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Studies of some acoustic resonant systems Greaves, Thomas 1986

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Studies of Some A c o u s t i c Resonant Syot by Thomas Greaves B.Sc.,Queen's University at Kingston,Ontario,1982 A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of M A S T E R O F SCIENCE in The Faculty of Graduate Studies Department of Physics We accept this thesis as conforming to the required standard The University of British Columbia January 1986 © Thomas Greaves, 1986 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of P M Y S / C S The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date ftfol i m Abstract ii The resonant frequency shift, Sf, of an acoustic resonator by a small spheroidal obstacle is calculated. For a spheroid whose dimensions are small compared to the wavelength of the sound waves it is shown that Sf can be expressed in terms of the spheroid geometry and the structure of the acoustic modes in the immediate vicinity of the spheroid, whereas in the usual method of calculating Sf one needs to know the detailed structure of the normal modes of the resonator. Interest in the topic is motivated by current studies of the acoustical levitation of materials in microgravity environements. The significance of the results for the study of the acoustical levitation of liquid droplets in neutral buoyancy environments is discussed. With a view to applying these results, the mechanical properties of an electro-acoustic transducer with an impedance well matched to liquids are measured. The transducer consists of neoprene sandwiched between two conducting plates. The dis-placement of the transducer under different driving conditions has been measured usng an interferometric technique with angstrom sensitivity. An unusual feature of the interferometer is the degree of sensitivity achievable without vibration isolation mountings or temperature control of the environment. Using these measurements a qualitative model is developed to explain the unexpected aspects of the dynam-ical behavior of the transducer. The thesis concludes with-suggestions for further development of the transducer system. Table of Contents iii Abstract ii Table of Contents iii List of Figures v List of Tables vi Acknowledgements vii I I N T R O D U C T I O N 1.1 Spheroids in Acoustic Fields 1 1.2 The E M E E T Transducer 2 1.3 Outline of Thesis 3 H R E S O N A N T F R E Q U E N C Y SHIFT D U E T O A SPHEROIDAL O B S T A C L E II. 1 Introduction 4 11.2 Theory .4 11.3 Spherical Obstacle 10 11.4 Flat Disc Obstacle 10 IH E X P E R I M E N T A L VERIFICATION O F SPHEROID C A L C U L A T I O N m . l Introduction ; 12 ni.2 Experimental System 12 HI.3 Results 14 HI.4 Comparison of Theory with Experimental Results 16 HI.5 Conclusions 17 • iv IV PROPERTIES O F T H E E M E E T T R A N S D U C E R rV.l Introduction 19 rV.2 Description of the Transducer Experiments 21 V M E A S U R E M E N T O F T H E SPRING C O N S T A N T O F N E O P R E N E V.l Introduction 22 V.2 Experimental System 22 V.3 Results 25 VI S U R F A C E S T R U C T U R E O F N E O P R E N E 28 Vn D Y N A M I C M E A S U R E M E N T O F T H E PROPERTIES O F N E O P R E N E Vn.l Introduction 31 VII.2 The Electric Stress 33 VII.3 The Interferometric Technique 33 VII.4 Results 37 VII.5 Conclusions 39 V7.I.6 Mechanical Impedance of the E M E E T Transducer 40 Vin E N E R G Y DENSITIES IN DIELECTRIC MATERIALS Vm.l Introduction 41 VHI.2 Parallel Plate Capacitor 41 VHL3 Parallel Plate Capacitor with a Dielectric 42 VHI.4 Parallel Plate Capacitor with a Dielectric and an Air Gap 42 VHI.S The E M E E T Transducer 44 VTJI.6 Comparison of the Model with the Experimental Results 47 VHI.7 Breakdown Effects in the E M E E T Transducer 48 IX DESIGN CONSIDERATIONS FOR T H E E M E E T T R A N S D U C E R IX. 1 Suggestions for Further Study into the Basic Physics of the E M E E T Trans-ducer 51 IX.2 Suggestions for Constructing a Practical Device 52 X SUMMARY OF THESIS RESULTS 54 APPENDIX I - EVALUATION OF / 3 56 APPENDIX II - O B L A T E SPHEROIDAL COORDINATES 59 References 61 vi List of Figures Figure Page [II.l] Oblate Spheroid 7 [in.l] Experimental System - Disc in an Acoustic Resonator 13 [III.2] Amplitude versus Frequency for 9 = 40° 15 [in.3] Resonant Frequency as a Function of 0 16 [IV. 1] Linear Second Order Mechanical System 19 [IV.2] Schematic Representation of Transducer System 21 [V.l] Transducer Apparatus 24 [V.2] Spring Constant Data 26 [VI. 1] Electron Micrograph of Neoprene 40x magnification 28 [VI.2] Electron Micrograph of Neoprene 400x magnification 29 [VI.3] Electron Micrograph of Neoprene 4000x magnification 29 [VTJ.l] Detail of Apparatus used in Dynamic Measurements 32 [VTI.2] Simulated Light Intensity Signal 36 [VTI.3] Frequency Response of E M E E T Transducer 38 [VIII.1J Three Parallel Plate Capacitors 42 [VIII.2] Simple Model for the E M E E T Transducer 46 [Vin.3] Apparatus to Study Breakdown Effects 46 [1.1] Section of a Spheroid in an Acoustic Resonator 57 vif List of Tables [lj Spring Constant of Neoprene 27 [2] Mechanical Properties of Neoprene 39 [3] Comparison of Measured and Predicted Properties of Neoprene 47 Acknowledgements viii I would like to thank my supervisor Dr. Frank Curzon for his support and di-rection of my work. I also appreciate Lorne Whitehead's (TIR Systems, Vancouver) guidance and insights into the behavior of the E M E E T transducer. I would like to thank Peter Haas for his help in designing the E M E E T transducer apparatus and Jack Bosma for his friendly and expert tutoring in the student shop. Credit is owing to Doug Plant for the acoustic resonator apparatus. Al Cheuck provided electronics support. I would like to thank Ernie Williams for his help in the glass blowing shop and all of my collegues in the Plasma Group for their friendly helpful suggestions. I am grateful to Helen Salter for her computing assistance. Judith Nairn did all the diagrams. Rob Clarke and Colin Bradley are to be credited with the initial work on the transducer. Finally I wish to thank Schlumberger Middle East S.A. ( Dubai, United Arab Emirates ) for granting and extending my leave of absence to work on this project. C H A P T E R I CHAPTER I: 1 I N T R O D U C T I O N 1-1 Spheroids in Acoustic Fields There has long been interest in the perturbing effects of spheroids in flow fields. Konig (1891)1 calculated the forces exerted on a spheroid in a laminar flow field and also considered the limiting case of a disc in such a field. Leung et al. (1982)4 measured the resonant frequency shift of a thin disc with its surface normal to the pressure gradient and showed that the shift varied sinusoidally along the axis of the resonator. These investigators did not show how the resonant frequency shift depends on the orientation of the disc with respect to the local pressure gradient. Recent interest in the topic is motivated by interest in the acoustic levitation of materials in low gravity environments.2 - 8 Trinh (1985)7 reported an ultrasonic levitation device to study surface waves on freely suspended liquids, supercooling of materials, the temperature and contamination dependence of surface tension and the optical diffraction properties of substances both on earth and in space-borne laboratories. One method of studying such systems on earth is to suspend a liquid drop in another liquid. Liquid drops suspended in liquids of comparable density, in a uniform gravitational field assume a spheroidal shape which motivates the spheroid perturbation calculation. To study the stability of such systems in acoustic fields requires an acoustic transducer with a good impedance match to liquids. This CHAPTER I: 2 motivates study of the E M E E T transducer. The resonance frequency shift of a resonator containing a rigid sphere has been calculated by Leung et al. (1982)4 using a Green's function method. This method requires complete knowledge of the mode structure of the unperturbed resonator. It is noted in their paper that sufficient detuning of the resonator may result in reduction of the levitation force. Curzon and Plant (1986)8 calculated the frequency shift of a resonator containing a spherical obstacle. Their result is expressed in terms of the local acoustical field rather than in the normal mode parameters of the resonator. The theory presented in Chapter II is a calculation of the resonant frequency shift of an acoustic resonator due to a perturbing spheroidal obstacle. The result is expressed in terms of the local acoustic field. 