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Two-port electroacoustic model of a piezoelectric composite circular plate

University of Florida Institutional Repository

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TWO PORT ELECTROACOUSTIC MODEL OF A PIEZOELECTRIC COMPOSITE CIRCULAR PLATE By SURYANARAYANA A.N. PRASAD A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGR EE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2002

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Copyright 2002 by Suryanarayana A.N. Prasad

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To my parents A.C. Nagaraja Prasad & A.N. Kamala

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iv ACKNOWLEDGMENTS First of all, I thank Dr. Sheplak for involving me in this research work. I also thank him for providing financial support for my graduate study at the Department of Aerospace Engineering, Mechanics and Engineering Science, University of Florida. I gratefully acknowledge support from NASA Langley Research Center (grant numbers NAG 1 2261 and NAG 1 2249 monitored by M.G. Jones and S. Gorton, respectively). I express my gratitude to my advisors Dr. Sheplak and Dr. Sankar for the guidance they provided during the complete course of my thesis work. I thank Dr. Cattafesta for providing useful tips, which led to the successful completion of the thesis. I thank Dr. Nishida and Dr. Sheplak for providing me insight into lumped element modeling and two port network modeling. I am grateful to Dr. Sankar for his lectures on plate theory, which provided the basis for this thesis. I extend my gratitude to Distinguished Professor Dr. Haftka for the thought provoking discussions about optimization studies on the piezoelectric unimorph disk transducer. I thank Mr. Stephen Horowitz and Mr. Quentin Ga llas for helping me on the experimental verification section. I appreciate Mr. Stephen Horowitz also for the useful discussions on optimization and two port modeling. I thank Mr. Sridhar Gururaj for helping me in submitting the thesis.

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v I thank all my profe ssors at the Indian Institute of Technology Madras and all my teachers at Shrine Vailankanni Senior Secondary School for providing the pedestal for graduate study. Above all, I gratefully acknowledge the effort of my parents in shaping my thinking with t heir love, affection and guidance. Last but not the least, I thank all my friends for their emotional support.

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vi TABLE OF CONTENTS Page ACKNOWLEDGMENT S ................................ ................................ ................................ .. iv LIST OF TABLES ................................ ................................ ................................ .............. ix LIST OF FIGURES ................................ ................................ ................................ ............. x ABSTRACT ................................ ................................ ................................ ...................... xv CHAPTERS 1 INTRODUCTION ................................ ................................ ................................ .......... 1 Static Displacement Behavior o f the Piezoelectric Unimorph Transducer ................. 2 Background ................................ ................................ ................................ .................. 3 Thesis Layout ................................ ................................ ................................ ............... 5 2 TWO PORT NETWORK MODELING ................................ ................................ ........ 7 Lumped Element Modeling ................................ ................................ ......................... 7 Two Port Network Modeling An Introduction ................................ ......................... 7 Two Port Model of Piezoelectric Transduction ................................ .......................... 9 Two Port Model of a 1 D Piezoelectric ................................ ................................ ....... 9 Two Port Electroa coustic Model of a Piezoelectric Composite Plate ....................... 11 Equivalent Circuit Representation and Parameter Extraction ................................ ... 12 3 MECHANICAL BEHAVIOR OF THE PIEZOELECTRIC COMPOSITE PLATE ... 18 Problem Formulation ................................ ................................ ................................ 18 Assumptions ................................ ................................ ................................ .............. 18 Equilibrium Equations ................................ ................................ ............................... 22 Strain Displacement Relationships ................................ ................................ ........... 23 Constitutive Equations ................................ ................................ ............................... 24 Governing Differential Equations ................................ ................................ .............. 27 General Solution ................................ ................................ ................................ ........ 28 The Problem of Piezoelectric Unimorph Disk Transducer ................................ ........ 28 Central Composite Plate ................................ ................................ ........................ 28 Outer Annular Plate ................................ ................................ ............................... 30

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vii Interface Compatibility Conditions ................................ ................................ ....... 32 Solution Techniques ................................ ................................ ................................ .. 32 Simple Analytical Solution ................................ ................................ .................... 33 N umerical Method to Obtain Constants ................................ ................................ 33 4 MODEL VERIFICATION ................................ ................................ ........................... 34 Theoretical Verification ................................ ................................ ............................. 34 Finite Element Model ................................ ................................ ................................ 36 Pressure Loading ................................ ................................ ................................ ... 36 Voltage Loading ................................ ................................ ................................ .... 38 Experimental Verification ................................ ................................ ......................... 40 Causes for Deviation of the Experimental Results from Theory ............................... 43 5 NON DIMENSIONALIZATION ................................ ................................ ................ 45 Buckingham p Theorem ................................ ................................ ............................ 45 Non Dimensional Deflection for Pressure Loading ................................ .................. 48 Non Dimensionalization for Voltage Loading ................................ .......................... 54 Non Dimensionaliza tion of Lumped Element and Two Port Network Parameters .. 60 6 CONCLUSIONS AND FUTURE WORK ................................ ................................ ... 87 APPENDICES A DETAILED DERIVATION OF THE GE NERAL SOLUTION FROM PLATE CONSTITUTIVE EQUATIONS ................................ ................................ ................. 89 B ANALYTICAL SOLUTION ................................ ................................ ....................... 94 Analytical Expression for Radial Defection 0 u and Slope q ................................ .... 94 Analytical Expression for Vertical Deflection 0 w .................................................... 99 Analytical Expression for Short Circuit Acoustic Compliance AS C ....................... 100 Analyti cal Expression for Effective Acoustic Piezoelectric Coefficient A d ........... 103 C MATLAB CODES ................................ ................................ ................................ ..... 105 Subroutines used by Program 1 ................................ ................................ ............... 105 Program 1: Progra m used to derive Response of a particular Piezoelectric Tranducer ................................ ................................ ................................ ................. 113 Subroutines used only by Program 2 ................................ ................................ ....... 115 Program 2: Program used to derive Response of a particular Piezoelectric Transducer ................................ ................................ ................................ ............... 116 Program 3: Program implementing Direct Solution of a particular Piezoelectric Transducer ................................ ................................ ................................ ............... 122

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viii D FINITE ELEMENT MODEL (ABAQUS) INPUT FILE ................................ .......... 125 Pressure Loading only (Norm alized Piezoelectric Patch Radius = 0.2) .................. 125 Voltage Loading only (Normalized Piezoelectric Patch Radius = 0.55) ................. 136 LIST OF REFERENCES ................................ ................................ ................................ 149 BIOGRAPHICAL SKETCH ................................ ................................ ........................... 151

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ix LIST OF TABLES Table Page 2.1: Conjugate power variables and corresponding dissipative and energy storage elements in various domains. ................................ ................................ ............................ 8 4.1: Properti es of the piezoelectric unimorph disk used in the finite element model ..................... 37 4.2: Properties of the piezoelectric bender APC 850 ................................ ................................ 41 4.3: Lumped element and two port parameters ................................ ................................ ........ 42

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x LIST OF FIGURES Figure Page 1.1: Cross sectional schematic of a clamped axisymmetric piezoelectric unimorph disk transducer ................................ ................................ ................................ ....................... 2 2.1: Schematic of a piezoelectric plate that can be approximated to be 1 D. ............................. 9 2.2: Equivalent two port circuit representation of the piezoelectric unimorph at low frequencies. ................................ ................................ ................................ .................... 12 2.3: Equivalent circuit representation in the electric domain of the piezoelectric unimorph. ................................ ................................ ................................ ....................... 13 2.4: Equivalent circuit representation in the electric domain of the piezoelectric unimorph when decoupled from the acoustic domain. ................................ ....................... 13 2.5: Another equivalent t wo port circuit representation of the piezoelectric unimorph at low frequencies. ................................ ................................ ................................ .............. 14 2.6: Equivalent circuit representation in the acoustic domain of the piezoelectric unimorph. ................................ ................................ ................................ ....................... 15 2.7: Equivalent circuit representation in the acoustic domain of the piezoelectric unimorph when electric behavior is decoupled. ................................ ................................ 15 2.8: Equivalent two port circuit representation of axisymmetric piezoelectric unimorph disk at frequencies com parable to that of the primary resonance ................................ ....... 16 3.1: An axisymmetric multi layered transversely isotropic circular plate with pressure load, radial load and a moment. All loads shown are considered positive. ......................... 19 3.2: Cross sectional view of the plate shown in Figure 3.1, showing the sign conventions and labels used. ................................ ................................ ................................ .............. 19 3.3: Top view of the plate shown in Figure 3.1 ................................ ................................ ......... 20 3.4: Enlarged isometric vi ew of the element shown in Figure 3.3 and Figure 3.2 with generalized forces acting on it. ................................ ................................ ......................... 20

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xi 3.5: An element of the multilayered transversely isotropic composite plate of length dr placed at a distanc e r from the center acted on by generalized forces. ............................. 21 3.6: Undeformed and deformed shape of an element of circumferential width f and length dr at a radial distance r from the center in the reference plane. ............................. 23 4.1: Comparison of maximum deflection for different radii of the piezoelectric material as predicted by the analytical solution and finite element model for pressure application. ................................ ................................ ................................ ..................... 38 4.2: Comparison of maximum deflection for different radii of the piezoelectric material as predicted by the analytical solution and finite element model for a unit voltage loading. ................................ ................................ ................................ ........................... 39 4.3: A schematic of the experimental setup showing the laser vibrometer focusing on the clamped PZT unimorph bender. ................................ ................................ ...................... 40 4.4: Measured displacement frequency response function obtained by convert ing velocity measurements using 1 j w integrating factor. ................................ ....................... 41 5.1: Non dimensional plot of the center deflection of a piezoelectric unimorph disc with 0.02 ps EE = subjected only to a pressu re load. ................................ ............................ 51 5.2: Non dimensional plot of the center deflection of a piezoelectric unimorph disc with 0.2 ps EE = subjected only to a pressure load. ................................ .............................. 52 5.3: Non dimensional plot of the center deflection of a piezoelectric unimorph disc with 0.4 ps EE = subjected only to a pressure load. ................................ .............................. 52 5.4: Non dimensional plot of the center deflection of a piezoelectric unimorph disc with 0.6 ps EE = subjected only to a pressure load. ................................ .............................. 53 5.5: Non dimensional plot of the center deflection of a piezoelectric unimorph disc with 0.8 ps EE = subjected only to a pressure load. ................................ .............................. 53 5.6: Non dimensional plot of the center deflection of a piezoelectric unimorph disc with 0.6 ps EE = and 2 0.02 s hR = subjected only to a pressure load. ................................ 54 5.7: Non dimensional plo t of the center deflection of a piezoelectric unimorph disc with 0.02 ps EE = subjected only to a voltage load. ................................ .............................. 57 5.8 Non dimensional plot of the center deflection of a piezoelectric unimorph disc with 0.2 ps EE = subjected only to a voltage load. ................................ ................................ 58

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xii 5.9: Non dimensional plot of the center deflection of a piezoelectric unimorph disc with 0.4 ps EE = subjected only to a voltage load. ................................ ................................ 58 5.10: Non dimensional plot of the center deflection of a piezoelectric unimorph disc with 0.6 ps EE = subjected only to a voltage load. ................................ ................................ 59 5.11: Non dimensional plot of the center deflection of a piezoelectric unimorph disc with 0.8 ps EE = subjected only to a voltage load. ................................ ................................ 59 5.12: Non dimensional plot of the center deflection of a piezoelectric unimorph disc with 0.6 ps EE = and 2 0.02 s hR = subjected only to a voltage load. ................................ .. 60 5.13: Non dimensional short circuit acoustic compliance plots for 0.02 ps EE = ................... 61 5.14: Non dimensional short circuit ac oustic compliance plots for 0.2 ps EE = ..................... 62 5.15: Non dimensional short circuit acoustic compliance plots for 0.4 ps EE = ..................... 62 5.16: Non dimensional short circuit ac oustic compliance plots for 0.6 ps EE = ..................... 63 5.17: Non dimensional short circuit acoustic compliance plots for 0.8 ps EE = ..................... 63 5.18: Non dimensional short circuit ac oustic compliance plots for 0.6 ps EE = and 2 0.02 s hR = ................................ ................................ ................................ ................ 64 5.19: Non dimensional acoustic mass plots for 0.02 ps EE = (Aluminum/PVDF). .................. 65 5.20: Non dimensional acoustic mass plots for 0.05 ps EE = (Aluminum/PVDF). .................. 66 5.21: Non dimensional acoustic mass plots for 0.2 ps EE = (Silcon/PZT). .............................. 66 5.22: Non dimensional acoustic mass plots for 0.4 ps EE = (Silcon/PZT). .............................. 67 5.23: Non dimensional acoustic mass plots for 0.6 ps EE = (Silcon/PZT). .............................. 67 5.24: Non dimensional acoustic mass pl ots for 0.6 ps EE = (Brass/PZT). .............................. 68 5.25: Non dimensional acoustic mass plots for 0.8 ps EE = (Brass/PZT). .............................. 68 5.26: Non dimensional acoustic mass plots for 0.6 ps EE = and 2 0.02 s hR = (Silcon/PZT). ................................ ................................ ................................ .................. 69

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xiii 5.27: Non dimensional effective acoustic piezoelectric coefficient plots for 0.02 ps EE = ................................ ................................ ................................ ............... 71 5.28: Non dimensio nal effective acoustic piezoelectric coefficient plots for 0.2 ps EE = ......... 71 5.29: Non dimensional effective acoustic piezoelectric coefficient plots for 0.4 ps EE = ......... 72 5.30: Non dimensional effective acoustic piezoelectric coefficient plots for 0.6 ps EE = ......... 72 5.31: Non dimensional effective acoustic piezoelectric coefficient plots for 0.8 ps EE = ......... 73 5.32: Non dimensional effective acoustic piezoelectric coefficient plots for 0.6 ps EE = and 2 0.02 s hR = ................................ ................................ ................................ .......... 73 5.33: Non dimensional A f for 0.02 ps EE = ................................ ................................ ........ 75 5.34: Non dimensional A f for 0.2 ps EE = ................................ ................................ .......... 76 5.35: Non dimensional A f for 0.4 ps EE = ................................ ................................ .......... 76 5.36: Non dimensional A f for 0.6 ps EE = ................................ ................................ .......... 77 5.37: Non dimensional A f for 0.8 ps EE = ................................ ................................ .......... 77 5.38: Non dimensional A f for 0.6 ps EE = and 2 0.02 s hR = ................................ ............. 78 5.39: Non dimensional A f for 0.02 ps EE = ................................ ................................ ........ 79 5.40: Non dimensional A f for 0.2 ps EE = ................................ ................................ .......... 80 5.41: Non dimensional A f for 0.4 ps EE = ................................ ................................ .......... 80 5.42: N on dimensional A f for 0.6 ps EE = ................................ ................................ .......... 81 5.43: Non dimensional A f for 0.8 ps EE = ................................ ................................ .......... 81 5.44: Non dimensional A f for 0.6 ps EE = and 2 0.02 s hR = ................................ ............. 82 5.45: K for 0.02 ps EE = ................................ ................................ ................................ .. 83 5.46: K for 0.2 ps EE = ................................ ................................ ................................ .... 84

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xiv 5.47: K for 0.4 ps EE = ................................ ................................ ................................ .... 84 5.48: K for 0.6 ps EE = ................................ ................................ ................................ .... 85 5.49: K for 0.8 ps EE = ................................ ................................ ................................ .... 85 5.50: K for 0.6 ps EE = and 2 0.02 s hR = ................................ ................................ ....... 86

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xv Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science TWO PORT ELECTROACOUSTIC MODEL OF A PIEZOELECTRIC COMPOSITE CIRCULAR PLATE By Suryanarayana A. N. Prasad December 2002 Chairman: Dr. Mark Sheplak Cochairman: Prof. Bhavani V. Sankar Major Department: Department of Mechanical and Aerospace Engineering. Accurate prediction of mode shape and the deflection field of a circular piezoelectric composit e transducer are important for the design of sensors and actuators. There is a need for a model, which lays emphasis on the physics of the problem and predicts the deflection field as a function of pressure and voltage loading. Such a theory would help in developing a non dimensionalization procedure, which could be used to extract non dimensional parameters for developing an optimization procedure for the design of a sensor or an actuator. This thesis presents the development of such a theory that predicts the central deflection and mode shape. Lumped element modeling (LEM) of the system, which is used to extract system parameters of an equivalent single degree of freedom (SDOF) system, is also discussed. The two port network representation that is used to develop an equivalent circuit representation of the piezoelectric transduction is presented. Non dimensionalization of the plate equations is also discussed in great detail.

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1 CHAPTER 1 INTRODUCTION CHAPTER 1 Commonly used electroacoustic 1 devices, such as microphones and headphones, use circular disk transducers that are piezoelectric composite plates. Recent devices, such as synthetic jet actuators, 2 3 used in flow control applications, can also be driven by piezoelectric composite circular plate. Micro fluidic pump drivers 4 represent another relatively new application for these devices. A design procedure, to find a set of system parameters for the optimal performance of these devices, helps to improve the performance of these devices and also widen its range of applications. Development of such a design procedure is possible only when the response of these devices to loadin g and change of system parameters is known. Although experimental techniques can be used to characterize piezoelectric transducers, analytical models that can predict the response of the transducers are helpful in understanding the effects of various param eters on the predicted response and also in optimizing the performance of the devices. Determining the dynamics of a single degree of freedom (SDOF) system would be an easier task than determining the dynamic response of a corresponding complex system. Su ch a SDOF system would be a compact macromodel that provides physical insight and accurately captures the energy behavior and dependence on the material properties. This system is obtained by lumped element modeling (LEM) of the piezoelectric unimorph disk transducer. The lumped element parameters of the system are

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2 obtained from the static deflection behavior of the system when subjected to pressure and voltage loading. This approximation will be valid in Fourier space and will simplify the problem of deter mining the complicated dynamics of the piezoelectric composite circular unimorph transducer into a static analysis of an axisymmetric piezoelectric multi layered isotropic composite circular plate. As the problem consists of various domains, a two port net work representation 1 5 is used to model the transduction from the input port to the output port. The parameters obtained from the LEM of the plate are fed into the two por t network model of the device. This two port model is used in developing an equivalent circuit of the device. The behavior of the device to loading is obtained from the transfer function of this equivalent circuit. CHAPTER 1 Static displacement behavior of the piez oelectric unimorph transducer P h p R 2 z Piezo Shim h s R 1 r V Figure 1 1 : Cross sectional schematic of a clamped axisymmetric piezoelectric unimorph disk transducer Figure 1 1 shows a cross section of a clamped circular piezoelectric unimorph composite plate subjected to a uniform transverse pressure loading P and/or a voltage V A piezoceramic material of thickness p h and radius 1 R is bonded on top of a shim

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3 material of outer radius 2 R and thickness s h The loading creates a transverse displacement field () wr and a radial displacement field (,) urz Background Previous work in the area of piezoelectric composite circular plates focused on structures that are symmetrically layered about the neutral axis such as bimorph transducers. 6 Because of midplane symmetry there is no bending extension coupling in bimorph transducers and this simplifies the analysis. However, many of the devices cannot be manufactured in this manner (for example, most pi ezoelectric unimorph transducers fabricated through micro electromechanical systems (MEMS) technology have geometry as shown in Figure 1 1 ). Hence there exists a need to extend the existing piezoelectric composite pl ate theory to the case of an axisymmetric multi layered transversely isotropic piezoelectric composite circular plate. Morris and Foster 4 developed an optimization procedure for a piezoelectric bimorph micropump dr iver (the same geometry as in Figure 1 1 ) using finite element method (FEM) with the help of ANSYS 7 software. They performed optimization of the micropump driver for both pinned and the fi xed case by identifying non dimensional p groups using the Buckingham theorem. They used a higher order routine in ANSYS to accomplish this task. They have developed some empirical equations for optimal radius ratio and thickness ratio for a particular set of materials for a particular aspect ratio. They also discussed edge support effects and effect of bond layer. Dobrucki and Pruchnicki 6 formulated the problem of a piezoelectric axisymmet ric bimorph and used FEM to solve the problem. They derived the equations that would determine the bending moment and extensional forces produced by the piezoelectric

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4 material on application of an electric field. They used average elastic parameters for an alyzing the composite plate. Use of the bimorph as a sensor was also discussed. They experimentally verified their results from the FEM solution. Verification of the theory with simpler geometry was also performed. They also have proved that on the rim of a clamped circular transducer the electric signal produced is zero. Stavsky and Loewy 8 solved numerically the dynamics of isotropic composite circular plates using Kirchoffs plate theory. They found the vibrations of the composi te plate to be analogous to the vibration of a homogenous shallow spherical shell. They also discussed effects due to material arrangement, radius, material and plate composition on frequency of vibration of the composite circular plate. They obtained a sy stem of equations of the 6 th order. The solution for this system of equations can be expressed in terms of Bessel functions, the argument for which is determined from the characteristic equation of order 3. They also discussed numerical examples showing th e effect that arises due to heterogeneity on vibration response of the composite to be significant. Adelman and Stavsky 9 formulated the problem of piezoelectric circular composite plates using Kirchoffs plate the ory. Static behavior and flexural extensional vibratory response of metal piezoceramic unimorphs and PZT 5H bimorphs possessing silver electrodes are solved numerically. Their formulation is identical to that of the formulation discussed in this thesis, ex cept that they use variables 1 E and 2 E that relate the fictitious force/moments generated to the electric field applied instead of the comprehensive equation for fictitious forces that describe the piezoele ctric transduction shown in Eq. ( 3 26 ) and Eq. ( 3 27 ) They also discussed numerical exa mples showing the effect of silver electrode on unimorph piezoelectric benders.

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5 Chang and Du 10 performed optimization of a unimorph disk transducer based on an electro elastic theory assuming free boundary conditio ns, which is non physical for most applications. They modified the existing Kirchoffs plate theory by adding a term to account for the piezoelectric layer. They assumed that the electric field variation in the thickness direction could be represented as a quadratic function and the electric charge to be equal and opposite on the top and bottom electrodes of the piezoelectric layer. Dumir et al. 11 obtained a non linear axisymmetric solution for the static and tran sient moderately large deflection of a laminated axisymmetric annular plate acted on by uniformly distributed ring loads by using first order shear deformation theory. Effect of inplane inertia was neglected while the rotary inertia was considered. The mat erial was treated to be orthotropic. They used a numeric technique called the Newmark b scheme in order to solve the governing differential equations. They simplified the solution and verified the same with the solution from the c lassical plate theory. Thesis Layout Chapter 2 of t his thesis presents a two port, lumped element model of an axisymmetric piezoelectric unimorph transducer with the geometry and loading described in Figure 1 1 I n LEM, the individual components of a piezoelectric unimorph are modeled as elements of an equivalent electrical circuit using conjugate power variables. The synthesis of the two port model required determination of the transverse static deflection field as a function of pressure and voltage loading. I n Chapter 3, c lassical laminated plate theory was used to derive the equations of equilibrium for circular laminated plates containing one or more piezoelectric layers. The equations were solved for a unimor ph device wherein the diameter of the piezoelectric

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6 layer was less than that of the shim ( ) 12 RR < An exact analytical static solution of the displacement field of the axisymmetric piezoelectric unimorph is determined. The solution for a nnular plate obtained using the classical plate theory matches with the solution provided in the paper by Dumir et al. 11 Chapter 4 verifies the result obtained from Chapter 3 by theoretical means and by a Finite Element Model. M ethods to estimate the model parameters are discussed and experimental verification is presented. Chapter 5 discusses how the governing equations are used in the non dimensionalization of the field variables, lumped element parameters and two port network parameters. This proves to be a simpler and more comprehensive option. The results corresponding to the work described in a Chapter are summarized at the end of the corresponding chapters. In addition, a summary of the main results is pr ovided in the conclusions (Chapter 6). Future work and concurrent work is also discussed in Chapter 6.

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7 CHAPTER 2 TWO PORT NETWORK MODELING 2 The piezoelectric composite plate actuator represents a coupled electro mechanical acoustic system with frequency dependent properties determined by device dimensions and material properties. The analysis a nd design of such a coupled domain transducer system is commonly performed using lumped element models. 1 This is justified, because the prediction by LEM matches the actual value to within 2%, as per Merhaut. 12 Lumped Element Modeling The main assumption employed in LEM is that the characteristic length scales of the governing physical phenomena are much larger than the largest geometric dimension. For example, for the vi bration of a piezoelectric plate, the bending wavelength and electromagnetic wavelength must be significantly larger than the device itself. If this assumption holds, then the temporal and spatial variations can be decoupled. This decoupling permits the go verning partial differential equations of the distributed system to be lumped into a set of coupled ordinary differential equations through the solution of the static equations. The individual components of a piezoelectric unimorph are modeled as element s of an equivalent electrical circuit using two port modeling. Two Port Network Modeling An Introduction Any linear conservative electroacoustic transduction can be modeled using the electrical analogy as a transformer or a gyrator with series and para llel impedances or

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8 admittances on each of its ports. 1 The transduction from one domain to another, which is a function of system parameters, is represented in terms of admittances and impedances in each domain. Ta ble 2 1 : Conjugate power variables 13 and corresponding dissipative and energy storage elements in various domains. Energy Domain Effort Variable Flow Variable Energ y Dissipater Kinetic Energy Storage Potential Energy Storage Displacement Mechanical Force Velocity Dashpot Mass Spring Displacement Acoustical Pressure Volume Velocity Vent Acoustic Mass Cavity Volume Displacement Electrical Voltage Current Resistor In ductor Capacitor Electric Charge Resistors are used to represent any dissipative element. Dissipative elements in other domains are shown in the fourth column of Ta ble 2 1 Inductors and capacitors are used to represent elements that store generalized kinetic energy and potential energy respectively. Corresponding elements in other domains are shown in fifth and sixth column of Ta ble 2 1 The conjugate power variables, 13 the effort and flow, are identified in each of the domains as shown in the second and third column of Ta ble 2 1 The product of the conjugate power variables is a measure of power. In impedance analogy, elements sharing co mmon flow are connected in series while elements sharing a common effort are connected in parallel.

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9 Two Port Model of Piezoelectric Transduction A piezoelectric transducer converts electric energy into strain that is realized as a displacement in the mecha nical domain. Usually a piezoelectric transduction is represented in tensor form as per IEEE standards as shown in Eq. ( 2 1 ) and Eq. ( 2 2 ) E S s T dE =+ ( 2 1 ) and T D d TE e =+ ( 2 2 ) where D is the dielectric displacement in 2 / Cm T is the stress in [ ] Pa S is the strain, E is the electric field in [ ] / Vm e is the permitivity in [ ] / C Vm s is the compliance in [ ] 1/ Pa and d is the piezoelectric coefficient in [ ] / CN A piezoelectric material responds with a strain field not only due to application of stress but also due to application of electric field. An application of stress creates a charge (due to piezoelectric transduction) in addition to the charge created due to the application of voltage acro ss the piezoelectric (a dielectric medium). Two Port Model of a 1 D Piezoelectric + V F,x 3-dir Fixed B.C. plate of area A plate thickness h p Figure 2 1 : Schematic of a piezoelectric plate that can be approximated to be 1 D. In case of a 1 D piezoelectric, the force F and voltage V act only in the 3 direction as shown in Figure 2 1 Application of the force not only gives rise to a deflection x but also creates a polarization represented by an electrical charge q ;

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10 P MS x CF = ( 2 3 ) and q dF = ( 2 4 ) In the above equati ons, P MS C and d are the short circuit mechanical compliance and effective mechanical piezoelectric coefficient (that is responsible for a strain in 3 direction due to application of electric field in the 3 d irection) of the piezoelectric material respectively and are given by 0 P V MS x C F = = ( 2 5 ) and 0 P x d V = = ( 2 6 ) Application of voltage creates a deformation x in addition to creating a polarization represented by an electric charge q ; x dV = ( 2 7 ) and EF qCV = ( 2 8 ) EF C is the electrical free capacitance of the 1 D piezoelectric that is given by 0 EF p kA C h e = ( 2 9 ) where k is the dielectric constant of the piezoelectric in the 3 direction due to application of an electric field in the 3 direction and 0 e is permitivity of free space. The transduction in the case of a 1 D piezoelectric in the static case, when subjected to both voltage and force load, is found by superimposing Eqs. ( 2 3 ) ( 2 4 ) ( 2 7 ) and ( 2 8 ) ;

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11 P MS EF Cd xF qV dC = ( 2 10 ) Two Port Electroacoustic Model of a Piezoelectric Composite Plate In the case of a piezoelectric unimorph disc, application of voltage creates bending and not an extension as in the case of the 1 D piezoelectric described above. Also the focus in this thesis is oriented towards an electroacoustic model ra ther than an electro mechanical model of the piezoelectric unimorph. Hence integration over the surface area of the unimorph disc needs to be performed to extend the electromechanical model of the 1 D piezoelectric to a piezoelectric unimorph disc transduc er. In the acoustic domain, a volume displacement is created in a piezoelectric transducer not only due to application of pressure but also due to application of voltage. Application of pressure creates a charge separation across the piezoelectric layer (d ue to piezoelectric transduction) in addition to the charge separation created due to the application of voltage. Hence the transduction of an axisymmetric piezoelectric unimorph disk in static case is expressed as A SA A EF Cd Vo lP dC qV D = ( 2 11 ) where Vol D is the volume displaced by the plate due to the application of pressu re P and voltage V ; 2 0 2 () R Vo l r wr dr p D= ( 2 12 ) and AS C is the short circuit acoustic compliance of the plate. Expression for the acoustic compliance is extracted from the equivalent SDOF system by equating the strain energy of the actual system to the potential energy of the SDOF system in the following manner:

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12 2 0 0 0 ( )2 R V AS V w r rdr Vol C PP p = = D == ( 2 13 ) In Eq. ( 2 11 ) A d is the effective acoustic piezoelectric coefficient, which is given by 2 0 0 0 ( )2 R P A P w r rdr Vol d VV p = = D == ( 2 14 ) Equivalent Circuit Representation and Parameter Extraction Assuming time harmonic function and differentiating both sides of Eq. ( 2 11 ) with respect to time yields the expression for conjugate power variable at low frequencies; A SA A EF j C jd QP j d jC iV ww ww = ( 2 15 ) where w is the radian frequency. The equivalent circuit of the piezoelectric unimorph at low frequencies takes the for m shown in Figure 2 2 + V + P i Q f A Q f A V + C EB 1: f A C AS Figure 2 2 : Equivalent two port circuit representation of the piezoelectric unimorph at low frequencies. In Figure 2 2 Q is the volume velocity and i is the current. The parameter A f appearing as the transformer turns ratio in the equivalent circuit representation is the

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13 electroacoust ic transduction coefficient, which is given by the ratio of the effective acoustic piezoelectric coefficient and the short circuit acoustic compliance of the plate; A A AS d C f = ( 2 16 ) When unloaded, the equivalent circuit representation in Figure 2 2 is represented as shown in Figure 2 3 The energy stored in the two circuit elements shown in Figure 2 3 should equal the energy stored in the circuit element shown in Figure 2 4 Hence the electroacoustic energy coupling factor, which is a qu antity that describes the fraction of energy converted from electric domain to the acoustic domain, is 22 22 2 A 2 AA 2 1 P.E. stored in 2 1 P.E. stored in 2 AS A S AS A E F E F A S EF EF CV CC d K C C CC CV f ff = = == ( 2 17 ) + V C EB f A 2 C AS Figure 2 3 : Equivalent circuit representation in the electric domain of the piezoelectric unimorph. + V C EF Figure 2 4 : Equivalent circuit representation in the electric domain of the piezoelectric unimorph when decoupled from the acoustic domain.

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14 The blocked electrical capacitance at the electrical side is obtained by e quating the energy storage in Figure 2 3 and Figure 2 4 ; 22 22 A 1 11 2 22 E B A S EF C V C V CV f += ( 2 18 ) Substituting Eq. ( 2 17 ) in Eq. ( 2 18 ) we ob tain 2 2 22 1 11 2 22 E B E F EF C V CV K CV += ( 2 19 ) which simplifies to ( ) 2 1 E B EF C CK =. ( 2 20 ) + P + V Q i f A i f A P + C AO 1: f A C EF Figure 2 5 : Another equivalent two port circuit representation of the piezoelectric unimorph at low frequencies. Transduction mechanism in the case of a piezoelectric unimorph at low frequencies has another equivalent circuit representation as shown in Figure 2 5 The parameter A f appearing as the transformer turns ratio in the equivalent circuit representation shown in Figure 2 5 is the acousto electric transduction coefficient, which is given by the rat io of the effective acoustic piezoelectric coefficient of the plate and free electrical capacitance of the piezoelectric layer;

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15 A A EF d C f = ( 2 21 ) When the electrical port in the equivalent circuit representation shown in Figure 2 5 is shorted, the equivalent circuit simplifies to the circuit shown in Figure 2 6 The energy stored in the two circuit elements shown in Figure 2 6 should equal the energy stored in the circuit element shown in Figure 2 7 Hence the energy coupling factor is de rived as 22 2 22 A 2 AA 2 1 P.E. stored in 2 1 P.E. stored in 2 EF E F E FA A S A S A S EF AS CP C Cd K C C CC CP f ff = = == ( 2 22 ) + P C AO f A 2 C EF Figure 2 6 : Equivalent circuit representation in the acoustic domain of the piezoelectric unimorph. + P C AS Figure 2 7 : Equivalent circuit representation in the acoustic domain of the piezoelectric unimorph when electric behavior is decoupled. The open acoustic compliance at the acoustic port is obtained by equating the energy storage in Figure 2 6 and Figure 2 7 ;

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16 2 222 A 1 11 2 22 E F A O AS C P C P CP f += ( 2 23 ) Substituting Eq. ( 2 22 ) in Eq. ( 2 23 ) we obtain 2 222 1 11 2 22 A S A O AS C P K C P CP += ( 2 24 ) which simplifies to ( ) 2 1 A O AS C CK =. ( 2 25 ) For higher frequencies the mass of the disk becomes important and leads to resonance. The circuit in Figure 2 8 will describe the system at higher frequencies, upto the fundamental natural frequency. + V + P i Q f A Q f A V + C EB 1 : f A C AS M A Figure 2 8 : Equivalent two port circuit representation of axisymmetric piezoelectric unimorph disk at frequencies comparable to that of the pri mary resonance The acoustic mass A M of the equivalent SDOF system is extracted from the deflection response of the system by equating the kinetic energy stored in the SDOF system to that of the actual system in the following manne r: ( ) ( ) 2 2 2 0 2 R AA M w r rdr Vol p r = D ( 2 26 ) where A r is the areal density of t he piezoelectric composite plate. For the geometry described in Figure 1 1 the value of the areal density remains a constant from zero radius

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17 to the end of the inner composite region. There is a change in value of the areal density at the interface and then remains a constant until the clamped edge. The volume displaced due to application of a load remains a constant independent of the region. The short circuit natural frequency s f in Hz of the system is given by 1 2 s A SA f CM p = ( 2 27 ) while the open circuit resonant frequency o f is given by 1 2 o A OA f CM p = ( 2 28 ) By substitu ting Eqs. ( 2 25 ) and ( 2 27 ) in Eq. ( 2 28 ) we obtain the relationship between the resonant frequencies as ( ) ( ) 22 1 2 11 s o A SA f f C K MK p == -, ( 2 29 ) which on simplification yields ( ) 2 1 os f Kf -= ( 2 30 ) The deflection field of the piezoelectric composite plate, due to unit applied pressure and due to unit applied voltage, is required to determine , and A SA C dK The deflection field is obtained by analyzing the mechanical behavior of the piezoelectric unimorph, which is modeled and discussed in Chapter 3 of the thesis.

