrum recorded to the side of the bar, why it must be the longitudinal resonance frequency. The measured longitudinal resonance frequencies are shown on table A.1. Here Young’s modulus, calculated using equation A.3, is also displayed. The measurement has 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] 100.10 99.75 49.85 50.00

9.80 30.01 10.00 30.00

25781 25656 51578 50094

E [P a] 7.57 · 1010 7.44 · 1010 7.51 · 1010 7.13 · 1010

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

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

APPENDIX A. MEASURING YOUNG’S MODULUS OF ALUMINUM BARS

calculated Young’s modulus. For this reason, and to produce a simulation model that has good use in design purposes, the optimal Young’s modulus has been found using simulations of the bars, with a range of values for Young’s modulus. This is described 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 Comsol simulation tool. This is done to have a better starting point of the simulations used to design the mechanical horn. The Young’s modulus found by this method may not be the “real” value for the material, but it will be the value that gives the most accurate 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 simulated with a sweep of Young’s modulus in Comsol. For each value of Young’s modulus the resonance frequency of the four bars is calculated and stored. After the sweep is complete the optimal value will be the one that gives the least error on the resonance frequencies. Since there are four measurements, the combined error of the resonance frequencies is calculated as the norm of the individual errors, the errors being the difference between the measured and the calculated resonance frequency. The error norm is calculated as the Euclidean norm given by s X |ai (E)|2 (A.4) kA (E)k = i

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 simulations, is E = 7.54·1010 . For this value the error norm has a value of kA (7.54 · 1010 )k = 76.6Hz.

APPENDIX A. MEASURING YOUNGâ&#x20AC;&#x2122;S MODULUS OF ALUMINUM BARS

(a)

(b)

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

APPENDIX A. MEASURING YOUNGâ&#x20AC;&#x2122;S MODULUS OF ALUMINUM BARS

(a) Bar with diameter 10mm

(b) Bar with diameter 30mm

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

APPENDIX A. MEASURING YOUNGâ&#x20AC;&#x2122;S MODULUS OF ALUMINUM BARS

(a) Bar with diameter 10mm

(b) Bar with diameter 30mm

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

APPENDIX A. MEASURING YOUNGâ&#x20AC;&#x2122;S MODULUS OF ALUMINUM BARS

Figure A.4 - Closer look at the frequency response for the 100x10mm bar. The close resonances for the transversal and longitudinal vibrations can be distinguished because the response is recorded both in front of and to the side of the aluminum bar.

APPENDIX B. OPTICAL EXPERIMENT FOR VIBRATIONAL DISPLACEMENT PATTERNS

Appendix B Optical experiment for vibrational displacement patterns In this appendix an experiment is described, that is used for visualizing small displacements of vibrating surfaces such as the surface of a radiating plate. The setup is based on optical speckles and the interference of these [22, 26]. The setup is based on 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 optical equipment 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 also needed. It is important that the object under test is very stable mounted, since a small 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.1 The 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. After the mirror the light passes trough a microscope objective, which expands the light beam. The expanded beam hits the plane side of the semi matte glass plate, and is spread from the matte side of the plate. Statistical optics say that the light spread from a matte glass plate, will make speckles which is a pattern popularly described as “light cigars”. An example of speckle a pattern is shown in figure B.2.

APPENDIX B. OPTICAL EXPERIMENT FOR VIBRATIONAL DISPLACEMENT PATTERNS

Figure B.1 - Setup for visualizing the vibrational displacement

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

The speckles emitted from the semi matte glass plate are reflected partially by the plane glass plate, and from the test object as well. The interference between these two reflections is what enables one to visualize the displacement of the test object. By recording a reference pattern of the resulting speckle interference pattern, with the USB camera without excitation of the test object, and subtracting this image from the patterns recorded with excitation, it is possible to visualize the displacement of the test object. When excitation is applied the areas where displacement occurs will have an intensity difference between the reference pattern and the excited pattern. Since an USB camera can only record absolute values of intensity, the areas with displacement is shown in the difference image as an illuminated area. An example of this is shown in figure B.3 which shows the vibration pattern of a brass radiation plate of radius 25mm. The figure also shows the reference image B.3(a) where the test object can still be seen.