1-2 The E M E E T Transducer The acronym ' E M E E T ' stands for enhanced elastomer electrostatic transducer and was coined by Lome Whitehead. In 1981 Whitehead and Clarke observed anomalous sound intensities emanating from a capacitor constructed of two alu-minum plates sandwiching a thin sheet of dental dam. A large A C potential was applied across the plates. The sound intensities were observed to be diminished only slightly when the device was submerged in water indicating that the trans-ducer impedance was well matched to that of water. The second half of this thesis is an investigation of the properties of the E M E E T transducer. CHAPTER I: 3 1-3 Thesis Outline In Chapter II the calculation of the resonant frequency shift of an acoustic res-onator due to a spheroidal obstacle is given. The limiting cases of a flat disc and a sphere are also considered. In Chapter DI a simple experiment is described in which the resonant frequency shift of an acoustic resonator due to a flat disc is mea-sured. The suitability of the disc as a diagnostic probe is discussed. The E M E E T transducer is introduced in Chapter IV. Chapter V is a discussion of the interfer-ometric measurement of the quasi static spring constant of neoprone which is the dielectric in the E M E E T transducer. Electron micrographs of the neoprene surface are presented in Chapter VI to explain the non linearity of the response discussed in Chapter V. Chapter VII is an account of the interferometric measurement of the frequency response of the transducer done to extend understanding of the quasi-static measurement described in Chapter V . The dependence of the amplitude of the transducer displacement on the frequency of driving electric field and the static bias pressure is measured. Chapter VIII is a review of the forces experienced by linear dielectric materials in electric fields. The review is presented to explain the forces measured in Chapter V U . Chapter IX summarizes the properties of the E M E E T transducer for future design considerations.Suggestions for further study are made. Chapter X is a summary of the thesis results. C H A P T E R H CHAPTER H: 4 Resonant Frequency Shift Due to a Spheroidal Obstacle H - l Introduction The theory presented below addresses the problem of calculating the resonance frequency shift of a resonator containing a spheroidal obstacle. The resonance frequency shift is expressed in terms of the local acoustic field. The limiting cases in which spheroids tend to be discs or spheres are also considered. H-2 Theory The initial assumption is that the pressure P in the resonator satisfies the wave equations V 2 P = ± ^ (1) c2 at2 K } i.e. in the case of the unperturbed resonator for a mode of frequency, w, V 2 P = - w 2 P / c 2 . (2) C H A P T E R H: 5 In the case of the perturbed resonator, for a mode of frequency u+6u, the perturbed pressure Pi satisfies V'Pl = -(u + &jfPl/c3 (3) where c is the speed of sound in air. At a rigid boundary, the displacement of the medium along the normal n to the boundary is zero. The corresponding boundary conditions are that n • V P = 0 in the case of the unperturbed resonator and n • V P i = 0 in the case of the perturbed resonator, i.e. Neumann boundary conditions. The assumption of a rigid boundary is equivalent to the notion that the acoustic impedance of the boundary greatly exceeds the acoustic impedance of the gas. Note that in the case of the perturbed resonator, the boundary of the obstacle is assumed to be rigid. Green's second identity is the following where V is any volume enclosed by a surface S. f ( $ V 2 t f - *V 2*) dV = / ( $ V t f ) « < i S - / (¥V*).<fS (4) Jv Js Js Assuming < 1 and applying Green's second identity to equations (1) and (2), then one obtains the result that J P i V P • dA' - j P V P i • dA! ~ J P P X dV' (5). In equation (5), dV' is a volume element of the volume V of the perturbed resonator and dA' is a surface element S' of the rigid boundaries (including the boundary of the perturbing obstacle) of the perturbed resonator. Upon application of the boundary conditions, equation (5) reduces to - j Pi V P »dS ~ j PPi dV' (6). C H A P T E R II: 6 In equation (6), dS is a surface element of the perturbing obstacle surface 5,in the perturbed resonator. See Fig. 1.1 To evaluate the left hand side of equation (6), the perturbed pressure Pi must be determined near the surface of the perturbing obstacle. The excess pressure P at the surface of the spheroid to first order is given by P = P(0) + V ( 0 ) P « r where P(0) is the excess pressure at the centre of the spheroid and V(0)P is the pressure gradient at the centre of the spheroid and the origin is chosen to be the spheroid center. A pressure gradient in the x z plane and at an angle 9 to the z axis may be written V P = ||VP||(sin0x + cos0z) where || VP|[ is the magnitude of the pressure gradient and x and 5 are unit vectors in their respective directions. In Fig. II. 1, V P is parallel to the Z' axis. It is not necessary to specify a y component for V P because of the rotational symmetry of the spheroid about the z axis. Expressed in oblate spheroidal coordinates9 V P • r = -ta|| VP|| [cos 9P°(ri)P?{ie) + sin 9 cos jP* [ri)P{ (tf)] (7). A discussion of oblate spheroidal coordinates is found in appendix II. P£{n) and are Legendre polynomials of the first and second kind respectively. It is assumed that the spheroidal obstacle is defined by the coordinate surface f = ft = constant. The boundary condition that the pressure gradient be tangential to the spheroid surface implies that terms of the sort AP?(v)Q0dk)+ BPl1(r,)Q\(ic) cos <f> C H A P T E R II: 7 F i g . I I . l Oblate Spheroid must be added to P to get Pi. A and B are complex coefficients adjusted so that (^) = 0. It must be noted that this approximation is only valid provided that the wavelength of the pressure disturbance is much greater than any dimension of the spheroid, ie. terms of order uPfift/c3 have been dropped where b is the interfocal separation of the spheroid. This condition determines the validity of the first order approximation. Thus and A = ta||VP||cot^ l °'(t f l , ) /f l? '( . - jb) (8) and B = ta l lVPIIs int f^ 'CiftJ / f l j 'Ct - fe ) (9)-The primed Legendre polynomials are the ( partial derivatives of the polynomials CHAPTER II: 8 evaluated at £ = f0- Thus the left hand side of equation (6) becomes Ix +12, where I^-jpVP.dS (10) and I7 = -Jp*VP*dS (11) where /* = M°0l )fl ! (*) +BP!(T,)Q\(ic)coa<f, (12). The value of / x is independent of the orientation of the obstacle and gives the 'volume' contribution to the resonance frequency shift. This is seen by converting the surface integral in equation (10) to a volume integral with Gauss's theorem and exploiting the vector identity V • ( P V P ) = ( V P ) 3 + P V 3 P . It follows that equation (10) can be written as h = -J(VP)3dr + J ^ ( V 3 P ) 3 d r . Since the integrands are scalar quantities and vary slowly over r, the obstacle vol-ume, then / 1 = r ( - ( V P ) 3 + ( £ ( V . V P ) ) 3 ) which is independent of the obstacle orientation but proportional to the obstacle volume r. The volume of a spheroid f = fo is r = 8ira3fo(l + f 3 )/3 (13). See appendix II. Evaluation of J 3 is straightforward. See Appendix I. I2 = ( - r /2 ) ( (VP) 3 ) ( rcos 3 0 + Asin 30). CHAPTER TL 9 With the above expressions for I\ and Jj, the resonance frequency shift can be evaluated from equation (6) to yield 2u6u c 3 f PPidV = r [ ( - (V • V P ) ) 3 - (VP) 3 (1 + (l/2)(rcos 3 9 + Asin 3 $))} J w where A and T are defined below _ arctan(l/fo) - (1/ft) arctan(l/fo)-?o/(tf + l) = - f t arctan(l/ft) + + 1) 2-(<r03/(l + fo 3))-foarctan(l/f 0)' The expression can be further simplified by noting that J PPX dV' ~ J P*dV which is consistent with the previously mentioned assumptions. With this approximation and some rearrangement we get V = {TfJw\& * V P ) 3 " ( V P ) 3 " ((V2))(VP)3(rcos3 0 + Asin 3 9)] (14). The resonance frequency shift is a consequence of two effects, a volume effect from Ii and a deflection effect from / j . The terms in equation (14) involving V and A are the deflection terms, the perturbing spheroid deflects the flow according to its orientation in the resonator. It is instructive to consider limiting cases of equation (14). One case of interest is that of a spherical obstacle in an axisymmetric resonator. Another is that of a flat disc obstacle. CHAPTER IL 10 LI-3 Spherical Obstacle In the limit as ? goes to infinity, the spheroid becomes a sphere i.e. lim r = lim A = 1. f — » 0 O f—»oo Assuming the pressure disturbance is of the form P = Pcos(kx) then equation(H) simplifies to where Vr is the resonator volume, k is the wave number,and z is the axis of symmetry which is the result found by Leung et a/.