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18 CHAPTER 3 MECHANICAL BEHAVIOR OF THE PIEZOELECTRIC COMPOSITE PLATE 3 For developing a two port electroacoustic model of the piezoelectric composite plate, the quantities defined in Eq. ( 2 13 ) ( 2 14 ) and Eq. ( 2 26 ) need to be determined. These quantities depend on the vertical deflection that is determined by analysis of the mechanical behavior of the axisymmetric piezoelectric composite plate. Problem Formulation Problem formulation is based on the classical Kirchoffs plate theory. 14 The fact that one of the layers is a piezoelectric is accounted for in the constitutive relation by adding an additional strain term due to the piezoelectric layer. Assumptions The assumptions made in developing a linear small deflection pl ate theory with piezoelectric effect are as follows: The plate (shown in Figure 3 1 ), is assumed to be in a state of plane stress 14 normal to the z axis. In other words, the normal stress zz s and the shear stress rz t are approximately equal to zero.

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19 R N r (R) M r (R) P (r) Figure 3 1 : An axisymmetric multi layered transversel y isotropic circular plate with pressure load, radial load and a moment. All loads shown are considered positive. The shear stresses z q t and r q t also vanishes due to the assumed axisymmetric nature of the pro blem. Stress measures, corresponding to the plane stress case, (i.e.) rr s and qq s exist. Only, shear stress in the transverse direction exists (i.e. 0 rz t ). Figure 3 2 delineates the active stress resultants on a multi layered composite circular plate acted on by a pressure load ( ) Pr R E n ,n n h n M N r (R) E 1 ,n 1 h 1 Plane of Reference M r (R) N r (R) P (r) Q r (R) Q r (R) +z z 2 z 1 Figure 3 2 : Cro ss sectional view of the plate shown in Figure 3 1 showing the sign conventions and labels used.

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20 The active stresses on the circular plate are shown in Figure 3 3 Figure 3 4 delineates the force, moment and shear force resultants acting on an infinitesimal element of a multi layered isotropic composite circular plate. R r r+dr N r (R) Figure 3 3 : Top view of the plate shown in Figure 3 1 P( r ) Figure 3 4 : Enlarged isometric view of the element shown in Figure 3 3 and Figure 3 2 with generalized forces acting on it.

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21 P (r) r r+dr Figure 3 5 : A n element of the multilayered transv ersely isotropic composite plate of length dr placed at a distance r from the center acted on by generalized forces. In Figure 3 4 and Figure 3 5 r N and N q are the force resultants in radial and circumferential directions respectively; 2 1 z r rr z N dz s = ( 3 1 ) and 2 1 z z N dz q qq s = ( 3 2 ) The moment resultants in radial and circumferential directions r M and M q are given by 2 1 z r rr z M zdz s = ( 3 3 ) and 2 1 z z M zdz q qq s = ( 3 4 ) r Q is the shear force resultant given by

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22 2 1 z r rz z Q dz t = ( 3 5 ) Equilibrium Equations For the case of the axisymmetric multi layered composite, the field variables 0 and u q are functions of radius alone (due to axisymmetry). Therefore, the partia l differentials equal the corresponding total differentials. Taking balance of forces in the radial direction, of the projection of the element (shown in Figure 3 4 ) in the r q plane, we obtain ( ) 1 0 r r dN NN d rr q + -= ( 3 6 ) Taking moment about the line tangent to r dr + of the e lement shown in Figure 3 4 we obtain ( ) 1 0 r rr dM Q MM d rr q + -= ( 3 7 ) T aking balance of the forces shown in Figure 3 5 in the vertical direction, we obtain ( ) 0 rr d QQ Pr d rr + += ( 3 8 ) The moment resultants are expressed in matrix form as 2 1 z r rr z M zdz M q qq s s = ( 3 9 ) where moments/moment resultants causing tension in top surface are considered positive. The force resultants are expressed in matrix form as 2 1 z r rr z N dz N q qq s s = ( 3 10 ) where radial forces/force resultants causing tension are considered positive.

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23 Equations ( 3 6 ) ( 3 8 ) are the equations of equilibrium 14 15 16 of an axisymmetric multi layered composite plate. The pressure load acting on the bottom surface is uniform. Therefore, these equations are suitably modified to yield the equilibrium equations of an axisymmetric pi ezoelectric unimorph. The resulting expressions are given by ( ) 0 r r NN dN drr q += ( 3 11 ) ( ) r r r MM dM Q drr q +=, ( 3 12 ) and 0. rr dQQ P drr ++= ( 3 13 ) Strain Displacement Relationships The field variables involved in the problem are the radial deflection in the reference p lane 0 u and slope q respectively. r u 0 (r) u 0 (r) + du 0 (r) f dr Figure 3 6 : Undeformed and deformed shape of an element of circumferential width f and length dr at a radial distance r from the center in the reference plane. The increment shown in Figure 3 6 contains a total derivative because of the fact that the reference plane displacements are just a function of radius due to axisymmetry,

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24 (i.e.) ( ) 00 u ur = and ( ) r qq = The strains are obtained from the field variables in the following manner. The radial strain measure fo r Kirchoffs plate theory is given by 0 r r r rr z e ek =+ ( 3 14 ) where 0 rr e and r k are the radial strain in the reference plane and curvature in the radial direction, respectively; 0 0 0 00 rr u d u u du r d r r dr e +== +( 3 15 ) and 2 2 r d wd d r dr q k = =. ( 3 16 ) The circumferential strain measure for Kirchoffs plate theory is given by 0 z q q q qq e ek =+ ( 3 17 ) where 0 qq e and q k are the circumferential strain in the reference plane and curvature in the circumferential direction respectively and are obtained by taking balance of force s acting on the element shown in Figure 3 6 ; ( ) 0 0 0 r ur u rr qq qq e q +== ( 3 18 ) and 1 dw r d rr q q k = =. ( 3 19 ) Constitutive Equations For an elastic plate, the st ress strain relations 14 15 are given by

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25 1 1 12 1 2 11 r r rr QQ QQ q q qq se se = ( 3 20 ) where [ ] 2 1 1 1 E Q n n n = ( 3 21 ) E is the Youngs modulus and n is the Poissons ratio of the layer. From Chapter 2, it can be understood that Eq. ( 3 22 ) represents the inverse constitutive relation of a piezoelectric material as per IEEE standards. E S s T dE =+ ( 3 22 ) Thus for an axisymmetric piezoelectric composite plate discussed above, the constitutive relations are given by [ ] 0 31 33 0 31 r rr rr d Q zE d q qq qq sk e sk e = +, ( 3 23 ) where 33 E is the uniform electric field across the thickness of the piezoelectric layer and 31 d is the piezoelectric c oefficient in the plane perpendicular to the direction of application of the electric field. In the above expression, number 3 represents the z direction and number 1 represents the radial direction. The plate constitutive equations are obtained by integr ating the constitutive equations of the individual layers through the thickness and are given by [ ] [ ] [ ] 0 P r r P N N AB N N q q ek = +( 3 24 ) and

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26 [ ] [ ] [ ] 0 P r r P M M BD M M q q ek = +, ( 3 25 ) where [ ] [ ] 2 1 z z A Q dz = is the extensional rigidity matrix, [ ] [ ] 2 1 z z B Q zdz = is the flexural extensional coupling matrix and [ ] [ ] 2 1 2 z z D Qz dz = is the flexural rigidity matrix The flexural extensional coupling matrix is zero, if the material layers are symmetrical about the reference plane (For example, a piezoelectric bimorph). The force resultants, P r N and P N q generated internally due to the application of 33 E across the piez oelectric layer are given by 2 31 1 1 12 33 31 1 2 11 1 z P r P z d QQ N E dz d QQ N q = ( 3 26 ) The moment resultants, P r M and P M q generated internally due to the application of 33 E across the piezoelectric layer are given by 2 31 1 1 12 33 31 1 2 11 1 z P r P z d QQ M E zdz d QQ M q = ( 3 27 ) From the form of Eq. ( 3 26 ) and Eq. ( 3 27 ) we can infer that the material is transversely isotropic; PPP r NNN q == ( 3 28 ) and PPP r MMM q == ( 3 29 )

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27 Substituting the strain displacement relationships and curvature displacement relationships shown in Eqs. ( 3 14 ) ( 3 19) in the plate constitutive relationships shown in Eqs. ( 3 24 ) and ( 3 25 ) we obtain 00 11121112 00 12111211 ()() ()() ()() ()() P r P durur drr AABB N N drrdrr N durur drr N AABB drrdrr q qq qq +-=+-( 3 30 ) and 00 11121112 00 12111211 ()() ()() ()() ()() P r P durur drr BBDD M M drrdrr M durur drr M BBDD drrdrr q qq qq +-=+-. ( 3 31 ) Governing Differential Equations Equations ( 3 30 ) and ( 3 31 ) are substituted in the plate equations of equilibrium shown in Eqs. ( 3 11 ) ( 3 13 ) The field variables in the resulting expression, are de coupled (see Appendix A) by introducing the variables a and 11 D defined as 11 11 B A a = ( 3 32 ) and 2 11 1111 11 B DD A =. ( 3 33 ) Assuming that the electric field is uniform and the material i s transversely isotropic is used to show that the internal force and moment resultants, generated by the piezo, has a constant value all along the reference plane. The resulting expression is simplified to yield the governing differential equations, which are

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28 2 2 2* 11 ( ) 1 ( ) () 2 d r d r r Pr d r r d r rD q qq + -= ( 3 34 ) and 2 0 00 2 2* 11 ( ) ( ) () 1 2 d u r d u r ur Pr d r r d rrD a + -= ( 3 35 ) General Solution The general solution to the above set of differential equations is determined as 3 2 01 11 16 a Pr u ar rD a =+( 3 36 ) and 3 2 1 11 1 16 b Pr br rD q =+, ( 3 37 ) where 1 a 2 a 1 b and 2 b are constants th at are determined by applying the boundary conditions. A detailed derivation of this solution is shown in Appendix A. The Problem of Piezoelectric Unimorph Disk Transducer The problem of circular piezoelectric unimorph disk transducer shown in Figure 1 1 is solved by combining the solution for deflection field of an annular plate of inner radius R 1 and outer radius R 2 with the solution for deflection field of the inner composite plate of radius R 1 To differentiate between the const ants involved in the solutions in the central composite region and outer annular region, symbols (1 ) (2) and are used to denote inner and outer regions respectively. Central Composite Plate In the central composite plate the central deflect ions of the plate is bounded:

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29 ( ) 0 0 u < ( 3 38 ) and ( ) 0 q < ( 3 39 ) Substituting Eq. ( 3 38 ) and Eq. ( 3 39 ) in the general solution, we obtain (1) 2 0 a = ( 3 40 ) and (1) 2 0 b = ( 3 41 ) Therefore the deflections are given by ( ) 3 (1 ) (1) 01 *(1) 11 16 Pr ur ar D a =( 3 42 ) and ( ) 3 (1 ) (1) 1 *(1) 11 1 16 Pr r br D q =. ( 3 43 ) In matrix form, Eq. ( 3 42 ) and Eq. ( 3 43 ) is written as ( ) ( ) (1) (1) 3 0 1 (1) *(1) (1) 11 1 1 16 ur a Pr r r D b a q =. ( 3 44 ) From Eq. ( 3 44 ) we obtain ( ) ( ) (1) 0 (1) 2 1 *(1) (1) (1) 11 1 3 1 16 d ur a Pr dr D b dr dr a q =. ( 3 45 ) Substituting Eqs. ( 3 44 ) ( 3 45 ) in Eqs. ( 3 30 ) ( 3 31 ) we obtain the following expression for generalized force resultants in the radial direction:

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30 ( ) ( ) (1) (1 ) (1 ) (1 ) (1 ) (1 ) (1 ) (1 ) (1 ) (1) 2 1 1 1 2 1 1 1 2 1 1 1 1 2 1 1 12 (1) *(1) (1 ) (1 ) (1 ) (1 ) (1 ) (1 ) (1 ) (1 ) (1) 11 1 1 1 2 1 1 1 2 1 1 1 1 2 1 1 12 33 16 33 P r P r Nr A A B B a B BAA N Pr Mr D B B DD b D D BB M aa aa + + -= ++ + -( 3 46 ) Substituting Eqs. ( 3 32 ) ( 3 33 ) in Eq. ( 3 46 ) we obtain ( ) ( ) ( ) ( ) ( ) ( ) (1 ) (1 ) (1 ) (1) (1) (1 ) (1 ) (1) 2 1 1 1 2 1 1 12 1 1 2 12 (1) *(1) (1 ) (1 ) *(1 ) (1) (1 ) (1 ) (1 ) (1) 11 1 1 2 1 1 12 1 1 1 2 1 1 12 16 3 P r P r A A BB Nr a BA N Pr Mr D b BDD M B B DD a a + -+ = +++ + -+ ( 3 47 ) At the interface, the deflections are given by ( ) ( ) (1) (1) 3 01 1 1 1 (1) *(1) (1) 1 11 1 1 16 uR a PR R R D b a q =( 3 48 ) and the generalized force resultants are given by ( ) ( ) ( ) ( ) ( ) ( ) (1 ) (1 ) (1 ) (1) (1) (1 ) (1 ) (1) 2 1 1 1 2 1 1 12 1 1 1 2 12 1 (1) *(1) (1 ) (1 ) *(1 ) (1) (1 ) (1 ) (1 ) (1) 1 11 1 1 2 1 1 12 1 1 1 2 1 1 12 16 3 P r P r A A BB NR a BA N PR MR D b BDD M B B DD a a + -+ = +-+ + -+ ( 3 49 ) Outer Annular Plate For the outer annular plate, [ ] B vanishes because of the symmetry about its neutral axis, which in turn implies that *(2 ) (2) 1 1 11 DD = and 0 a = As the outer annular plate is isotropic (2 ) (2) 1 2 11 s DD n = and (2 ) (2) 1 2 11 s AA n = The solution in the annular region is obtained from the general solution by applying fixed condition at the clamped edge and interface matching condi tion at the inner edge. The fixed condition at the clamped edge is applied by making ( ) 02 0 uR = ( 3 50 ) and ( ) 2 0 R q = ( 3 51 )

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31 Substituting Eqs. ( 3 50 ) and ( 3 51 ) in the general solution, we obtain (2 ) (2 )2 2 12 a aR =( 3 52 ) and 4 (2 ) (2 )2 2 2 22 (2) 11 16 PR b bR D = -. ( 3 53 ) Substituting back Eqs. ( 3 52 ) and ( 3 53 ) into the general solution, we obtain ( ) ( ) (2) (2) 2 0 1 2 4 (2) (2) (2) 3 2 11 1 0 16 ur a R P r R r rD b r r q = -. ( 3 54 ) Taking the first derivative of Eq. ( 3 54 ) with respect to the radius, we obtain ( ) ( ) (2) 0 (2) 2 1 2 4 2 (2) (2) 2 (2) 2 11 1 2 0 1 16 3 d ur a R P dr R rD b r dr r dr q = ++ ( 3 55 ) Substituting 1 rR = in Eq. ( 3 54 ) w e obtain the deflections at the interface, which are given by ( ) ( ) (2) (2) 2 01 1 2 4 1 3 (2) (2) (2 )2 1 1 1 11 1 1 0 16 uR a R P R R R R RD b R q = -. ( 3 56 ) The generalized force resultants in the outer annular region are given by ( ) ( ) ( ) ( ) ( ) ( ) (2 ) (2) 00 (2 ) (2) (2) 1 1 12 (2) (2) (2 ) (2 ) (2) 1 1 12 1 r r d u r ur AA Nr d rr Mr dr D Dr d rr q q + = -. ( 3 57 )

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32 Substituting Eq. ( 3 54 ) and Eq. ( 3 55 ) in Eq. ( 3 57 ) we obtain a relationship for generalized force resultants in the radial direction only in terms of the constants involved in the analytical solution for deflection; ( ) ( ) ( ) ( ) ( ) ( ) (2) (2 ) (2) 2 2 1 11 2 4 (2) 2 (2 ) (2) 2 1 11 4 0 11 16 31 r ss r ss Nr Aa R Pr R Mr r Db r nn nn =+ + -+ + +. ( 3 58 ) The generalized force resultants at the interface are given by ( ) ( ) ( ) ( ) ( ) ( ) (2) (2 ) (2) 22 1 1 11 21 4 (2) 2 (2 ) (2 )2 1 1 1 11 4 1 0 11 31 16 r ss ss r NR Aa R PR R MR R Db R nn nn =+ + -+ + +. ( 3 59 ) Interface Compatibility Conditions At the interface, deflections and the resultants must match the following conditions: ( ) ( ) (1 ) (2) 0 101 uRuR = ( 3 60 ) ( ) ( ) (1 ) (2) 11 RR qq = ( 3 61 ) ( ) ( ) (1 ) (2) 11 rr NRNR = ( 3 62 ) and ( ) ( ) (1 ) (2) 11 rr MRMR = ( 3 63 ) From the above derivation, it can be noted that the piezoelectric effect does not appear explicitly in the general solution but is introduced by the interface matching conditions of the generalized force resultant s shown in Eq. ( 3 62 ) and ( 3 63 ) Solution Techniques The expressions continued to incre ase in complexity. Analytical expression for the deflection field in terms of basic parameters is huge and does not provide any

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33 physical insight and is hard to evaluate or code. Hence two different techniques are employed to obtain the solution. Simple Ana lytical Solution The first technique is to obtain a simple analytical solution by making valid assumptions based on reality. The solution is simplified by introducing constants, which are functions of interface deflections. Such a solution is useful for th e purpose of non dimensionalization. Such an effort to obtain an analytical expression is discussed in Appendix B. The MATLAB code implementing the same is attached in Appendix C. Numerical Method to Obtain Constants The second technique is to code up the above equations in matrix form in MATLAB. The equivalent stiffness matrix of each of the regions is evaluated. The matrices are assembled to form an equivalent global stiffness matrix. The deflections and generalized force resultants at the interface are f ound by inverting the stiffness matrix. The constants and hence the deflection field is found from the evaluated interface deflections. The MATLAB code is attached in Appendix C of the thesis.

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34 CHAPTER 4 MODEL VERIFICATION CHAPTER 4 Verification of the analytical form of the solution obtained in Chapter 3 is required before using it to develop any desig n procedure. The following section deals with the verification of the solution obtained from Chapter 3 by theoretical and experimental means in addition to verifying it with a finite element model. Theoretical Verification To verify the theory, the solutio n for a classical plate obtained from the piezoelectric composite plate theory should match exactly the classical plate solution for a homogenous plate. The solution obtained from the theory will be equal to that of the classical plate solution if the equi valent flexural rigidity 11 D defined in Chapter 3 is same as the flexural rigidity D of the classical plate. The flexural rigidity of a classical circular plate is given by ( ) 3 2 1 21 Eh D n = ( 4 1 ) The equivalent flexural rigidity 11 D of the classical plate is given by 2 11 1 1 11 11 B DD A =. ( 4 2 ) Substituting the expression for flexural rigidity in Eq. ( 4 2 ) for a homogenous plate, we obtain

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35 2 2 *2 11 2 2 1 1 1 zh zh z zh z z E zdz E D z dz E dz n n n + + + =. ( 4 3 ) Pulling the constants out of the integral in the above expression, we obtain 2 *2 11 2 1 zh zh z zh z z zdz E D z dz dz n + + + =. ( 4 4 ) Computing the definite integrals in the above expression, we obtain 2 2 3 11 2 2 13 zh zh z zh z z z Ez D z n + + + =, ( 4 5 ) which on further simplification yields ( ) ( ) 2 2 2 3 3 11 2 2 13 z hz z hz E D h n ++=. ( 4 6 ) Using elementary arithmetic identities, the above expression is simplified to yield ( ) 2 2 23 11 2 2 33 1 34 zhh E z h z hh D n + ++ =, ( 4 7 ) which is simplified further to yield ( ) 2 23 3 22 11 2 44 1 34 z h z hh Eh D z h zh n ++ = ++. ( 4 8 ) The above expression on simplification yields

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36 ( ) 3 33 2 2 22 11 2 2 1 34 1 21 E h h Eh D z h z h z h z hD n n = ++--== ( 4 9 ) which implies that the equivalent flexural rigidity appearing in the piezoelectric circular composite plate theory reduces to flexural rigidity of a classical circular plate for a homogenous plate. Finite Element Model The fin ite element model of the piezoelectric unimorph disk was made for several extents of piezoelectric patch in ABAQUS CAE, 17 both for pressure loading and voltage loading. The geometry and the material properties us ed corresponding to Brass/PZT bender (Shim = Brass, Piezo = PZT) are tabulated in Table 4 1 A sample input file for each of the following cases of applied loading is attached in Appendix D of the thesis. Pressure Loading A short circuit across the PZT was modeled by applying an equipotential boundary condition on the top and bottom surfaces of the piezoelectric patch. A pressure load equivalent of 1000 Pa was applied on the bottom surface of the shim layer. The shim was meshed wi th 8 noded linear axisymmetric brick elements while the piezoelectric layer was meshed with similar brick elements with piezoelectric stresses. The geometry was scaled by a factor of 1000 (this is done in order to avoid numerical truncation errors in the s olver) and the pressure loading was diminished by a factor of 1000 in order that the output deflections are directly in meters. The maximum static deflection ( ) 0 0 w for each case was determined both from the analytical solution (obtain ed from the MATLAB code attached in Appendix C) and the finite element

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37 model. The results were plotted as a function of normalized radius of the piezoelectric patch used as shown in Figure 4 1 Table 4 1 : Properties of the piezoelectric unimorph disk used in the finite element model Geometrical properties Outer radius (radius of the mount) 500 m m Radius of the piezoelectric layer 0 500 m m Thickness of the shim 5 m m T hickness of the piezoelectric layer 2 m m Mechanical properties Youngs modulus of the shim 90 Gpa Poissons ratio of the shim 0.3 Density of the shim 8700 kg/m 3 Youngs modulus of the piezoelectric layer 30 Gpa Poissons ratio of the piezoelectric la yer 0.3 Density of the piezoelectric layer 7500 kg/m 3 Electric and Dielectric properties Relative permitivity of the piezoelectric layer 1000 Piezoelectric constant responsible for an extension in 1 direction due to application of field in the 3 direct ion ( 31 d ) 50 pC/N Plot of the center deflection of a circular piezoelectric unimorph obtained from the analytical solution, described in Eqs. (B 31 ) and (B. 37 ) matches the solution obtained from the finite element model to within 1%. Furthermore, mesh refinement studies

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38 indicated that the finer the mesh, the lesser the deviation of the solution obtained from the finite element model to that obtained from the theory. (The case shown in Figure 4 1 corresponds to the case where spacing between the nodes was 0.001 mm.) 0 0.2 0.4 0.6 0.8 1 5.5 6 6.5 7 7.5 8 8.5 9 9.5 x 10 -7 Normalized Piezoelectric Patch Radius R 1 /R 2 Center Deflection (m) Analytical FEM Figure 4 1 : Comparison of maximum deflection for different radii of the piezoelectric material as predicted by the analytical solution and finite element model for pressure application. Voltage Loading The finite element model mad e for pressure loading was modified to yield a finite element model of the piezoelectric unimorph disk subjected to voltage loading. In this case, the pressure loading was reduced to zero. A potential boundary condition of unit strength was applied on the top surface of the piezoelectric layer while the bottom surface

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39 was retained at zero potential. Other parameters were retained at the same value as in the previous case. Figure 4 2 shows that the plot of the center deflection of a c ircular piezoelectric unimorph obtained from the analytical solution, described in Eqs. (B 31 ) and (B. 37 ) matches the solution obtained from the finite element model made for voltage loading to within 1%. Mesh refinement studies indicated lesser deviation with a finer mesh similar to the case with pressure loading. (The case shown in Figure 4 2 corresponds to the case where spacing between the nodes was 0.001 mm.) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 x 10 -7 Normalized Piezoelectric Patch Radius R 1 /R 2 Center Deflection (m) Analytical FEM Figure 4 2 : Comparison of maximum deflection for different radii of the piezoelectric material as predicte d by the analytical solution and finite element model for a unit voltage loading.

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40 Experimental Verification In order to further validate the model, experiments were conducted in the Dynamics and Control Laboratory at the University of Florida. Scanning Laser Vibrometer Piezoelectric composite circular plate PZT Incident laser Reflected laser Brass V Figure 4 3 : A schematic of the experimental setup showing the laser vibrometer focusing on the clamped PZT unimorph bender. A periodic chirp signal of 5 V amplitude with frequency ranging from ~100 Hz to 4000 Hz was applied across the circular piezoelectric unimorph (APC International Ltd. Model APC850) to determine its natural frequency and mode shape. A Polytech PI laser scanning laser vibrometer (MSV200), shown in Figure 4 3 was used to measure the transverse deflection of the clamped piezoelectric composite plate due to the application of voltage across the piezoelectric material. The geometry and material properties of the piezoelectric unimorph b ender used for experiment are shown in Table 4 2

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41 Table 4 2 : Properties of the piezoelectric bender APC 850 Geometric properties Mechanical properties Electric and Dielectric propertie s 2 R 11.7 mm s E 89.6 GPa r e 1750 1 R 10.0 mm s n 0.324 31 d 175 pC/N s h 0.221 mm s r 8700 kg/m 3 p h 0.234 mm p E 63 GPa p n 0.31 p r 7700 kg/m 3 Experiments were then performed by applying a voltage of 5 VAC at 100 Hz via conductive copper tape attached to the two sides of the composite plate and the laser was scanned across the surface. Frequency (Hz) 100 1000 10000 m m/V 0.01 0.1 1 10 100 Figure 4 4 : Measured displacement fre quency response function obtained by converting velocity measurements using 1 j w integrating factor.

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42 The 100 Hz test frequency was very small compared to the measured natural frequency of approximately 3360 Hz as shown in the freque ncy response of the piezoelectric unimorph disk transducer found in Figure 4 4 As the experiment was conducted at such a low frequency, mass effect was neglected and the mode shape obtained should approximate the st atic mode shape. Table 4 3 : Lumped element and two port parameters Parameter Theory Experiment Error % f 142 P aV AS C 1.40e 013 42 m s kg A M 13800 4 k gm A d 1.98e 011 3 mV EF C 20.8 nF EB C 18.0 nF K 0.37 s f 3620 Hz f 3360 Hz 7.6 15.9 o f 3890 Hz ( ) 0 0 w 0.107 mV m 0.0923 mV m 15.9 The res ults from the experiments were compared with the corresponding analytical solutions. They show matching of the mode shape to a considerable extent. However the deflection is off by 15.9%. The natural frequency, obtained from the

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43 experiment (around 3360 Hz ) does not lie between short circuit and open circuit resonant frequencies, obtained from the theory (see extracted lumped and two port network parameter table shown in Table 4 3 ). It is expected that, the natural frequency lie withi n this range. This is because, in the case of a piezoelectric with a short (potential difference across the terminals is zero), the natural frequency will correspond to short circuit resonant frequency and a piezoelectric in static case (zero current) woul d correspond to the open circuit case. Since the experiment is performed in a dynamic condition rather than static and with a value of voltage applied across the terminals, the resonant frequency is expected to lie in between the open circuit and the short circuit value. Table 4 3 shows the value of the lumped element and two port network parameters obtained from the theory. It is found that, the values obtained deviates from the theory to a considerable extent. Causes for Deviation of the Experimental Results from Theory It should be noted that the theory neglects the bond layer between the piezoceramic patch and the brass shim and thus assumes that a perfect bond exists between the piezoceramic and the shim. In this case, the estima ted bond layer thickness is 25 m m (1 mil), which must be accounted for. In addition, the axisymmetric assumption implies that the circular piezoceramic patch is bonded in the center of the shim. Commercial unimorphs exhibit some non uniformity in this regard. Furthermore, the piezoceramic patch has a thin metal electrode layer (e.g., silver) of a different radius. Fringing field effects are not modeled in the present calculations. Silver electrode also effects in mechanically stiffe ning the plate, which is not accounted

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44 for in this calculation. An accurate analysis would involve three sections viz. a three layered inner composite disc, a two layered composite annular plate and an outer annular plate. Finally, it should be emphasized that an ideal clamped boundary is difficult to achieve in practice. In the current experimental setup, thick clamp plates are used in conjunction with several bolts uniformly spaced around the circumference.

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45 CHAPTER 5 NON DIMENSIONALIZATION CHAPTER 5 The analytical expressions for the transverse deflections derived in Chapter 3 of the thesis, are too co mplicated to obtain any sort of a physical insight into scaling. To facilitate design of a piezoelectric unimorph using the analytical solution, a non dimensional representation of the transverse deflection is necessary. Such a representation of the transv erse deflection would prove to be a good design tool. The Poissons ratio of the shim materials and piezoelectric materials are close to each other with a value around 0.3. Hence, an assumption that the Poissons ratio of the shim and piezoelectric materi al are the same, would simplify the problem. Furthermore, the governing equations contain terms that are either ratios of the Poissons ratio or ratios of the difference of unity and square of Poissons ratio. These quantities are still closer to unity. Th erefore, in order to simplify the non dimensionalization procedure, the effect of Poissons ratio is ignored. Morris and Foster 4 have also neglected the effect of Poissons ratio. Non dimensionalization is carried out using the Buckingham p Theorem. 18 Buckingham p Theorem The independent parameters involved in this simplified problem are p E s E p h s h 1 R 2 R 31 d V P 0 w and 0 u which are 11 independent varia bles. The three dimensions involved in the problem are that of length L force F and voltage V Therefore there should exist eight (11 3 = 8) independent non dimensional variable s.

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46 To design the best piezoelectric disc for a particular shim, it is better to non dimensionalize the variables with respect to the shim variables. This leaves s E to be the basic dimension to non dimensionalize variables with unit s of pressure and leaves s h or 2 R to be the basic dimension to non dimensionalize variables with units of length. Choosing 2 R for normalizing the length scale, we obtain p s E E s P E 1 2 R R 2 s h R 2 p h R 31 2 dV R 0 2 w R and 0 2 u R to be the 8 non dimensional variables. T he basic field variables can be expressed in non dimensional form as 0 31 1 01 2 22 22 ,,,,, pp s ss Eh w h dV R P wf R EERR RR == ( 5 1 ) and 0 31 1 02 2 22 22 ,,,,, pp s ss Eh u h dV R P uf R EERR RR == ( 5 2 ) However, such a representation will not provide any so rt of physical insight that would facilitate design. Hence a non dimensional form of the primary variables, which could be expressed in the best possible form, is required. Morris and Foster 4 represented the optim al radius ratio ( ) opt r and thickness ratio ( ) opt t (i.e. their field variables) as functions of aspect ratio a Youngs modulus ratio b and a ratio D defined in Eq. ( 5 3 ) ; ( ) ( ) 1 opt rg a ab = ( ) ( ) 2 opt tg ab = and

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47 ( ) ( ) * 3 opt t gD a = where 1 2 R r R = p s t t t = and (2) 11 3 2 p D D ER = ( 5 3 ) They numerically found the functional dependence of their field variables on the fore mentioned ratios and developed some empirical relationships. Since the problem discussed in this thesis, is not directly concerned with the optimization of a particular device, the field variables are chosen to be 0 u and 0 w A non dimensional representation of these field variables, are conventionally obtained by non dimensionalizing the plate equations. However, non dimensionalizing the plate equations is als o a complex task. Therefore, governing equations that provide physical insight are used for the purpose of non dimensionalization. The response of the central composite plate, when subjected to a pressure P and voltage V, is represented by ( ) (1 ) (1) 01 ur ar = ( 5 4 ) ( ) 3 (1 ) (1) 2 (1) 11 16 Pr r ar D q =, ( 5 5 ) ( ) ( ) (1 ) (1 ) (1) 1 11 1 P r Nr A aN n =+( 5 6 ) and ( ) ( ) ( ) 2 (1 ) (1 ) (1) 1 12 3 1 16 P r Pr M r D aM n n + = -+ +, ( 5 7 )

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48 where the number (1) in superscript indicates deflections and generalized forces of the central composite plate along an axis where coupling matrix vanishes. For the outer annular plate, [ ] B vanishes because of the symmetry about its neutral axis. Hence the response of the outer annular plate is given by ( ) (2) (2 ) (2) 1 01 b ur ar r =+ ( 5 8 ) ( ) (2) 3 (2 ) (2) 2 2 (2) 11 16 b Pr r ar rD q = ++ ( 5 9 ) ( ) ( ) ( ) (2) (2 ) (2 ) (2 ) (2) 1 1 1 1 11 2 11 r b Nr A aA r nn =+-( 5 10 ) and ( ) ( ) ( ) ( ) 2 (2) (2 ) (2 ) (2 ) (2) 2 1 1 2 11 2 3 11 16 r Pr b Mr D aD r n nn + =+--+ ( 5 11 ) where the number (2) in superscript indicates deflections and generalized forces of the outer annular plate. Non Dimensional Deflection for Pressure Loading The non dimensional verti cal deflection ( ) 0 wr is obtained from the expression shown in Eq. ( 5 12 ) The symbol in the superscript indicates a non dimensional quantity. ( ) 1 1 (1 ) (2) 0 0 R r R w r d r dr qq =+ ( 5 12 ) Substituting 2 r r R = in Eq. ( 5 12 ) we obtain

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49 ( ) 1 (1 ) (2) 02 0 w r R d r dr h h qq =+ ( 5 13 ) The non dimensional form of the slope is obtained by dividing the slope with the forcing term in Eq. ( 5 9 ) to yield (1) (1) 3 2 (2) 11 16 P PR D q q = and (2) (2) 3 2 (2) 11 16 P PR D q q = ( 5 14 ) The letter P in the superscript is used to represent non dimensio nal parameters that are obtained when the piezoelectric unimorph is subjected to pressure loading alone. Even though the slope by itself is non dimensional, the above form is more useful because it does not vary with change in loading and the overall dimen sions. Substituting Eq. ( 5 14 ) in Eq. ( 5 13 ) we obtain ( ) ** 1 0 (1 ) (2) 3 2 2 0 (2) 11 1 16 PP wr d r dr PR R D h h qq =+ ( 5 15 ) Eq. ( 5 15 ) implies that the non dimensionalized vertical deflection is given by ( ) ( ) 0 0 4 2 (2) 11 16 P wr wr PR D = ( 5 16 ) The non dimensional vertical displacement obtained in Eq. ( 5 16 ) is not dependent on the aspect ratio of the shim (see Figure 5 4 and Figure 5 6 ). It is dependent on the ratios of the radius, thickness and Youngs modulus. ( ) 1 0 2 ,, pp P ss Eh R wrf EhR = ( 5 17 ) The non dimensional form of the field variable shown in Eq. ( 5 17 ) is much simpler than the form indicated in Eq. ( 5 1 ) The best way to represent the whole set of deflections of a piezoelectric unimorph in a compact form is to plot the non dimensional

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50 center deflection against the radius ratio 12 RR for different values of ps hh for a particular ps EE as shown in Figure 5 1 Figure 5 5 The value of 2 / s hR used in the first five plots of each of the non dimensional variables discussed in the section is 0.01. Commercially available disc benders manufactured by APC International limited have the Youngs modulus ratio varying between 0.6 and 0.8. The Youngs modulus of PZT deposited in MEMS level device can be as low as 30 GPa. The Youngs modulus of the shim layer (Silicon) is around 150 GPa resulting in a Youngs modulus ratio of 0.2. Since, Silicon is a moderately anisotropic material with an aniso tropic coefficient 19 of 1.57 (which is close to unity. For an isotropic material, the value of anisotropic coefficient is unity), the Silicon layer is treated as transversely isotropic in this analysis. Hence val ues of 0.02, 0.2, 0.4, 0.6 and 0.8 were selected for the ratio of the Youngs Modulii. The Youngs modulus ratio value was taken to as small as 0.02 to accommodate PVDF Aluminum benders. In order to show that these non dimensional variables do not vary w ith change in aspect ratio 2 / s Rh a plot of the non dimensional variable with value of Youngs modulus ratio at 0.6 and the value of 2 / s hR at 0.02. The 4 th and 6 th plots of each of the non dimensional variables d iscussed in the following section are exactly the same even though the aspect ratios are different. This proves the non dependent nature of the non dimensional variables on aspect ratio. The denominator is multiplied with a factor of 0.25, to make its valu e equal to that of the central deflection of a clamped classical circular plate acted upon by a pressure load;

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51 ( ) ( ) 0 0 4 2 (2) 11 64 P wr wr PR D = ( 5 18 ) The expression for non dimensional vertical deflection is represented in terms of ratios (mentioned in the beginning of this chapter) as ( ) ( ) ( ) 0 0 4 2 3 2 16 1 21 P ss wr wr PR Eh n = ( 5 19 ) which on simplification yields ( ) ( ) ( ) 0 2 0 3 2 2 3 1 4 P ss wr R wr R P Eh n = ( 5 20 ) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.75 0.8 0.85 0.9 0.95 R 1 /R 2 w 0 P (0) h p /h s = 0.00001 0.3 0.6 0.9 1.2 E p /E s = 0.02 Figure 5 1 : Non dimensional plot of the center deflection of a piezoelectric unimorph disc with 0.02 ps EE = subjected only to a pressure load.