APPENDIX B. OPTICAL EXPERIMENT FOR VIBRATIONAL DISPLACEMENT PATTERNS

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

APPENDIX C. DATA SHEET FOR LANGEVIN TRANSDUCER

Appendix C Data sheet for Langevin transducer On the next page the data sheet for the Langevin transducer used in this project can be seen. The transducer is made by a Swiss company called M.P. Interconsulting. The specific transducer used in this project has a resonance frequency of f = 29.9kHz.

MPI-3025F-30H

Type

419

(g)

Weight

154

᧤mm᧥

Length

Maximal-

30/25

(mm)

/ceramic-

M10X35

(mm)

Bolt

Joint

(Fs±0.5kHz)

Frequency

(!)

min.

Impedance

5500

(C±10%PF)

Capacitance

Input

300

(W)

power

Al7075

Metal

mass

Front

APPENDIX D. DATA SHEET FOR ALUMINUM EN AW 2011

Appendix D Data sheet for aluminum EN AW 2011 On the next page the data sheet for the aluminum alloy used for the fabricated parts can be seen. The data sheet does not contain a value of Youngâ&#x20AC;&#x2122;s modulus, and for that reason it has been necessary to determine this value experimentally. The experiment for determining Youngâ&#x20AC;&#x2122;s modulus for the aluminum alloy, is described in appendix A

2011 - ALUMINIUM ALLOY Typical Analysis (Ave. values %)

5.5

0.7

0.4

0.3

92.0

NEAREST STANDARD

ISO

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.

MECHANICAL PROPERTIES

FABRICATION PROPERTIES

PHYSICAL PROPERTIES

Temper

Tensile strength MPa

0.2% Proof stress MPa

Elong.A5 %

Elong.A50 %

Hardness Vickers

310-365

260-290

10-15

100

320-350

270-270

10-18

310-395

230-300

8-12

10-12

115

370-420

275-315

10-13

10-12

120

Machinability

Excellent

Deep drawing

Poor

Spinning

Poor

Extruding

Good

Density

2.84 (kg/dm3)

Melting point (Liquidus)

645oC

Melting point (Solidus, Eutectic) Coefficient of thermal expansion

540oC

Thermal conductivity

177-W m-1K-1

Specific heat capacity

863 JKg-1 K-1

Electrical resistivity

38 nΩ m

Electrical conductivity

45% IACS

23-µmm-1K-1

APPENDIX E. WORK SKETCH OF DESIGNED HORN

Appendix E Work sketch of designed horn In this appendix a work sketch of the horn designed for experimental validation can be seen.

APPENDIX E. WORK SKETCH OF DESIGNED HORN

Figure E.1 - Horn designed for experimental validation

APPENDIX F. EQUIPMENT LIST

Appendix F Equipment list This appendix consists of a list of equipment used throughout this project. The used Pulse system consist of the following • Br¨ uel & 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 Br¨ uel & Kjær 1/8” microphone used is used in the following setup • Br¨ uel & Kjær Type 4138 - 1/8” Pressure-field Microphone Br¨ uel & Kjær Type 2670 - preamplifier for 1/4” microphone Br¨ uel & Kjær Type UA-0160 - 1/8” to 1/4” Adapter The Force hammer used is used in a combination of the following equipment • Br¨ uel & Kjær Type 8203 - Force hammer Br¨ uel & Kjær Type 2647 A - Charge to DeltaTron Converter Also used are: • Agilent 34405 A 5¡ Digit Multimeter • Kikusui 65M Hz Oscilloscope cos 5060 • Yamaha Natural Sound Amplifier AX-396 • Uniphase 1135/P 632.8nm Helium-Neon laser • Br¨ uel & Kjær Type 4231 - Calibrator for condenser microphone