(1982)4 using the Green's function method. The result also agrees with the result obtained by Curzon and Plant (1985)8. HI-4 Flat Disc Obstacle In the limit as f goes to zero, the spheroid shrinks to a disc. In this case, equation (10) is trivially zero, the 'volume' term makes no contribution for a flat disc. The sole contribution to the resonance frequency shift comes from equation (11) tu/u = (r/Vr) ((5/4)cos(*x) - (1/4)) (15) (11). For f = 0, equation (12) simplifies to Pt = - ( 2 a / » ) | | V P | | c M ^ 0 ( i | ) f l ? ( t f ) and CHAPTER II: 11 thus 73 = -(8/3)a3(VP)3cos30. Finally, Su 4 c3 a 3 i~ , * •Z—zZTrrrf"*"*' (16)' Upon first inspection of equation (14) it would seem that as r tends to zero then both the volume and deflection contributions to the resonant frequency shift vanish. However, as r tends to zero it goes in direct proportion to f which can be seen from equation (13). Furthermore, as ? tends to zero, then T diverges as —1/f, thus the product r r remains finite, preserving the deflection contribution to the resonance frequency shift. All other terms including the A term vanish. These results were verified experimentally by the methods described in the next chapter. C H A P T E R HI 12 C H A P T E R m Experimental Verification of Spheroid Calculation LTI-1 Introduction A simple experiment was conducted to test some of the predictions of the theory presented in the previous chapter. The resonance frequency shift caused by the introduction of a flat disc into an axially symmetric resonator was measured. The experiment was designed to test the suitability of the disc as a diagnostic probe to determine the local pressure gradient in an acoustic resonator. III-2 Experimental System The resonator is a plywood box ( 0.438 m x 0.438* m x 1.18 m ) arranged with the longest dimension vertical. The plywood walls of the box are 19.1 mm thick. An 8ft, 10 Watt , 150 mm diameter speaker is suspended in the bottom of the box by strings attached to the walls. A 8 mm diameter, electrostatic microphone is mounted in styrofoam in the inside top lid of the box. See Fig. U L l . CHAPTER HI 13 0.438m —i F i g . LTI.l Experimental System R = resonator; Pr = probe; S = loudspeaker; M = microphone; C = counter;F = frequency generator, Fc = external frequency control; O = dual trace oscillocope A 150 mm diameter, 1.5 mm thick aluminum disc is used to perturb the resonant frequency. The disc is mounted on a rigid rod which passes through a hole in the side of the resonator. The orientation of the disc is determined by a protractor mounted on the resonator around the hole and a pointer fixed to the support rod. The speaker is driven at frequencies ranging from 420 Hz to 470 Hz. The resonance frequency is m 434 Hz. At this frequency the long axis of the resonator is spanned 3/2 wavelengths of the pressure distribution. The perturbing obstacle is located near a displacement anti-node. The frequency generator is a Hewlett Packard 3312A. The frequency is controlled externally using 6 volt cell. The cell insures the introduction CHAPTER m 14 of very little noise and the control is simple to operate. The resonance condition is determined by feeding the signal generator and microphone outputs into a dual trace oscilloscope and adjusting the signal frequency to maximize the microphone output. The frequency was determined by measuring the period of the signal with a Hewlett Packard 5314A Universal Counter averaging over 10 periods of the signal. This method produced measurements of the resonant frequency consistent to 1 part in 5000. The resonant frequency was measured as a function of the orientation angle of the disc in the resonator, ie. the angle between the plane of the disc and the long axis of the resonator. 111-3 Results Fig. IH.2 is a plot of the the microphone output versus the frequency with the disc angle 9 = 40". The solid curve is ( ( ( 2 * / ) » - ( 2 * / o ) ) » + 7 a (2*/)a)* where A = Forcing amplitude (arbitrary units), /o = the resonant frequency (Hz), and 7 = the damping coefficient (Hz) From these fits were extracted the resonance frequencies as a function of the disc angle 9. For the data shown in Fig. HI.2 the following values were extracted using a x1 minimization fit; A = 6.3 ± 0 . 1 x 10* , f0 = 432.5 ± 0 . 1 Hz , and 7 = 6 . 6 6 ± 0 . 0 8 H z . The uncertainties correspond to excursions of one standard deviation in x2-CHAPTER IE 15 60 » i i i I i . i i I i . i i , I I I , ! i , , ,1 400 410 420 430 440 4S0 4S0 470 Frequency [Hz] F i g . III . 2 Amplitude versus frequency for disc angle 9 = 40° Fig. III.3 is a plot of the resonant frequency as a function of disc angle. The solid line in Fig. III.3 is the best fit to a function of the form / = / o [ l - S c o s 2 ( 0 + /?)] (17) where / is the resonant frequency, /o is the unperturbed resonance frequency, 9 is the disc orientation angle and /? is an offset angle. The offset angle is included to account for a constant offset between the protractor reading and the actual disc orientation with respect to the resonator axis. Equation (17) may be rearranged to show that the fractional shift in the resonant frequency Sf/fia given by 5/ / / = £ c o s 2 ( 0 + /J). The x2 minimization fit yielded a value for ^  = —10.8 ± 0.8° and a value of B = 0.0054 ± 0 . 0 0 0 1 . and a value of f0 = 4 3 4 . 1 6 ± 0 . 0 4 Hz. The uncertainties correspond to excursions of one standard deviation in the x 2 fit. CHAPTER HI 16 -100 -so o so Disc Angle [ 9 deg. ] Fig. in.3 Resonant frequency as a function of orientation angle of perturbing disc. III-4 Comparison of Theory with Experimental Results Equation (16) of chapter II predicts the fractional shift in the resonant frequency to be given by 6u 4 c 3 a 3 ,«_—.» i „ , , where c is the speed of sound in air, w is the resonant frequency (rads/sec), a is the radius of the perturbing disc, P is the excess pressure in the resonator, dV is a volume element of the resonator, and 9 is angle between the pressure gradient and the normal to the plane of the disc. If one assumes the spatial dependence of the longitudinal pressure disturbance to be given by P = PQ cos(Jbx) then the wave number k = 3ir/L corresponds to a mode of frequency 434 Hz, where L is the length of the resonator. CHAPTER m 17 Substitution of the above values in equation (16) yields a value of — = 0.0050 cos2 9 u whereas the experiment yields Sui „ — = 0.0054 cos3 9. The discrepancy with the experimentally measured value is approximately seven per cent which is within the experimental uncertainty. TJI-5 Conclusions The simple experiment vindicates the theory of the resonant frequency shift caused by the introduction of a perturbing disc in an acoustic resonator. The disc probe serves as a diagnostic probe for the measurement of the relative magnitude and direction of the pressure gradient within the resonator. The fre-quency shift, Sf, is completely specified without reference to detailed knowledge of the mode structure of the resonator. This statement comes with the caveats that the resonant frequency shift must be small enough to disallow coupling to other modes and that the scale length of the perturbing obstacle be small compared to the wavelength of the pressure disturbance. The recent interest in acoustic levitation of materials in microgravity environ-ments motivates the study of the effects of perturbing obstacles on the resonant modes of acoustic resonators. It is difficult to simulate microgravity environments on earth, however one method would be to study neutral buoyancy systems. Such a system might consist of a liquid drop suspended in another liquid of comparable CHAPTER ID 18 density. A density gradient could be established in the host liquid. The vertical position of the suspended drop would be determined by the equilibrium matching of the buoyancy, gravitational and acoustic forces. To study the stability of such systems in acoustical fields requires a sonic trans-ducer with a good impedance match to liquids. A good candidate for such a trans-ducer is the E M E E T transducer. The specific acoustic impedance