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52 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.3 0.4 0.5 0.6 0.7 0.8 0.9 R 1 /R 2 w 0 P (0) h p /h s = 0.00001 0.3 0.6 0.9 1.2 E p /E s = 0.2 Figure 5 2 : Non dimensional plot of the center deflection of a piezoelectric unimorph disc with 0.2 ps EE = subjected only to a pressure load. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 R 1 /R 2 w 0 P (0) h p /h s = 0.00001 0.3 0.6 0.9 1.2 E p /E s = 0.4 Figure 5 3 : Non dimensional plot of the center deflection of a piezoelectric unimorph disc with 0.4 ps EE = subjected only to a pressure load.

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53 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 R 1 /R 2 w 0 P (0) h p /h s = 0.00001 0.3 0.6 0.9 1.2 E p /E s = 0.6 Figure 5 4 : Non dimensional plot of the center deflection of a piezoelectric unimorph disc with 0.6 ps EE = subjected only to a pressure load. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 R 1 /R 2 w 0 P (0) h p /h s = 0.00001 0.3 0.6 0.9 1.2 E p /E s = 0.8 Figure 5 5 : Non dimensional plot of the center deflection of a piezoelectric unimorph disc with 0.8 ps EE = subjected only to a pressure load.

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54 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 R 1 /R 2 w 0 P (0) h p /h s = 0.00001 0.3 0.6 0.9 1.2 E p /E s = 0.6 Figure 5 6 : Non dimensional plot of the center deflection of a piezoelectric unimorph disc with 0.6 ps EE = and 2 0.02 s hR = subjected only to a pressure load. Non Dimensionalization for Voltage Loading Equations ( 5 4 ) ( 5 11 ) are modified for the case when voltage alone is applied in the following manner: ( ) (1 ) (1) 01 ur ar = ( 5 21 ) ( ) (1 ) (1) 2 r ar q = ( 5 22 ) ( ) ( ) (1 ) (1 ) (1) 1 11 1 P rr Nr A aN n =+, ( 5 23 ) ( ) ( ) (1 ) (1 ) (1) 1 12 1 P rr M r D aM n = -+, ( 5 24 ) ( ) (2) (2 ) (2) 1 01 b ur ar r =+ ( 5 25 )

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55 ( ) (2) (2 ) (2) 2 2 b r ar r q =+ ( 5 26 ) ( ) ( ) ( ) (2) (2 ) (2 ) (2 ) (2) 1 1 1 1 11 2 11 r b Nr A aA r nn =+-( 5 27 ) and ( ) ( ) ( ) (2) (2 ) (2 ) (2 ) (2) 2 1 1 2 11 2 11 r b Mr D aD r nn =+-. ( 5 28 ) The expression for non dimensional vertical deflection ( ) 0 V wr is obtained from the expression ( ) 1 1 (1 ) (2) 0 0 R r R w r d r dr qq =+ ( 5 29 ) Substituting 2 r r R = in Eq. ( 5 29 ) we obtain ( ) (1 ) (2) 02 0 r w r R d r dr h h qq =+ ( 5 30 ) The non dimensional form of the slope when subjected to voltage loading alone is obtained by dividing the slope with an equivalent of the forcing term found in Eq. ( 5 24 ) to yield (1) (1) 2 (2) 11 V P r MR D q q = and (2) (2) 2 (2) 11 V P r MR D q q = ( 5 31 ) The letter V in the superscript is used to represent non dimensional parameters that are obtained when the piezoelectric unimorph is subjected to voltage loading alone. Even though the slope by itself is no n dimensional, the above form is more useful because it

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56 does not vary with change in loading, piezoelectric constant, relative permitivity of the piezoelectric and the overall dimensions. Substituting Eq. ( 5 31 ) in Eq. ( 5 30 ) we obtain ( ) ** 1 0 (1 ) (2) 2 0 2 (2) 11 1 VV P r wr d r dr R MR D h h qq =+ ( 5 32 ) Equation ( 5 32 ) implies that the non dimensional vertical deflection is given by ( ) ( ) 0 0 2 2 (2) 11 V P r wr wr MR D = ( 5 33 ) The non dimensional vertical displacement obtained in Eq. ( 5 33 ) is also independent of aspect ratio and is only a function of the radius ratio, thickness ratio and Youngs modulus ratio, which is much simpler to represent than the form i ndicated in Eq. ( 5 1 ) ; ( ) 1 0 2 ,, pp V ss Eh R wrf EhR = ( 5 34 ) The best way to represent the whole set of deflections is to plot it in the same manner as in the case of pressure loading. The denominator was multiplied by a factor of 1 1 n + to make its value equal to that of the deflection obtained for a classical circular plate acted on by a moment equal to piezoelectric couple shown in Eq. ( 3 27 ) The revised expression is given by ( ) ( ) ( ) 0 0 2 2 (2) 11 1 V P r wr wr MR D n = + ( 5 35 ) The expression for non dimensional vertical defle ction is represented in terms of ratios (mentioned in the beginning of this chapter) in the following manner:

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57 ( ) ( ) ( ) ( ) ( ) 0 0 2 3 12 3 2 1 1 1 21 V p ss ps s s pp ss wr wr E Eh dV h hR E h Eh Eh n n n = + -+ + ( 5 36 ) (i.e.) ( ) ( ) 0 0 2 2 31 2 2 1 1 21 pp V ss pp s ss Eh wr Eh wr R Eh dV R E h hR + = + ( 5 37 ) The var iation of the center deflection of a unimorph subjected only to a voltage load, with variation in non dimensional ratios is delineated in Figure 5 7 Figure 5 12 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.1 0.15 0.2 0.25 R 1 /R 2 w 0 V (0) h p /h s =0.00001 0.3 0.6 0.9 1.2 E p /E s =0.02 Figure 5 7 : Non dimensional plot of the center deflection of a piezoelectric unimorph disc with 0.02 ps EE = subjected only to a voltage load.

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58 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.1 0.15 0.2 0.25 R 1 /R 2 w 0 V (0) h p /h s =0.00001 0.3 0.6 0.9 1.2 E p /E s =0.2 Figure 5 8 Non dimensional plot of the center deflection of a piezoelectric unimorph disc with 0.2 ps EE = subjected only to a voltage load. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.1 0.15 0.2 0.25 R 1 /R 2 w 0 V (0) h p /h s =0.00001 0.3 0.6 0.9 1.2 E p /E s =0.4 Figure 5 9 : Non dimensional plot of the center deflection of a piezoelectric unimorph d isc with 0.4 ps EE = subjected only to a voltage load.

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59 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.1 0.15 0.2 0.25 R 1 /R 2 w 0 V (0) h p /h s =0.00001 0.3 0.6 0.9 1.2 E p /E s =0.6 Figure 5 10 : Non dimensional plot of the center deflection of a piezoelectric unimorph disc with 0.6 ps EE = subjected only to a voltage load. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.1 0.15 0.2 0.25 R 1 /R 2 w 0 V (0) h p /h s =0.00001 0.3 0.6 0.9 1.2 E p /E s =0.8 Figure 5 11 : Non dimensional plot of the center deflection of a piezoelectric unimorph disc with 0.8 ps EE = subjected only to a voltage load.

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60 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.1 0.15 0.2 0.25 R 1 /R 2 w 0 V (0) h p /h s =0.00001 0.3 0.6 0.9 1.2 E p /E s =0.6 Figure 5 12 : Non dimensional plot of the center deflection of a piezoelectric unimorph disc with 0.6 ps EE = and 2 0.02 s hR = subjected only to a voltage load. To further enable design and optim ization of a circular piezoelectric unimorph, non dimensional lumped element and two port network parameters are needed. Non Dimensionalization of Lumped Element and Two Port Network Parameters Some of the lumped element and two port network parameters are obtained from the transverse deflection by expressions described in Chapter 2 of this thesis. The non dimensional parameters of these variables, has the same dependence on the corresponding dimensional variables, as the transverse deflection. In other wor ds, the equivalent non dimensional parameter is obtained from the corresponding dimensional parameter by dividing it with the value of the parameter for a classical circular plate. T he non dimensional short circuit acoustic compliance is given by

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61 ( ) AS AS AS Shim C C C = ( 5 38 ) where ( ) AS Shim C represents the short circuit acousti c compliance of the shim alone; ( ) ( ) 62 2 3 1 16 s AS Shim ss R C Eh pn = ( 5 39 ) Above expression for ( ) AS Shim C in terms of non dimensional ratios is given by ( ) ( ) 4 2 2 2 2 1 1 16 s A Ss Shim ss Rh R P C E hP p n =. ( 5 40 ) The non dimensional acoustic compliance is also a function of the Youngs modulus ratio, thickness ratio and radius ratio. The best way to represent these variables will be similar to that of the deflections. Figure 5 13 Figure 5 18 delineates such plots of short circuit acoustic compliance of the piezoelectric unimorph. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.75 0.8 0.85 0.9 0.95 R 1 /R 2 C A S h p /h s = 0.00001 0.3 0.6 0.9 1.2 E p /E s = 0.02 Figure 5 13 : Non dimensional short circuit aco ustic compliance plots for 0.02 ps EE =

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62 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.3 0.4 0.5 0.6 0.7 0.8 0.9 R 1 /R 2 C A S h p /h s = 0.00001 0.3 0.6 0.9 1.2 E p /E s = 0.2 Figure 5 14 : Non dimensional short circuit acoustic compliance plots for 0.2 ps EE = 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 R 1 /R 2 C A S h p /h s = 0.00001 0.3 0.6 0.9 1.2 E p /E s = 0.4 Figure 5 15 : Non dimensional short circuit acoustic compliance plots for 0.4 ps EE =

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63 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 R 1 /R 2 C A S h p /h s = 0.00001 0.3 0.6 0.9 1.2 E p /E s = 0.6 Figure 5 16 : Non dimensional short circuit acoustic compliance plots for 0.6 ps EE = 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 R 1 /R 2 C A S h p /h s = 0.00001 0.3 0.6 0.9 1.2 E p /E s = 0.8 Figure 5 17 : Non dimensional short circuit acoustic compliance plots for 0.8 ps EE =

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64 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 R 1 /R 2 C A S h p /h s = 0.00001 0.3 0.6 0.9 1.2 E p /E s = 0.6 Figure 5 18 : Non dimensional short circuit acoust ic compliance plots for 0.6 ps EE = and 2 0.02 s hR = The non dimensional acoustic mass is given by ( ) A A A Shim M M M = ( 5 41 ) where ( ) A Shim M represents the acoustic mass of the shim alone; ( ) ( ) 2 2 1.8 A Shim A Shim M R r p = ( 5 42 ) where ( ) A Shim r represents the areal density of the shim layer. The non dimensional mass like the deflections are a function of the usual three ra tios. In addition it is also a function of density ratio. Density ratio, like Youngs Modulus ratio, is unique for a certain set of material. Aluminum/PVDF benders have

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65 Youngs modulus ratio of 0.02 0.05. Therefore the density ratio of Aluminum/PVDF bend ers (i.e. 1760/2700) is chosen for the plots shown in Figure 5 19 Figure 5 20 The Silicon/PZT benders have their Youngs Modulus ratio varying from 0.2 0.6. Therefore the density ratio of Silicon/PZT b enders (i.e. 7500/2300) is chosen for the plots shown in Figure 5 21 Figure 5 23 and Figure 5 26 The Brass/PZT benders have their Youngs Modulus ratio varying from 0.6 0.8. Therefore the density ratio of Brass/PZT (i.e. 7500/8700) benders produced by APC International Ltd. is chosen for the plots shown in Figu re 5 24 Figure 5 25 Figure 5 26 like the plots of previous non dimensional ratios, has all ratios similar to Figure 5 23 except the aspect ratio. The plots appear the same indicating that the non dimensional acoustic mass does not vary with change in aspect ratio. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 R 1 /R 2 M A E p /E s =0.02 h p /h s =0.00001 0.3 0.6 0.9 1.2 Figure 5 19 : Non dimensional acoustic mass plots for 0.02 ps EE = (Aluminum/PVDF).

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66 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 R 1 /R 2 M A E p /E s =0.05 h p /h s =0.00001 0.3 0.6 0.9 1.2 Figure 5 20 : Non dimensiona l acoustic mass plots for 0.05 ps EE = (Aluminum/PVDF). 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.5 2 2.5 3 3.5 4 4.5 R 1 /R 2 M A E p /E s =0.2 h p /h s =0.00001 0.3 0.6 0.9 1.2 Figure 5 21 : Non dimensional acoustic mass plots for 0.2 ps EE = (Silcon/PZT).

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67 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.5 2 2.5 3 3.5 4 4.5 R 1 /R 2 M A E p /E s =0.4 h p /h s =0.00001 0.3 0.6 0.9 1.2 Figure 5 22 : Non dimensional acoustic mass plots for 0.4 ps EE = (Silcon/PZT). 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.5 2 2.5 3 3.5 4 4.5 R 1 /R 2 M A E p /E s =0.6 1.2 0.9 0.6 0.3 h p /h s =0.00001 Figure 5 23 : Non dimensional acoustic mass plots for 0.6 ps EE = (Silcon/PZT).

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68 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 R 1 /R 2 M A E p /E s =0.6 h p /h s =0.00001 0.3 0.6 0.9 1.2 Figu re 5 24 : Non dimensional acoustic mass plots for 0.6 ps EE = (Brass/PZT). 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 R 1 /R 2 M A E p /E s =0.8 h p /h s =0.00001 0.3 0.6 0.9 1.2 Figure 5 25 : Non dimensional acoustic mass plots for 0.8 ps EE = (Brass/PZT).

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69 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.5 2 2.5 3 3.5 4 4.5 R 1 /R 2 M A E p /E s =0.6 1.2 0.9 0.6 0.3 h p /h s =0.00001 Figure 5 26 : Non dimensional acoustic mass plots for 0.6 ps EE = and 2 0.02 s hR = (Silcon/PZT). Unlike the parameters that were non dimensionalize d by quantities possessing a physical significance, some of the two port network parameters do not have quantities for the shim alone described. We can obtain the non dimensional form of the two port network parameters by finding the equivalent parameters for the shim alone from first principles. The effective acoustic piezoelectric coefficient as defined in Chapter 2 is obtained by computing the volume displaced by a unit application of voltage due to the vertical deflection field, analytical expression fo r which is obtained in Chapter 3. Therefore, the non dimensional form is derived by evaluating the volume displaced due to a displacement field obtained by the application of a moment equal to the piezoelectric couple generated on the shim of radius 2 R ;

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70 ( ) A A A Shim d d d = ( 5 43 ) where ( ) A Shim d repre sents the effective acoustic piezoelectric coefficient of the piezoelectric with dimensions of the shim that is obtained from the following expression: ( ) ( ) ( ) 2 22 2 (2) 11 0 2 1 p R r A Shim MRr rdr D d V p n + = ( 5 44 ) Substituting for the terms appearing in the above expression, we obtain ( ) ( ) ( ) ( ) 2 22 3 12 3 0 2 1 2 1 1 21 p R s ss A Shim E d V h Rr rdr Eh d V n p n n + = ( 5 45 ) Pulling the constants out of the integral in the above expression, we obtain ( ) ( ) 2 31 22 2 3 0 24 R ps A Shim ss Edh d R r rdr Eh p =. ( 5 46 ) The above expression is simplified as ( ) 2 2 2 2 31 6 p A Shim ss E R d Rd Eh p = ( 5 47 ) which can be written as ( ) 3 2 3 12 2 2 6 p s A Shim ss E d V Rh R d R E hV p = ( 5 48 )

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71 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.01 0.02 0.03 0.04 0.05 0.06 0.07 R 1 /R 2 d A h p /h s = 0.00001 0.3 0.6 1.2 0.9 E p /E s = 0.02 Figure 5 27 : Non dimensional effective acoustic piezoelectric coefficient plots for 0.02 ps EE = 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 R 1 /R 2 d A h p /h s = 0.00001 0.3 0.6 1.2 0.9 E p /E s = 0.2 Figure 5 28 : Non dimensional effective acoustic piezoelectric coefficient plots for 0.2 ps EE =

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72 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 R 1 /R 2 d A h p /h s = 0.00001 0.3 0.6 1.2 0.9 E p /E s = 0.4 Figure 5 29 : Non dimensional effective acoustic piezoelectric coefficient plots f or 0.4 ps EE = 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 R 1 /R 2 d A h p /h s = 0.00001 0.3 0.6 1.2 0.9 E p /E s = 0.6 Figure 5 30 : Non dimensional effective acoustic piezoelectric coefficient plots for 0.6 ps EE =

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73 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 R 1 /R 2 d A h p /h s = 0.00001 0.3 0.6 1.2 0.9 E p /E s = 0.8 Figure 5 31 : Non dimensional effective acoustic piezoelectric coefficient plots for 0.8 ps EE = 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 R 1 /R 2 d A h p /h s = 0.00001 0.3 0.6 1.2 0.9 E p /E s = 0.6 Figure 5 32 : Non dimensional effective acoustic piezoelectric coefficient plots for 0.6 ps EE = and 2 0.02 s hR =

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74 The best way to represent this non dimensional parameter is the same as that of the previously obtained parameters (see Figure 5 27 Figure 5 32 ). It is found th at A d is directly proportional to the square of the aspect ratio. Non dimensional electrical free capacitance was defined in the same manner as the ratio of the electrical free capacitance of the piezoelectric composite circular p late to capacitance of a piezoelectric disc with the dimensions of the shim. (i.e.) ( ) EF EF EF Shim C C C = ( 5 49 ) where ( ) 2 2 EF Shim s R C h ep = ( 5 50 ) (i.e.) 2 2 1 1 2 2 2 p EF p s s R R h R C h R h h ep ep == ( 5 51 ) Now that the non dimensional form of all the independent two port parameters and lumped p arameters are known, non dimensional representation of other parameters can be found directly by replacing the form of variables visible in the expressions with the corresponding non dimensional variables. Hence the non dimensional representation for A f is given by * A A AS d C f = ( 5 52 ) Eq. ( 5 52 ) implies that

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75 ( ) ( ) AS shim AA A shim C d ff = ( 5 53 ) Substituting Eqs. ( 5 40 ) and ( 5 47 ) in Eq. ( 5 53 ) we obtain ( ) 4 2 2 2 2 3 2 3 12 2 2 1 1 16 6 s ss AA p s ss Rh R P E hP E d V Rh R R E hV p n ff p = ( 5 54 ) which on simplification, gives ( ) 31 2 2 2 1 9 61 p s AA ss E dV RE P V R P Eh ff n = ( 5 55 ) The plots of non d imensional A f are shown in Figure 5 33 Figure 5 38 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 R 1 /R 2 f A h p /h s = 0.00001 0.3 0.6 0.9 1.2 E p /E s = 0.02 Figure 5 33 : Non dimensional A f for 0.02 ps EE =

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76 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 R 1 /R 2 f A h p /h s = 0.00001 0.3 0.6 0.9 1.2 E p /E s = 0.2 Figure 5 34 : Non dimensional A f for 0.2 ps EE = 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 R 1 /R 2 f A h p /h s = 0.00001 0.3 0.6 0.9 1.2 E p /E s = 0.4 Figure 5 35 : Non dimensional A f for 0.4 ps EE =

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77 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 R 1 /R 2 f A h p /h s = 0.00001 0.3 0.6 0.9 1.2 E p /E s = 0.6 Figure 5 36 : Non dimensional A f for 0.6 ps EE = 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.01 0.02 0.03 0.04 0.05 0.06 0.07 R 1 /R 2 f A h p /h s = 0.00001 0.3 0.6 0.9 1.2 E p /E s = 0.8 Figure 5 37 : Non dimens ional A f for 0.8 ps EE =

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78 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 R 1 /R 2 f A h p /h s = 0.00001 0.3 0.6 0.9 1.2 E p /E s = 0.6 Figure 5 38 : Non dimensional A f for 0.6 ps EE = and 2 0.02 s hR = The non dimensio nal representation for A f is obtained in the same manner as that of A f and is given by * A A EF d C f = ( 5 56 ) Substituting Eqs. ( 5 43 ) and ( 5 49 ) in Eq. ( 5 56 ) we obtain ( ) ( ) A A shim A EF EF shim d d C C f = ( 5 57 ) which on simplification, gives ( ) ( ) EF shim AA A shim C d ff = ( 5 58 ) Substituting Eqs. ( 5 48 ) and ( 5 50 ) in Eq. ( 5 58 ) we obtain

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79 2 2 3 2 3 12 2 2 6 s AA p s ss R h E d V Rh R RE hV ep ff p = ( 5 59 ) Multiplying both sides of the above equation with the denominator of right hand side of Eq. ( 5 59 ) we obtain 3 2 31 2 2 6 p s AA ss E d Vh R RE hV ff e = ( 5 60 ) which on simplification, gives 2 31 22 31 6 p s AA sss E dE RR PV EhE d VP ff e = ( 5 61 ) Figure 5 39 Figure 5 44 depicts the variation of non dimensional A f with variation in non dimensional ratios. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.05 0.1 0.15 0.2 0.25 0.3 R 1 /R 2 f A E p /E s =0.02 h p /h s =0.00001 0.3 0.9 1.2 0.6 Figure 5 39 : Non dimensional A f for 0.02 ps EE =

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80 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 R 1 /R 2 f A E p /E s =0.2 h p /h s =0.00001 0.3 0.9 1.2 0.6 Figure 5 40 : Non dimensional A f for 0.2 ps EE = 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.01 0.02 0.03 0.04 0.05 0.06 R 1 /R 2 f A E p /E s =0.4 h p /h s =0.00001 0.3 0.9 1.2 0.6 Figure 5 41 : Non dimensional A f for 0.4 ps EE =

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81 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 R 1 /R 2 f A E p /E s =0.6 h p /h s =0.00001 0.3 0.9 1.2 0.6 Figure 5 42 : Non dimensional A f for 0.6 ps EE = 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 R 1 /R 2 f A E p /E s =0.8 h p /h s =0.00001 0.3 0.9 1.2 0.6 Figure 5 43 : Non dimensional A f for 0.8 ps EE =

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82 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 R 1 /R 2 f A E p /E s =0.6 h p /h s =0.00001 0.3 0.9 1.2 0.6 Figure 5 44 : Non dimensional A f for 0.6 ps EE = and 2 0.02 s hR = The coupling coefficient represents the ideal fraction of energy transduced to the other domain and by definition is non dimensional. The coefficient provides an indication of the electroacoustic energy conversion for the unimorph, but does not yield the actual value because it does not account for electrical and mechanical losses as well as electrical and mechanical loads. 1 Therefore the equivalent representation for K will be referred to as the universal representation for the coupling coefficient, since this representation is independent of the electric and dielectric properties of the piezoelectric layer. It is denoted by K and is given by ( ) 2 * ** A A S EF d K CC = ( 5 62 ) which on simplification yields

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83 * ** A A S EF d K CC = ( 5 63 ) Substituting Eqs. ( 5 40 ) ( 5 48 ) and ( 5 50 ) in Eq. ( 5 63 ) we obtain ( ) 2 31 2 1 4 1 p s s E dE KK E ne = ( 5 64 ) It is noted that K is directly proportional to 31 / d e Even though Youngs modulus r atio appears in the relationship, nothing can be said about the dependency of K on Youngs modulus ratio. Since these parameters depend on the fore mentioned parameters, the best possible representation of these non dimensional pa rameters is in no way different from the representation of those basic parameters. Figure 5 45 Figure 5 50 depict the variation of K with variation in non dimensional ratios. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 R 1 /R 2 K E p /E s =0.02 h p /h s =0.00001 0.3 0.6 0.9 1.2 Figure 5 45 : K for 0.02 ps EE =

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84 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 R 1 /R 2 K E p /E s =0.2 h p /h s =0.00001 0.3 0.6 0.9 1.2 Figure 5 46 : K for 0.2 ps EE = 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 R 1 /R 2 K E p /E s =0.4 h p /h s =0.00001 0.3 0.6 0.9 1.2 Figure 5 47 : K for 0.4 ps EE =

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85 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.01 0.02 0.03 0.04 0.05 0.06 R 1 /R 2 K E p /E s =0.6 h p /h s =0.00001 0.3 0.6 0.9 1.2 Figure 5 48 : K for 0.6 ps EE = 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 R 1 /R 2 K E p /E s =0.8 h p /h s =0.00001 0.3 0.6 0.9 1.2 Figure 5 49 : K for 0.8 ps EE =

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86 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.01 0.02 0.03 0.04 0.05 0.06 R 1 /R 2 K E p /E s =0.6 h p /h s =0.00001 0.3 0.6 0.9 1.2 Figure 5 50 : K for 0.6 ps EE = and 2 0.02 s hR = The non dimensional variables obtained in this chapter are dependent only on the thickness ratio, Youngs modulus ratio and the ratio of the radii of the shim and the piezoelectric material. The plots shown in Figure 5 1 Figure 5 50 depict variation of the different non dimensional parameters with respect to these non dimensional ratios. Therefore these charts can be used in creating a procedure for design of the best piezoelectric unimorph disc for a chosen device.

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87 CHAPTER 6 CONCLUSIONS AND FUTURE WORK Two port electroacoustic model of a piezoelectric composite circular plate has been developed in this thesis. The analysis developed for the mechanical behavior of an axisymmetric piezoelectric multi layer ed composite circular plate in this thesis is for any axisymmetric circular piezoelectric transducer. The solution converges to already existing classical plate solutions upon simplification. FEM indicates that the center deflection matches with the theor y to within 1% as predicted by Merhaut. 12 The experiments with the Brass PZT benders using laser vibrometry indicate that the modeshape is predicted to a considerable level of accuracy. However the natural freque ncy and center deflection are off by 15.9%. This is attributed to the problem with clamping, bond layer thickness and presence of a silver electrode. The experimentally measured frequency lies close to the predicted natural frequency range indicating the v alidity of the lumped assumption. A set of non dimensional parameters was identified for the problem of piezoelectric axisymmetric composite disc transducers. These parameters are useful in representing the deflection field of the composite plate in a com plete manner for a wide range of system parameters. These parameters also have been useful in determining non dimensional two port network variables, some of which are directly the system parameters that need to be optimized for maximum performance of the piezoelectric transducer for the device of our choice.

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88 Some of the non dimensional two port network parameters do not vary even with change in material properties in addition to not varying with change in the overall dimension. Based on the device requirem ent and design constraints the parameter that needs to be extremized can be pulled out of the non dimensional charts discussed in Chapter 5. The value of the non dimensional parameter, the corresponding radius ratio, thickness ratio and Youngs modulus rat io can be read from the charts in Chapter 5. The non dimensional parameters so read can be converted to their actual form by using the relationship between the corresponding dimensional and non dimensional forms in Chapter 5. Thus the device specifications can be determined. If the shim is designed, the charts can be used to find the best piezoelectric patch that would make the device function optimally. The non dimensional parameters also facilitate a possibility of development of a higher order design pro cedure for determining an optimal geometry of the piezoelectric material for maximum actuation/ sensing/ energy reclamation/ impedance tuning. A synthetic jet using PZT Brass disc as the actuating element and a MEMS based PZT mi crophone are being designed at the Interdisciplinary Microsystems Group at the University of Florida using a similar procedure. Future work will involve extending the theory for plates with large initial in plane stresses. A large deflection analysis of th e circular piezoelectric composite plate will extend the usage to high displacement actuators like the THUNDER 20 and RAINBOW. 20

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89 APPENDIX A DETAILED DERIVATION OF THE GEN ERAL SOLUTION FROM PLATE CONSTITUTIVE EQUATIONS Rewriting Eq. ( 3 30 ) from Chapter 3, we have 00 1 1 1 2 1 1 12 00 2 1 2 2 2 1 22 ( ) () ( ) () ( ) () ( ) () P r r P d u r ur d rr A A BB N N d rr d rr N d u r ur d rr N A A BB d rr d rr q q qq qq + -=+ -. (A. 1 ) Rewriting Eq. ( 3 31 ) from Chapter 3, we have 00 1 1 1 2 1 1 12 00 2 1 2 2 2 1 22 ( ) () ( ) () ( ) () ( ) () P r r P d u r ur d rr B B DD M M d rr d rr M d u r ur d rr M B B DD d rr d rr q q qq qq + -=+-. (A. 2 ) Restating equations of equilibrium from Chapter 3, we have ( ) r r r MM dM Q d rr q + =, (A. 3 ) ( ) 0 r r NN dN d rr q += (A. 4 ) and ( ) 0 rr d QQ Pr d rr + += (A. 5 ) For the case described in Figure 1 1 Eq. (A. 5 ) becomes () rr d QQ P rP d rr + = =. (A. 6 )

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90 Multiplying both sides by r (integration factor) and integrating once with respect to the radius r we obtain 2 () 2 r Pr rQPrrdrPrdrc =-=-=+ (A. 7 ) Applying the boundary condition that, () 2 r Prc Qr r =+ at rR = (A. 8 ) we obtain 2 () 22 r PRPR QR R p p -== (A. 9 ) whic h implies 0 c = Hence () 2 r Pr Qr = (A. 10 ) Substituting back Eq. (A. 10) in Eq. (A. 3), we obtain ( ) 1 0 2 r r dM Pr MM drr q -+-= (A. 11 ) Substituting Eq. (A. 1) in Eq. (A.4), we obtain 00 11121112 00 11121112 00 21222122 ()() ()1 () ()() ()1 0 () 1 ()() ()1 () p r p r p durur ddr AABBrN drdrrdrr durur dr AABBrN drrdrr durur dr r AABBrN drrdrr q q q q q q q +---+ = +---+--, (A. 12 )

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91 which is simplified as 2 2 000 1111 2222 ()()() 1()1()() 0 durdurur drdrr AB drrdrrdrrdrr qqq ++-++= (A. 13 ) This simplification occurs because the piezoelectric properties and the field are uniform on the r q plane making ( ) ( ) 1 ppp rr d NNN drr q +go to z ero. Similarly substituting Eq. (A. 2 ) in Eq. (A. 11), we obtain 2 2 000 1111 2222 ()()() 1()1()() 2 durdurur drdrrPr BD drrdrrdrrdrr qqq ++-++= (A. 14 ) Substituting s re = in Eq. (A. 13 ) and Eq. (A. 14 ) we obtain ( ) ( ) ( ) 2 11011 10 DAusBs q --= (A. 15 ) and ( )( ) ( ) 3 2 11011 1 2 s Pe DBusDs q --= (A. 16 ) Eq. (A. 15 ) and Eq. (A. 16 ) can be written as: ( ) [ ] 2 1 10 Dy -= (A. 17 ) and ( )[ ] 3 2 2 1 2 s Pe Dy -= (A. 18 ) where, ( ) ( ) 111011 yAusBs q =, (A. 19 ) and ( ) ( ) 211011 yBusDs q =. (A. 20 )

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92 We know that, general sol ution to the above equations are given by 1 ss y a e be =+ (A. 21 ) and 2 ss p y c e d ey = ++ (A. 22 ) (i.e.) 1 b y ar r =+ (A. 23 ) and 2 p d y c ry r =++ (A. 24 ) If 1 y and 2 y (which are linear combinations of 0 u and q ) were just functions of r and 1 r then 0 u and q will also be in the same form as 1 y and 2 y which results in the following expression of the primary variables: 2 01 a u a r particula r solution r =++ (A. 25 ) and 2 1 b b r particula r solution r q =++ (A. 26 ) The particular solution to the equation ( ) [ ] 3 2 2 1 2 s Pe Dy -= (A. 27 ) is given by

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93 ( ) 3 33 2 r 2 1 6 16 31 ss p P e P eP y = == (A. 28 ) Thi s correction for particular solution is found by inverting the relationship between 12 yy and 0 u q in the following manner: ( ) 1 1 11 0 3 2 1 1 11 1 1 1 1 11 0 1 16 DB u Pr BA A DB q = (A. 29 ) The quantities 2 11 1 1 11 11 B DD A -= (A. 30 ) and 11 11 B A a = (A. 31 ) are defined to simplify the solution. The resulting expression for solution is given by 3 2 01 11 16 a Pr u ar rD a =+, (A. 32 ) and 3 2 1 11 1 16 b Pr br rD q =+, (A. 33 ) which is same as the solutions shown in Eq. ( 3 36 ) and Eq. ( 3 37 ) of the equa tion of Chapter 3.