APPENDIX G. CD CONTAINING THE THESIS AND VIDEOS OF LEVITATION EXPERIMENTS

Appendix G CD containing the thesis and videos 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 with a brass disc attached to the surface. Levitation.wmv - Fine tuning of the reflector and levitation of polystyrene balls • 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, with crude 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

BIBLIOGRAPHY

Bibliography [1] Dale Ensminger and Foster B. Stulen. Ultrasonics : data, equations and their practical uses. CRC Press, 2009. [2] J.A. Gallego-Juarez, G. Rodriguez-Corral, and L. Gaete-Garreton. An ultrasonic transducer for high power applications in gases. Ultrasonics, 16(6):267 – 271, 1978. [3] J. A. Gallego-Juárez, G. Rodr´ıguez-Corral, E. Riera-Franco de Sarabia, C. Campos-Pozuelo, F. Vázquez-Mart´ınez, and V. M. Acosta-Aparicio. A macrosonic system for industrial processing. Ultrasonics, 38(1-8):331 – 336, 2000. [4] J. A. Gallego-Juárez, G. Rodr´ıguez-Corral, E. Riera-Franco de Sarabia, F. Vázquez-Mart´ınez, C. Campos-Pozuelo, and V. M. Acosta-Aparicio. Recent developments in vibrating-plate macrosonic transducers. Ultrasonics, 40(18):889 – 893, 2002. [5] Michel Barran. Biography of paul langevin. http://scienceworld.wolfram. com/biography/Langevin.html, confirmed in December 2009. [6] D. Chacón, G. Rodr´ıguez-Corral, L. Gaete-Garretón, E. Riera-Franco de Sarabia, and J.A. Gallego-Juárez. A procedure for the efficient selection of piezoelectric 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 experimental validation. The Journal of the Acoustical Society of America, 111(6):2675–2680, 2002. [8] Christopher R. Field and Alexander Scheeline. Design and implementation of an efficient acoustically levitated drop reactor for in stillo measurements. Review of Scientific Instruments, 78(12):125102, 2007. [9] C. Campos-Pozuelo, A. Lavie, B. Dubus, G.Rodr´ıguez-Corral, and J.A. GallegoJuárez. Numerical study of air-borne acoustic field of stepped-plate high-power ultrasonic transducers. Acta Acustica united with Acustica, 84(6):1042–1046, November/December 1998. Torrough deduction of plane, torsional, transversal(called bending) and other waves.

BIBLIOGRAPHY

[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 Berlin Heidelberg, 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.nd edition, 2007. [15] T. G. Wang and C. P. Lee. Nonlinear Acoustics, chapter 6. Radiation pressure and 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 geometric parameters. Phys. Rev. E, 66(2):026605, Aug 2002. [18] W. J. Xie, C. D. Cao, Y. J. L¨ u, Z. Y. Hong, and B. Wei. Acoustic method for levitation of small living animals. Applied Physics Letters, 89(21), November 2006. [19] Louis V. King. On the acoustic radiation pressure on spheres. Proceedings of the 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 in an ideal fluid. Soviet Physics Doklady, 6:773–+, March 1962. [21] Dale Ensminger. Ultrasonics - Fundamentals, Technology, applications. Marcel Dekker, Inc. New York and Basel, 2nd edition, revised and expanded edition, 1988. [22] S.J. Frank L. Pedrotti and Leno S. Predotti. Introduction to optics. Prentice hall, 2.nd edition, 1993. [23] Ren´e Skov Hansen. A compact espi system for displacement measurements ofspecular reflecting or optical rough surfaces. Optics and Lasers in Engineering, 41:73–80, 2004. [24] N Kudo, H Ouchi, K Yamamoto, and H Sekimizu. A simple schlieren system for visualizing a sound field of pulsed ultrasound. Journal of Physics: Conference Series, 1(1), 2004. [25] Ren´e Skov Hansen. Holographic and Speckle Interferometry. PhD thesis, The Engineering College of Odense, 1997.

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