of the E M E E T transducer is similar to that of water. A discussion of the properties of the E M E E T transducer follows. C H A P T E R I V CHAPTER IV: 19 Properties of the E M E E T Transducer rV-1 Intoduction: Physical M o d e l of the Transducer The model used to describe the E M E E T transducer is that of a linear, second order, mechanical system. See Fig. IV. 1. F cos at F i g . IV.1 A Second Order Linear System m is the mass, r is the damping coefficient of the dashpot, k is the spring constant, and F is the applied force. CHAPTER IV: The differential equation which desribes such systems is simply 20 <Px dx m—f + r— + kz = F cos ut (18) at* at for which the solution is given by; _ F s i n ( w t - a ) w[(,* +(m«-*/w) 9)*] ( 1 Q ) _ Fam(ut-a) K ' where mu — Ar/w tana = ; — r Z m is the mechanical impedance of the system.1 0 The velocity is maximized for the angular frequency u = u0 = y/k/m. The Q of the system may be shown to be Q — muo/r. For Q >^ 1 , Q = Ar/A0 where Ar is the displacement at resonance and Ao is the displacement in the limit as the frequency of the applied force F goes to zero. In this sense Q is a measure of the amplification of the system. At low frequencies the system is said to be 'spring loaded' and the kx term in equation (18) dominates. The displacement is independent of the mass and damping coefficient at low frequencies. At high frequencies the displacement is said to be mass loaded and the displacement is inversely proportional to the square of the applied frequency and independent of the damping coefficient and spring constant. Knowledge of F,k,r, and m specifies the system response at a given frequency. Fig. IV.2 is a schematic representation of the experimental system. Neoprene (1 mm thick) is sandwiched between two flat conducting plates. The bottom plate is fixed. The upper plate has mass m and can move. The mass of the upper plate is much greater than that of the neoprene. The applied force F may be mechanical or electrical or a combination of both. CHAPTER IV: 21 neoprene /7T77 F i g . IV.2 Schematic Representation of Experiment rV-2 Description of Transducer Experiments Chapter V is a discussion of the quasi static measurement of the spring constant Jfc for which the applied force is purely mechanical. A uniform, known pressure is applied to the upper plate. The displacement of the plate is measured as the pressure is slowly released. Chapter VII is a discussion of dynamic measurements of kt r, and F. In this case the applied force is electrical and time varying. The displacement of the mass is measured as a function of the frequency of the applied force. The swept frequency range spans the displacement resonance frequency. CHAPTER V: 22 C H A P T E R V M E A S U R E M E N T OP T H E S P R I N G C O N S T A N T OP N E O P R E N E V - l Introduction Neoprene and other synthetic rubbers are non-linear materials, that is the dis-placement of these materials is not directly proportional to the applied stress. An account of the measurement of the quasi static spring response of neoprene follows. By quasi static it is to be understood that the displacement always is in equilib-rium with the applied stress. This equivalent to neglecting the damping and inertial terms in equation (18). The nonlinearities are expected to be particularly apparent in the quasi static measurement as the displacements are large. It is relatively easy to apply large mechanical stresses and measure the corresponding displacements. It is less easy to apply large electrical stresses. V-2 Experimental System The apparatus used to measure the spring constant is shown in Fig. V . l . Neo-prene was compressed between two optical flats each coated with a 1 micron thick-ness of aluminum. One of the optical flats was fixed to a massive (25 kg) steel base. The second optical flat was fixed to a light (0.250 kg ) aluminum piston. The piston CHAPTER V: 23 was sealed inside a nylon cylinder with two ' O ' rings. Ah interferometric technique was used to measure the displacement of the neoprene as a function of applied pres-sure. Compressed air was used to apply the force to the neoprene through the rigid piston. The use of a fluid assured the uniformity of the applied force. To the rear face of the piston was attached a front facing mirror which served as the mirror for one arm of a Michelson interferometer. The second arm of the interferometer included a pressure compensation cylinder to equalize the pressure in both paths of the interferometer. The pressurized path lengths were equalized to 1.0 mm or 1.2% of the pressurized length. It was necessary to pressure compensate both interferometer paths because the refractive index of air, n, is pressure dependent. The pressure dependence of the refractive index of air is given in the following equation13 where n§TP is the refractive index of air at 15°C, a is the thermal expansion co-efficient of the gas, P is the pressure of the gas in mm of mercury,and n 0 is the refractive index of air at 0 ° C and 760 mm of mercury. At atmospheric pressure and 15°C, [TISTP — 1) — 0.3 x 10~3. It is easily shown that a pressure change of 0.50 MPa (~ 5 atmospheres) in a pathlength of 0.20 meters results in an apparent pathlength change of 3 x 10~4 meters or 475 wavelengths of a HeNe laser. This apparent pathlength change is much larger than the expected displacement of neo-prene subject to a 0.50 MPa pressure change. A pressure compensation cell was installed in the interferometer for this reason. The pressure vessel was pressurized to ~ 800 kPa and then opened to atmo-spheric pressure on a time scale of minutes. The displacement was determined by counting the interferometer fringes recorded on a strip chart recorder. The pressure vessel itself has inherent elasticity which was determined by re-DIGITAL STORAGE OSCILLOSCOPE BUFFER PHOTOMULTIPLIER SPECTRUM ANALYSER BUFFER POWER SUPPLY H t N * LASER PRESSURE COMPENSATION CELL ABEAM SPLITTER 3 FRONT FACING MIRROR PHOTOMULTIPLIER POWER SUPPLY PISTON AND CYLINOER ASSEMBLY 0 o VALVE REGULATOR AIR CHAPTER V: 25 moving the neoprene from its position between the optical flats and repeating the measurement. This measurement is called the system response. V - 3 Results Fig. V.2 is a summary of the data. The system response seems to be linear for pressures greater than 60 kPa. For pressures less than 60 kPa both the system response and the neoprene response are not very repeatable. This behavior may be attributed to to the small 'springiness' of the ' O ' ring seals on the piston. A minimum pressure is required to insure the optical flats are in direct contact with the neoprene. For pressures less than 60 kPa the interferometer become misaligned indicating a wobbling of the piston on its ' O ' ring seals. Each fringe corresponds to a pathlength change of A in the moving arm of the interferometer, where A is the wavelength of the HeNe laser (632.8 nm) used as the light source in the Interferometer. This corresponds to a piston motion of A/2. The fringe counting technique does not yield displacement information directly. For pressures greater than 60 kPa the system response and the neoprene response were repeatable. The displacement corresponding to pressures of 60 kPa was chosen to be the origin. For each of the runs shown in Fig. V.2 a displacement offset has been added to give zero displacement at 60 kPa. The apparent neoprene response curve in Fig. V.2 has to be corrected for the system elasticity to obtain the true neoprene response. The system elasticity couples with the neoprene elasticity as, _L - -L + i -where ka is the apparent spring constant, ks is the neoprene spring constant,and 30 25 Q> 20 c <5 15 E O O Q- 10 D i s p l a c e m e n t v e r s u s A p p l i e d P r e s s u r e — r NEOPRENE RESPONSE SYSTEM RESPONSE 200 400 600 Applied Pressure [kPa] 800 Legend Run #1 X Run #2 • RUN 3 • RUN #4 B RUN * 5 *•••>• J b mm mm* ii • n u n J B • o run £ 0 _ « cn « O o o co O p H M CHAPTER V: 27 k, is the system spring constant, l/k, is the slope of the system response curve in Fig. V.2 and is seen to be independent of the applied pressure for pressures greater than 60 kPa. l/ka is the slope of the neoprene response curve in Fig. V.2. Neoprene is seen to be highly non-linear as ka depends strongly on the applied pressure. It was suspected that the non linearity of the neoprene response is a surface effect, one possibility being a fibrous surface. For this reason electron micrographs of