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94 APPENDIX B ANALYTICAL SOLUTION An analytical solution is obtained in terms of variables that are functions o f primary variables in the following manner. Analytical Expression for Radial Defection 0 u and Slope q Equations ( 3 44 ) ( 3 47 ) ( 3 56 ) and ( 3 58 ) is simplified by defining the following constants: ( ) (1 ) (1) 1 1 1 12 eq A AA =+ ( ) (1 ) (1) 1 1 1 12 eq B BB =+ ( ) (1 ) (1) 1 1 1 12 eq D DD =+ (1) 12 1 *(1) 11 A f D = (1) 12 2 *(1) 11 B f D = (1) 12 3 *(1) 11 D f D = 2 1 16 PR P = ( ) ( ) 2 2 4 2 1 11 ss R f R nn = + +, and ( ) ( ) 4 2 5 4 1 31 ss R f R nn = + +. (B. 1 ) In the inner composite region, generalized forces at t he interface described in Eq. ( 3 49 ) simplifies to the form shown in Eq. (B. 2 ) by using the above constants; ( ) ( ) (1) (1) 21 1 1 11 (1) (1) 23 1 1 11 3 e q eq P r e q eq P r ff NR A Ba N P ff MR B Db M a a = +-+ (B. 2 ) Rewriting Eq. ( 3 48 ) we have ( ) ( ) (1) (1) 3 01 1 1 1 (1) *(1) (1) 1 11 1 1 16 uR a PR R R D b a q =.

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95 In the outer annular region, generalized forces at the interface described in Eq. ( 3 59 ) on substituting the con stants described in Eq. (B. 1 ) reduces to ( ) ( ) (2) (2)(2) 1 111 4 (2) (2)(2) 5 1 111 0 0 0 r r NR Aa fP f MR Db =+ (B. 3 ) Rewriting Eq. (3.56), we obtain ( ) ( ) (2) (2) 2 0 1 2 4 (2) (2) (2) 3 2 11 1 0 16 ur a R P r R r rD b r r q =-. Substituting Eqs. ( 3 48 ) and ( 3 56 ) in interface conditions shown in Eqs. ( 3 60 ) & ( 3 61 ) we obtain ( ) ( ) (2) (1)(2) 32 01 11 12 4 11 3 (2) *(1)(2) (1)(2)2 1 1 11111 11 1 0 1 1616 uR aa PRR P RR R R R DRD bb R a q -==-. (B. 4 ) Dividing both s ides of the above equation by 1 R and combining terms containing P we obtain *(1) (1)(2) 2 11 11 2 2 (1)(2) 4 1 11 2 (2)4*(1) 11111 1 11 1 D aa R P R bb R DRD a =--. (B. 5 ) The above expression in matrix form is simplified to yield 2 (1)(2) 2 11 2*(1) 111 1 R P aa RD a =-+ (B. 6 ) and 24 ** (1)(2) 22 11 2(2)4*(1) 111111 11 RR PP bb RDRD =---+ (B. 7 )

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96 Substituting Eqs. (B. 2 ) (B. 3 ) in interface compatibility conditions shown in Eqs. ( 3 62 ) ( 3 63 ) we obtain (1 ) (2 ) (2) 21 ** 1 1 1 1 11 4 (1 ) (2 ) (2) 2 35 1 1 1 1 11 0 3 e q eq P e q eq P ff A B a Aa N P fP f ff B D b Db M a a + + =+ ++ -. (B. 8 ) Rewriting the above expression, we obtain (2 ) (2 ) (1) 21 1 1 1 1 11 4 (2 ) (2 ) (1) 2 35 1 1 1 1 11 3 e q eq P e q eq P ff A a A Ba N fP f ff D b B Db M a a = ++ +-(B. 9 ) Eq. (B. 9 ) is simplified b y defining the following constants: 121 tff a =(B. 10 ) and 22 35 3 tf ff a = + +. (B. 11 ) Substituting Eqs. (B. 10 ) (B. 11 ) in Eq. (B. 9 ) we obtain (1 ) (1 )* (2) 1 1 1 11 1 (2) 1 14 e q e qP A a B b P tN a Af += (B. 12 ) and (1 ) (1 )* (2) 1 1 1 12 1 (2) 1 14 e q e qP D b B a P tM b Df -+ = (B. 13 ) Substituting for (1) 1 a and (1) 1 b from Eqs. (B. 6 ) (B. 7 ) in Eqs. (B. 12 ) and (B. 13 ) we obtain 2 24 ** (2 ) (2 )* 2 22 1 1 1 11 2 (1 ) 2 (2 ) 4 *(1) 1 1 1 1 1 1 1 11 (2) 1 (2) 1 14 1 11 e q e qP R RR P PP A a B b P tN R D R D RD a Af a ++ += (B. 14 ) and

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97 2 42 ** (2 ) (2 )* 2 22 1 1 1 12 2 (2 ) 4 *(1 ) 2 *(1) 1 1 1 1 1 1 1 11 (2) 1 (2) 1 14 1 11 e q e qP R RR P PP D b B a P tM R D R D RD b Df a ++ -+ = (B. 15 ) The above expressions are simplified to yield 2 4* ** (2 )* 2 21 1 11 2 (2 ) 4 *(1 ) *(1) 1 1 1 1 1 1 11 (2) 1 2 (2) 2 1 1 41 2 1 11 1 eq e qP eq R R PA PP B b P tN R D R DD a R A fA R a + +-+ = -(B. 16 ) and 2 4* (2 )* 2 1 21 1 12 2 *(1 ) (2 ) 4 *(1) 1 1 1 1 1 1 11 (2) 1 2 (2) 2 1 1 41 2 1 11 1 e q eq e qP eq R P D R PD P B a P tM R DD RD b R D fD R a + + -+ = -. (B. 17 ) Eqs. (B. 14 ) (B. 17 ) are simplified by defini ng the following constants: 2 (2) 2 1 1 1 41 2 1 1 eq R g A fA R = -, (B. 18 ) 2 2 21 2 1 1 eq R gB R =(B. 19 ) and 2 (2) 2 3 1 1 41 2 1 1 eq R g D fD R = -. (B. 20 ) Substituting Eqs. (B. 18 ) (B. 20 ) in Eqs. (B. 14 ) (B. 17 ) we obtain 4 ** (2 )* 12 11 2 11 (2 ) 4 *(1 ) *(1) 1 1 1 1 1 11 (2) 1 1 1 e q e q eq P P B R P B PA g b P tN D R DD a g a + +-+ = (B. 21 ) and

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98 * 4* (2 )* 1 121 2 12 *(1 ) (2 ) 4 *(1) 1 1 1 1 1 11 (2) 1 3 1 e q e q eq P P B P D R PD g a P tM DD RD b g a + -+ = (B. 22 ) Substituting Eq. (B. 22 ) in Eq. (B. 21 ) we obtain 4 11 21 22 (1 ) (2 ) 4 *(1 )* 1 1 1 1 1 11 (2) 1 2 4 3 12 1 2 11 31 (2 ) 4 *(1 ) *(1) 1 1 1 1 1 11 1 1 e q e q eq P e q e q eq P B D RD M gt D D R DP P a g gg B R BA N gt D R D PD a a + + -+ = + -+ (B. 23 ) Similarly substituting Eq. (B. 21 ) in Eq. (B. 22 ) we obtain 4 1 2 11 21 (2 ) 4 *(1 ) * (1) 1 1 1 1 1 11 (2) 1 2 4 3 12 11 21 12 *(1 ) (2 ) 4 *(1 )* 1 1 1 1 1 11 1 1 e q e q eq P e q e q eq P B R BA N gt D R D PD P b g gg B D RD M gt D D R DP a a + + -+ = + -+ (B. 24 ) Substituting Eq. (B. 24 ) in Eq. (B. 6 ) we obtain 4 2 11 21 2 22 *(1 ) (2 ) 4 *(1 )* 2 1 1 1 1 1 11 1 (1) 1 2 (1) 4 3 1 2 11 1 2 11 31 (2 ) 4 *(1 ) *(1) 1 1 1 1 1 11 1 1 1 e q e q eq P e q e q eq P B D RD M R gt P D D R DP R P a g g gD B R BA N gt D R D PD a a a + + -+ =+ + -+ (B. 25 ) Substituting Eq. (B. 23 ) in Eq. (B. 7 ) we obtain 4 2 1 2 11 2 21 (2 ) 4 *(1 ) *(1) 2 4 1 1 1 1 1 11 1 (1) 2 1 2 (2 )4 4 3 1 2 1 11 1 1 21 12 *(1 ) (2 ) 4 *(1 )* 1 1 1 1 1 11 1 1 1 1 e q e q eq P e q e q eq P B R BA N R gt P D R D PD R R P b g g g DR B D RD M gt D D R DP a a + + -+ = -+ -+ *(1) 11 P D + (B. 26 ) Equations (B. 23 ) (B. 26 ) gives the expression for the constants appearing in the solution shown in Eqs. ( 3 44 ) and ( 3 56 ) Hence we have an analytical solution for the primary field variables radial defection ( ) 0 ur and the slope ( ) r q

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99 Analytical Expression for Vertical D eflection 0 w Vertical deflection ( ) 0 wr is calculated by integrating the slope ( ) r q with respect to the radius in each of the regions and by applying the fixed boundary condition at the c lamp in the outer annular region and by applying the compatibility condition that the deflection should be continuous at the interface in the central composite region. Therefore, in the outer annular region, ( ) ( ) (2 ) (2) 0 w r r d r Const q =+ (B. 27 ) (i.e.) ( ) ( ) ( ) ( ) 44 2 2 (2 ) (2 )2 012 (2) 11 4 ln ln 2 64 P R rr r wr b R r const D = ++ (B. 28 ) By applying the clamped boundary condition that ( ) (2) 02 0 wR = (B. 29 ) in the expression for ( ) (2) 0 wr gi ven in Eq. (B. 28 ) we obtain the expression for vertical deflection in the outer annular region, which is given by ( ) 4 44 22 22 2 (2 ) (2 )2 2 0 12 (2) 2 11 4 ln ln 2 64 r P R rR R rR r wr bR RD -+ = -+ (B. 30 ) Hence the vertical deflection at the interface is given by ( ) 4 44 1 2 12 22 2 (2 ) (2 )2 1 21 01 12 (2) 2 11 4 ln ln 2 64 R P R RR R R RR wR bR RD -+ = -+ (B 31 ) In the inner composite region the vertical deflection is given by ( ) ( ) (1 ) (1) 0 w r r d r Const q =+ (B. 32 )

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100 (i.e.) ( ) 24 (1 ) (1) 01 *(1) 11 2 64 r Pr wr b const D = -+ (B. 33 ) Thus the vertical deflection at the interface is given by ( ) 24 (1 ) (1) 11 011 (1) 11 2 64 R PR wR b const D = -+ (B. 34 ) By applying the interface compatibility condition that ( ) ( ) (1 ) (2) 0101 wRwR = (B. 35 ) in Eqs. (B. 34 ) and (B 31 ) we obtain 4 44 1 2 12 2 2 42 2 (2 ) 2 (1) 1 2 1 11 1 21 (2 ) *(1) 2 1 1 11 4 ln ln 2 6 4 6 42 R P R RR R R R R P RR cons t b Rb R DD -+ = + +. (B. 36 ) By substituting the constant obtained in Eq. (B. 36 ) in Eq. (B. 33 ) we have the expression for vertical deflection in the central composite region as ( ) ( ) 44 2 2 22 1 (1 ) (2 )2 1 1 21 1 12 (1) 1 12 (1) 0 4 44 1 2 12 2 (2) 11 ln 2 6 42 4 ln 64 P rR r R R RR b bR DR wr R P R RR R D -+ -+ = -+ (B. 37 ) Analytical Expression for Short Circuit Acoustic Compliance AS C The expression for short circuit acous tic compliance is obtained by finding the ratio of the volume displaced by the unimorph due to applied pressure to the magnitude of the pressure load (i.e. 0 V = );

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101 12 1 (1 ) (2) 00 0 0 ( ) 2 ( )2 RR R V AS wr rd rwr rdr C P pp = + = (B. 38 ) Substituting Eqs. (B. 37 ) and (B. 30 ) in Eq. (B. 38 ) we obtain ( ) 1 44 2 2 22 1 (1 ) (2 )2 1 1 21 1 12 *(1) 1 12 4 44 1 0 2 12 2 (2) 11 4 2 22 2 (2 )2 2 12 2 ln 2 6 42 4 ln 2 64 4 ln ln 2 R AS P rR r R R RR b bR DR rdr R P R RR R C D P r P Rr R rR r bR R p -+ -+ + -+ = -+ 2 1 44 2 (2) 11 0 64 R R V R rdr D = + (B. 39 ) The above expression on simplification yields ( ) ( ) 1 2 1 2 22 4 21 (1 ) (2) 1 11 4 44 1 2 12 44 1 2 *(1 ) (2) 1 1 11 0 4 44 22 2 (2) 11 88 4 ln 2 6 4 64 4 ln 64 R AS R R RR R bb R P R RR P rR R C rdr P DD r P R rR R rdr D p ++ -+ = ++ -+ 0 V = (B. 40 ) which is simplified furt her to yield the following expression:

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102 ( ) ( ) 1 2 1 2 22 4 21 (1 ) (2) 1 11 4 44 1 2 12 54 1 2 *(1 ) (2) 1 1 11 0 4 54 22 2 (2) 11 88 4 ln 2 6 4 64 4 ln 64 R AS R R RR R bb R r P R RR P r rR R C dr P DD r P R r r rR R dr D p ++ -+ = ++ -+ 0 V = (B. 41 ) Computing the definite integrals found in the above expression, we obtain ( ) 2 1 2 22 4 *4 21 (1 ) (2) 11 11 (1) 11 4 44 1 2 12 2 (2) 11 4 2 6 6 24 1 2 1 2 1 12 2 (2) 11 8 8 12 1 2 l n 33 2 24 1 2 l n 43 384 AS R R RR R PR bb D R P R RR R C PD R P R R R R RR R D p + ++ -+ =+ ++ 0 V = (B. 42 ) The above expression on simplification yields the analytical expression for the short circuit a coustic compliance of the piezoelectric unimorph given by ( ) ( ) 2 2 2 44 62 4 *4 2 1 12 21 (1 ) (2) 11 11 *(1 ) (2) 1 1 11 0 32 2 8 8 1 2 12 AS V R R P RR RR R PR C bb P DD p = += + ++ (B. 43 ) where

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103 4 1 2 11 21 (2 ) 4 *(1 ) *(1) 1 1 1 1 1 11 (2) 1 2 4 3 12 11 21 12 *(1 ) (2 ) 4 (1) 1 1 1 1 1 11 1 1 e q e q eq e q e q eq B R BA gt D R DD P b g gg B D RD gt D D RD a a + -+ = +, (B. 44 ) and 4 2 1 2 11 2 21 (2 ) 4 *(1 ) *(1) 2 4 ** 1 1 1 1 1 11 1 (1) 2 1 2 (2 )4 4 3 1 2 1 1 1 11 1 1 21 12 *(1 ) (2 ) 4 *(1) 1 1 1 1 1 11 1 1 1 1 e q e q eq e q e q eq B R BA R gt P D R DD R R PP b g g g D RD B D RD gt D D RD a a + -+ = -+ +*(1) (B. 45 ) Analytical Expression for Effective Acoustic Piezoelectric Coefficient A d The expression for effective acoustic piezoelectric coefficient is obtained by finding the ratio of the volume displaced by the unimorph due to an applied voltage with zero pressure loading; 12 1 (1 ) (2) 00 0 0 ( ) 2 ( )2 RR R P A wr rd rwr rdr d V pp = + = (B. 46 ) Substituting Eqs. (B. 37 ) and (B. 30 ) in Eq. (B. 46 ) we obtai n 1 2 1 2 2 22 (1 ) (2 )2 1 1 21 112 2 0 22 (2 )2 2 12 2 0 ln 22 2 ln 2 R A R R P r R R RR bb R rdr R d V rR r b R rdr R p = -+ -+ = (B. 47 ) Grouping terms containing r we obtain 1 2 1 3 2 22 (1 ) (2 )2 1 1 21 112 2 0 32 (2 )2 2 12 2 0 ln 22 2 ln 2 R A R R P r r R R RR b b rR dr R d V r rR r b R r dr R p = -+ -+ = (B. 48 )

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104 Computing the definite integrals found in the above expression, we obtain 4 2 22 (1 ) (2 )2 1 11 21 1 12 2 44 22 (2) 2 1 2 11 1 2 0 ln 8 22 2 ln 82 A P R R R RR b bR R d V RR R RR b R p = -+ -+ = + (B. 49 ) The above expression on simplification yields the analytical expression for the effective acoustic piezoelectric coefficient of the piezoelectric unimorph given by ( ) 2 22 4 21 (1 ) (2) 1 11 0 2 88 A P RR R d bb V p = =+ (B. 50 ) where, (2) 21 1 2 3 12 PP g N gM b g gg + = (B. 51 ) 2 (2) 2 12 1 22 3 1 21 1 PP g N g MR b g g gR + =, (B. 52 ) 31 1 p P p E N dV n = (B. 53 ) and ( ) 31 1 p P ps p E M dV hh n =+ (B. 54 ) Analytical e xpression for acoustic mass was too complicated and hence was obtained numerically.

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105 APPENDIX C MATLAB CODES Three MATLAB codes were programmed to implement the electroacoustic modeling discussed in the thesis. First code is used to derive the response of a particular system. The second code is used to extract curves shown in Chapter 5 of the thesis. The third code uses the direct analytical expressions obtained in Chapter 3. The MATLAB codes are split up into different modules, which are referred to as su b routines from the three main programs. The three main programs direct the tasks to different sub routines and output the necessary parameters. The sub routines and the main programs used are presented in this appendix. Subroutines used by Program 1 shim. m %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Material Properties of the Shim % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Es = 89.63e9; %Young's modulus of shim(Brass) vs = 0.324; %Poisson's ratio of shim ts = 2.0574e 004; %thickness of the shim layer(annular) R2 = 0.923*0.0254/2; %radius of the shim layer mu = 1; %const of proportionality of shear stress vs. shear strain densitys = 8700; %Density of the shim rhos = densitys*ts; %areal density of shim % %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% piezo.m %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Material Properties of the piezoelectric material % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %

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106 Ep = 63.00E9; %Young's modulus of the p iezoelectric material(PZT 5H)(APC855) vp = 0.33; %Poissons ratio of the piezoelectric material tp = 2.3368e 004; %thickness of the piezoelectric material R21 = 0.724; %piezoelectric material radii expressed as a fraction of shim radius R1 = R21*R 2; %Piezoelectric radius d31 = 2.7e 10; %electromechanical transduction const of the piezoelectric material densityp = 7700; %Density of the piezoelectric material rhop = densityp*tp; %Areal Density of the piezoelectric material epsilon0 = 8.85 E 12; %permitivity of free space in F/m dielectricconstant = 3250; %relative permitivity of the piezoelectric material epsilon = dielectricconstant*epsilon0; %absolute permitivity of the piezoelectric material P = 1; %Pressure in Pa applied on the top surface of the structure V = 0; %Voltage in V applied Ef = V/tp; %electric field in V applied in direction 3(Z) % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% customisedpiezo.m %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Material Properties of the piezoelectric material % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Ep = input(' \ nEnter Youngs modulus E of piezoelectric material layer (Eshim=89.63E9) in Pascals : '); vp = input(' \ nEnter Poisson''s ratio of piezoelectric material(Poisson''s ratio of shim=0.324) : '); tp = input(' \ nEnter Piezoelectric material Thickness(shim thickness=0.00023) in meters : '); R21 = input(' \ nEnter Radius of piezoelectric layer as a fraction of rad ius of shim(0.0127m): '); d31 = input(' \ nEnter d31 (electromagnetic transduction coefficient) for the piezoelectric : '); Densityp = input(' \ nEnter Density of piezoelectric(shim=16000kg/m3) : '); epsilon0 = input(' \ nEnter Relat ive Permitivity(Dielectric Constant of Piezoelectric : '); epsilon = epsilon*8.85e 12; %absolute permitivity of piezoelectric rhop = rhop*tp; %Areal Density R1 = R2*R21; %Radius of piezoelectric % reading in electrical and mechanical l oads P = input(' \ nEnter Pressure exerted on the composite section : '); Ef = input(' \ nEnter strength of Electric field applied across the piezoelectric : '); % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% initialise.m %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Initializing variables % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %

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107 Cp = 0; Fp = 0; num1=1400; r=linspace(0,1,num1+1); sr=size(r); U0=zeros(sr); theta=zeros(sr); % %%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%% adshim.m %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Computing A,D for the outer annular plate % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % AA = Es*ts/(1 (vs^2)); DD = Es*ts^3/(12*(1 (vs^2))); % %%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% abdpiezo1.m %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Computing A,B,D for the central composite section % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Constitutive Relations for isotropic circular plates Qs = [1 vs; vs 1].*(Es/(1 (vs^2))); Qp = [1 vp; vp 1].*(Ep/(1 (vp^2))); Qsi= [1 vsi; vsi 1].*(Esi/(1 (vsi^2))); % Computing A,B,D for the central composite section % taking reference plane as center of the shim layer z1 = ts/ 2; %distance of bottom of shim layer from reference zi = ts/2; %distance of interface from reference zi2 = ts/2 + tp; %distance of top of the piezoelectric layer from reference z2 = ts/2 + tp + tsi/2; %distance of top of silver fr om reference A = Qs.*(zi z1) + Qp.*(zi2 zi) + Qsi.*(zi2 zi); B = Qs.*((zi^2 z1^2)/2) + Qp.*((zi2^2 zi^2)/2) + Qsi.*((zi2^2 zi^2)/2); D = Qs.*((zi^3 z1^3)/3) + Qp.*((zi2^3 zi^3)/3) + Qsi.*((zi2^3 zi^3)/3); % Computing Delta(determinant o f matrix mapping defined variables y1,y2 to U0 theta

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108 delta=A(1,1)*D(1,1) ((B(1,1))^2); % Computing fictitious forces due to piezoelectric Cp=Ef* (Ep/(1 vp)) d31 (zi2^2 zi^2)/2; Fp=Ef* (Ep/(1 vp)) d31 (zi2 zi); % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% solver.m %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Actual Module, which sets up the system of equations and superposes the solutions % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % annular; (Sub routine call a nnular.m) sets up system of equations for the annular section composite; (Sub routine call composite.m) sets up system of equations for the composite section % Computing deflections and Forces at the interface by superposition if R1~=0 if R1~=R2 Keff=(KMat1 KMat2); Peff= (PMat1 PMat2); XMat=inv(Keff)*(Peff); end; end; % Finding constants in the system of equations for Deflection in the central composite region if R1~=0 Const1=inv(amat1)*(XMat f1); end; % Finding consta nts in the system of equations for Deflection in the annular region if R1~=R2 Const2=inv(amat2)*(XMat f2); FMat=bmat2*Const2+g2; end; % Initializing deflections to zero r=linspace(0,1,num1+1); sr=size(r); U0=zeros(sr); theta=zeros(sr); % Compu ting deflections at discrete num1 number of points jj=floor(R21*num1)+1; %finding the argument at the interface for i=1:num1+1 rad1 = r(i)*R2; if (i
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109 U0(i) = Const1(1)*rad1 B(1,1)*P*(rad1^3)/(16*delta); theta( i) = Const1(2)*rad1 A(1,1)*P*(rad1^3)/(16*delta); else U0(i) = Const2(1)*(rad1 (R2^2)/rad1); funcr = (rad1^3) (R2^4)/rad1; theta(i) = Const2(2)*(rad1 (R2^2)/rad1) P*funcr/(16*DD); end; end; % Determi ning vertical deflection by integration of theta at num1 number of points W0=zeros(sr); for i=jj+1:num1+1 W0(i)=( (R2^2)*log(r(i)) + ( (r(i)^2 1)*0.5*(R2^2) ) )*Const2(2) + (P*(R2^4)*log(r(i))/(16*DD)) P*((r(i)^4) 1)*(R2^4)/(64*DD); end; % Verti cal deflection at the interface Wint=( (R2^2)*log(R21)+((R21^2 1)*0.5*(R2^2)))*Const2(2) + (P*(R2^4)*log(R21)/(16*DD)) P*((R21^4) 1)*(R2^4)/(64*DD); WConst2=Wint ( (Const1(2)*(R1^2)/2) (A(1,1)*P*(R1^4))/(64*delta) ); W0(jj)=Wint; for i=1:jj 1 W0(i ) = (Const1(2)*((r(i)*R2)^2)/2) (A(1,1)*P*((r(i)*R2)^4))/(64*delta)+WConst2; end; % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% annular.m %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Module, which sets up the system of equations fo r the outer annular plate % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Compute stiffness due to annular plate KMat2 and % Setting up the system of equations KMat2*XMat=PMat2 if R1~=R2 R12=1/R21; factor=(1+(R12^2))+vs*(1 (R12^2)); b mat2=[factor*AA 0;0 factor*DD]; factor2= (R2^2 R1^2)/R1; amat2=[factor2 0;0 factor2]; func = (R1^3) (R2^4)/R1; f2=[0; P*func/(16*DD)]; difffunc = 3*(R1^2) + (R2^4)/(R1^2); g2=[0;(P/16)*(difffunc+vs*func/R1)]; KMat2=bmat2*inv(amat2); %stiffness matrix due to annular plate; PMat2=g2 bmat2*inv(amat2)*f2;

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110 else XMat=[0;0]; PMat2=[0;0]; KMat2=[0 0;0 0]; end; % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% composite.m %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% % % Module, which sets up the system of equations for the central composite plate % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Compute stiffness due to composite section KMat2 and % Setting up the system of equations KMa t1*XMat=PMat2 if R1~=0 amat1=[R1 0;0 R1]; %Deflection XMat= amat*Consts + f temp1=P*(R1^3)/(16*delta); f1=temp1*[ B(1,1); A(1,1)]; bmat1=[A(1,1)+A(1,2) (B(1,1)+B(1,2));B(1,1)+B(1,2) (D(1,1)+D(1,2))]; temp2=P*R1*R1/(16*delta); temp3= B(1,1) A(1,2) + A(1,1) B(1,2); temp5= B(1,1) B(1,2) + 3*delta + A(1,1) D(1,2); fict =[Fp;Cp]; g1=[temp2*temp3;temp2*temp5]; KMat1=bmat1*inv(amat1); %stiffness matrix (FMat=KMat*Xmat+PMat) KMat1=KMat1*[1 0;0 1]; PMat1=g1 bmat1*in v(amat1)*f1; PMat1=PMat1+fict; PMat1=[1 0;0 1]*PMat1; else XMat=[0;0]; PMat1=[0;0]; KMat1=[0 0;0 0]; end; % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% integ1.m %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Lumped Ele ment Modeling Parameter Extraction % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Computing Strain energy and hence compliance % Numerical integration using Trapezoidal rule temp1=r(1)*W0(1)*0.5+r(num1+1)*W0(num1+1)*0.5;

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111 for i=2:num1 temp 1=temp1+W0(i)*r(i); end; temp1=((temp1^2)^0.5)*R2*(R2/num1); se=pi*P*(temp1); %strain energy Area=pi*(R2^2); %total Area Weff=se/(0.5*P*Area); %Deflection volume velocity convention Cme=Weff/(P*Area); %Compliance volume velocity convention wr= W0 (1)/Weff; %ratio of the Displacement in the two conventions Cms=Cme*(wr^2); %Compliance max deflection convention % Computing Mass % Numerical integration using Trapezoidal rule temp3=(r(1)*(W0(1)^2)+r(jj)*(W0(jj)^2)); temp4=(r(jj)*(W0(jj)^2)+r(num1+1 )*(W0(num1+1)^2)); for i=2:jj 1 temp3=temp3+2*(W0(i)^2)*r(i); end; for i=jj+1:num1 temp4=temp4+2*(W0(i)^2)*r(i); end; temp3=temp3*R2*(R2/num1)*0.5*(rhos+rhop); temp4=temp4*R2*(R2/num1)*0.5*rhos; Mms=2*pi*(temp3+temp4)/(W0(1)^2); %Mass max d eflection Convention Mme=Mms*(wr^2); % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% oparam1.m %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Lumped Element Modeling Other Miscellaneous Parameter Extraction % %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Area1=pi*R1*R1; %Area of the inner composite section Area=pi*R2*R2; %Total Area actmass=rhos*Area+(rhop)*Area1; %Actual Mass of the plate massratio=Mme/actmass; %Effective Mass/Actual Mass frequency=(1/(2*pi))*(C me*Mme)^( 0.5); %Resonant Frequency sensitivity=Cme*frequency; % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% plotdef.m %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Sub routine to plot deflection curves %

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112 %%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%% % figure(1); plot(r,U0); Title(' Radial Deflection Vs Radius '); ylabel('Deflection u_0 ----->') xlabel('r / R2 ---->'); hold on; figure(2); plot(r,theta); Title(' Angular Deflection Vs Radius '); ylabel('Angular Deflection \ theta ----->') xlabel('r / R2 ---->'); hold on; figure(3); plot(r,W0); Title(' Vertical Deflection Vs Radius '); ylabel('Deflection w_0 ----->') xlabel('r / R2 ---->'); hold on; % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% plotdef1.m %%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Sub routine to plot deflection curves % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % figure(1); plot(r,U0,'r'); Title(' Radial Deflection Vs Radius red(V=1,P=0) blue(P=1,V=0) '); ylabel('Deflect ion u_0 ----->') xlabel('r / R2 ---->'); hold on; figure(2); plot(r,theta,'r'); Title(' Angular Deflection Vs Radius red(V=1,P=0) blue(P=1,V=0) '); ylabel('Angular Deflection \ theta ----->') xlabel('r / R2 ---->'); hold on; figure(3); plot(r,W0,'r'); T itle(' Vertical Deflection Vs Radius red(V=1,P=0) blue(P=1,V=0) '); ylabel('Deflection w_0 ----->') xlabel('r / R2 ---->'); hold on;% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% integG1.m

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113 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Two port Electromechanical model parameter extraction % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Computing Strain energy and hence compliance % Numerical integration using Trapezoidal rule temp1=r(1)*W0(1)*0.5+r(num1+1)*W0(num1+1)*0.5; for i=2:num1 temp1=temp1+W0(i)*r(i); end; temp1=((temp1^2)^0.5)*R2*(R2/num1); se=pi*(temp1); Area=pi*(R2^2); %total Area Weff=se/(0.5*Area); %Deflection volume velocity convention G=Weff/(V); %Transduction Impedance volume velocity de=W0(1); %transduction capacitance leverage= G/d31; %how much of d31 acts in the other direction wr= W0(1)/Weff; %ratio of the Displacement in the two conventions Cme=Cme*(wr^2); %Compliance max deflection convention Area1=pi*(R1^2); %Area of the piezo Ce f=epsilon*Area1/tp; %Electrical Free Capacitance % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Program 1: Program used to derive Response of a particular Piezoelectric Tranducer MAIN PROGRAM 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Programmed By % Suryanarayana A.N. Prasad % Dept. of AeMES % University of Florida % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % M A I N P R O G R A M 1 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % clear all; shim; %Reading shim properties(Subroutine call shim.m) choice = input(' \ nFor customised input enter 1, For standard set of input enter 2 : '); if (choice==1) customizedpiezo; %Reading piezo properties(Subroutine call customizedpiezo.m) else piez o; %Reading piezo props from console(Subroutine call piezo.m) end; initialise; %initialising sampling etc.(Subroutine call initialise.m) adshim; %(subroutine call adshim.m)

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114 % Response of the piezoelectric unimorph to unit pressure loading t si=0; abdpiezo1; %A,B,D Matrix extraction (subroutine call abdpiezo1.m) solver; %(subroutine call solver.m) % Lumped Element Modeling parameter extraction integ1; %(subroutine call integ1.m) oparam1; %(subroutine call oparam.m) Miscella neous parameters % output W0(1) %Centre Deflection(Maximum) plotdef; %(subroutine call plotdef.m) M1=FMat(2) %Interface Bending Moment F1=FMat(1) %Interface Radial Force Cme %Equivalent Mechanical Compliance Mme %Equivalent Mechanical Mass M assratio %Massratio frequency %Short circuit resonant Frequency sensitivity %Gain Bandwidth product dV1=Weff*Area %Volume displacement % Response of the piezoelectric unimorph to unit voltage loading % Initializing Voltage Loading P=0; V=1; Ef=V/t p; abdpiezo1; %(subroutine call abdpiezo1.m) solver; %(subroutine call solver.m) code to setup and solve the equations % output plotdef1; %(subroutine call plotdef1.m) deflection plot % Two port parameter extraction integG1; %(subrouti ne call integG1.m) two port network parameters % output Weff %Effective Displacement W0(1) %Maximum Displacement De = G %Transduction Compliance phi = Weff/Cme %phi phi1 = De/Cef %phi prime Cef %Electrical free Compliance K=De/((Cme*Cef)^0. 5) %lector acoustic coupling factor Ca=Cme*Area*Area %Acoustic Compliance Ma=Mme/Area/Area %Acoustic Mass dV2=Weff*Area %Volume displacement