neoprene surfaces were made. A discussion of this work is the subject of the following chapter. It should be noted that hysteresis effects, if any, are not readily observable as only the neoprene response to decreasing pressure was measured. Table I is a summary of the spring constant data extracted from Fig. V.2. TABLE 1 t B I A S PRESSURE NEOPRENE SPRING CONSTANT ( k P a ) (xlO3 N/m) 129 2.4 245 5.2 313 6.3 Table 1 Spring Constant of Neoprene As the applied pressure is the independent variable not the applied force, to obtain the spring constant (corrected for system elasticity) from Fig. V.2 then one must multiply kff by the area of the neoprene sample (1.78 x 10 - 3 m 3) to obtain the spring constant. This is the spring constant which is tabulated in Table 1. CHAPTER VI 28 C H A P T E R V I S U R F A C E S T R U C T U R E O F N E O P P R E N E To gain physical insight into the properties of neoprene, particularly its spring constant, electron micrographs of several samples of neoprene of nominal thickness 1.0 mm were made. F i g . VI .1 Neoprene 40X magnification Figures V. 1, V.2 and V.3 show the microscopic structure of neoprene at 40X,400X,| and 4000X magnification. The surface is rough on a scale of 1O~0 meters. There is a seemingly random distribution of 'blocks' of neoprene piled upon one CHAPTER VI 29 F i g . VI.2 Neoprene 400X magnification another. These blocks provide a surface that is pocked with cavities of micrometer dimensions. Fig . VI .3 Neoprene 4000X magnification CHAPTER VI 30 The thickness of the nominally 1.0 mm thick neoprene was measured using a digital micrometer with a precision of 0.001 mm. A sample of 10.0 x 1 0 _ 3 m 2 area was measured. The mean thickness was measured to be 0.819 mm with a standard deviation of 0.005 mm. The measurement location was selected randomly over the sample area. The measurement is consistent with the notion of surface irregularities with a height scale of tens of micrometers. It is suggested that the low pressure, small displacement springiness of neoprene is due to the compression of the surface irregularities. For compression beyond a few micrometers, the spring constant of neoprene increases rapidly. The increased springiness of the compressed neoprene at higher pressures is due to its bulk com-pressibility. Knowledge of the spring constant of neoprene determines its behavior at low frequencies. The high frequency behavior is determined by the damping coefficient and mass of the system. To determine the mechanical impedance of neoprene the piston in Fig. V . l was subjected to an electrical stress, the frequency of which was easily controlled and measured. This is the topic of the following chapter. CHAPTER VII: 31 C H A P T E R V H Dynamic Measurement of the Properties of the E M E E T transducer VII-1 Introduction A resonance technique was used to measure the spring constant, damping co-efficient, and force on the transducer. The apparatus is shown in Fig. VTJ.l and is similar to the apparatus described in the previous chapter. For the dynamic measurement, the applied force has two components. The first component is due to a mechanically applied bias pressure. The second component of the force is electrical in origin. The two conducting plates depicted in Fig. IV.2 are subjected to sinusoidal voltage of known frequency and magnitude. The displacement of the piston (Fig. VU.1) which serves as the upper plate in Fig. IV.2 is measured as a function of the frequency of the applied voltage using an interferometric technique. The interferometric technique is discussed in detail in the following section. The measurements are made for several bias pressures. The properties of neoprene are found to be strongly dependent on the bias pressure. CHAPTER VH: F i g . V I I . l Detail of Apparatus Used in Dynamic Measurements 32 FUNCTION GENERATOR AUDIO TRANSFORMER PISTON FRONT FACING MIRROR TEFLON SLEEVE | OPTICAL FLAT CHAPTER VH: 33 Vn-2 The Electric Stress There are theoretical reasons to suppose that the electric force is proportional to the square of the applied voltage. See chapter VIII. Equation (18) can be written <Px dx , - v . , 9 i, .» m — + r — + kx = CV2 cos2(wi*) at* at where C is a yet undetermined coefficient. The right hand side of the equation can be written as — (l + co.(2w«)). For this reason one expects the the displacement to be harmonic in time but at twice the frequency of the applied stress. VLT- 3 The Interferometric Technique A Michelson type interferometer was used to measure the displacement of the piston shown in Fig. VTJ.l. See Born and Wolf 1 2 for a complete description of the theory of operation of a Michelson interferometer. The light intensity at the photomultiplier (PMT) in Fig. V . l is given by ,=w(f?) where A is the pathlength difference for the two arms of the interferometer and A is the wavelength (632.8 nm) of the light source (HeNe laser).' For the experimental system A = 2xp p cos 2u>t + x 0 where u is the frequency of the applied voltage, xpp is the peak to peak displacement of the piston and XQ is the steady state pathlength difference between the two arms CHAPTER VII: 34 of the interferometer. The factor of 2 in the above expression accounts for the fact that changes in the light path are equal to twice the displacement of the mirror on the piston. Furthermore the light source must have a coherence length greater than xo for the analysis to be valid. Typical coherence lengths for a HeNe laser are the order of meters which is greater than any of the interferometer dimensions, so this condition is easily satisfied. Thus the photomultiplier light intensity is given by where <p = 2?rio/A is the steady state phase difference. The behavior of this function depends strongly on the relative magnitudes of xpp and A. There are two terms in the cosine argument of equation (20). The first term is time dependent, ie. the (4jr/A)xpp cos(2u/<) term fluctuates with angular frequency 2w. The second term <p is nominally static. In the case that xpp > A/2, the light intensity / attains its maximum amplitude IQ with angular frequency of 2u. The frequency with which / attains its maximum is independent of <p. In the experiment xpp <£. A and the <p term assumes more importance. For <p = riff where n is an integer then to first order Thus to first order the light intensity -has no time dependence for <p = rnr. For <p = njr/2 then (20) (21) (22) * T + (-ir(V2)(4»r/A)*ppCos(2o ,0 CHAPTER VH- 35 to first order. In this case, the light intensity has two components, one static and one fluctuating.The fluctuating term / / is given by Equation (23) can rearranged to give (where the amplitude of / / is implied) X p p ~ 2 * / 0 Fig. VLT.2 is a computer generated simulation that shows how the light intensity signal / depends on the phase angle <p. The solid curve in Fig. VTJ.2 is the function where <p(t) = IT sin (0.005:). In this case x p p / A = 1/40 , t ranges from 0 to 250 (arbitrary units), thus <p(t) assumes values from 0 to jr. The extreme left hand side of Fig. VII.2 corresponds to n = 0 in equation (21). The middle of Fig. VLT.2 cor-responds to n = 1/2 in equation (22). It is obvious that the fluctuating component of the signal achieves its maximium there. The right side of Fig. VII.2 corresponds to n = 1 and the fluctuating component of the signal assumes its minimum value. To determine xpp,Ij and IQ must be determined. To this end, <p was modu-lated from 0 to 2ff at a frequency small compared to the displacement frequency 2w. Experimentally the <p modulation was achieved in a crude but effective way. Blowing on the 'fixed' arm of the interferometer changed the temperature of the arm sufficiently for the path length to change by several wavelengths of the HcNe source. The thermal expansion and contraction was slow (~ 0.2 Hz) compared to the displacement frequency of the piston (kHz). The amplitude Io was determined by measuring the peak to peak displacement of the signal / . This was accomplished using the peak storage mode of the digital storage scope. The peak values of / correspond to the phase <p assuming the values / / = (V2)(4ir/A)xppCos(2w<) (23). CHAPTER VII: 35 Phase Angle [ <p rads ] F i g . VII . 2 Simulated Light Intensity Signal of integer multiples of ir, ie. I swinging from 0 to Jo as in equation (21) and as shown in Fig. VTI.2. The envelope of the extrema of the light intensity signal J 0 was stored. The width of the envelope is proportional to IQ and stays constant for at least 10 minutes. If was determined by making peak to peak measurements of the component of the signal / at the angular frequency 2w ie. / / . This was accomplished using the spectrum analyser which measures the amplitude spectrum. The peak values of / / occur for <p assuming values of half integer multiples of ic as in equation (22). The spectrum analyser was also operated in a peak detection mode with a sampling rate ~ 2 Hz which is high compared to the modulation frequency of <p. The driving frequency of the transducer was swept slowly (10 kHz in 5 minutes) using the Hewlett Packard function generator and 70 and / / were determined over this range. CHAPTER VII: 37 IQ was typically a volt and If typically a millivolt. The sensitivty of the interfer-ometer is revealed here as the spectrum analyser used to measure / / has microvolt precision. The independent measurements of the fluctuating and static components of the light intensity signal allows displacement measurements of A/1000. Mechan-ical noise in the interferometer and room in which the experiment was conducted were the limiting features. The output of the photomultiplier was buffered with a unity gain amplifier to avoid loading of the photomultiplier by the oscilloscope and spectrum analyser. This is necessary because the photomultiplier is a 'current' device and the spectrum analyser and scope are 'voltage ' devices. VII-4 Results The displacement of the piston was measured over a range of frequencies from 1.5 kHz to 12.0 kHz. The displacement information was obtained from the light intensity by the above method. Measurements were conducted for three bias pressures. The data are summa-rized in Fig. Vn.3. The solid lines are the best fits to the amplitude function of a damped forced oscillator. From these fits are extracted the spring constants, damping coefficients and ap-plied forces. All displacements shown are peak to peak displacements consequently the forces are the peak to peak forces. These results are tabulated in Table 2. D i s p l a c e m e n t v e r s u s F r e q u e n c y Legend A 129 k P a x 245 k P a a 313 k P a 129 k P a f i t  245 k P a f i t 313 k P a f i t Frequency [Hz] C H A P T E R VII : 39 TABLE 2 BIAS PRESSURE | SPRING CONSTANT j DAMPING COEFFICIENT (kPa) (x.108 N/m) (kg/S) PEAK TO PEAK force (N) 129 5.32 ± 0.07 1550 ± 80 0.098 ± 004 245 6.02 ± 0.09 1470 ± 175 0.088 ± 0.008 313 7.46 ± .09 820 ± 150 0.041 ± 0.005 Table 2 Mechanical Properties of Neoprene VII-5 Conclusions The spring constants obtained by the dynamic measurement agree with those obtained by the low frequency method (described in chapter V) to within a factor of two. A discussion of possible reasons for the discrepancy is reserved for the conclusion of the following chapter. The following chapter is an account of the forces experienced by dielectrics in electric fields. Discussion of the forces tabulated in Table 2. is also reserved for the conclusion of chapter VIII. The strong bias pressure dependence of the spring constants, damping coeffi-cients and forces is to be noted. It is instructive to compare the measured values of the spring constants with published values. A survey of the literature yielded no values for neoprene however one source 1 3 quotes values of 23 x l O 8 Pa and 0.05 x l O 8 Pa for the Young's Moduli of hard and soft rubber respectively. Assuming a one dimensional approximation is CHAPTER VH: 40 valid then the spring constant k is related to YM, the Young's Modulus by YM = k(L/A) where L is the length of the sample and A is the cross sectional area. Applying the experimental values then YM is found to range from 2.5 x 10s to 3.4 x 108 Pa. The measured values of the spring constants seem reasonable in this light. VII.6 Mechanical Impedance of the E M E E T Transducer The mechanical impedance of the E M E E T transducer is given by equation (19) Zm = (r* + [mu>-k/u,)*yf* The Q of the system is small so for frequencies below resonance the response is very flat, as shown in Fig. VTI.3. In this case Zm k/v. For a bias pressure of 129 kPa and a frequency of 5000 Hz then Zm ~ 10 x 10a kg /sm 2 The specific acoustic impedance of water is ~ 1.5 x 10a kg/sm 2 . For lower bias pressures the impedance match of the E M E E T transducer to liquids can be expected to be even better as the spring constant will be smaller. CHAPTER VIE: 41 C H A P T E R V I H E N E R G Y D E N S I T I E S I N D I E L E C T R I C M A T E R I A L S V m - 1 Introduction To explain the experimental results presented in chapter VU, a review of the theory of the forces experienced by linear dielectric materials subjected to electric fields follows. Sections VTfl-2 through VTH-4 deal with ideal parallel plate capacitors while section VTII-5 deals with the specific case of the E M E E T transducer. Three parallel plate capacitors are shown in Fig. V m . l . The electrodes are separated by a distance r and have a surface area S. The dielectric for capacitor I is air. The dielectric for capacitor II has a relative permittivity er. The dielectric in capacitor III is the same as for capacitor II with the addition of a thin film of air of thickness t. Vm-2 Parallel Plate Capacitor For capacitor I the electric field Ei is given simply by Et = V/r where V is the applied voltage. Fringing effects have been neglected. The electric CHAPTER VIB: 42 F i g . V1TJ.1 Three Parallel Plate Capacitors displacement Di is given by Di = e0Et where eo is the permittivity of free space. A consequence of Gauss' law is that the free surface charge density cr/, on the electrodes equals the the electric displacement Di, thus oi = e0V/r. The energy density Wj in the gap is given by 1 1 V - 3 VIU-3 Parallel Plate Capacitor with a Dielectric For capacitor LT the electric field En is the same as in case I, viz. En = V/r. CHAPTER VIE: 43 The electric displacement is given by Dn - €reoEn = erDi in the gap and it vanishes in the electrode . The energy density in the dielectric is given by Wu = \EUDU = ^r<ro^ (24). Introduction of the dielectric increases the energy density of the capacitor by a factor of the relative permittivity of the dielectric, provided the potential across the electrodes is unchanged. Vm-4 Parallel Plate Capacitor with a Dielectric and an A i r G a p Assuming the free charge density at the thin air film dielectric interface is zero then the normal component of the electric displacement is conserved, i.e., COEAIR = €r€0ED where EAIR is the electric field in the air film and EQ is the electric field in the dielectric. The potential across the electrodes is obtained by integrating the electric field with the result that V = tEAIR + {r-t)ED = (t(l-(l/er)) + (r/€r))EAIR . E V  A I R < ( l - ( l / € r ) ) - ( r / £ r ) trV/r. Thus DAIR = trtoV/r. CHAPTER Vffl : 44 Similarly the dielectric field is found to be ED=V/r and the electric displacement is found to be DD = eR€0V/r. The energy density in the air gap is given by The energy density in the dielectric ii (25). Note that the energy density in the air gap is larger than the energy density in the dielectric by a factor of c r. VLT-5 The E M E E T Transducer In Chapter VTJ it was shown that the peak to peak forces experienced by the moving electrode (piston) ranged from 0.04 N to 0.1 N depending on the bias pres-sure. These forces correspond to peak to peak pressures ranging from 22.5 Pa to If the transducer is modelled as a parallel plate capacitor filled with a dielectric (case LT) then the electostatic pressure is given by equation (24). 