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115 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Subroutines used only by Program 2 shim.m %%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Material Properties of the Shim % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Es = 89.63e9; %youngs modulus of shim(Brass) vs = 0.324; %poissons ratio of shim R2 = 2.54e 3/2; %radius of the shim lay er ts = R2*TSR2; %thickness of the shim layer(annular) mu = 1; %const of proportionality of shear stress vs shear strain densitys = 8700; %Density of the shim rhos = densitys*ts; %areal density of shim % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%% piezo2.m %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Material Properties of the piezo % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Ep = Es*EPES; %Youngs modulus of the piezo vp = 0.3; %Poissons ratio of the piezo material tp = ts*TPTS; %thickness of the piezo R1 = R21*R2; %Piezoelectric radius d31 = 5.0e 11; %electromechanical transduction const of the piezo densityp = 7600; %Density of the piezo rhop = densityp*tp; %Areal Density of the piezo epsil on0 = 8.85E 12; %permitivity of free space in F/m dielectricconstant = 1000; %relative permitivity of the piezo epsilon = dielectricconstant*epsilon0; %absolute permitivity of the piezo P = 1; %Pressure in Pa applied on the top surface of the struc ture V = 0; %Voltage in V applied Ef = V/tp; %electric field in V applied in direction 3(Z) silver; %load properties of silver(Sub routine call silver.m) % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% silver.m %%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Material Properties of the silver electrode % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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116 %properties obtained from http://www.web elements.com/webelements/elements/text/Ag/phys.html Esi=83E9; vsi=0.37; tsi=0.0011*0.0254; densitysi=10490; Rsi=0.718*0.0254; rhosi=densitysi*tsi*(Rsi/0.02)^2; % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% abdpiezo2.m %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%% % % Computing A,B,D for the central composite section % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Constitutive Relations for isotropic circular plates Qs = [1 vs; vs 1].*(Es/(1 (vs^2))); Qp = [1 vp; vp 1].*(Ep/(1 ( vp^2))); % Computing A,B,D for the central composite section % taking reference plane as centre of the shim layer z1 = ts/2; %distance of bottom of shim layer from reference zi = ts/2; %distance of interface from reference zi2 = ts/2 + tp; %distance of top of the piezo layer from reference z2 = ts/2 + tp + tsi/2; %distance of top of silver from reference A = Qs.*(zi z1) + Qp.*(zi2 zi); B = Qs.*((zi^2 z1^2)/2) + Qp.*((zi2^2 zi^2)/2; D = Qs.*((zi^3 z1^3)/3) + Qp.*((zi2^ 3 zi^3)/3; % Computing Delta(determinant of matrix mapping defined variables y1,y2 to U0 theta delta=A(1,1)*D(1,1) ((B(1,1))^2); % Computing fictitious forces due to piezo Cp=Ef* (Ep/(1 vp)) d31 (zi2^2 zi^2)/2; Fp=Ef* (Ep/(1 vp)) d31 (zi2 zi); % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Program 2: Program used to derive Response of a particular Piezoelectric Transducer MAIN PROGRAM 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Programmed By

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117 % Suryanarayana A.N. Prasad % Dept. of AeMES % University of Florida % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % M A I N P R O G R A M 2 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % clear all; count=0; EPES = 30/150; TSR2 = 1/112.5; rstep=0.005 ; tstep=0.3; tmax=1.3; tmin=0.00001; for TPTS=tmin:tstep:tmax count=count+1; count2=0; for R21=0:rstep:1 if R21==0 R21=0.000001; else if R21==1 R21=0.999999; end; end; count2=count2+1; % Analysis with unit pressure loading shim2; %Reading shim properties piezo2; %Reading piezo props from console initialise2; %initialising sampling etc. .(Subroutine call initialise.m) adshim; %(subroutine call adshim.m) abdpiezo1; %(subroutine call abdpiezo1.m) solver; %(subroutine call solver.m) integ1; %(subroutine call integ1.m) oparam1; %(subroutine call oparam1.m) Wnd1=W0(1)*64*DD/R2/P/(R2^3); %Non di mensional deflection for applied pressure WN1(count,count2)= Wnd1; %Non dimensional deflection for applied pressure Ca=Cme*Area*Area; %Acoustic Compliance Ma=Mme/Area/Area; %Acoustic Mass CND(count,count2)=Cme*Area *3*64*DD/R2^4; %Non dimensional Compliance MND(count,count2)=Mme/(1.8*rhos*Area); %Non dimensional Mass freqND(count,count2)=1/sqrt(CND(count,count2)*MND(count,count2)); %C1nd1=Const1*16*DD/R2^3 %C2nd1=Const2*16*DD/R2^3 % Analysis with unit voltage loading piezo; P=0;V=1;Ef=V/tp; abdpiezo2; %(subroutine call abdpiezo2.m) solver; %(subroutine call solver.m) integG1; %(subroutine call integG1.m) Wnd2 = W0(1)*DD*2*(1+vs)/Cp/R2/R2; %Non dimensional deflection for applied voltage

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118 WN2(count,count2) = Wnd2; %Non dimensional deflection for applied voltage phi(count,count2) = G*Area/Ca; %acoustic phi (pascals/Volt or C/m^3) phiND( count,count2) = leverage*TSR2^2/CND(count,count2); %acoustic phi (ND)(retains shape of phi) %phiND1(count,count2) = Wnd2/CND(count,count2); %acoustic phi (ND2) phiprime(count,count2) = G*Area/Cef; %acoustic phi' (m^3/C) K(count,c ount2) = G*Area/(Ca*Cef)^0.5; %K (ND) k=K(count,count2); %phiprimeND(count,count2) = leverage*TSR2^2; %acoustic phi' (ND) phiprimeND1(count,count2) = leverage*TSR2^2*TPTS/R21^2; %acoustic phi' (ND2)(retains shape of phi') % phiprimeND2(count,count2) = Wnd2*TPTS/R21^2; %acoustic phi' (ND3) %phiprimeND3(count,count2) = leverage*TSR2^2/R21^2; %acoustic phi' (ND4) %phiprimeND4(count,count2) = Wnd2/R21^2; %acoustic phi' (ND3) KND(count,count2) = sqrt(ph iprimeND1(count,count2)*phiND(count,count2)); %Universal K %optim(count,count2)=phi(count,count2)*Ca/Cef/Area/Area/(1 k^2)*TSR2; %C1nd2=Const1*DD/Cp/R2 %C2nd2=Const2*DD/Cp/R2 chargeV(count,count2)=Cef*(1 k^2)*V; if (count2>1) %if (R21<0.999999) inchargedensityV(count,count2 1)=(chargeV(count,count2) chargeV(count,count2 1))/rstep/ 2/pi/(R21 rstep/2); radii2(count,count2 1)=R21 rstep/2; %else %chargedensity(count,count2)=(charge(count,count2) charge(count,count2 1))/rstep/2/pi/R21; %end; %else %chargedensity(count,count2)=(charge(count,count2+1) charge(count,count2))/rstep/2/pi/R21; end; chargedensityV(count,count2)=chargeV(count,count2)/Area/R21/R21; chargeP(count,count2)=phi(count,count2)*Ca*Cef*(1 k^2)/(Cef*(1 k^2)+phi(count,count2)^2*Ca); chargedensityP(count,count2)=chargeP(count,count2)/Area/R21/R21; if (co unt2>1) %if (R21<0.999999) inchargedensityP(count,count2 1)=(chargeP(count,count2) chargeP(count,count2 1))/rstep/2/pi/(R21 rstep/2); end; radii(count2)=R21; end; figure(1); plot(radii,WN 1(count,:),'k'); hold on; drawnow; figure(2) plot(radii,WN2(count,:),'k'); hold on; drawnow; figure(3); plot(radii,CND(count,:),'k'); hold on; drawnow; figure(4)

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119 plot(radii,MND(count,:),'k'); hold on; drawnow; figure(5); plot(radii,phiND(count,:),'k'); hold on; drawnow; figure(6) %plot(radii,phiprimeND(count,:),'k'); %hold on; %drawnow; %figure(11); %plot(radii,phiND1(count,:),'k'); %hold on; %drawnow; %figure(12) plot(radii,phiprimeND1(count,:),'k'); hold on; drawnow; %figure(13) %plot(radii,phiprimeND2(count,:),'k'); %hold on; %drawnow; %figure(14) %plot(radii,phiprimeND3(count,:),'k'); %hold on; %drawnow; %figure(15) %plot(radii,phiprimeND4(count,:),'k'); %hold on; %drawnow; figure(7); plot(radii,phi(count,:),'k'); hold on; drawnow; figure(8) plot(radii,phiprime(count,:),'k'); hold on; drawnow; figure(9) plot(radii,K(count,:),'k'); hold on; drawnow; figure(10); %plot(radii,optim(count,:),'k'); %hold on; %drawnow; %figure(16) plot(radii,KND(count,:),'k'); hold on; drawnow; figure(11) plot(radii,chargeV(coun t,:),'k'); hold on; drawnow; figure(12)

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120 plot(radii2,inchargedensityV(count,:),'k'); hold on; drawnow; figure(13) plot(radii,freqND(count,:),'k'); hold on; drawnow; figure(14) plot(radii,chargedensityV(count,: ),'k'); hold on; drawnow; figure(15) plot(radii,chargeP(count,:),'k'); hold on; drawnow; figure(16) plot(radii,chargedensityP(count,:),'k'); hold on; drawnow; figure(17) plot(radii2,inchargedensityP(count,:), 'k'); hold on; drawnow; end; figure(1); %title('plot of non dimensional center deflection on application of unit pressure against Radius of piezo patch for Ep/Es=61/89.63 for different tp/ts'); xlabel('R_1/R_2'); ylabel('w(0)/(P*R_2^4/64D_2)'); fig ure(2); %title('plot of non dimensional center deflection on application of unit voltage against Radius of piezo patch for Ep/Es=61/89.63 for different tp/ts'); xlabel('R_1/R_2'); ylabel('w(0)*2*D_2*(1+ \ nu_s)/C_pR_2^2'); figure(3); %title('plot of non dime nsional center deflection on application of unit pressure against Radius of piezo patch for Ep/Es=61/89.63 for different tp/ts'); xlabel('R_1/R_2'); ylabel('Non Dimensional C_A'); figure(4); %title('plot of non dimensional center deflection on application of unit voltage against Radius of piezo patch for Ep/Es=61/89.63 for different tp/ts'); xlabel('R_1/R_2'); ylabel('Non Dimensional M_A'); figure(5); xlabel('R_1/R_2'); ylabel('Non Dimensional \ phi (leverage/C_N_D)'); figure(6); %xlabel('R_1/R_2'); %ylabel( 'Non Dimensional \ phi''(leverage)'); %figure(11); %xlabel('R_1/R_2'); %ylabel('Non Dimensional \ phi (W_N_D/C_N_D)'); %figure(12); xlabel('R_1/R_2'); ylabel('Non Dimensional \ phi'' (leverage/C_E_F_N_D)');

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121 %figure(13); %xlabel('R_1/R_2'); %ylabel('Non Dimens ional \ phi'' (W_N_D/C_E_F_N_D)'); %figure(14); %xlabel('R_1/R_2'); %ylabel('Non Dimensional \ phi'' (Leverage/C_E_F_N_D_2)'); %figure(15); %xlabel('R_1/R_2'); %ylabel('Non Dimensional \ phi'' (W_N_D/C_E_F_N_D_2)'); figure(7); xlabel('R_1/R_2'); ylabel('phi') ; figure(8); xlabel('R_1/R_2'); ylabel('phi'''); figure(9); xlabel('R_1/R_2'); ylabel('K'); figure(10); %title('plot of non dimensional center deflection on application of unit pressure against Radius of piezo patch for Ep/Es=61/89.63 for different tp/ts') ; %xlabel('R_1/R_2'); %ylabel('OPTIMAL PARAMETER'); %figure(16); %title('plot of non dimensional center deflection on application of unit pressure against Radius of piezo patch for Ep/Es=61/89.63 for different tp/ts'); xlabel('R_1/R_2'); ylabel('Universal K'); figure(11); xlabel('R_1/R_2'); ylabel('Q(charge) applied V'); figure(12); xlabel('R_1/R_2'); ylabel('incremental charge density V applied'); figure(13); xlabel('R_1/R_2'); ylabel('Non Dimensional Frequency'); figure(14); xlabel('R_1/R_2'); ylabel('Q/A applied V'); figure(15); xlabel('R_1/R_2'); ylabel('Q(charge) applied P'); figure(16); xlabel('R_1/R_2'); ylabel('Q/A applied P'); figure(17); xlabel('R_1/R_2'); ylabel('incremental charge density P applied'); % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%

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122 Program 3: Program implementing Direct Solution of a Particular Piezoelectric Transducer directsolution.m %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Programmed By % Suryanarayana A.N. Prasad % Dept. of AeMES % University of Flori da % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % M A I N P R O G R A M direct solution % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % shimg2; piezog2; initialise; abdpiezo2; Dstar=delta/A(1,1); alpha=B(1,1)/A(1,1); adshim; % Intermediate constants defined in chapter 3 Aeq=A(1,1)+A(1,2); Beq=B(1,1)+B(1,2); Deq=D(1,1)+D(1,2); f1=A(1,2)/Dstar; f2=B(1,2)/Dstar; f3=D(1,2)/Dstar; Pstar=P*R1^2/16; R21=R2/R1; f4=(1+vs)+R21^2*(1 vs); f5=(3+vs)+R21^4*(1 vs); %f5= f5; t1=(f2 f1*a lpha); t2=( alpha*f2+3+f3 f5); g1=AA*f4 (1 R21^2)*Aeq; g2=(1 R21^2)*Beq; g3=DD*f4 (1 R21^2)*Deq; temp=Pstar/(g3*g1 g2^2); Ar1=Aeq/Dstar; Ar2=Aeq/DD; Br1=Beq/Dstar; Br2=Beq/DD; Dr1=Deq/Dstar; Dr2=Deq/DD;

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123 % Analytical expression for constants temp2=(+Br 1*alpha+Dr2*(1 R21^4) Dr1+t2+Cp/Pstar); temp3=(Br2*(1 R21^4) Br1+t1+Fp/Pstar+alpha*Ar1); a12=temp*(g2*temp2+g3*temp3); a11=a12*(1 R21^2)+Pstar*alpha/Dstar; % temp4=( Br2*(1 R21^4)+Br1 t1 Fp/Pstar alpha*Ar1); % temp5=( Br1*alpha Dr2*(1 R21^4)+Dr1 t2 Cp/Psta r); b12= temp*(g1*temp2+g2*temp3); b11=b12*(1 R21^2)+Pstar/Dstar Pstar*(1 R21^4)/DD; interface_i=round(num1/R21); r=0:R2/(num1 1):R2; for i=1:num1 if i<=interface_i u0P(i)=a11*r(i) alpha*P*(r(i)^3)/16/Dstar; thetaP(i)=b11*r(i) P*(r(i)^ 3)/16/Dstar; w0P(i)=b11*((r(i)^2 R1^2)/2) P*(r(i)^4 R1^4)/64/Dstar+b12*((R1^2 R2^2)/2 R2^2*log(R1/R2))+P*(4*R2^4*log(R1/R2) R1^4+R2^4)/64/DD; else u0P(i)=a12*(r(i) R2^2/r(i)); thetaP(i)=b12*(r(i) R2^2/r(i)) P*((r(i)^3) (R2^4)/r( i))/16/DD; w0P(i)=b12*((r(i)^2 R2^2)/2 R2^2*log(r(i)/R2))+P*(4*R2^4*log(r(i)/R2) r(i)^4+R2^4)/64/DD; end; end; % Evaluation of short circuit acoustical compliance temp4=Pstar/12/Dstar; temp5=(R1^4); temp6=(b12*(((R2^2) (R1^2))^2)/8 (b11*(R1^4 ))/8); temp7=Pstar/12/DD; temp8= (R1^4)+(3*(R2^4)) 2*(R2^6)/(R1^2); Cas=2*pi*(temp6+temp4*temp5+temp7*temp8)/P; % output module figure(1); plot(r/R2,u0P,'b'); xlabel('Normalized Radius r/R_2'); ylabel('radial displacement u_0(r)'); figure(2); hold on; pl ot(r/R2,thetaP,'b'); xlabel('Normalized Radius r/R_2'); ylabel('slope \ theta (r)'); figure(3); hold on; plot(r/R2,w0P,'b'); xlabel('Normalized Radius r/R_2'); ylabel('Deflection w_0(r)'); P=0;V=1;Ef=V/tp; abdpiezo2; % Analytical expressions for constants

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124 b12v= (g2*Fp+g1*Cp)/(g3*g1 g2^2); b11v=b12v*(1 R21^2); a12v=(g3*Fp+g2*Cp)/(g3*g1 g2^2);; a11v=a12v*(1 R21^2); for i=1:num1 if i<=interface_i u0V(i)=a11v*r(i) alpha*P*(r(i)^3)/16/Dstar; thetaV(i)=b11v*r(i); w0V(i)=b11v*((r(i)^2 R1^2)/2)+b12v*((R1^2 R2^2)/2 R2^2*log(R1/R2)); else u0V(i)=a12v*(r(i) R2^2/r(i)); thetaV(i)=b12v*(r(i) R2^2/r(i)); w0V(i)=b12v*((r(i)^2 R2^2)/2 R2^2*log(r(i)/R2)); end; end; % Evaluation of effective acoustic piezoelectric coefficient temp6v=(b12v*(R2^2 R1^2)^2/8 b11v*R1^4/8); da=2*pi*(temp6v)/V; % output module figure(1); plot(r/R2,u0V,'r'); xlabel('Normalized Radius r/R_2'); ylabel('radial displacement u_0(r)'); figure(2); hold on; plot(r/R2,thetaV,'r'); figure(3); ho ld on; plot(r/R2,w0V,'r'); % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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125 APPENDIX D FINITE ELEMENT MODEL (ABAQUS) INPUT FILE Pressure Loading only (Normalized Piezoelectric Patch Radius = 0.2) *Heading ** Job name: piezoplate_pre_R21_0_2_for_thesi s Model name: Model 1 *Preprint, echo=YES, model=YES, history=YES, contact=YES ** ** PARTS ** *Part, name=pzt *End Part *Part, name=shim *End Part ** ** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, name=shim 1, part=shim *Node 1, 0 ., 0. 2, 0.025, 0. 3, 0.05, 0. 4, 0.075, 0. 5, 0.1, 0. 6, 0.125, 0. 7, 0.15, 0. 8, 0.175, 0. 9, 0.2, 0. 10, 0.225, 0. 11, 0.25, 0. 12, 0.275, 0. 13, 0.3, 0. 14, 0.325, 0. 15, 0.35, 0. 16, 0.375, 0. 17, 0.4, 0. 18, 0.425, 0. 19, 0.45, 0. 20, 0.475, 0. 21, 0.5, 0. 22, 0., 0. 001 23, 0.025, 0.001 24, 0.05, 0.001 25, 0.075, 0.001 26, 0.1, 0.001 27, 0.125, 0.001 28, 0.15, 0.001

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126 29, 0.175, 0.001 30, 0.2, 0.001 31, 0.225, 0.001 32, 0.25, 0.001 33, 0.275, 0.001 34, 0.3, 0.001 35, 0.325, 0.001 36, 0.35, 0.001 37, 0.375, 0.001 38, 0.4, 0.001 39, 0.425, 0.001 40, 0.45, 0.001 41, 0.475, 0.001 42, 0.5, 0.001 43, 0., 0.002 44, 0.025, 0.002 45, 0.05, 0.002 46, 0.075, 0.002 47, 0.1, 0.002 48, 0.125, 0.002 49, 0.15, 0.002 50, 0.175, 0.002 51, 0.2, 0.002 52, 0.225, 0.002 53, 0.25, 0.002 54, 0.275, 0.002 55, 0.3, 0.002 56, 0.325, 0.002 57, 0.35, 0.002 58, 0.375, 0.002 59, 0.4, 0.002 60, 0.425, 0.002 61, 0.45, 0.002 62, 0.475, 0.002 63, 0.5, 0.002 64, 0., 0.003 65, 0.02 5, 0.003 66, 0.05, 0.003 67, 0.075, 0.003 68, 0.1, 0.003 69, 0.125, 0.003 70, 0.15, 0.003 71, 0.175, 0.003 72, 0.2, 0.003 73, 0.225, 0.003 74, 0.25, 0.003 75, 0.275, 0.003 76, 0.3, 0.003 77, 0.325, 0.003 78, 0.35, 0.003 79, 0.375, 0.003 80, 0.4, 0.003 81, 0.425, 0.003 82, 0.45, 0.003 83, 0.475, 0.003 84, 0.5, 0.003 85, 0., 0.004

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127 86, 0.025, 0. 004 87, 0.05, 0.004 88, 0.075, 0.004 89, 0.1, 0.004 90, 0.125, 0.004 91, 0.15, 0.004 92, 0.175, 0.004 93, 0.2, 0.004 94, 0.225, 0.004 95, 0.25, 0.004 96, 0.275, 0.004 97, 0.3, 0.004 98, 0.325, 0.004 99, 0.35, 0.004 100, 0.375, 0.004 101, 0.4, 0.004 102, 0.425, 0.004 103, 0.45, 0.004 104, 0.475, 0.004 105, 0.5, 0.004 106, 0., 0.005 107, 0.025, 0.005 108, 0.05, 0.005 109, 0.075, 0.005 110, 0.1, 0.005 111, 0.125, 0.005 112, 0.15, 0.005 113, 0.175, 0.005 114, 0.2, 0.005 115, 0.225, 0.005 116, 0.25, 0.005 117, 0.275, 0.005 118, 0.3, 0.005 119, 0.325, 0.005 120, 0.35, 0.005 121, 0.375, 0.005 122, 0.4, 0.005 123, 0.425, 0.005 124, 0.45, 0.005 125, 0.475, 0.005 126, 0.5, 0.005 127, 0.0125, 0. 128, 0.025, 0.0005 129, 0.012 5, 0.001 130, 0., 0.0005 131, 0.0375, 0. 132, 0.05, 0.0005 133, 0.0375, 0.001 134, 0.0625, 0. 135, 0.075, 0.0005 136, 0.0625, 0.001 137, 0.0875, 0. 138, 0.1, 0.0005 139, 0.0875, 0.001 140, 0.1125, 0. 141, 0.125, 0.0005 142, 0.1125, 0.001

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128 143, 0.1375, 0. 144, 0.15, 0.0005 145, 0.1375, 0.001 146, 0.1625, 0. 147, 0.175, 0.0005 148, 0.1625, 0.001 149, 0.1875, 0. 150, 0.2, 0.0 005 151, 0.1875, 0.001 152, 0.2125, 0. 153, 0.225, 0.0005 154, 0.2125, 0.001 155, 0.2375, 0. 156, 0.25, 0.0005 157, 0.2375, 0.001 158, 0.2625, 0. 159, 0.275, 0.0005 160, 0.2625, 0.001 161, 0.2875, 0. 162, 0.3, 0.0005 163, 0.2875, 0.001 164, 0.3125, 0. 165, 0.325, 0.0005 166, 0.3125, 0.001 167, 0.3375, 0. 168, 0.35, 0.0005 169, 0.3375, 0.001 170, 0.3625, 0. 171, 0.375, 0.0005 172, 0.3625, 0.001 173, 0.3875, 0. 174, 0.4, 0.0005 175, 0.3875, 0.001 176, 0.4125, 0. 177, 0.425, 0.0005 178, 0.4125, 0.001 179, 0.4375, 0. 180, 0.45, 0.0005 181, 0.4375, 0.001 182, 0.4625, 0. 183, 0.475, 0.0005 184, 0.4625, 0.001 185, 0.4875, 0. 186, 0.5, 0.0005 187, 0.4875, 0.001 188, 0.025, 0.0015 189, 0.0125, 0.002 190, 0., 0.0015 191, 0.05, 0.0015 192, 0.0375, 0.002 193, 0.07 5, 0.0015 194, 0.0625, 0.002 195, 0.1, 0.0015 196, 0.0875, 0.002 197, 0.125, 0.0015 198, 0.1125, 0.002 199, 0.15, 0.0015

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129 200, 0.1375, 0.002 201, 0.175, 0.0015 202, 0.1625, 0.002 203, 0.2, 0.0015 204, 0.1875, 0.002 205, 0.225, 0.0015 206, 0.2125, 0.002 207, 0.25, 0.0015 208, 0.2375, 0.002 209, 0.275, 0.0015 210, 0.2625, 0.002 211, 0.3, 0.0015 212, 0.2875, 0.002 213, 0.325, 0.0015 214, 0.3125, 0. 002 215, 0.35, 0.0015 216, 0.3375, 0.002 217, 0.375, 0.0015 218, 0.3625, 0.002 219, 0.4, 0.0015 220, 0.3875, 0.002 221, 0.425, 0.0015 222, 0.4125, 0.002 223, 0.45, 0.0015 224, 0.4375, 0.002 225, 0.475, 0.0015 226, 0.4625, 0.002 227, 0.5, 0.0015 228, 0.4875, 0.002 229, 0.025, 0.0025 230, 0.0125, 0.003 231, 0., 0.0025 232, 0.05, 0.0025 233, 0.0375, 0.003 234, 0.075, 0.0025 235, 0.0625, 0.003 236, 0.1, 0.0025 237, 0.0875, 0.003 238, 0.125, 0.0025 239, 0.1125, 0.003 240, 0.15, 0.0025 241, 0.1375, 0.003 242, 0.175, 0.0025 243, 0.1625, 0.003 244, 0.2, 0.0025 245, 0.1875, 0.003 246, 0.225, 0.0025 247, 0.2125, 0.003 248, 0.25, 0.0025 249, 0.2375, 0.003 250, 0.275, 0.0025 251, 0.2625, 0.003 252, 0.3, 0.0025 253, 0.2875, 0.003 254, 0.325, 0.0025 255, 0.3125, 0.003 256, 0.35, 0.0025

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130 257, 0.337 5, 0.003 258, 0.375, 0.0025 259, 0.3625, 0.003 260, 0.4, 0.0025 261, 0.3875, 0.003 262, 0.425, 0.0025 263, 0.4125, 0.003 264, 0.45, 0.0025 265, 0.4375, 0.003 266, 0.475, 0.0025 267, 0.4625, 0.003 268, 0.5, 0.0025 269, 0.4875, 0.003 270, 0.025, 0.0035 271, 0.0125, 0.004 272, 0., 0.0035 273, 0.05, 0.0035 274, 0.0375, 0.004 275, 0.075, 0.0035 276, 0.0625, 0.004 277, 0.1, 0.0035 278, 0.0875, 0. 004 279, 0.125, 0.0035 280, 0.1125, 0.004 281, 0.15, 0.0035 282, 0.1375, 0.004 283, 0.175, 0.0035 284, 0.1625, 0.004 285, 0.2, 0.0035 286, 0.1875, 0.004 287, 0.225, 0.0035 288, 0.2125, 0.004 289, 0.25, 0.0035 290, 0.2375, 0.004 291, 0.275, 0.0035 292, 0.2625, 0.004 293, 0.3, 0.0035 294, 0.2875, 0.004 295, 0.325, 0.0035 296, 0.3125, 0.004 297, 0.35, 0.0035 298, 0.3375, 0.004 299, 0.375, 0.0035 300, 0.3625, 0.004 301, 0.4, 0.0035 302, 0.3875, 0.004 303, 0.425, 0.0035 304, 0.4125, 0.004 305, 0.45, 0.0035 306, 0.4375, 0.004 307, 0.475, 0.0035 308, 0.4625, 0.004 309, 0.5, 0.0035 310, 0.4875, 0.004 311, 0.025, 0.0045 312, 0.0125, 0.005 313, 0., 0.0045

PAGE 146

131 314, 0.05, 0.0045 315, 0.0375, 0.005 316, 0.075, 0.0045 317, 0.0625, 0.005 318, 0.1, 0.0045 319, 0.0875, 0.005 320, 0.125, 0.0045 321, 0.112 5, 0.005 322, 0.15, 0.0045 323, 0.1375, 0.005 324, 0.175, 0.0045 325, 0.1625, 0.005 326, 0.2, 0.0045 327, 0.1875, 0.005 328, 0.225, 0.0045 329, 0.2125, 0.005 330, 0.25, 0.0045 331, 0.2375, 0.005 332, 0.275, 0.0045 333, 0.2625, 0.005 334, 0.3, 0.0045 335, 0.2875, 0.005 336, 0.325, 0.0045 337, 0.3125, 0.005 338, 0.35, 0.0045 339, 0.3375, 0.005 340, 0.375, 0.0045 341, 0.3625, 0.005 342, 0.4, 0.0 045 343, 0.3875, 0.005 344, 0.425, 0.0045 345, 0.4125, 0.005 346, 0.45, 0.0045 347, 0.4375, 0.005 348, 0.475, 0.0045 349, 0.4625, 0.005 350, 0.5, 0.0045 351, 0.4875, 0.005 *Element, type=CAX8R 1, 1, 2, 23, 22, 127, 128, 129, 130 2, 2, 3, 24, 23, 131, 132, 133, 128 3, 3, 4, 25, 24, 134, 135, 136, 132 4, 4, 5, 26, 25, 137, 1 38, 139, 135 5, 5, 6, 27, 26, 140, 141, 142, 138 6, 6, 7, 28, 27, 143, 144, 145, 141 7, 7, 8, 29, 28, 146, 147, 148, 144 8, 8, 9, 30, 29, 149, 150, 151, 147 9, 9, 10, 31, 30, 152, 153, 154, 150 10, 10, 11, 32, 31 155, 156, 157, 153 11, 11, 12, 33, 32, 158, 159, 160, 156 12, 12, 13, 34, 33, 161, 162, 163, 159 13, 13, 14, 35, 34, 164, 165, 166, 162 14, 14, 15, 36, 35, 167, 168, 169, 165 15, 15, 16, 37, 36, 170, 171, 172, 168 16, 16, 17, 38, 37, 173, 174, 175, 171 17, 17, 18, 39, 38, 176, 177, 178, 174 18, 18, 19, 40, 39, 179, 180, 181, 177

PAGE 147

132 19, 19, 20, 41, 40, 182, 183, 184, 180 20, 20, 21, 42, 41, 185, 186, 187, 183 21, 22, 23, 44, 43, 129, 188, 189, 190 22, 2 3, 24, 45, 44, 133, 191, 192, 188 23, 24, 25, 46, 45, 136, 193, 194, 191 24, 25, 26, 47, 46, 139, 195, 196, 193 25, 26, 27, 48, 47, 142, 197, 198, 195 26, 27, 28, 49, 48, 145, 199, 200, 197 27, 28, 29, 50, 49, 148, 201, 202, 199 28, 29, 30, 51, 50, 151, 203, 204, 201 29, 30, 31, 52, 51, 154, 205, 206, 203 30, 31, 32, 53, 52, 157, 207, 208, 205 31, 32, 33, 54, 53, 160, 209, 210, 207 32, 33, 34, 55, 54, 163, 211, 212, 209 33, 34, 35, 56, 55, 166, 213, 214, 211 34, 35, 36, 57, 56, 169, 215, 216, 213 35, 36, 37, 58, 57, 172, 217, 218, 215 36, 37, 38, 59, 58, 175, 219, 220, 217 37, 38, 39, 60, 59, 178, 221, 222, 219 38, 39, 40, 61, 60, 181, 223, 224, 221 39, 40, 41, 62, 61, 18 4, 225, 226, 223 40, 41, 42, 63, 62, 187, 227, 228, 225 41, 43, 44, 65, 64, 189, 229, 230, 231 42, 44, 45, 66, 65, 192, 232, 233, 229 43, 45, 46, 67, 66, 194, 234, 235, 232 44, 46, 47, 68, 67, 196, 236, 237, 234 45, 47, 48, 69, 68, 198, 238, 239, 236 46, 48, 49, 70, 69, 200, 240, 241, 238 47, 49, 50, 71, 70, 202, 242, 243, 240 48, 50, 51, 72, 71, 204, 244, 245, 242 49, 51, 52, 73, 72, 206, 246, 247, 244 50, 52, 53, 74, 73, 208, 248, 249, 246 51, 53, 54, 75, 74, 210, 250, 251, 248 52, 54, 55, 76, 75, 212, 252, 253, 250 53, 55, 56, 77, 76, 214, 254, 255, 252 54, 56, 57, 78, 77, 216, 256, 257, 254 55, 57, 58, 79, 78, 218, 258, 259, 256 56, 58, 59, 80, 79, 220, 260, 261, 258 57 59, 60, 81, 80, 222, 262, 263, 260 58, 60, 61, 82, 81, 224, 264, 265, 262 59, 61, 62, 83, 82, 226, 266, 267, 264 60, 62, 63, 84, 83, 228, 268, 269, 266 61, 64, 65, 86, 85, 230, 270, 271, 272 62, 65, 66, 87, 86, 233, 273, 274, 270 63, 66, 67, 88, 87, 235, 275, 276, 273 64, 67, 68, 89, 88, 237, 277, 278, 275 65, 68, 69, 90, 89, 239, 279, 280, 277 66, 69, 70, 91, 90, 241, 281, 282, 279 67, 70, 71, 92, 91, 243, 283, 284, 281 68, 71, 72, 93, 92, 245, 2 85, 286, 283 69, 72, 73, 94, 93, 247, 287, 288, 285 70, 73, 74, 95, 94, 249, 289, 290, 287 71, 74, 75, 96, 95, 251, 291, 292, 289 72, 75, 76, 97, 96, 253, 293, 294, 291 73, 76, 77, 98, 97, 255, 295, 296, 293 74, 77, 78, 99, 98 257, 297, 298, 295 75, 78, 79, 100, 99, 259, 299, 300, 297

PAGE 148

133 76, 79, 80, 101, 100, 261, 301, 302, 299 77, 80, 81, 102, 101, 263, 303, 304, 301 78, 81, 82, 103, 102, 265, 305, 306, 303 79, 82, 83, 104, 103, 267, 307, 308, 305 80, 83, 84, 105, 104, 269, 309, 310, 307 81, 85, 86, 107, 106, 271, 311, 312, 313 82, 86, 87, 108, 107, 274, 314, 315, 311 83, 87, 88, 109, 108, 276, 316, 317, 314 84, 88, 89, 110, 109, 278, 318, 319, 316 85, 89, 90, 111, 110, 280, 320, 321, 318 86, 9 0, 91, 112, 111, 282, 322, 323, 320 87, 91, 92, 113, 112, 284, 324, 325, 322 88, 92, 93, 114, 113, 286, 326, 327, 324 89, 93, 94, 115, 114, 288, 328, 329, 326 90, 94, 95, 116, 115, 290, 330, 331, 328 91, 95, 96, 117, 116, 292, 332, 333, 330 92, 96, 97, 118, 117, 294, 334, 335, 332 93, 97, 98, 119, 118, 296, 336, 337, 334 94, 98, 99, 120, 119, 298, 338, 339, 336 95, 99, 100, 121, 120, 300, 340, 341, 338 96, 100, 101, 122, 121, 302, 342, 343, 340 97, 101, 102, 123, 122, 304, 344, 345, 342 98, 102, 103, 124, 123, 306, 346, 347, 344 99, 103, 104, 125, 124, 308, 348, 349, 346 100, 104, 105, 126, 125, 310, 350, 351, 348 ** Region: (shim:Picked) *Elset, elset=_I1, internal, generate 1, 100, 1 ** Section: shim *Solid Section, elset =_I1, material=shim 1., *End Instance ** *Instance, name=pzt 1, part=pzt 0., 0.005, 0. *Node 1, 0., 0. 2, 0.025, 0. 3, 0.05, 0. 4, 0.075, 0. 5, 0.1, 0. 6, 0., 0.001 7, 0.025, 0.001 8, 0.05, 0.001 9, 0.075, 0.001 10, 0.1, 0.001 11, 0., 0.002 12, 0.025 0.002 13, 0.05, 0.002 14, 0.075, 0.002 15, 0.1, 0.002 16, 0.0125, 0. 17, 0.025, 0.0005 18, 0.0125, 0.001 19, 0., 0.0005 20, 0.0375, 0. 21, 0.05, 0.0005

PAGE 149

134 22, 0.0375, 0.001 23, 0.0625, 0. 24, 0.075, 0.0005 25, 0.0625, 0.001 26, 0.0875, 0. 27, 0.1, 0.0005 28, 0.0875, 0.001 29, 0.025, 0.0015 30, 0.0125, 0.002 31, 0., 0.0015 32, 0.05, 0.0015 33, 0.0375, 0.0 02 34, 0.075, 0.0015 35, 0.0625, 0.002 36, 0.1, 0.0015 37, 0.0875, 0.002 *Element, type=CAX8RE 1, 1, 2, 7, 6, 16, 17, 18, 19 2, 2, 3, 8, 7, 20, 21, 22, 17 3, 3, 4, 9, 8, 23, 24, 25, 21 4, 4, 5, 10, 9, 26, 27, 28, 24 5, 6, 7, 12, 11, 18, 29, 30, 31 6, 7, 8, 13, 12, 22, 32, 33, 29 7, 8, 9, 14, 13, 25, 34, 35, 32 8, 9, 10, 15, 14, 28, 36, 37, 34 ** Region: (pzt:Picked) *Elset, elset=_I1, internal, generate 1, 8, 1 ** Section: pzt *Solid Section, elset=_I1, material=pzt 1., *End Instance *Nset, nset=_G31, internal, instance=pzt 1 11, 12, 13, 14, 15, 30, 33, 35, 37 *Elset, elset=_G31, internal, instance=pzt 1, generate 5, 8, 1 *Nset, nset=_G32, internal, instanc e=pzt 1 1, 2, 3, 4, 5, 16, 20, 23, 26 *Elset, elset=_G32, internal, instance=pzt 1, generate 1, 4, 1 *Nset, nset=_G33, internal, instance=shim 1 21, 42, 63, 84, 105, 126, 186, 227, 268, 309, 350 *Elset, elset=_G33, internal, instance=shim 1, generate 20, 100, 20 *Nset, nset=_G34, internal, instance=shim 1 1, 22, 43, 64, 85, 106, 130, 190, 231, 272, 313 *Nset, nset=_G34, internal, instance=pzt 1 1, 6, 11, 19, 31 *Elset, elset=_G34, internal, instance=shim 1, generate 1, 81, 20 Elset, elset=_G34, internal, instance=pzt 1 1, 5 *Elset, elset=__G29_S1, internal, instance=pzt 1, generate 1, 4, 1 *Surface, type=ELEMENT, name=_G29, internal __G29_S1, S1 *Elset, elset=__G30_S3, internal, instance=shim 1, generate

PAGE 150

135 81, 100, 1 *Surf ace, type=ELEMENT, name=_G30, internal __G30_S3, S3 *Elset, elset=__G39_S1, internal, instance=shim 1, generate 1, 20, 1 *Surface, type=ELEMENT, name=_G39, internal __G39_S1, S1 ** Constraint: Constraint 1 *Tie, name=Constraint 1, adjust=yes _G30, _G2 9 *End Assembly ** ** MATERIALS ** *Material, name=pzt *Density 7500., *Dielectric 8.85e 09, *Elastic 3e+10, 0.3 *Piezoelectric, type=E 0., 0., 0., 0., 0., 0., 5e 11, 0. 5e 11, 0., 0., 0., 0., 0., 0., 0. 0., 0. *Material, name=shim *Density 2500., *Elastic 9e+10, 0.3 ** ** BOUNDARY CONDITIONS ** ** Name: axisymmetric Type: Symmetry/Antisymmetry/Encastre *Boundary _G34, ZSYMM ** Name: fixed Type: Symmetry/Antisymmetry/Encastre *Boundary G33, ENCASTRE ** ---------------------------------------------------------------** ** STEP: Voltage_loading_only ** *Step, name=Voltage_loading_only, perturbation *Static ** ** BOUNDARY CONDITIONS ** ** Name: bottom_pzt Type: Electric potential *Boun dary _G32, 9, 9 ** Name: top_pzt Type: Electric potential *Boundary _G31, 9, 9 ** ** LOADS **

PAGE 151

136 ** Name: negative_pressure_load Type: Pressure *Dsload _G39, P, 1. ** ** OUTPUT REQUESTS ** *Restart, write, frequency=1 ** ** FIELD OUTPUT: F Output 1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H Output 1 ** *Output, history, variable=PRESELECT *El Print, freq=999999 *Node Print, freq=999999 *End Step Voltage Loading only (Normalized Piezoelectric Patch Radius = 0.55) *Heading ** Job name: piezoplate_pot_R21_0_55_for_thesis Model name: Model 1 *Preprint, echo=YES, model=YES, history=YES, contact=YES ** ** PARTS ** *Part, name=pzt *End Part *Part, name=shim *End Part ** ** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, name=shim 1, part=shim *Node 1, 0., 0. 2, 0.025, 0. 3, 0.05, 0. 4, 0.075, 0. 5, 0.1, 0. 6, 0.125, 0. 7, 0.15 0. 8, 0.175, 0. 9, 0.2, 0. 10, 0.225, 0. 11, 0.25, 0. 12, 0.275, 0. 13, 0.3, 0.

PAGE 152

137 14, 0.325, 0. 15, 0.35, 0. 16, 0.375, 0. 17, 0.4, 0. 18, 0.425, 0. 19, 0.45, 0. 20, 0.475, 0. 21, 0.5, 0. 22, 0., 0.001 23, 0.025, 0.001 24, 0.05, 0.001 25, 0.075, 0.001 26, 0.1, 0.001 27, 0.125, 0.001 28, 0.15, 0.0 01 29, 0.175, 0.001 30, 0.2, 0.001 31, 0.225, 0.001 32, 0.25, 0.001 33, 0.275, 0.001 34, 0.3, 0.001 35, 0.325, 0.001 36, 0.35, 0.001 37, 0.375, 0.001 38, 0.4, 0.001 39, 0.425, 0.001 40, 0.45, 0.001 41, 0.475, 0.001 42, 0.5, 0.001 43, 0., 0.002 44, 0.025, 0.002 45, 0.05, 0.002 46, 0.075, 0.002 47, 0.1, 0.002 48, 0.125, 0.002 49, 0.15, 0.002 50, 0.175, 0.002 51, 0.2, 0.002 52, 0.225, 0.002 53, 0.25, 0.002 54, 0.275, 0.002 55, 0.3, 0.002 56, 0.325, 0.002 57, 0.35, 0.002 58, 0.375, 0.002 59, 0.4, 0.002 60, 0.425, 0.002 61, 0.45, 0.002 62, 0.475, 0.002 63, 0.5, 0.002 64, 0., 0.003 65, 0.025, 0.003 66, 0.05, 0.003 67, 0.075, 0.003 68, 0.1, 0.003 69, 0.125, 0.003 70, 0.15, 0.003

PAGE 153

138 71, 0.175 0.003 72, 0.2, 0.003 73, 0.225, 0.003 74, 0.25, 0.003 75, 0.275, 0.003 76, 0.3, 0.003 77, 0.325, 0.003 78, 0.35, 0.003 79, 0.375, 0.003 80, 0.4, 0.003 81, 0.425, 0.003 82, 0.45, 0.003 83, 0.475, 0.003 84, 0.5, 0.003 85, 0., 0.004 86, 0.025, 0.004 87, 0.05, 0.004 88, 0.075, 0.004 89, 0.1, 0.004 90, 0.125, 0.004 91, 0.15, 0.004 92, 0.175, 0.0 04 93, 0.2, 0.004 94, 0.225, 0.004 95, 0.25, 0.004 96, 0.275, 0.004 97, 0.3, 0.004 98, 0.325, 0.004 99, 0.35, 0.004 100, 0.375, 0.004 101, 0.4, 0.004 102, 0.425, 0.004 103, 0.45, 0.004 104, 0.475, 0.004 105, 0.5, 0.004 106, 0., 0.005 1 07, 0.025, 0.005 108, 0.05, 0.005 109, 0.075, 0.005 110, 0.1, 0.005 111, 0.125, 0.005 112, 0.15, 0.005 113, 0.175, 0.005 114, 0.2, 0.005 115, 0.225, 0.005 116, 0.25, 0.005 117, 0.275, 0.005 118, 0.3, 0.005 119, 0.325, 0.005 120, 0.35, 0.005 121, 0.375, 0.005 122, 0.4, 0.005 123, 0.425, 0.005 124, 0.45, 0.005 125, 0.475, 0.005 126, 0.5, 0.005 127, 0.0125, 0.

PAGE 154

139 128, 0 .025, 0.0005 129, 0.0125, 0.001 130, 0., 0.0005 131, 0.0375, 0. 132, 0.05, 0.0005 133, 0.0375, 0.001 134, 0.0625, 0. 135, 0.075 0.0005 136, 0.0625, 0.001 137, 0.0875, 0. 138, 0.1, 0.0005 139, 0.0875, 0.001 140, 0.1125, 0. 141, 0.125, 0.0005 142, 0.1125, 0.001 143, 0.1375, 0. 144, 0.15, 0.0005 145, 0.1375, 0.001 146, 0.1625, 0. 147, 0.175, 0.0005 148, 0.1625, 0.001 149, 0.1875, 0. 150, 0.2, 0.0005 151, 0.1875, 0.001 152, 0.2125, 0. 153, 0.225, 0.0005 154, 0.2125, 0.001 155, 0.2375, 0. 156, 0.25, 0.00 05 157, 0.2375, 0.001 158, 0.2625, 0. 159, 0.275, 0.0005 160, 0.2625, 0.001 161, 0.2875, 0. 162, 0.3, 0.0005 163, 0.2875, 0.001 164, 0.3125, 0. 165, 0.325, 0.0005 166, 0.3125, 0.001 167, 0.3375, 0. 168, 0.35, 0.0005 169, 0.3375, 0.001 170, 0.3625, 0. 1 71, 0.375, 0.0005 172, 0.3625, 0.001 173, 0.3875, 0. 174, 0.4, 0.0005 175, 0.3875, 0.001 176, 0.4125, 0. 177, 0.425, 0.0005 178, 0.4125, 0.001 179, 0.4375, 0. 180, 0.45, 0.0005 181, 0.4375, 0.001 182, 0.4625, 0. 183, 0.475, 0.0005 184, 0.4625, 0.001

PAGE 155

140 185, 0.4875, 0. 186, 0.5, 0.0005 187, 0.4875, 0.001 188, 0.025, 0.0015 189, 0.0125, 0.002 190, 0., 0.0015 191, 0.05, 0.0015 192, 0. 0375, 0.002 193, 0.075, 0.0015 194, 0.0625, 0.002 195, 0.1, 0.0015 196, 0.0875, 0.002 197, 0.125, 0.0015 198, 0.1125, 0.002 199, 0.15 0.0015 200, 0.1375, 0.002 201, 0.175, 0.0015 202, 0.1625, 0.002 203, 0.2, 0.0015 204, 0.1875, 0.002 205, 0.225, 0.0015 206, 0.2125, 0.002 207, 0.25, 0.0015 208, 0.2375, 0.002 209, 0.275, 0.0015 210, 0.2625, 0.002 211, 0.3, 0.0015 212, 0.2875, 0.002 213, 0.325, 0.0015 214, 0.3125, 0.002 215, 0.35, 0.0015 216, 0.3375, 0.002 217, 0.375, 0.0015 218, 0.3625, 0.002 219, 0.4, 0.0015 220, 0.3875, 0.0 02 221, 0.425, 0.0015 222, 0.4125, 0.002 223, 0.45, 0.0015 224, 0.4375, 0.002 225, 0.475, 0.0015 226, 0.4625, 0.002 227, 0.5, 0.0015 228, 0.4875, 0.002 229, 0.025, 0.0025 230, 0.0125, 0.003 231, 0., 0.0025 232, 0.05, 0.0025 233, 0.0375, 0.003 234, 0.075, 0.0025 2 35, 0.0625, 0.003 236, 0.1, 0.0025 237, 0.0875, 0.003 238, 0.125, 0.0025 239, 0.1125, 0.003 240, 0.15, 0.0025 241, 0.1375, 0.003

PAGE 156

141 242, 0.175, 0.0025 243, 0.1625, 0.003 244, 0.2, 0.0025 245, 0.1875, 0.003 246, 0.225, 0.0025 247, 0.2125, 0.003 248, 0.25, 0.0025 249, 0.2375, 0.003 250, 0.275, 0.0025 251, 0.2625, 0.003 252, 0.3, 0.0025 253, 0.2875, 0.003 254, 0.325, 0.0025 255, 0.3125, 0.003 256, 0.35, 0.0025 257, 0.3375, 0.003 258, 0.375, 0.0025 259, 0.3625, 0.003 260, 0.4, 0.0025 261, 0.3875, 0.003 262, 0.425, 0.0025 263, 0.4125 0.003 264, 0.45, 0.0025 265, 0.4375, 0.003 266, 0.475, 0.0025 267, 0.4625, 0.003 268, 0.5, 0.0025 269, 0.4875, 0.003 270, 0.025, 0.0035 271, 0.0125, 0.004 272, 0., 0.0035 273, 0.05, 0.0035 274, 0.0375, 0.004 275, 0.075, 0.0035 276, 0.0625, 0.004 277, 0.1, 0.0035 278, 0.0875, 0.004 279, 0.125, 0.0035 280, 0.1125, 0.004 281, 0.15, 0.0035 282, 0.1375, 0.004 283, 0.175, 0.0035 284, 0.1625, 0.0 04 285, 0.2, 0.0035 286, 0.1875, 0.004 287, 0.225, 0.0035 288, 0.2125, 0.004 289, 0.25, 0.0035 290, 0.2375, 0.004 291, 0.275, 0.0035 292, 0.2625, 0.004 293, 0.3, 0.0035 294, 0.2875, 0.004 295, 0.325, 0.0035 296, 0.3125, 0.004 297, 0.35, 0.0035 298, 0.3375, 0.004

PAGE 157

142 2 99, 0.375, 0.0035 300, 0.3625, 0.004 301, 0.4, 0.0035 302, 0.3875, 0.004 303, 0.425, 0.0035 304, 0.4125, 0.004 305, 0.45, 0.0035 306, 0.4375, 0.004 307, 0.475, 0.0035 308, 0.4625, 0.004 309, 0.5, 0.0035 310, 0.4875, 0.004 311, 0.025, 0.0045 312, 0.0125, 0.005 313, 0., 0.0045 314, 0.05, 0.0045 315, 0.0375, 0.005 316, 0.075, 0.0045 317, 0.0625, 0.005 318, 0.1, 0.0045 319, 0.0875, 0.005 320, 0 .125, 0.0045 321, 0.1125, 0.005 322, 0.15, 0.0045 323, 0.1375, 0.005 324, 0.175, 0.0045 325, 0.1625, 0.005 326, 0.2, 0.0045 327, 0.1875 0.005 328, 0.225, 0.0045 329, 0.2125, 0.005 330, 0.25, 0.0045 331, 0.2375, 0.005 332, 0.275, 0.0045 333, 0.2625, 0.005 334, 0.3, 0.0045 335, 0.2875, 0.005 336, 0.325, 0.0045 337, 0.3125, 0.005 338, 0.35, 0.0045 339, 0.3375, 0.005 340, 0.375, 0.0045 341, 0.3625, 0.005 342, 0.4, 0.0045 343, 0.3875, 0.005 344, 0.425, 0.0045 345, 0.4125, 0.005 346, 0.45, 0.0045 347, 0.4375, 0.005 348, 0.475, 0.00 45 349, 0.4625, 0.005 350, 0.5, 0.0045 351, 0.4875, 0.005 *Element, type=CAX8R 1, 1, 2, 23, 22, 127, 128, 129, 130 2, 2, 3, 24, 23, 131, 132, 133, 128 3, 3, 4, 25, 24, 134, 135, 1 36, 132

PAGE 158

143 4, 4, 5, 26, 25, 137, 138, 139, 135 5, 5, 6, 27, 26, 140, 141, 142, 138 6, 6, 7, 28, 27, 143, 144, 145, 141 7, 7, 8, 29, 28, 146, 147, 148, 144 8, 8, 9, 30, 29, 149, 150, 151, 147 9, 9, 10, 31, 30, 152 153, 154, 150 10, 10, 11, 32, 31, 155, 156, 157, 153 11, 11, 12, 33, 32, 158, 159, 160, 156 12, 12, 13, 34, 33, 161, 162, 163, 159 13, 13, 14, 35, 34, 164, 165, 166, 162 14, 14, 15, 36, 35, 167, 168, 169, 165 15, 15, 16, 37, 36, 170, 171, 172, 168 16, 16, 17, 38, 37, 173, 174, 175, 171 17, 17, 18, 39, 38, 176, 177, 178, 174 18, 18, 19, 40, 39, 179, 180, 181, 177 19, 19, 20, 41, 40, 182, 183, 184, 180 20, 20, 21, 42, 41, 185, 186, 187, 183 21, 22, 2 3, 44, 43, 129, 188, 189, 190 22, 23, 24, 45, 44, 133, 191, 192, 188 23, 24, 25, 46, 45, 136, 193, 194, 191 24, 25, 26, 47, 46, 139, 195, 196, 193 25, 26, 27, 48, 47, 142, 197, 198, 195 26, 27, 28, 49, 48, 145, 199, 200, 197 27, 28, 29, 50, 49, 148, 201, 202, 199 28, 29, 30, 51, 50, 151, 203, 204, 201 29, 30, 31, 52, 51, 154, 205, 206, 203 30, 31, 32, 53, 52, 157, 207, 208, 205 31, 32, 33, 54, 53, 160, 209, 210, 207 32, 33, 34, 55, 54, 163, 211, 212, 209 33, 34, 35, 56, 55, 166, 213, 214, 211 34, 35, 36, 57, 56, 169, 215, 216, 213 35, 36, 37, 58, 57, 172, 217, 218, 215 36, 37, 38, 59, 58, 175, 219, 220, 217 37, 38, 39, 60, 59, 178, 221, 222, 219 38, 39, 40, 61, 60, 181, 22 3, 224, 221 39, 40, 41, 62, 61, 184, 225, 226, 223 40, 41, 42, 63, 62, 187, 227, 228, 225 41, 43, 44, 65, 64, 189, 229, 230, 231 42, 44, 45, 66, 65, 192, 232, 233, 229 43, 45, 46, 67, 66, 194, 234, 235, 232 44, 46, 47, 68, 67, 196, 236, 237, 234 45, 47, 48, 69, 68, 198, 238, 239, 236 46, 48, 49, 70, 69, 200, 240, 241, 238 47, 49, 50, 71, 70, 202, 242, 243, 240 48, 50, 51, 72, 71, 204, 244, 245, 242 49, 51, 52, 73, 72, 206, 246, 247, 244 50, 52, 53, 74, 73, 208, 248, 249, 246 51, 53, 54, 75, 74, 210, 250, 251, 248 52, 54, 55, 76, 75, 212, 252, 253, 250 53, 55, 56, 77, 76, 214, 254, 255, 252 54, 56, 57, 78, 77, 216, 256, 257, 254 55, 57, 58, 79, 78, 218, 258, 259, 256 56, 58 59, 80, 79, 220, 260, 261, 258 57, 59, 60, 81, 80, 222, 262, 263, 260 58, 60, 61, 82, 81, 224, 264, 265, 262 59, 61, 62, 83, 82, 226, 266, 267, 264 60, 62, 63, 84, 83, 228, 268, 269, 266

PAGE 159

144 61, 64, 65, 86, 85, 230, 270, 271, 272 62, 65, 66, 87, 86, 233, 273, 274, 270 63, 66, 67, 88, 87, 235, 275, 276, 273 64, 67, 68, 89, 88, 237, 277, 278, 275 65, 68, 69, 90, 89, 239, 279, 280, 277 66, 69, 70, 91, 90, 241, 281, 282, 279 67, 70, 71, 92, 91, 243, 283, 2 84, 281 68, 71, 72, 93, 92, 245, 285, 286, 283 69, 72, 73, 94, 93, 247, 287, 288, 285 70, 73, 74, 95, 94, 249, 289, 290, 287 71, 74, 75, 96, 95, 251, 291, 292, 289 72, 75, 76, 97, 96, 253, 293, 294, 291 73, 76, 77, 98, 97, 255 295, 296, 293 74, 77, 78, 99, 98, 257, 297, 298, 295 75, 78, 79, 100, 99, 259, 299, 300, 297 76, 79, 80, 101, 100, 261, 301, 302, 299 77, 80, 81, 102, 101, 263, 303, 304, 301 78, 81, 82, 103, 102, 265, 305, 306, 303 79, 82, 83, 104, 103, 267, 307, 308, 305 80, 83, 84, 105, 104, 269, 309, 310, 307 81, 85, 86, 107, 106, 271, 311, 312, 313 82, 86, 87, 108, 107, 274, 314, 315, 311 83, 87, 88, 109, 108, 276, 316, 317, 314 84, 88, 89, 110, 109, 278, 318, 319, 316 85, 89, 9 0, 111, 110, 280, 320, 321, 318 86, 90, 91, 112, 111, 282, 322, 323, 320 87, 91, 92, 113, 112, 284, 324, 325, 322 88, 92, 93, 114, 113, 286, 326, 327, 324 89, 93, 94, 115, 114, 288, 328, 329, 326 90, 94, 95, 116, 115, 290, 330, 331, 328 91, 95, 96, 117, 116, 292, 332, 333, 330 92, 96, 97, 118, 117, 294, 334, 335, 332 93, 97, 98, 119, 118, 296, 336, 337, 334 94, 98, 99, 120, 119, 298, 338, 339, 336 95, 99, 100, 121, 120, 300, 340, 341, 338 96, 100, 101, 122, 121, 302, 342, 343, 340 97, 101, 102, 123, 122, 304, 344, 345, 342 98, 102, 103, 124, 123, 306, 346, 347, 344 99, 103, 104, 125, 124, 308, 348, 349, 346 100, 104, 105, 126, 125, 310, 350, 351, 348 ** Region: (shim:Picked) *Elset, elset=_I1, internal, generate 1, 100, 1 ** Section: shim *Solid Section, elset=_I1, material=shim 1., *End Instance ** *Instance, name=pzt 1, part=pzt 0., 0.005, 0. *Node 1, 0., 0. 2, 0.025, 0. 3, 0.05, 0. 4, 0.075, 0. 5, 0.1, 0. 6, 0.125, 0.

PAGE 160

145 7, 0.15, 0. 8, 0.175, 0. 9, 0.2, 0. 10, 0.225, 0. 11, 0 .25, 0. 12, 0.275, 0. 13, 0., 0.001 14, 0.025, 0.001 15, 0.05, 0.001 16, 0.075, 0.001 17, 0.1, 0.001 18, 0.125, 0.001 19, 0.15, 0.001 20, 0.175, 0.001 21, 0.2, 0.001 22, 0.225, 0.001 23, 0.25, 0.001 24, 0.275, 0.001 25, 0., 0.002 26, 0.025, 0.002 27, 0.05, 0.002 28, 0.075, 0.002 29, 0.1, 0.002 30, 0.125, 0.002 31, 0.15, 0.002 32, 0.175, 0.002 33, 0.2, 0.002 34, 0.225, 0.002 35, 0.25, 0.002 36, 0.275, 0.002 37, 0.0125, 0. 38, 0.025, 0.0005 39, 0.0125, 0.00 1 40, 0., 0.0005 41, 0.0375, 0. 42, 0.05, 0.0005 43, 0.0375, 0.001 44, 0.0625, 0. 45, 0.075, 0.0005 46, 0.0625, 0.001 47, 0.0875, 0. 48, 0.1, 0.0005 49, 0.0875, 0.001 50, 0.1125, 0. 51, 0.125, 0.0005 52, 0.1125, 0.001 53, 0.1375, 0. 5 4, 0.15, 0.0005 55, 0.1375, 0.001 56, 0.1625, 0. 57, 0.175, 0.0005 58, 0.1625, 0.001 59, 0.1875, 0. 60, 0.2, 0.0005 61, 0.1875, 0.001 62, 0.2125, 0. 63, 0.225, 0.0005

PAGE 161

146 64, 0.2125, 0.001 65, 0.2375, 0. 66, 0.25, 0.0005 67, 0.2375, 0.001 68, 0.2625, 0. 69, 0.275, 0.0005 70, 0.2625, 0.001 71, 0.025, 0.0015 72, 0.0125, 0.002 73, 0., 0.0015 74, 0.05, 0.0015 75, 0.0 375, 0.002 76, 0.075, 0.0015 77, 0.0625, 0.002 78, 0.1, 0.0015 79, 0.0875, 0.002 80, 0.125, 0.0015 81, 0.1125, 0.002 82, 0.15, 0.0015 83, 0.1375, 0.002 84, 0.175, 0.0015 85, 0.1625, 0.002 86, 0.2, 0.0015 87, 0.1875, 0.002 88, 0.225, 0.0015 89, 0.2125, 0.002 90, 0.25, 0.0015 91, 0.2375, 0.002 92, 0.275, 0.0015 93, 0.2625, 0.002 *Element, type=CAX8RE 1, 1, 2, 14, 13, 37, 38, 39, 40 2, 2, 3, 15, 14, 41, 42, 43, 38 3, 3, 4 16, 15, 44, 45, 46, 42 4, 4, 5, 17, 16, 47, 48, 49, 45 5, 5, 6, 18, 17, 50, 51, 52, 48 6, 6, 7, 19, 18, 53, 54, 55, 51 7, 7, 8, 20, 19, 56, 57, 58, 54 8, 8, 9, 21, 20, 59, 60, 61, 57 9, 9, 10, 22, 21, 62, 63, 64, 60 10, 10, 11, 23, 22, 6 5, 66, 67, 63 11, 11, 12, 24, 23, 68, 69, 70, 66 12, 13, 14, 26, 25, 39, 71, 72, 73 13, 14, 15, 27, 26, 43, 74, 75, 71 14, 15, 16, 28, 27, 46, 76, 77, 74 15, 16, 17, 29, 28, 49, 78, 79, 76 16, 17, 18, 30, 29, 52, 80, 81, 78 17, 18, 19, 31, 30, 55, 82, 83, 80 18, 19, 20, 32, 31, 58, 84, 85, 82 19, 20, 21, 33, 32, 61, 86, 87, 84 20, 21, 22, 34, 33, 64, 88, 89, 86 21, 22, 23, 35, 34, 67, 90, 91, 88 22, 23, 24, 36, 35, 70, 92, 93, 90 ** Region: (pzt:Picked) *Elset, elset=_I1, internal, generate 1, 22, 1 ** Section: pzt

PAGE 162

147 *Solid Section, elset=_I1, material=pzt 1., *End Instance *Nset, nset=_G31, internal, instance=pzt 1 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 72, 75, 77, 79 81, 83, 85, 87, 89, 91, 93 *Elset, elset=_G31, internal, instance=pzt 1, gen erate 12, 22, 1 *Nset, nset=_G32, internal, instance=pzt 1 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 37, 41, 44, 47 50, 53, 56, 59, 62, 65, 68 *Elset, elset=_G32, internal, instance=pzt 1, generate 1, 11, 1 *Nset, nset=_G33, internal, inst ance=shim 1 21, 42, 63, 84, 105, 126, 186, 227, 268, 309, 350 *Elset, elset=_G33, internal, instance=shim 1, generate 20, 100, 20 *Nset, nset=_G34, internal, instance=shim 1 1, 22, 43, 64, 85, 106, 130, 190, 231, 272, 313 *Nset, nset=_G34, in ternal, instance=pzt 1 1, 13, 25, 40, 73 *Elset, elset=_G34, internal, instance=shim 1, generate 1, 81, 20 *Elset, elset=_G34, internal, instance=pzt 1 1, 12 *Elset, elset=__G29_S1, internal, instance=pzt 1, generate 1, 11, 1 *Surface, type=EL EMENT, name=_G29, internal __G29_S1, S1 *Elset, elset=__G30_S3, internal, instance=shim 1, generate 81, 100, 1 *Surface, type=ELEMENT, name=_G30, internal __G30_S3, S3 *Elset, elset=__G39_S1, internal, instance=shim 1, generate 1, 20, 1 *Surface, t ype=ELEMENT, name=_G39, internal __G39_S1, S1 ** Constraint: Constraint 1 *Tie, name=Constraint 1, adjust=yes _G30, _G29 *End Assembly ** ** MATERIALS ** *Material, name=pzt *Density 7500., *Dielectric 8.85e 09, *Elastic 3e+10, 0.3 *Piezoelectric, type =E 0., 0., 0., 0., 0., 0., 5e 11, 0. 5e 11, 0., 0., 0., 0., 0., 0., 0. 0., 0. *Material, name=shim *Density

PAGE 163

148 2500., *Elastic 9e+10, 0.3 ** ** BOUNDARY CONDITIONS ** ** Name: axisymmetric Typ e: Symmetry/Antisymmetry/Encastre *Boundary _G34, ZSYMM ** Name: fixed Type: Symmetry/Antisymmetry/Encastre *Boundary _G33, ENCASTRE ** ---------------------------------------------------------------** ** STEP: Voltage_loading_only ** *Step, name=Volta ge_loading_only, perturbation *Static ** ** BOUNDARY CONDITIONS ** ** Name: bottom_pzt Type: Electric potential *Boundary _G32, 9, 9 ** Name: top_pzt Type: Electric potential *Boundary _G31, 9, 9, 1. ** ** LOADS ** ** Name: pressure_load Type: Pressu re *Dsload _G39, P, 1e 12 ** ** OUTPUT REQUESTS ** *Restart, write, frequency=1 ** ** FIELD OUTPUT: F Output 1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H Output 1 ** *Output, history, variable=PRESELECT *El Print, freq=999999 *Node Print, freq=999999 *End Step

PAGE 164

149 LIST OF REFERENCES 1. Rossi, M., Acoustics and Electroacoustics, Artech House, Norwood, MA, Chapter 5, pp. 245 308, 1988. 2. Chen, F. J., Yao, C., Beeler, G. B., Bryant, R. G. and Fox, R. L., Development of Synthetic Jet A ctuators for Active Flow Control at NASA Langley, AIAA Paper 2000 2405, June 2000. 3. Gallas, Q., Mathew, J., Kashyap, A., Holman, R., Carroll, B., Nishida, T., Sheplak, M. and Cattafesta, L., Lumped Element Modeling of Piezoelectric Driven Synthetic Jet Ac tuators, AIAA Paper 2002 0125, January 2002. 4. Morris, C. J. and Foster, F. K., Optimization of a circular piezoelectric bimorph for a micropump driver, J. Micromech. Microeng. Vol. 10, pp. 459 465, Feb 2000. 5. Prasad, S., Sankar, B.V., Cattafesta, L. N., Horowitz, S., Gallas, Q., Sheplak, M., Two Port Electroacoustic Model of an Axisymmetric Piezoelectric Composite Plate, AIAA Paper 2002 1365, April 2002. 6. Dobrucki, B. and Pruchnicki, P., Theory of piezoelectric axisymmetric bimorph, Sensors and Actuato rs A, Vol. 58, pp. 203 212, 1997. 7. Ansys Inc., http://www.ansys.com/ansys/index.htm , Aug 1, 2002. 8. Stavsky, Y. and Loewy, R., Axisymmetric Vibrations of Isotropic Composite Circular Plates, J. Acoust Soc. Am. Vol. 49, No. 5(2), pp. 819 822, 1971. 9. Adelman, N. T. and Stavsky, Y., Flexural extensional behavior of composite piezoelectric circular plates, J. of Acoust. Soc. Am. Vol. 67(3), pp. 819 822, Mar 1980. 10. Chang, S. H. and Du, B. C., Optimizati on of axisymmetric bimorphic disk transducers, J. Acoust. Soc. Am. Vol. 109(1), pp. 194 202, Jan 2001. 11. Dumir, P.C., Joshi, S., Dube, G. P., Geometrically nonlinear axisymmetric analysis of thick laminated annular plate using FSDT, Composites: Part B Vol. 32, pp. 1 10, 2001. 12. Merhaut, J., Theory of Electroacoustics, McGraw Hill, New York, NY, Chapter 2, pp 57 108, 1981.

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150 13. Senturia, S.D., Microsystems Design, Kluwer Academic Publishers, Boston, MA, Nov 2000. 14. Timoshenko S. P. and Krieger, S. W., Theo ry of Plates and Shells, McGraw Hill, New York, NY, 1959. 15. Reddy, J. N., Theory and Analysis of Elastic Plates, Taylor & Francis, London, 1999. 16. Timoshenko, S. P. and Goodier, J. N., Theory of Elasticity, McGraw Hill, New York, NY, 1990. 17. Abaqus, "Abaqu s/CAE Product Description Version 6.2," http://www.hks.com/products/p_abcae.html Aug 1, 2002. 18. Fox, R. W. and McDonald, A. T., Introduction to Fluid Mechanics, 4 th ed., John Wiley & Sons, Inc., N ew York, NY, 1992 19. Madou, M. J., Fundamentals of Microfabrication, CRC Press, Boca Raton, FL, Oct 1997. 20. Wise, S. A., Displacement properties of RAINBOW and THUNDER piezoelectric actuators, Sensors and Actuators A Vol. 69, pp. 33 38, 1998.

PAGE 166

151 BIOGRAPHICA L SKETCH Suryanarayana A.N. Prasad was born on the 12 th of March 1979, in Bombay, Maharashtra, India. He completed his schooling from Shrine Vailankanni Senior Secondary School, Madras, India, in 1996. He obtained his baccalaureate degree (B. Tech. Naval A rchitecture) from Indian Institute of Technology Madras, Tamil Nadu, India, in 2000. He joined the Department of Aerospace Engineering, Mechanics and Engineering Science, University of Florida, U.S.A in the term fall 2000 with a graduate research scholar ship and is currently pursuing his master's degree with a major in aerospace engineering and a minor in electrical and computer engineering.