56.2 Pa. (24) C H A P T E R VIII: 45 In the experiment V = 500cos(wt), r = 0.82mm, er ~ 7 thus the peak to peak pres-sure Wu ~ 11.5 Pa. The value chosen for er is the value quoted in the literature 1 4 for a 1 kHz measurement. If the transducer is modelled as a parallel plate capacitor filled with a dielectric plus a small air gap (case IH) then the electrostatic pressure in the air gap has a peak to peak value of The E M E E T transducer seems to be an intermediate case between models LT and HI. This is perhaps not unreasonable. At high bias pressures the air gaps between the electrode and the neoprene ( as seen in the electron micrographs ) are diminished in size and the system tends towards model H . This suggests a model for the E M E E T transducer where the dielectric is con-sidered to be constructed of independent blocks of neoprene. Futhermore the shear stresses between the blocks are neglected. See Fig. VT1I.2. The area of the blocks in contact with the electrodes is Ait the area of the air gaps is Ao, and the total surface area is A = A\ + A 2 . Application of a quasistatic force F results in an electrode displacement Ar and where fcj; is the spring constant of the »™ block, Au is the apparent spring constant of the system, and ftp is the mean spring constant of the blocks. The displacement Ax is small compared to r , the electrode separation. For an electrically applied force Fe then WAIR = CrWtI ~ 77Pa. (25) F / A x = ]Tfc r <Ai ~ ^ A r r A , = kx A\ = k^A (26) (27) CHAPTER VTJL 46 USTOM F I X E D CLECTROOC F i g . Vni.2 Model of E M E E T Transducer which is shown in sections 3 and 4 of this chapter. Substituting equation (26) in equation (27) then w = h%{&1-<')+€') (28) where W is the electrostatic pressure at the neoprene electrode interface. In the case of kT = kA ie. no air gaps, then 1 V 7 thus equation (28) can be written as kA kT (29) Thus the spring constants and electrostatic pressures are expected to scale according to equation (29) with this model. CHAPTER VIII: 47 Vm.6 Comparison of the M o d e l W i t h Experimental Results In equation (2d) the measured quantities are kA and W. For a bias pressure of 313 kPa then kA = 7.5 x 108 N/m and W = 21.9 kPa. For er = 9.5 then Pe = 15.6 Pa and &r = 7.8 x 10s N/m. These results can be checked against the measurements made at bias pressures of 245 kPa and 129 kPa. The comparison is summarized in Table 3. TABLE 3 BIAS PRESSURE MEASURED SPRING CONSTANT PREDICTED SPRING CONSTANT (kP2) (xlO 8 N/m) xlO 8 N/m 129 6.02 5.88 245 5.32 5.54 Table 3 Comparison of Measured and Predicted Properties of Neoprene Equation (29) is sensitive to the value of c r. In the literature er is found to be 7 at 1 kHz. Another source 1 5 suggests that er is frequency dependent from 100 Hz to 1kHz and ranges from 10 to 6 over this frequency range. For simplicity the model assumes er constant. Equation (28) is a quasi-static approximation which justifies the value of 9.5 used for er. The assumption has been made that &r is independent of the bias pressure. The model assumes the non-linearity of the neoprene spring response to be entirely a surface effect. The agreement of the data with the fit to a forced oscillator function C H A P T E R VIII: 48 indicates that the frequency dependence of K? is small The small discrepancy between the values predicted from the model and the measured values tabulated in Table 3, suggests that the bias pressure dependence of is also small. VIII.7 Breakdown Effects in the E M E E T Transducer In the model assumed in the previous section the electric field in the air gap is given by E^IR = eRV/r. For the experimental conditions then < E A J R >RM,~ 4.1 x 10eV/m. Naively one expects air to breakdown at this field strength. If breakdown occurs then one expects luminous flashes. A simple experiment was devised to make a preliminary investigation of this aspect of the transducer. The apparatus is shown below in Fig. VIII.3. F i g . VIII.3 A p p a r a t u s to S tudy Breakdown Effects CHAPTER Vffl: 49 Neoprene was sandwiched between two glass plates each coated with a thin (500 A) deposit of aluminum to make an electrode. The coating was thin to enable light transmission through the electrode. The plates were clamped loosely around the perimeter. Observations When fields comparable to those applied in the E M E E T transducer (Chapter IV) were applied to the neoprene, breakdown was observed in the form of light pulses. The photomultiplier output consisted of pulses coincident with the applied field attaining its peak value. The pulses seemed to appear when the applied field attained a threshhold value and were generated for as long as the applied field was above the threshhold. At higher fields the pulses could be observed with the naked eye in a darkened room. They seemed to be distributed uniformly over the electrode surface. The pulses were short in duration (~ 10~6 seconds). The pulse shape suggested that the breakdown is like electrodeless breakdown10. No breakdown of the dielectric was observed for these field strengths. The effect was not diminished when thin Mylar sheets sandwiched the neoprene. According to Paschen's scaling law for electrical breakdown of air in gaps be-tween planar electrodes17 there is a minimum breakdown voltage for a given pres-sure and gap length. Reduction of the gap length for a fixed pressure results in an increase in the breakdown voltage. For air at atmospheric pressure the minimum breakdown voltage ~ 275 volts for a gap length of 70 micrometers. For gaps smaller than 70 micrometers the expected breakdown voltage is higher than 275 volts. The breakdown voltage increases rapidly as the gap is reduced. A possible explanation of the observed breakdown effects in the E M E E T trans-ducer is that the air gaps in the neoprene electrode surface are breaking down. The electron micrographs presented in Chapter VI indicate there is a distribution of CHAPTER VUJ: 50 gap lengths over the neoprene surface. The observed threshold effect is perhaps explained by the largest gaps breaking down first. As the field strength is increased then more and more gaps achieve the breakdown threshold. The uniformity of the spatial distribution of the light pulses observed with the naked eye is consistent with spatial distribution of the gaps seen in the electron micrographs. CHAPTER IX: 51 C H A P T E R LX D E S I G N C O N S I D E R A T I O N S F O R T H E E M E E T T R A N S D U C E R L X - 1 Suggestions for Further Study into the Basic Physics of the Trans-ducer Further research on the behavior of the E M E E T transducer should be directed along the following lines. The behavior of the device with low bias pressures has yet to be quantified. The apparatus described in Chapters V and VII is unsuitable for measurements with a bias pressure of less than 60 kPa. This is due to the resistance offered by the piston seals. One possibility would be to replace the piston assembly with aneroid bellows. Larger displacements (~ micrometers) can be expected of the device for smaller bias pressures. The apparatus shown in Fig. VHI.3 was used to make an interferometric measurement of the transducer displacement. A front facing mirror was glued to one of the electrodes to serve as one mirror in a Michelson type interferometer. Displacements of order of a wavelength of the HeNe laser were observed for an applied peak to peak voltage of 1000 V . In this simple experiment the bias pressure is not well defined as the device is clamped' mechanically around its periphery, however one can be certain that the bias pressure is less than 60 kPa. A second area of obvious interest is the study of breakdown phenomena in the transducer. The air pockets formed by the roughness of the neoprene surface seems capable of supporting high energy densities on account of their small dimensions. CHAPTER IX: 52 Breakdown of these gaps is staved off in accordance with Paschen's law. The break-down seems to be like electrodeless breakdown. The threshold of breakdown and its frequency dependence remains to be quantified. LX-2 Suggestions for Constructing a Practical Device The first question to be asked is why one would want to construct the E M E E T transducer. There is no apparent shortage of acoustic transducers. Acoustic trans-ducers are typically electromagnetic (moving coil) devices or piezoelectric or mag-netorestrictive devices. Electromagnetic devices offer large displacements (of order of mm) but can provide only small pressures. Futhermore inductive devices are large and power consumptive. Piezoelectric devices offer large pressures but only small displacements (angstroms). The E M E E T transducer is an intermediate de-vice offering relatively large pressures and moderate displacements. Furthermore the E M E E T ' s mechanical impedance is well matched to liquids. To optimize the performance of the E M E E T transducer it is desirable to reduce the spring constant as much as possible. One way might be to deposit an elastomer material in the form of small beads or drops on a flat plate. The surface density of the drops could be kept small thus reducing the effective spring constant of the material. It would also be desirable to keep the elastomer bead size less than 50 microm-eters to keep the air gaps a comparable dimension. Small air gaps are capable of supporting high fields before breakdown of the air. Even higher breakdown fields could be achieved