Permanent Link: http://ufdc.ufl.edu/UFE0000594/00001

Material Information

Title: Two-port electroacoustic model of a piezoelectric composite circular plate
Physical Description: xv, 151 p.
Language: English
Creator: Prasad, Suryanarayana A. N. ( Dissertant )
Sheplak, Mark ( Thesis advisor )
Sankar, Bhavani V. ( Thesis advisor )
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2002
Copyright Date: 2002

Subjects

Subjects / Keywords: Mechanical and Aerospace Engineering thesis, M.S   ( local )
Dissertations, Academic -- UF -- Mechanical and Aerospace Engineering   ( local )

Notes

Abstract: Accurate prediction of mode-shape and the deflection field of a circular piezoelectric composite transducer are important for the design of sensors and actuators. There is a need for a model, which lays emphasis on the physics of the problem and predicts the deflection field as a function of pressure and voltage loading. Such a theory would help in developing a non-dimensionalization procedure, which could be used to extract non-dimensional parameters for developing an optimization procedure for the design of a sensor or an actuator. This thesis presents the development of such a theory that predicts the central deflection and mode-shape. Lumped element modeling (LEM) of the system, which is used to extract system parameters of an equivalent single degree of freedom (SDOF) system, is also discussed. The two-port network representation that is used to develop an equivalent circuit representation of the piezoelectric transduction is presented. Non-dimensionalization of the plate equations is also discussed in great detail.
Subject: axisymmetric, composit, dimensionalization, electroacoustic, element, isotropic, lumped, model, modeling, multilayered, non, piezoelectric, plate, port, theory, two, unimorph
General Note: Title from title page of source document.
General Note: Includes vita.
Thesis: Thesis (M.S.)--University of Florida, 2002.
Bibliography: Includes bibliographical references.
General Note: Text (Electronic thesis) in PDF format.

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0000594:00001

Permanent Link: http://ufdc.ufl.edu/UFE0000594/00001

Material Information

Title: Two-port electroacoustic model of a piezoelectric composite circular plate
Physical Description: xv, 151 p.
Language: English
Creator: Prasad, Suryanarayana A. N. ( Dissertant )
Sheplak, Mark ( Thesis advisor )
Sankar, Bhavani V. ( Thesis advisor )
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2002
Copyright Date: 2002

Subjects

Subjects / Keywords: Mechanical and Aerospace Engineering thesis, M.S   ( local )
Dissertations, Academic -- UF -- Mechanical and Aerospace Engineering   ( local )

Notes

Abstract: Accurate prediction of mode-shape and the deflection field of a circular piezoelectric composite transducer are important for the design of sensors and actuators. There is a need for a model, which lays emphasis on the physics of the problem and predicts the deflection field as a function of pressure and voltage loading. Such a theory would help in developing a non-dimensionalization procedure, which could be used to extract non-dimensional parameters for developing an optimization procedure for the design of a sensor or an actuator. This thesis presents the development of such a theory that predicts the central deflection and mode-shape. Lumped element modeling (LEM) of the system, which is used to extract system parameters of an equivalent single degree of freedom (SDOF) system, is also discussed. The two-port network representation that is used to develop an equivalent circuit representation of the piezoelectric transduction is presented. Non-dimensionalization of the plate equations is also discussed in great detail.
Subject: axisymmetric, composit, dimensionalization, electroacoustic, element, isotropic, lumped, model, modeling, multilayered, non, piezoelectric, plate, port, theory, two, unimorph
General Note: Title from title page of source document.
General Note: Includes vita.
Thesis: Thesis (M.S.)--University of Florida, 2002.
Bibliography: Includes bibliographical references.
General Note: Text (Electronic thesis) in PDF format.

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0000594:00001


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TWO-PORT ELECTROACOUSTIC MODEL OF A PIEZOELECTRIC COMPOSITE
CIRCULAR PLATE
















By

SURYANARAYANA A.N. PRASAD


A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF
FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2002




























Copyright 2002

by

Suryanarayana A.N. Prasad





























To my parents

A.C. Nagaraja Prasad & A.N. Kamala















ACKNOWLEDGMENTS


First of all, I thank Dr. Sheplak for involving me in this research work. I also

thank him for providing financial support for my graduate study at the Department of

Aerospace Engineering, Mechanics and Engineering Science, University of Florida. I

gratefully acknowledge support from NASA Langley Research Center (grant numbers

NAG-1-2261 and NAG-1-2249 monitored by M.G. Jones and S. Gorton, respectively).

I express my gratitude to my advisors Dr. Sheplak and Dr. Sankar for the

guidance they provided during the complete course of my thesis work. I thank Dr.

Cattafesta for providing useful tips, which led to the successful completion of the thesis. I

thank Dr. Nishida and Dr. Sheplak for providing me insight into lumped element

modeling and two-port network modeling. I am grateful to Dr. Sankar for his lectures on

plate theory, which provided the basis for this thesis. I extend my gratitude to

Distinguished Professor Dr. Haftka for the thought-provoking discussions about

optimization studies on the piezoelectric unimorph disk transducer.

I thank Mr. Stephen Horowitz and Mr. Quentin Gallas for helping me on the

experimental verification section. I appreciate Mr. Stephen Horowitz also for the useful

discussions on optimization and two-port modeling. I thank Mr. Sridhar Gururaj for

helping me in submitting the thesis.









I thank all my professors at the Indian Institute of Technology Madras and all

my teachers at Shrine Vailankanni Senior Secondary School for providing the pedestal

for graduate study.

Above all, I gratefully acknowledge the effort of my parents in shaping my

thinking with their love, affection and guidance.

Last but not the least, I thank all my friends for their emotional support.
















TABLE OF CONTENTS


Page

A C K N O W L E D G M E N T S .................................................................................................. iv

LIST OF TABLES ................ ................................... .......... ......... ix

LIST OF FIGURES ............................... .... ...... ... ................ x

A B S T R A C T ................................................... .................... ............... xv

CHAPTERS
1 IN TR O D U C TIO N ............ ... .. ................ .................................... .... ............

Static Displacement Behavior of the Piezoelectric Unimorph Transducer...............2
B ack g ro u n d .......................................................... ................ 3
Thesis L ayout ................................................................. ..... ......... 5

2 TW O-PORT NETW ORK M ODELING .............................................. .................. 7

Lumped Element M odeling ........................................................ 7
Two-Port Network Modeling An Introduction................................................. 7
Two-Port Model of Piezoelectric Transduction .....................................................9
Tw o-Port M odel of a 1-D Piezoelectric.................................................. ............... 9
Two-Port Electroacoustic Model of a Piezoelectric Composite Plate..................... 11
Equivalent Circuit Representation and Parameter Extraction.............. ................12

3 MECHANICAL BEHAVIOR OF THE PIEZOELECTRIC COMPOSITE PLATE... 18

P rob lem F orm u nation ................. .................................................... ...... ......... 18
A ssu m p tio n s ...................... .. ............. .. ................................................. 1 8
E equilibrium E qu nations ....................................................................... ..................22
Strain-D isplacem ent R relationships ........................................ ....................... 23
C onstitutive E equations ........................................ ................... .. ...... 24
G governing D differential Equations.................................... ......................... .. ......... 27
General Solution..... ...... ..... ................. ............... 28
The Problem of Piezoelectric Unimorph Disk Transducer.......................................28
C central C om posite P late ............................................................. .....................2 8
Outer A nnular Plate .......................... ........ ... ....... ..................... 30









Interface Com patibility Conditions ............................................ ............... 32
Solution Techniques ............................................................. 32
Sim ple A nalytical Solution............................................................... ............... 33
Numerical Method to Obtain Constants............. .......................................33

4 M ODEL VERIFICA TION ....................................... ...... ................... ............... 34

Theoretical V erification.................................................. ............................... 34
Finite Elem ent M odel .................................. ... .. ..... ............ 36
P pressure L loading ............................................ ................... .. ......36
Voltage Loading .................................. .. .. .. ...... .. ............38
E xperim ental V erification .................................................................................. ... 40
Causes for Deviation of the Experimental Results from Theory.............................43

5 NON-DIM EN SIONALIZATION ........................................ .......................... 45

Buckingham Theorem ............................ .......................... ..... ............... 45
Non-Dimensional Deflection for Pressure Loading ...............................................48
Non-Dimensionalization for Voltage Loading................................... ..... .......... 54
Non-Dimensionalization of Lumped Element and Two-Port Network Parameters.. 60

6 CONCLUSIONS AND FUTURE WORK ..... .................. ...............87



APPENDICES
A DETAILED DERIVATION OF THE GENERAL SOLUTION FROM PLATE
CON STITU TIVE EQU A TION S .................................................... .................. 89

B ANALYTICAL SOLUTION ............................................................ ............. .94

Analytical Expression for Radial Defection u0 and Slope 0..................................94
Analytical Expression for Vertical Deflection w .................................................99
Analytical Expression for Short-Circuit Acoustic Compliance C .......................100
Analytical Expression for Effective Acoustic Piezoelectric Coefficient d ...........103

C M ATLAB CODES ............................................................ ..................... 105

Subroutines used by Program 1 ....................................... ....... .............. 105
Program 1: Program used to derive Response of a particular Piezoelectric
Tranducer ............................... .............. .................. ........... 113
Subroutines used only by Program 2 ................................................... ......... .. ........ 115
Program 2: Program used to derive Response of a particular Piezoelectric
T ran sdu cer .................. ........................... ........ ............ ...... ...... ............. ... .116
Program 3: Program implementing Direct Solution of a particular Piezoelectric
T ransducer ........... ................................................... . ... ..... ....... 122











D FINITE ELEMENT MODEL (ABAQUS) INPUT FILE .......................................125

Pressure Loading only (Normalized Piezoelectric Patch Radius = 0.2).................. 125
Voltage Loading only (Normalized Piezoelectric Patch Radius = 0.55)................. 136

LIST OF REFEREN CES ............................................................ .................. 149

BIOGRAPHICAL SKETCH .................................. ................. .......... ..... 151
















LIST OF TABLES


Table Page


2.1: Conjugate power variables and corresponding dissipative and energy storage
elem ents in various dom ains.................................................................................... 8

4.1: Properties of the piezoelectric unimorph disk used in the finite element model.....................37

4.2: Properties of the piezoelectric bender APC 850..................................... ............... 41

4.3: Lumped element and two-port parameters..................................... ..........................42
















LIST OF FIGURES


Figure Pagi

1.1: Cross-sectional schematic of a clamped axisymmetric piezoelectric unimorph disk
tran sdu cer .................................. ................. ........... ............................ .

2.1: Schematic of a piezoelectric plate that can be approximated to be 1-D. ........................... 9

2.2: Equivalent two-port circuit representation of the piezoelectric unimorph at low
frequencies. ................................................................ ........ .......... 12

2.3: Equivalent circuit representation in the electric domain of the piezoelectric
unim orph. ........ ..... ............. .................................... ........................... 13

2.4: Equivalent circuit representation in the electric domain of the piezoelectric
unimorph when decoupled from the acoustic domain ....................................................13

2.5: Another equivalent two-port circuit representation of the piezoelectric unimorph at
low frequencies..................... ... ......... ... ............... .......... ...... .......... 14

2.6: Equivalent circuit representation in the acoustic domain of the piezoelectric
unim orph. ........ ... ............... .................................... ........................... 15

2.7: Equivalent circuit representation in the acoustic domain of the piezoelectric
unim orph w hen electric behavior is decoupled............................................ .................... 15

2.8: Equivalent two-port circuit representation of axisymmetric piezoelectric unimorph
disk at frequencies comparable to that of the primary resonance ........... .............. 16

3.1: An axisymmetric multi-layered transversely isotropic circular plate with pressure
load, radial load and a moment. All loads shown are considered positive.......................19

3.2: Cross-sectional view of the plate shown in Figure 3.1, showing the sign conventions
and labels used. ................................................................................... 19

3.3: Top view of the plate shown in Figure 3.1 ...................... ....... ............................ 20

3.4: Enlarged isometric view of the element shown in Figure 3.3 and Figure 3.2 with
generalized forces acting on it ............................. ........... ................ ............... 20









3.5: An element of the multilayered transversely isotropic composite plate of length dr
placed at a distance r from the center acted on by generalized forces............................21

3.6: Undeformed and deformed shape of an element of circumferential width 0 and
length dr at a radial distance r from the center in the reference plane..............................23

4.1: Comparison of maximum deflection for different radii of the piezoelectric material
as predicted by the analytical solution and finite element model for pressure
application ........................................................... ................ 3 8

4.2: Comparison of maximum deflection for different radii of the piezoelectric material
as predicted by the analytical solution and finite element model for a unit voltage
loading............................... .. ......... .. .. ........................... ................. 39

4.3: A schematic of the experimental setup showing the laser vibrometer focusing on the
clam ped PZT unim orph bender. ............................................. .............................. 40

4.4: Measured displacement frequency response function obtained by converting
velocity measurements using 1/jco integrating factor......................................................41

5.1: Non-dimensional plot of the center deflection of a piezoelectric unimorph disc with
Ep E, = 0.02 subjected only to a pressure load ......................................................51

5.2: Non-dimensional plot of the center deflection of a piezoelectric unimorph disc with
Ep /E = 0.2 subjected only to a pressure load .......... ............................................. 52

5.3: Non-dimensional plot of the center deflection of a piezoelectric unimorph disc with
Ep E, = 0.4 subjected only to a pressure load .......... ............................................. 52

5.4: Non-dimensional plot of the center deflection of a piezoelectric unimorph disc with
E p /E = 0.6 subjected only to a pressure load...........................................................53

5.5: Non-dimensional plot of the center deflection of a piezoelectric unimorph disc with
E I/E = 0.8 subjected only to a pressure load...........................................53

5.6: Non-dimensional plot of the center deflection of a piezoelectric unimorph disc with
E /E, = 0.6 and h, R2 = 0.02, subjected only to a pressure load.......................... 54

5.7: Non-dimensional plot of the center deflection of a piezoelectric unimorph disc with
Ep /E = 0.02 subjected only to a voltage load ......... ....... ......... ........ ............. 57

5.8 Non-dimensional plot of the center deflection of a piezoelectric unimorph disc with
Ep E, = 0.2 subjected only to a voltage load.................................... ....... ......... 58









5.9: Non-dimensional plot of the center deflection of a piezoelectric unimorph disc with
E p /E = 0.4 subjected only to a voltage load.............................................................58

5.10: Non-dimensional plot of the center deflection of a piezoelectric unimorph disc with
E /E, = 0.6 subjected only to a voltage load.............................................................59

5.11: Non-dimensional plot of the center deflection of a piezoelectric unimorph disc with
Ep /E = 0.8 subjected only to a voltage load .................. ......... ................... 59

5.12: Non-dimensional plot of the center deflection of a piezoelectric unimorph disc with
Ep E, = 0.6 and h, R2 = 0.02, subjected only to a voltage load.............................60

5.13: Non-dimensional short-circuit acoustic compliance plots for E /E, = 0.02. ...................61

5.14: Non-dimensional short-circuit acoustic compliance plots for E p/E = 0.2 ...................62

5.15: Non-dimensional short-circuit acoustic compliance plots for E /E = 0.4 ...................62

5.16: Non-dimensional short-circuit acoustic compliance plots for E p/E = 0.6 ...................63

5.17: Non-dimensional short-circuit acoustic compliance plots for Ep/E, = 0.8 .....................63

5.18: Non-dimensional short-circuit acoustic compliance plots for E I/E = 0.6 and
h,/R 2 = 0.02 ...........................................................................64

5.19: Non-dimensional acoustic mass plots for EIE, = 0.02 (Aluminum/PVDF) ..................65

5.20: Non-dimensional acoustic mass plots for E /E = 0.05 (Aluminum/PVDF). ..................66

5.21: Non-dimensional acoustic mass plots for E /E = 0.2(Silcon/PZT). ............................66

5.22: Non-dimensional acoustic mass plots for E IE = 0.4(Silcon/PZT). .............................67

5.23: Non-dimensional acoustic mass plots for E IE, = 0.6(Silcon/PZT) ............................67

5.24: Non-dimensional acoustic mass plots for E IE = 0.6 (Brass/PZT). ............................68

5.25: Non-dimensional acoustic mass plots for E IE = 0.8 (Brass/PZT). .............................68

5.26: Non-dimensional acoustic mass plots for E /E = 0.6 and h/R2 = 0.02
(Silcon/PZT)................................... ................. .............. ...........69









5.27: Non-dimensional effective acoustic piezoelectric coefficient plots for
E /E = 0 .02 .................................................................................... .................. 7 1

5.28: Non-dimensional effective acoustic piezoelectric coefficient plots for E /E = 0.2..........71

5.29: Non-dimensional effective acoustic piezoelectric coefficient plots for E I/E = 0.4..........72

5.30: Non-dimensional effective acoustic piezoelectric coefficient plots for E I/E = 0.6..........72

5.31: Non-dimensional effective acoustic piezoelectric coefficient plots for E /Es = 0.8..........73

5.32: Non-dimensional effective acoustic piezoelectric coefficient plots for Ep/Es = 0.6
and h = 0 .02 ................................................................ ................................ 73


5.33: Non-dimensional A, for E /E


Non-dimensional

Non-dimensional

Non-dimensional

Non-dimensional

Non-dimensional

Non-dimensional

Non-dimensional

Non-dimensional

Non-dimensional

Non-dimensional

Non-dimensional

K* for E/E, =

K* for E /E, =


A for Epr/E

A for Ep/E




' for E /E




A for E/IE

0' for EIE,

A0 for EP/E

'A' for E /E,

'A' for E/E,
' for EIE,


0.02 ................................................. ............. 75

0.2.............. ....... ...............76

0.4.................. ...... ...............76

0.6.................. ...... ...............77

0.8 .................. ........... ......................77

0.6 and h,/t = 0.02 ...........................................78

0.02 ...................... .............................................79

0.2 ........... ..... ..................................... .......... 80

0.4 .................. ..................................... .......... 80

0.6 ...................... ..... ............ .............. 81

0.8.................. ...... ...............81

0.6 and hj/R = 0.02 .............................................82


5.34:

5.35:

5.36:

5.37:

5.38:

5.39:

5.40:

5.41:

5.42:

5.43:

5.44:

5.45:

5.46:


0.02 .................................. .................. ................. 83

0.2 .................................................................. .... ......... 84









5.47: K for E /E = 0.4 .......................... .................... ............. ............. 84

5.48: K for E E = 0.6 ......................................................... 85

5.49: K for E p/E = 0.8 .................... .. ...................... ............. ............. 85

5.50: K* for EP/E = 0.6 and h /R2 = 0.02 ..................................... ... ............... 86














Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

TWO-PORT ELECTROACOUSTIC MODEL OF A PIEZOELECTRIC COMPOSITE
CIRCULAR PLATE

By

Suryanarayana A. N. Prasad

December 2002

Chairman: Dr. Mark Sheplak
Cochairman: Prof. Bhavani V. Sankar
Major Department: Department of Mechanical and Aerospace Engineering.

Accurate prediction of mode-shape and the deflection field of a circular

piezoelectric composite transducer are important for the design of sensors and actuators.

There is a need for a model, which lays emphasis on the physics of the problem and

predicts the deflection field as a function of pressure and voltage loading. Such a theory

would help in developing a non-dimensionalization procedure, which could be used to

extract non-dimensional parameters for developing an optimization procedure for the

design of a sensor or an actuator.

This thesis presents the development of such a theory that predicts the central

deflection and mode-shape. Lumped element modeling (LEM) of the system, which is

used to extract system parameters of an equivalent single degree of freedom (SDOF)

system, is also discussed. The two-port network representation that is used to develop an

equivalent circuit representation of the piezoelectric transduction is presented. Non-

dimensionalization of the plate equations is also discussed in great detail.














CHAPTER 1
INTRODUCTION


Commonly used electroacoustic' devices, such as microphones and headphones,

use circular disk transducers that are piezoelectric composite plates. Recent devices, such

as synthetic jet actuators,2'3 used in flow control applications, can also be driven by

piezoelectric composite circular plate. Micro-fluidic pump drivers4 represent another

relatively new application for these devices.

A design procedure, to find a set of system parameters for the optimal

performance of these devices, helps to improve the performance of these devices and also

widen its range of applications. Development of such a design procedure is possible only

when the response of these devices to loading and change of system parameters is known.

Although experimental techniques can be used to characterize piezoelectric transducers,

analytical models that can predict the response of the transducers are helpful in

understanding the effects of various parameters on the predicted response and also in

optimizing the performance of the devices.

Determining the dynamics of a single degree of freedom (SDOF) system would

be an easier task than determining the dynamic response of a corresponding complex

system. Such a SDOF system would be a compact macromodel that provides physical

insight and accurately captures the energy behavior and dependence on the material

properties. This system is obtained by lumped element modeling (LEM) of the

piezoelectric unimorph disk transducer. The lumped element parameters of the system are









obtained from the static deflection behavior of the system when subjected to pressure and

voltage loading. This approximation will be valid in Fourier space and will simplify the

problem of determining the complicated dynamics of the piezoelectric composite circular

unimorph transducer into a static analysis of an axisymmetric piezoelectric multi-layered

isotropic composite circular plate.

As the problem consists of various domains, a two-port network representation1'5

is used to model the transduction from the input port to the output port. The parameters

obtained from the LEM of the plate are fed into the two-port network model of the

device. This two-port model is used in developing an equivalent circuit of the device. The

behavior of the device to loading is obtained from the transfer function of this equivalent

circuit.


Static displacement behavior of the piezoelectric unimorph transducer

Piezo
\ h l
h Shim R2 ---
Sr
+ \,

V Z
V --------------- ---------- ------------------------- -----




Figure 1.1: Cross-sectional schematic of a clamped axisymmetric piezoelectric unimorph
disk transducer



Figure 1.1 shows a cross-section of a clamped circular piezoelectric unimorph

composite plate subjected to a uniform transverse pressure loading P and/or a voltage

V. A piezoceramic material of thickness hp and radius R, is bonded on top of a shim









material of outer radius /R and thickness h,. The loading creates a transverse

displacement field w(r) and a radial displacement field u(r,z).

Background

Previous work in the area of piezoelectric composite circular plates focused on

structures that are symmetrically layered about the neutral axis such as bimorph

transducers.6 Because of midplane symmetry there is no bending-extension coupling in

bimorph transducers and this simplifies the analysis. However, many of the devices

cannot be manufactured in this manner (for example, most piezoelectric unimorph

transducers fabricated through micro-electromechanical systems (MEMS) technology

have geometry as shown in Figure 1.1). Hence there exists a need to extend the existing

piezoelectric composite plate theory to the case of an axisymmetric multi-layered

transversely isotropic piezoelectric composite circular plate.

Morris and Foster4 developed an optimization procedure for a piezoelectric

bimorph micropump driver (the same geometry as in Figure 1.1) using finite element

method (FEM) with the help of ANSYS7 software. They performed optimization of the

micropump driver for both pinned and the fixed case by identifying non-dimensional r

groups using the Buckingham theorem. They used a higher-order routine in ANSYS to

accomplish this task. They have developed some empirical equations for optimal radius

ratio and thickness ratio for a particular set of materials for a particular aspect ratio. They

also discussed edge support effects and effect of bond layer.

Dobrucki and Pruchnicki6 formulated the problem of a piezoelectric axisymmetric

bimorph and used FEM to solve the problem. They derived the equations that would

determine the bending moment and extensional forces produced by the piezoelectric









material on application of an electric field. They used average elastic parameters for

analyzing the composite plate. Use of the bimorph as a sensor was also discussed. They

experimentally verified their results from the FEM solution. Verification of the theory

with simpler geometry was also performed. They also have proved that on the rim of a

clamped circular transducer the electric signal produced is zero.

Stavsky and Loewy8 solved numerically the dynamics of isotropic composite

circular plates using Kirchoffs plate theory. They found the vibrations of the composite

plate to be analogous to the vibration of a homogenous shallow spherical shell. They also

discussed effects due to material arrangement, radius, material and plate composition on

frequency of vibration of the composite circular plate. They obtained a system of

equations of the 6th order. The solution for this system of equations can be expressed in

terms of Bessel functions, the argument for which is determined from the characteristic

equation of order 3. They also discussed numerical examples showing the effect that

arises due to heterogeneity on vibration response of the composite to be significant.

Adelman and Stavsky9 formulated the problem of piezoelectric circular composite

plates using Kirchoffs plate theory. Static behavior and flexural-extensional vibratory

response of metal-piezoceramic unimorphs and PZT-5H bimorphs possessing silver

electrodes are solved numerically. Their formulation is identical to that of the formulation

discussed in this thesis, except that they use variables E, and E2 that relate the fictitious

force/moments generated to the electric field applied instead of the comprehensive

equation for fictitious forces that describe the piezoelectric transduction shown in Eq.

(3.26) and Eq. (3.27). They also discussed numerical examples showing the effect of

silver electrode on unimorph piezoelectric benders.









Chang and Du1o performed optimization of a unimorph disk transducer based on

an electro-elastic theory assuming free boundary conditions, which is non-physical for

most applications. They modified the existing Kirchoffs plate theory by adding a term to

account for the piezoelectric layer. They assumed that the electric field variation in the

thickness direction could be represented as a quadratic function and the electric charge to

be equal and opposite on the top and bottom electrodes of the piezoelectric layer.

Dumir et al." obtained a non-linear axisymmetric solution for the static and

transient moderately large deflection of a laminated axisymmetric annular plate acted on

by uniformly distributed ring loads by using first-order shear deformation theory. Effect

of inplane inertia was neglected while the rotary inertia was considered. The material was

treated to be orthotropic. They used a numeric technique called the Newmark- / scheme

in order to solve the governing differential equations. They simplified the solution and

verified the same with the solution from the classical plate theory.

Thesis Layout

Chapter 2 of this thesis presents a two-port, lumped-element model of an

axisymmetric piezoelectric unimorph transducer with the geometry and loading described

in Figure 1.1. In LEM, the individual components of a piezoelectric unimorph are

modeled as elements of an equivalent electrical circuit using conjugate power variables.

The synthesis of the two-port model required determination of the transverse static

deflection field as a function of pressure and voltage loading.

In Chapter 3, classical laminated plate theory was used to derive the equations of

equilibrium for circular laminated plates containing one or more piezoelectric layers. The

equations were solved for a unimorph device wherein the diameter of the piezoelectric









layer was less than that of the shim (R, < R2). An exact analytical static solution of the

displacement field of the axisymmetric piezoelectric unimorph is determined. The

solution for annular plate obtained using the classical plate theory matches with the

solution provided in the paper by Dumir et al. 1

Chapter 4 verifies the result obtained from Chapter 3 by theoretical means and by

a Finite Element Model. Methods to estimate the model parameters are discussed and

experimental verification is presented.

Chapter 5 discusses how the governing equations are used in the non-

dimensionalization of the field variables, lumped element parameters and two-port

network parameters. This proves to be a simpler and more comprehensive option.

The results corresponding to the work described in a Chapter are summarized at

the end of the corresponding chapters. In addition, a summary of the main results is

provided in the conclusions (Chapter 6). Future work and concurrent work is also

discussed in Chapter 6.














CHAPTER 2
TWO-PORT NETWORK MODELING


The piezoelectric composite plate actuator represents a coupled electro-

mechanical-acoustic system with frequency dependent properties determined by device

dimensions and material properties. The analysis and design of such a coupled-domain

transducer system is commonly performed using lumped element models.' This is

justified, because the prediction by LEM matches the actual value to within 2%, as per

Merhaut.12

Lumped Element Modeling

The main assumption employed in LEM is that the characteristic length scales of

the governing physical phenomena are much larger than the largest geometric dimension.

For example, for the vibration of a piezoelectric plate, the bending wavelength and

electromagnetic wavelength must be significantly larger than the device itself. If this

assumption holds, then the temporal and spatial variations can be decoupled. This

decoupling permits the governing partial differential equations of the distributed system

to be "lumped" into a set of coupled ordinary differential equations through the solution

of the static equations. The individual components of a piezoelectric unimorph are

modeled as elements of an equivalent electrical circuit using two-port modeling.


Two-Port Network Modeling An Introduction

Any linear conservative electroacoustic transduction can be modeled using the

electrical analogy as a transformer or a gyrator with series and parallel impedances or









admittances on each of its ports.' The transduction from one domain to another, which is

a function of system parameters, is represented in terms of admittances and impedances

in each domain.




Table 2.1: Conjugate power variables13 and corresponding dissipative and energy storage
elements in various domains.

Energy Effort Flow Energy Kinetic Potential Displacement
Domain Variable Variable Dissipater Energy Energy
Storage Storage

Mechanical Force Velocity Dashpot Mass Spring Displacement

Acoustical Pressure Volume Vent Acoustic Cavity Volume
Velocity Mass Displacement

Electrical Voltage Current Resistor Inductor Capacitor Electric
Charge



Resistors are used to represent any dissipative element. Dissipative elements in

other domains are shown in the fourth column of Table 2.1. Inductors and capacitors are

used to represent elements that store generalized kinetic energy and potential energy

respectively. Corresponding elements in other domains are shown in fifth and sixth

column of Table 2.1. The conjugate power variables,13 the effort and flow, are identified

in each of the domains as shown in the second and third column of Table 2.1. The

product of the conjugate power variables is a measure of power. In impedance analogy,

elements sharing common flow are connected in series while elements sharing a common

effort are connected in parallel.









Two-Port Model of Piezoelectric Transduction

A piezoelectric transducer converts electric energy into strain that is realized as a

displacement in the mechanical domain. Usually a piezoelectric transduction is

represented in tensor form as per IEEE standards as shown in Eq. (2.1) and Eq. (2.2).

S =sET + dE, (2.1)
and

D= dT + TE, (2.2)
where D is the dielectric displacement in [C/m2], T is the stress in [Pa], S is the

strain, E is the electric field in [V/m], e is the permitivity in [C/Vm], s is the

compliance in [1/ Pa] and d is the piezoelectric coefficient in [C / N].

A piezoelectric material responds with a strain field not only due to application of

stress but also due to application of electric field. An application of stress creates a charge

(due to piezoelectric transduction) in addition to the charge created due to the application

of voltage across the piezoelectric (a dielectric medium).

Two-Port Model of a 1-D Piezoelectric

plate of area A
3-dir
t FVAE A v
plate F
thickness h.
P Fixed B.C.
Figure 2.1: Schematic of a piezoelectric plate that can be approximated to be 1-D.



In case of a 1-D piezoelectric, the force F and voltage V act only in the 3-

direction as shown in Figure 2.1. Application of the force not only gives rise to a

deflection x but also creates a polarization represented by an electrical charge q ;









x = C F (2.3)

and
q =dF (2.4)

In the above equations, C, and d are the short-circuit mechanical compliance

and effective mechanical piezoelectric coefficient (that is responsible for a strain in 3-

direction due to application of electric field in the 3-direction) of the piezoelectric

material respectively and are given by


c, =v0 (2.5)

and

d = P= (2.6)
V
Application of voltage creates a deformation x in addition to creating a

polarization represented by an electric charge q ;


x=dV (2.7)

and

q =CEFV, (2.8)

CEF is the electrical free capacitance of the 1-D piezoelectric that is given by


CEF (2.9)
hP

where k is the dielectric constant of the piezoelectric in the 3-direction due to application

of an electric field in the 3-direction and eo is permitivity of free space.

The transduction in the case of a 1-D piezoelectric in the static case, when

subjected to both voltage and force load, is found by superimposing Eqs. (2.3), (2.4),

(2.7) and (2.8);










{} s [CMS c F]{ (2.10)
q d CEFV,
Two-Port Electroacoustic Model of a Piezoelectric Composite Plate

In the case of a piezoelectric unimorph disc, application of voltage creates

bending and not an extension as in the case of the 1-D piezoelectric described above.

Also the focus in this thesis is oriented towards an electroacoustic model rather than an

electro-mechanical model of the piezoelectric unimorph. Hence integration over the

surface area of the unimorph disc needs to be performed to extend the electromechanical

model of the 1-D piezoelectric to a piezoelectric unimorph disc transducer.

In the acoustic domain, a volume displacement is created in a piezoelectric

transducer not only due to application of pressure but also due to application of voltage.

Application of pressure creates a charge separation across the piezoelectric layer (due to

piezoelectric transduction) in addition to the charge separation created due to the

application of voltage. Hence the transduction of an axisymmetric piezoelectric unimorph

disk in static case is expressed as



AVol= [Cs rd F] (2.11)
q d, CEF V

where AVol is the volume displaced by the plate due to the application of pressure P

and voltage V;

R2
AVol= 27rw(r)dr, (2.12)
0

and Cs is the short-circuit acoustic compliance of the plate. Expression for the acoustic

compliance is extracted from the equivalent SDOF system by equating the strain energy

of the actual system to the potential energy of the SDOF system in the following manner:









R2
f w(r)\v=o 2nrrdr
AVol v0
CAS 0 P
P v=oc P
In Eq. (2.11), dA is the effective acoustic piezoelectric coefficient, which is given by


(2.13)


R2
f w(r) o2rrdr
AVol P=0
dA -0_ =J w -0r- (2.14)
d V P- V
Equivalent Circuit Representation and Parameter Extraction

Assuming time harmonic function and differentiating both sides of Eq. (2.11)

with respect to time yields the expression for conjugate power variable at low

frequencies;


Q j(CAS jCd A (215)
\ \= \ [, (2.15)

where Co is the radian frequency. The equivalent circuit of the piezoelectric unimorph at

low frequencies takes the form shown in Figure 2.2.

iQ 1:A Q
CAS
+ +


V CEB B AV P



Figure 2.2: Equivalent two-port circuit representation of the piezoelectric unimorph at
low frequencies.



In Figure 2.2, Q is the volume velocity and i is the current. The parameter OA

appearing as the transformer turns ratio in the equivalent circuit representation is the






13


electroacoustic transduction coefficient, which is given by the ratio of the effective

acoustic piezoelectric coefficient and the short-circuit acoustic compliance of the plate;


A -A (2.16)
CAS
When unloaded, the equivalent circuit representation in Figure 2.2 is represented

as shown in Figure 2.3. The energy stored in the two circuit elements shown in Figure 2.3

should equal the energy stored in the circuit element shown in Figure 2.4. Hence the

electroacoustic energy-coupling factor, which is a quantity that describes the fraction of

energy converted from electric domain to the acoustic domain, is



K2 d(2.17)
K2 P.E. stored in O A 2 CASVJ YCAS (217)
P.E. stored in CF 1C FV2 CEF CASCEF
2




V CEB =



Figure 2.3: Equivalent circuit representation in the electric domain of the piezoelectric
unimorph.







V -C
-V-EF



Figure 2.4: Equivalent circuit representation in the electric domain of the piezoelectric
unimorph when decoupled from the acoustic domain.