by replacing the air in the gaps with sulfur hexafluoride gas. The usual problem with electrostatic devices is arcing across the electrodes. CHAPTER IX: 53 Larger displacements of the transducer can be achieved by sandwiching layers of the transducer material together. C H A P T E R X: 54 C H A P T E R X S U M M A R Y O F T H E S I S R E S U L T S The achievements reported in this thesis can be grouped into two areas; the work on the resonant frequency shifts caused by the introduction of a spheroidal obstacle in an acoustic resonator, and the work on the properties of the E M E E T transducer. The two areas are bridged by the possibiliites of applying the E M E E T transducer to study the acoustic levitation of spheroidal liquid drops in other liquids. The resonant frequency shift caused by the introduction of a spheroidal obstacle in an acoustic resonator is calculated for the case that the wavelength of the pressure disturbance is much longer than any of the spheroid dimensions. The result is expressed in terms of the spheroid geometry and structure of the acoustic modes in the immediate vicinity of the spheroid. The usual calculation requires detailed knowledge of the normal .modes of the resonator. The limiting cases of a spheroid viz. a flat disc and a sphere have also been considered. In the case of the spheroid, the calculation agrees with the results obtained by Curzon and Plant (1986) 8 and Leung et al (1982) 4. A simple experiment was conducted to test the disc calculation. The discrepancy between the measured and predicted resonance frequency shift was less than 8%. The result suggests that a disc obstacle can serve as diagnostic probe to measure the relative magnitude and direction of pressure gradients in acoustic resonator. CHAPTER X: 55 The mechanical properties of the E M E E T transducer have been measured using a novel interferometric technique. The technique is novel because small displace-ments of order of an angstrom are measured without taking elaborate measures to control temperature or to achieve vibration isolation. The electronics used in the experiment are sophisticated; a digital storage scope and an audio frequency spectrum analyser. A simple model is developed to explain the dynamical properties of the trans-ducer. The transducer is interesting itself for three main reasons. Firstly, the mechanical impedance of the device is well matched to liquids which invites a host of applications. Secondly, the transducer seems to support much higher energy densities than initially expected. The small air gaps at the rough neoprene surface allow high fields without arcing, the usual bane of capacitative acoustic devices. Thirdly, the device can be constructed at low cost which is always a consideration in any practical device. References 56 [1] W. K6nig,Ann.Phys.,43, 43 (1891) [2] M . Barmatz,Proc. Symposium on Materials Processing in the Reduced Gravity Environment of Space, Boston.MA, Nov. 1981,Elsevier,NY,25 (1982) [3] A . Young, M . Lee, I. Feng, D. Elleman and T . G . Wang, Proc. Symposium on Materials Processing in the Reduced Gravity Environment of Space, Boston, MA.Nov. 1981, Elsevier, NY, 67 (1982) [4] E . Leung, C P . Lee, N.Jacobi and T . G . Wang, J . Acoust. Soc. Am.,72, 615 (1982) [5] M . El-Raheb and P. Wagner,J. Acoust. Soc. Am. , 72, 1046 (1982) [6] M . Barmatz, J .L. Allen and M . Gaspar, J . Acoust. Soc. Am. , 73, 725 (1983) [7] E . H . Trinh, Rev. Sci. Instrumen., 56, 2059 (1985) [8] F .L . Curzon and D. Plant, A m . J . Phys., (in press) (1986) [9] P . M . Morse and H . Feshbach,Methods of Theoretical Physics, Part II, McGraw-Hill, New York, 1953, pp. 1292-1294 [10] H.J . Pain,Tie Physics of Vibrations and Waves, Wiley and Sons Ltd., pp. 52-55 (1976) [11] American Institute of Physics Handbook, 3rd ed., McGraw-Hill, pp.6-11 (1982) [12] M . Born and E. Wolf,Principles of Optics, 4th ed., Pergamon Press, pp.256-259 (1970) [13] L . E . Kinsler, A.R. Frey, A . B . Copens and J .V. Sanders, Fundamentals of Acous-tics, 3rd ed., Wiley and Sons, p. 461 (1982) 57 [14] Handbook of Chemistry and Physics, 52nd ed., The Chemical Rubber Co., (1972) [15] D . A . Parfeniuk, An Investigation of the Effect of Conductive Shells on Elec-trodeless Breakdown, M.A.Sc. Thesis (1982) [16] D . Friedmann, F .L . Curzon, and J. Young, Appl. Phys. Lett., 38, 414 (1981) [17] L . B . Loeb, Fundamental Processes of Electrical Discharges in Gases, Wiley and Sons, p.411 (1939) APPENDIX I: A P P E N D I X I - Evaluat ion of I2 1-2 is given by the expression I2 = - J P* V P • dS Recall equations (12),(8),(9),and (7), P f = AP?{t,)Qi(*) + BPll(n)Q\ ft) cos * A = ia\\VP\\coaOPP\ify)/Ql'{ify) B = ia\\VP\\sm$Pll\ito)/Q\'{i!0) P = P(0) + V P • r = P(0) - ia\\VP\\{coB9P?{r,)P?{is) + sinf/VfaW (if)cos In oblate spheroidal coordinates , the pressure gradient is given by „ „ i (dP\ A i /ap\ A I /<9P\ f where / i f and A,, and h$ are the oblate spheroidal scale factors. The differential surface vector is given by dS = (hnhfidridfys Thus VP.dS = +f^+(^) drjd* « f V. Of / Co = -ia 2 ||VP||(f 2 + l) x (coS9P?(r})P?'(i$0) + s i n ^ c o s^^^f/)^ 1'^)) <M0. A P P E N D I X I: 59 F i g . 1.1 Section of Spheroid in an Acoustic Resonator with the substitution of the scale factors and the substitution of the <; partial differential of equation (7). h is the sum of four integrals, ha> • • • i hd-ha = f'dt, (%a3||VP||2(ft + l ) 2 <K ) / 2 i W) J-l Jo -2* hb= f drj f ^ a 3 | | V P | | 2 ( f t + i f P f ' i i ^ / t f ' & o ) J-i Jo x cos6sin 6cos^P1°(f?)P11(r7)i2?(:Yo)/,i1'(*ro) he = j\j2Jd4>a*\\VP\\2{h + l ) 2 ^ ' ( » V o ) / f i i ' ( t V o ) x cos6sin6cos$P?{r})P?{Ti)Q\{\eQ)P?\is) I2d= I drjf ^ I I V P I I ^ o + l)2 '^^ )/^ !'^ ) J-i Jo x sin 3 6 cos2 4>Pi (r,)P} (r,)Q\ (tft)Pi'(<ft) APPENDIX I: CO / 2 b and J 2 c vanish in the integration over The above expressions are simplified by making use of the following identities. Pf'too) = i Pil{iSo) = iy/$ + i Pil\i*>) = »'&/\[$f+~l fii(»'fo) = f 0 arctan(l/f 0 ) - 1 = arctan(l/<r0) - W(tf + 1) fli(Ko) = ( W f r o + 1)) ~ \Ao+1 arctan(l/ft>)] fii'fao) = (l/\Ao + l ) (2 - (f0a/(fo + 1)) " fcarctan(l/fo)). With the above substitutions h = - ( T / 2 ) ( ( V P ) 3 ) ( r c o s 2 0 + Asin 3 0) where and r arctan(l/fo) - (1/fo) arctan(l/f0) - fo/(foa + 1) ^ _ - f t arctan(l/ft) + fo9/(^ + 1) 2 - (Co /(I + fo2)) ~ fo arctan(l/ft)' A P P E N D I X II: 61 A P P E N D I X II - O B L A T E SPHEROIDAL COORDINATES A detailed discussion of oblate spheroidal coordinates can be found in Morse and Feshbach. T h e notation used in this article deviates very little from their notation. Oblate spheroidal coordinates are useful because they are one of the eleven coordinate systems for which Laplace's equation separates. T h e y are of particular interest as spheroids tend to discs or spheres in limiting cases. Oblate spheroidal coordinates are obtained from confocal elliptic coordinates by rotation about their minor axes, i.e. an oblate spheroid is the volume carved out by rotation of an ellipse about its minor axis. See F i g . II.1. T h e relation between oblate spheroidal coordinates and cartesian coordinates is given by the following relations: z = a$T); x = a v ^ + l X l - r ^ c o s f , y = av% 2 + 1)(1 ->7 2)sin <f>. where f goes from 0 to oo, rj goes from —1 to +1, and <f> goes from o to 2ir. T h e surface <; = 0 is a disc of radius o, in the x — y plane centred at the origin. T h e surface f = ft = positive constant is a flattened spheroid of thickness through the axis of 2fto with an equatorial radiu3 of + T h e surface q = +1 is the positive z axis. T h e surface r\ = —1 is the negative z axis. T h e surfaces TJ = constant are hyperboloids of one sheet asymptotic to the cone of angle c o s - 1 r\ with respect to the z axis which is the axis of the cone. See F i g . II.1. T h e allowed solutions for the <f> equation for periodic boundary conditions are cosmtp and s i n m $ where m is a positive integer. T h e allowed solutions for the TJ A P P E N D I X D: 62 equation are, P%(ij), Legendre polynomials o f the first kind. T h e allowed solutions for the £ equation are, P£(if) and fia(ij) Legendre poly-nomials of the first and second kind. Note the imaginary arguments. T h e scale factors are given by the following relations: h« = a)JT=^> H = avV + iXi-'j2). T h e volume r,of a spheroid f = ft =constant is . + 1 /.2ir /•Co r + l />2* d{ dr) dih^hj, = 87ra 3ft(l + ft2)/3. Jo J-i Jo 

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