The blocked electrical capacitance at the electrical side is obtained by equating

the energy storage in Figure 2.3 and Figure 2.4;


SCEV2 + CASV2 = CEFV2. (2.18)
22 2
Substituting Eq. (2.17) in Eq.(2.18), we obtain

1 1 1
ICEB2 I+-CV2K2 = EFV2, (2.19)
2 2 2
which simplifies to


CEB CEF-K2). (2.20)




Q 'Ai 1:'A EF 1
SEF



P --C A AP V
-- -AO



Figure 2.5: Another equivalent two-port circuit representation of the piezoelectric
unimorph at low frequencies.



Transduction mechanism in the case of a piezoelectric unimorph at low

frequencies has another equivalent circuit representation as shown in Figure 2.5. The

parameter 0 appearing as the transformer turns ratio in the equivalent circuit

representation shown in Figure 2.5 is the acousto-electric transduction coefficient, which

is given by the ratio of the effective acoustic piezoelectric coefficient of the plate and free

electrical capacitance of the piezoelectric layer;






15


S-A d (2.21)
CEV
When the electrical port in the equivalent circuit representation shown in Figure

2.5 is shorted, the equivalent circuit simplifies to the circuit shown in Figure 2.6. The

energy stored in the two circuit elements shown in Figure 2.6 should equal the energy

stored in the circuit element shown in Figure 2.7. Hence the energy-coupling factor is

derived as



2 P.E. stored in 2CEF 2 2ECFP2
K_ = (2.22)
P.E. stored in CAS 1A2 CAS CASF
2 AS




P A CA A A2 EF



Figure 2.6: Equivalent circuit representation in the acoustic domain of the piezoelectric
unimorph.






P --C
-- CAS



Figure 2.7: Equivalent circuit representation in the acoustic domain of the piezoelectric
unimorph when electric behavior is decoupled.



The open acoustic compliance at the acoustic port is obtained by equating the

energy storage in Figure 2.6 and Figure 2.7;









CEFP22 + CAOP = -CASP2. (2.23)
2 2 2
Substituting Eq. (2.22) in Eq.(2.23), we obtain


CAsp2K2 +I CAP2= CAsP2, (2.24)
2 2 2
which simplifies to


O CAS (I-K2). (2.25)
For higher frequencies the mass of the disk becomes important and leads to

resonance. The circuit in Figure 2.8 will describe the system at higher frequencies, upto

the fundamental natural frequency.

i AQ A CAS MA Q
+ N- +-- +


C EB AV P



Figure 2.8: Equivalent two-port circuit representation of axisymmetric piezoelectric
unimorph disk at frequencies comparable to that of the primary resonance



The acoustic mass MA of the equivalent SDOF system is extracted from the

deflection response of the system by equating the kinetic energy stored in the SDOF

system to that of the actual system in the following manner:


M -A JPA(r) rdr, (2.26)
(A Vol) 0

where PA is the real density of the piezoelectric composite plate. For the geometry

described in Figure 1.1, the value of the areal density remains a constant from zero-radius









to the end of the inner composite region. There is a change in value of the real density at

the interface and then remains a constant until the clamped edge. The volume displaced

due to application of a load remains a constant independent of the region.

The short-circuit natural frequency f, in Hz of the system is given by

1
f, = (2.27)

while the open-circuit resonant frequency f, is given by

1
fo= (2.28)

By substituting Eqs. (2.25) and (2.27) in Eq. (2.28), we obtain the relationship

between the resonant frequencies as

1 f
1 = (2.29)
24Cs (lI-K2)MA (-K2)

which on simplification yields


f (- K2) = (2.30)

The deflection field of the piezoelectric composite plate, due to unit-applied

pressure and due to unit-applied voltage, is required to determine CAS, dA, and K. The

deflection field is obtained by analyzing the mechanical behavior of the piezoelectric

unimorph, which is modeled and discussed in Chapter 3 of the thesis.














CHAPTER 3
MECHANICAL BEHAVIOR OF THE PIEZOELECTRIC COMPOSITE PLATE



For developing a two-port electroacoustic model of the piezoelectric composite

plate, the quantities defined in Eq. (2.13), (2.14) and Eq. (2.26), need to be determined.

These quantities depend on the vertical deflection that is determined by analysis of the

mechanical behavior of the axisymmetric piezoelectric composite plate.

Problem Formulation

Problem formulation is based on the classical Kirchoff's plate theory.14 The fact

that one of the layers is a piezoelectric is accounted for in the constitutive relation by

adding an additional strain term due to the piezoelectric layer.

Assumptions

The assumptions made in developing a linear small deflection plate theory with

piezoelectric effect are as follows:

S The plate (shown in Figure 3.1), is assumed to be in a state of plane stress14

normal to the z-axis. In other words, the normal stress oz and the shear stress

zr, are approximately equal to zero.
































Figure 3.1: An axisymmetric multi-layered transversely isotropic circular plate with
pressure load, radial load and a moment. All loads shown are considered positive.




The shear stresses z,, and zro also vanishes due to the assumed axisymmetric


nature of the problem. Stress measures, corresponding to the plane stress case,


(i.e.) o, and oa, exist. Only, shear stress in the transverse direction exists


(i.e. zr, 0). Figure 3.2 delineates the active stress resultants on a multi-


layered composite circular plate acted on by a pressure load P (r) .


Q,(R)
A //


Plane of
Reference


Figure 3.2: Cross-sectional
conventions and labels used.


En, Vn, hn
I
I
I
I
I
I
/


?r(R)


I I i .i--

__ __ __ ________ __
iI

1 /Z

/ P (r)
EI,v,,h,
view of the plate shown in Figure 3.1, showing the sign


I


i









* The active stresses on the circular plate are shown in Figure 3.3. Figure 3.4

delineates the force, moment and shear force resultants acting on an

infinitesimal element of a multi-layered isotropic composite circular plate.






r dr




r N(/R)








Figure 3.3: Top view of the plate shown in Figure 3.1


X + TLdr
Sar



+ Q dTr
ar


M, + r dr
ar


N, +- d 0 -4' I
SMo+ dO 'P(r)

Figure 3.4: Enlarged isometric view of the element shown in Figure 3.3 and Figure 3.2
with generalized forces acting on it.
















N, + r dr




M, \M, + dr


\ Qr, + dr

r P(r)


r+dr


Figure 3.5: An element of the multilayered transversely isotropic composite plate of
length dr placed at a distance r from the center acted on by generalized forces.




In Figure 3.4 and Figure 3.5, N, and No are the force resultants in radial and


circumferential directions respectively;


z2
N, = a,,drr
zl


z2
No = J adz.


(3.1)





(3.2)


The moment resultants in radial and circumferential directions AM and M, are given by


z2
M, = ,, zdz
zl


z2
M, = a rzdz.
zl


(3.3)


(3.4)


Q, is the shear force resultant given by









z2
Q, = dz. (3.5)
zl
Equilibrium Equations

For the case of the axisymmetric multi-layered composite, the field variables

u0 and 0 are functions of radius alone (due to axisymmetry). Therefore, the partial

differentials equal the corresponding total differentials. Taking balance of forces in the

radial direction, of the projection of the element (shown in Figure 3.4) in the r-0 plane,

we obtain

dN 1
dN I-(N. -N,) = 0. (3.6)
dr r

Taking moment about the line tangent to r + dr of the dement shown in Figure 3.4, we

obtain


SQr +Q (Mr -Mo)= 0. (3.7)
dr r
Taking balance of the forces shown in Figure 3.5 in the vertical direction, we obtain


+ P(r)+ =0. (3.8)
dr r
The moment resultants are expressed in matrix form as


r = zdz, (3.9)
MeO zi (30ee0
where moments/moment resultants causing tension in top surface are considered positive.

The force resultants are expressed in matrix form as



[r = dz (3.10)

where radial forces/force resultants causing tension are considered positive.









Equations (3.6) (3.8) are the equations of equilibrium14,15'16 of an axisymmetric

multi-layered composite plate. The pressure load acting on the bottom surface is uniform.

Therefore, these equations are suitably modified to yield the equilibrium equations of an

axisymmetric piezoelectric unimorph. The resulting expressions are given by


dN, ( N, No )
dN+ =0, (3.11)
dr r


d + (M-Or, (3.12)
dr r

and

d + +P= 0. (3.13)
dr r

Strain-Displacement Relationships

The field variables involved in the problem are the radial deflection in the

reference plane u, and slope 0 respectively.



/.. \ \
S"U (r) du0 (r) 0








Figure 3.6: Undeformed and deformed shape of an element of circumferential width 0
and length dr at a radial distance r from the center in the reference plane.



The increment shown in Figure 3.6 contains a total derivative because of the fact

that the reference plane displacements are just a function of radius due to axisymmetry,










(i.e.) uo = u (r) and 0 =(r). The strains are obtained from the field variables in the

following manner.

The radial strain measure for Kirchoff s plate theory is given by


E, = e + zK (3.14)


where eo and Kr are the radial strain in the reference plane and curvature in the radial

direction, respectively;


0 U + du uo duo
err (3.15)
r +dr -r dr
and

d2w dO
Kr d (3.16)
dr2 dr
The circumferential strain measure for Kirchoff's plate theory is given by


eo, = e0 + zK0, (3.17)


where eo and Kc are the circumferential strain in the reference plane and curvature in

the circumferential direction respectively and are obtained by taking balance of forces

acting on the element shown in Figure 3.6;


o (r+uo)O-rO (3._u
= (3.18)
rO r
and
ldw 0
KC = (3.19)
r dr r
Constitutive Equations

For an elastic plate, the stress-strain relations14,15 are given by








arr Q11 Q112 r
Co LQ2 Q11 v (3.20)

where


[Q E { }, (3.21)

E is the Young's modulus and v is the Poisson's ratio of the layer.

From Chapter 2, it can be understood that Eq. (3.22) represents the inverse constitutive

relation of a piezoelectric material as per IEEE standards.

S = sET + dE (3.22)
Thus for an axisymmetric piezoelectric composite plate discussed above, the constitutive

relations are given by


{ =rr [Q +z E33 ,d (3.23)
('00 =K" d31


where E33 is the uniform electric field across the thickness of the piezoelectric layer and

d31 is the piezoelectric coefficient in the plane perpendicular to the direction of

application of the electric field. In the above expression, number 3 represents the z-

direction and number 1 represents the radial direction.

The plate constitutive equations are obtained by integrating the constitutive

equations of the individual layers through the thickness and are given by


No =[AlEo +[B][c-N(3.24)










r =[B[e] +[D][c] (3.25)


Z2 Z2
where [A]= [Q]dz is the extensional rigidity matrix, [B]= [Q]zdz is the flexural-
1 Z1

'2
extensional coupling matrix and [D]= [Q]z dz is the flexural rigidity matrix. The


flexural-extensional coupling matrix is zero, if the material layers are symmetrical about

the reference plane (For example, a piezoelectric bimorph). The force resultants, N, and

Nf generated internally due to the application of E33 across the piezoelectric layer are

given by


r =N] | 3[ 12 1 dz. (3.26)
N 12 Q 1

The moment resultants, MAf and MA generated internally due to the application of E,3

across the piezoelectric layer are given by


L3 zdz. (3.27)
S zi /Q12 Q11 d,,

From the form of Eq. (3.26) and Eq. (3.27), we can infer that the material is transversely

isotropic;


NP = NP= NP (3.28)
and


MH, = M=P = 3
r = 0o


(3.29)









Substituting the strain-displacement relationships and curvature-displacement

relationships shown in Eqs. (3.14) (3.19) in the plate constitutive relationships shown in

Eqs. (3.24) and (3.25), we obtain


du,(r) u,(r) d6(r) 6(r)
Nr dr r NP(3
(3.30)
N 1 du0(r) u0(r) d6O(r) 0(r) NP
dr r dr r

and

du,(r) u,(r) d6(r) 6(r)
dr r r dr r J a
M du, (r) u,(r) d6(r) 6(r) M'
dr r dr r

Governing Differential Equations

Equations (3.30) and (3.31) are substituted in the plate equations of equilibrium

shown in Eqs. (3.11) (3.13). The field variables in the resulting expression, are de-

coupled (see Appendix A) by introducing the variables a and DI* defined as


B.
a= -B (3.32)
All
and


D*, =DI D l (3.33)

Assuming that the electric field is uniform and the material is transversely isotropic is

used to show that the internal force and moment resultants, generated by the piezo, has a

constant value all along the reference plane. The resulting expression is simplified to

yield the governing differential equations, which are









d20(r) 1 dO(r) O(r) Pr
+ (3.34)
dr r dr r 2 D0


and


d2u (r) 1 duo(r) uo(r) Pra
r2-- (3.35)
dr r dr r 2D11


General Solution

The general solution to the above set of differential equations is determined as


a2 a Pr')
S= ar+ Pr3 (3.36)
r D,1, 16)


and


0= blr+ (3.37)
r Dl 16


where a,, a2, b, and b2 are constants that are determined by applying the boundary

conditions. A detailed derivation of this solution is shown in Appendix A.

The Problem of Piezoelectric Unimorph Disk Transducer

The problem of circular piezoelectric unimorph disk transducer shown in Figure

1.1 is solved by combining the solution for deflection field of an annular plate of inner

radius R1 and outer radius R2, with the solution for deflection field of the inner composite

plate of radius R1. To differentiate between the constants involved in the solutions in the

central composite region and outer annular region, symbols (1)and (2) are used to denote

inner and outer regions respectively.

Central Composite Plate

In the central composite plate the central deflections of the plate is bounded:









(3.38)


u (0) < K0
and

0(0) Substituting Eq. (3.38) and Eq. (3.39) in the general solution, we obtain


(3.39)


(3.40)


(3.41)


Therefore the deflections are given by


(1) (1) i Pr
uo (r)= a, r ---- 16

and


0" (r) = bl)r 16 .P

In matrix form, Eq. (3.42) and Eq. (3.43) is written as

(1) (r)J a ,(J 16Pr311
0 )(r) bl) 16D,1,(') 1


(3.42)


(3.43)


(3.44)


From Eq. (3.44), we obtain


dr al l)

dr
du6'1 (r) l^J
dO(l (r). I b)


3Pr2 fa2
16D;*(1) 1 "


Substituting Eqs. (3.44) (3.45) in Eqs. (3.30) (3.31), we obtain the following


expression for generalized force resultants in the radial direction:


(3.45)









L' (r) A') + A -B ) B a ]2 3B +B ^-34a-A ~a NP~
Mp(r) 11B +BI(1) -D 1 1 16D 1 ( 3D1 +D 12 -3aB1) -aB2 MP v
(3.46)
Substituting Eqs. (3.32) (3.33) in Eq. (3.46), we obtain


N(') (r) (A() +A 1) -(B(11) + B)) a1) Pr2 B) -Aa NP
M) (r) (B +B) (D + D) b(J 16D) -aB)+3D *(1) +D LM _P
(3.47)
At the interface, the deflections are given by

(u1) (R I) a PR P' 3
R(3.48)
0(1) (RI) bl( 16D,* 1) I
and the generalized force resultants are given by
FN(1 (RI) (A,( + ) (B1(l + Bi(2 a PR12 2
1N2 A +A ) -1B)+ ) I B,") AI2a NP
LrM1 (Ri) (B +B(1) (DI) + DIM)) bl() (1) aB1) -3D *('1) +D() LMP
12 \2 )11 1 1
(3.49)
Outer Annular Plate

For the outer annular plate, [B] vanishes because of the symmetry about its

neutral axis, which in turn implies that D*2) = D2), and a = 0. As the outer annular

plate is isotropic D =) ,D(2) and A2) =v_,AQ(2)

The solution in the annular region is obtained from the general solution by

applying fixed condition at the clamped edge and interface matching condition at the

inner edge. The fixed condition at the clamped edge is applied by making

u (R2) =0 (3.50)
and


0(R2)= 0.


(3.51)










Substituting Eqs. (3.50) and (3.51) in the general solution, we obtain

a2 = -aR(2)
and

PR4
b 2 = -b 2 R 2
16D1b2
Substituting back Eqs. (3.52) and (3.53) into the general solution, we obtain


uO (r) R 212) P R4
{ou2(r) 2 r )Lb 2J 16(2){3 }2
Taking the first derivative of Eq. (r) with respect to the radius, we obtain

Taking the first derivative of Eq. (3.54) with respect to the radius, we obtain


cdh (r) Rk\ a

de2 (r) [r2)[bl
dr


0
R4 .
r 2


Substituting r


R in Eq. (3.54), we obtain the deflections at the interface, which are


given by


(2 )
0 (2) (Rj


r 0
fD1
^ R1J


(3.56)


The generalized force resultants in the outer annular region are given by


Nr) (r)
) (r)]


S (r) (r)
A () + A (Ar)
dr r
dr r (


(3.52)


(3.53)


(3.54)


(3.55)


(3.57)


C1:Z) )








Substituting Eq. (3.54) and Eq. (3.55) in Eq. (3.57), we obtain a relationship for

generalized force resultants in the radial direction only in terms of the constants involved

in the analytical solution for deflection;


N,(2) A) R2 _( Pr204(5
(l+v)+ (1-v) + (3+ R4 ~ .(3.58)
M2)(r) 2 -D2)b(2) 16 1(3+V)+ (1- )

The generalized force resultants at the interface are given by


2M2(R) 2 2~) 1+(2) PR)+ 4

L _'R
L= 1-i {111/(}-v) .(3.59)

Interface Compatibility Conditions

At the interface, deflections and the resultants must match the following

conditions:

u ) (O ) (2) (=), (3.60)

e0 (R) 60(2) (R,), (3.61)

NA() ( A)= N(2) (R,) (3.62)
and

"M ( ) = M/ 2) (R,). (3.63)
From the above derivation, it can be noted that the piezoelectric effect does not

appear explicitly in the general solution but is introduced by the interface matching

conditions of the generalized force resultants shown in Eq. (3.62) and (3.63).

Solution Techniques
The expressions continued to increase in complexity. Analytical expression for

the deflection field in terms of basic parameters is huge and does not provide any









physical insight and is hard to evaluate or code. Hence two different techniques are

employed to obtain the solution.

Simple Analytical Solution

The first technique is to obtain a simple analytical solution by making valid

assumptions based on reality. The solution is simplified by introducing constants, which

are functions of interface deflections. Such a solution is useful for the purpose of non-

dimensionalization. Such an effort to obtain an analytical expression is discussed in

Appendix B. The MATLAB code implementing the same is attached in Appendix C.

Numerical Method to Obtain Constants

The second technique is to code up the above equations in matrix form in

MATLAB. The equivalent stiffness matrix of each of the regions is evaluated. The

matrices are assembled to form an equivalent global stiffness matrix. The deflections and

generalized force resultants at the interface are found by inverting the stiffness matrix.

The constants and hence the deflection field is found from the evaluated interface

deflections. The MATLAB code is attached in Appendix C of the thesis.















CHAPTER 4
MODEL VERIFICATION


Verification of the analytical form of the solution obtained in Chapter 3 is

required before using it to develop any design procedure. The following section deals

with the verification of the solution obtained from Chapter 3 by theoretical and

experimental means in addition to verifying it with a finite element model.

Theoretical Verification

To verify the theory, the solution for a classical plate obtained from the

piezoelectric composite plate theory should match exactly the classical plate solution for

a homogenous plate. The solution obtained from the theory will be equal to that of the

classical plate solution if the equivalent flexural rigidity D,, defined in Chapter 3 is same

as the flexural rigidity D of the classical plate. The flexural rigidity of a classical circular

plate is given by


Eh3
D 2). (4.1)
12(1 v2

The equivalent flexural rigidity D1, of the classical plate is given by


n2
D* = D B11 (4.2)
A11

Substituting the expression for flexural rigidity in Eq. (4.2) for a homogenous plate, we

obtain









z+h 2
z+h E v2
Dl' = z2dz z+h
Sl- dz
1V

Pulling the constants out of the integral in the above expression, we obtain


z+h 2

D1 = -v2 z 2dz ;+h
f1 dz


Computing the definite integrals in the above expression, we obtain


E
D,,l = v


Z2 z+h 2
+z h
3 z z+h


which on further simplification yields


E
l-v2


(z +h)3 -z3
3


(z+h)2 z22
2
h


Using elementary arithmetic identities, the above expression is simplified to yield


1= E 3z2 h+3zh2 + h3 (2z + h)2 h
DAr =to yield
1-v2 3 4

which is simplified further to yield


1 E h3 2 2 (4z2h+4zh2 +h3
1-v2 3 4
y;~


The above expression on simplification yields


(4.3)


(4.4)


(4.5)


(4.6)


(4.7)


(4.8)










D = -+ z2h+zh2 -z2h -zh2 h 2 2) D, (4.9)
1-v2 3 4 12( -v2

which implies that the equivalent flexural rigidity appearing in the piezoelectric circular

composite plate theory reduces to flexural rigidity of a classical circular plate for a

homogenous plate.

Finite Element Model

The finite element model of the piezoelectric unimorph disk was made for several

extents of piezoelectric patch in ABAQUS CAE,17 both for pressure loading and voltage

loading. The geometry and the material properties used corresponding to Brass/PZT

bender (Shim = Brass, Piezo = PZT) are tabulated in Table 4.1. A sample input file for

each of the following cases of applied loading is attached in Appendix D of the thesis.


Pressure Loading

A short-circuit across the PZT was modeled by applying an equipotential

boundary condition on the top and bottom surfaces of the piezoelectric patch. A pressure

load equivalent of 1000 Pa was applied on the bottom surface of the shim layer. The

shim was meshed with 8-noded linear axisymmetric brick elements while the

piezoelectric layer was meshed with similar brick elements with piezoelectric stresses.

The geometry was scaled by a factor of 1000 (this is done in order to avoid numerical

truncation errors in the solver) and the pressure loading was diminished by a factor of

1000 in order that the output deflections are directly in meters. The maximum static

deflection wo (0) for each case was determined both from the analytical solution

(obtained from the MATLAB code attached in Appendix C) and the finite element









model. The results were plotted as a function of normalized radius of the piezoelectric

patch used as shown in Figure 4.1.




Table 4.1: Properties of the piezoelectric unimorph disk used in the finite element model

Geometrical properties

Outer radius (radius of the mount) 500 rtm

Radius of the piezoelectric layer 0-500 rtm

Thickness of the shim 5 |tm

Thickness of the piezoelectric layer 2 rtm

Mechanical properties

Young's modulus of the shim 90 Gpa

Poisson's ratio of the shim 0.3

Density of the shim 8700 kg/m3

Young's modulus of the piezoelectric layer 30 Gpa

Poisson's ratio of the piezoelectric layer 0.3

Density of the piezoelectric layer 7500 kg/m3

Electric and Dielectric properties

Relative permitivity of the piezoelectric layer 1000

Piezoelectric constant responsible for an -50 pC/
extension in 1-direction due to application of
field in the 3-direction (d1 )



Plot of the center deflection of a circular piezoelectric unimorph obtained from

the analytical solution, described in Eqs. (B.31) and (B.37) matches the solution obtained

from the finite element model to within 1%. Furthermore, mesh refinement studies







38


indicated that the finer the mesh, the lesser the deviation of the solution obtained from the

finite element model to that obtained from the theory. (The case shown in Figure 4.1

corresponds to the case where spacing between the nodes was 0.001 mm.)



x10-7
9.5-
9 Analytical
9 FEM

8.5

c 8-

7.5

a 7.- n
7-
c
C

6.5-

6

5.5 -
0 0.2 0.4 0.6 0.8 1
Normalized Piezoelectric Patch Radius R /R2

Figure 4.1: Comparison of maximum deflection for different radii of the piezoelectric
material as predicted by the analytical solution and finite element model for pressure
application.




Voltage Loading

The finite element model made for pressure loading was modified to yield a finite

element model of the piezoelectric unimorph disk subjected to voltage loading. In this

case, the pressure loading was reduced to zero. A potential boundary condition of unit

strength was applied on the top surface of the piezoelectric layer while the bottom surface









was retained at zero potential. Other parameters were retained at the same value as in the

previous case.

Figure 4.2 shows that the plot of the center deflection of a circular piezoelectric

unimorph obtained from the analytical solution, described in Eqs. (B.31) and (B.37)

matches the solution obtained from the finite element model made for voltage loading to

within 1%. Mesh refinement studies indicated lesser deviation with a finer mesh similar

to the case with pressure loading. (The case shown in Figure 4.2 corresponds to the case

where spacing between the nodes was 0.001 mm.)


x 10-7


0.2 0.4 0.6 0.8
Normalized Piezoelectric Patch Radius R /R


Figure 4.2: Comparison of maximum deflection for different radii of the piezoelectric
material as predicted by the analytical solution and finite element model for a unit voltage
loading.









Experimental Verification

In order to further validate the model, experiments were conducted in the

Dynamics and Control Laboratory at the University of Florida.


PZT


Piezoelectric composite circular plate


Brass


Figure 4.3: A schematic of the experimental setup showing the laser vibrometer focusing
on the clamped PZT unimorph bender.



A periodic chirp signal of 5 V amplitude with frequency ranging from -100 Hz to

4000 Hz was applied across the circular piezoelectric unimorph (APC International Ltd.

Model APC850) to determine its natural frequency and mode shape. A Polytech PI laser

scanning laser vibrometer (MSV200), shown in Figure 4.3, was used to measure the

transverse deflection of the clamped piezoelectric composite plate due to the application

of voltage across the piezoelectric material. The geometry and material properties of the

piezoelectric unimorph bender used for experiment are shown in Table 4.2.


Scanning Laser
Vibrometer







41


Table 4.2: Properties of the piezoelectric bender APC 850

Geometric properties Mechanical properties Electric and Dielectric properties


R2 11.7 mm E, 89.6 GPa 1750


R, 10.0 mm v, 0.324 d31 -175 pC/N


0.221 mm


8700 kg/m3


hp 0.234 mm Ep 63 GPa


vp 0.31


Pp 7700 kg/m3

Experiments were then performed by applying a voltage of 5 VAC at 100 Hz via


conductive copper tape attached to the two sides of the composite plate and the laser was


scanned across the surface.


10




1





0.1


n n1


-I. -


100 1000

Frequency (Hz)
Figure 4.4: Measured displacement frequency response function
velocity measurements using 1/jco integrating factor.


10000


obtained by converting


. .. ........ .. .. .. .. ........ .. .. .. ,,, ,, ,, ,, ,,, ,, ,,, ................ .
ill-- -- -





:- -- --- -


I I I I i I


S. .. .' .1 i '









The 100 Hz test frequency was very small compared to the measured natural

frequency of approximately 3360 Hz, as shown in the frequency response of the

piezoelectric unimorph disk transducer found in Figure 4.4. As the experiment was

conducted at such a low frequency, mass effect was neglected and the mode shape

obtained should approximate the static mode shape.


Table 4.3: Lumped element and two-port parameters

Parameter Theory Experiment Error %

0 142 Pa/V

CAS 1.40e-013 m4S2 kg

MA 13800 kg/m4

dA 1.98e-011 m3/V

CEF 20.8nF

CEB 18.0 nF

K 0.37

f, 3620 Hz

f 3360 Hz 7.6 15.9

f, 3890 Hz

w0(0) 0.107 pm/V 0.0923 !pm/V 15.9



The results from the experiments were compared with the corresponding

analytical solutions. They show matching of the mode shape to a considerable extent.

However the deflection is off by 15.9%. The natural frequency, obtained from the









experiment (around 3360 Hz) does not lie between short-circuit and open-circuit resonant

frequencies, obtained from the theory (see extracted lumped and two-port network

parameter table shown in Table 4.3). It is expected that, the natural frequency lie within

this range. This is because, in the case of a piezoelectric with a short (potential difference

across the terminals is zero), the natural frequency will correspond to short-circuit

resonant frequency and a piezoelectric in static case (zero current) would correspond to

the open-circuit case. Since the experiment is performed in a dynamic condition rather

than static and with a value of voltage applied across the terminals, the resonant

frequency is expected to lie in between the open-circuit and the short-circuit value. Table

4.3 shows the value of the lumped element and two-port network parameters obtained

from the theory. It is found that, the values obtained deviates from the theory to a

considerable extent.

Causes for Deviation of the Experimental Results from Theory

It should be noted that the theory neglects the bond layer between the

piezoceramic patch and the brass shim and thus assumes that a perfect bond exists

between the piezoceramic and the shim. In this case, the estimated bond layer thickness is

25 pm (1 mil), which must be accounted for.

In addition, the axisymmetric assumption implies that the circular piezoceramic

patch is bonded in the center of the shim. Commercial unimorphs exhibit some non-

uniformity in this regard.

Furthermore, the piezoceramic patch has a thin metal electrode layer (e.g., silver)

of a different radius. Fringing field effects are not modeled in the present calculations.

Silver electrode also effects in mechanically stiffening the plate, which is not accounted






44


for in this calculation. An accurate analysis would involve three sections viz. a three-

layered inner composite disc, a two-layered composite annular plate and an outer annular

plate.

Finally, it should be emphasized that an ideal clamped boundary is difficult to

achieve in practice. In the current experimental setup, thick clamp plates are used in

conjunction with several bolts uniformly spaced around the circumference.














CHAPTER 5
NON-DIMENSIONALIZATION


The analytical expressions for the transverse deflections derived in Chapter 3 of

the thesis, are too complicated to obtain any sort of a physical insight into scaling. To

facilitate design of a piezoelectric unimorph using the analytical solution, a non-

dimensional representation of the transverse deflection is necessary. Such a

representation of the transverse deflection would prove to be a good design tool.

The Poisson's ratio of the shim materials and piezoelectric materials are close to

each other with a value around 0.3. Hence, an assumption that the Poisson's ratio of the

shim and piezoelectric material are the same, would simplify the problem. Furthermore,

the governing equations contain terms that are either ratios of the Poisson's ratio or ratios

of the difference of unity and square of Poisson's ratio. These quantities are still closer to

unity. Therefore, in order to simplify the non-dimensionalization procedure, the effect of

Poisson's ratio is ignored. Morris and Foster4 have also neglected the effect of Poisson's

ratio. Non-dimensionalization is carried out using the Buckingham 7t Theorem. 8

Buckingham x Theorem

The independent parameters involved in this simplified problem are E, E,, h,,

hs, R1, R2, d31, V, P, w0 and u0, which are 11 independent variables. The three

dimensions involved in the problem are that of length L, force F and voltage V.

Therefore there should exist eight (11-3 = 8) independent non-dimensional variables.








To design the best piezoelectric disc for a particular shim, it is better to non-

dimensionalize the variables with respect to the shim variables. This leaves E, to be the

basic dimension to non-dimensionalize variables with units of pressure and leaves h, or

R] to be the basic dimension to non-dimensionalize variables with units of length.

E P R h h d3 V
Choosing R] for normalizing the length scale, we obtain E, P h, hp d3V
E, E, R2' R2 R2 R2

w and to be the 8 non-dimensional variables.
R2 R2

The basic field variables can be expressed in non-dimensional form as


._wo (E, P h hp d31
R2 E,E,R2'R2R2 R2
and

U* U o E PP hk hP d31V
u0 2 =. (5.2)
SR E E, R'R '2 R ) 2
However, such a representation will not provide any sort of physical insight that

would facilitate design. Hence a non-dimensional form of the primary variables, which

could be expressed in the best possible form, is required.

Morris and Foster4 represented the optimal radius ratio (r* ) and thickness ratio


(t )op(i.e. their field variables) as functions of aspect ratio a, Young's modulus ratio 3

and a ratio D* defined in Eq. (5.3);




(r*)opt, (a,
(t)Lt= 2(a )'










*opt 3 D
(t g3(D*),
a
where


r*= R, t= t and D (5.3)
R2 t EpR3
They numerically found the functional dependence of their field variables on the fore-

mentioned ratios and developed some empirical relationships. Since the problem

discussed in this thesis, is not directly concerned with the optimization of a particular

device, the field variables are chosen to be u0 and w0.

A non-dimensional representation of these field variables, are conventionally

obtained by non-dimensionalizing the plate equations. However, non-dimensionalizing

the plate equations is also a complex task. Therefore, governing equations that provide

physical insight are used for the purpose of non-dimensionalization.

The response of the central composite plate, when subjected to a pressure P and

voltage V, is represented by


u^ (r)= a(')r, (5.4)


Pr3
0(1) (r) = a'r 16 (5.5)



N (r)= A(I ( +v)a(') -NP (5.6)


and

Pr2 (3+v)
M'M (r) = -D (1 +v)a' +r2(3 (5.7)
16









where the number (1) in superscript indicates deflections and generalized forces of the

central composite plate along an axis where coupling matrix vanishes.

For the outer annular plate, [B] vanishes because of the symmetry about its

neutral axis. Hence the response of the outer annular plate is given by


(2) =2b)
u,(r) =a2r +-, (5.8)
r


() (2 b () Pr3
02)(r) =a"r + 2 (5.9)
r 16D,2'


b,(2)
N2 (r) = A (1+v)a 2)- A (1-v) b (5.10)
r

and

b2) Pr2(3+v)
M (r)= D( (1 + v)a2) D1 (1 v ) -2 + (5.11)
r 16


where the number (2) in superscript indicates deflections and generalized forces of the

outer annular plate.

Non-Dimensional Deflection for Pressure Loading

The non-dimensional vertical deflection ,, (7) is obtained from the expression

shown in Eq. (5.12). The symbol in the superscript indicates a non-dimensional

quantity.

R r
wO(r) = jO(dr + O(2dr. (5.12)
0 R,
S r
Substituting r =- in Eq. (5.12), we obtain
R2










w(r = Rj0( )dr +fj(dr. (5.13)
0
The non-dimensional form of the slope is obtained by dividing the slope with the forcing

term in Eq. (5.9) to yield


0( ( ) and 0 2)P* 0 (5.14)

16D 16D,12)
The letter P in the superscript is used to represent non-dimensional parameters that are

obtained when the piezoelectric unimorph is subjected to pressure loading alone. Even

though the slope by itself is non-dimensional, the above form is more useful because it

does not vary with change in loading and the overall dimensions.

Substituting Eq.(5.14) in Eq. (5.13), we obtain
wO0 (r) 1 ( d7 *1
( 1 -J 0 dr +j( d-r. (5.15)
R2 PR2 0
16D(12)
Eq. (5.15) implies that the non-dimensionalized vertical deflection is given by

w Pr) (r) (5.16)
PR4
16D(12
The non-dimensional vertical displacement obtained in Eq. (5.16) is not

dependent on the aspect ratio of the shim (see Figure 5.4 and Figure 5.6). It is dependent

on the ratios of the radius, thickness and Young's modulus.


w* ( r)=f (5.17)

The non-dimensional form of the field variable shown in Eq. (5.17) is much

simpler than the form indicated in Eq. (5.1). The best way to represent the whole set of

deflections of a piezoelectric unimorph in a compact form is to plot the non-dimensional









center deflection against the radius ratio R,1/R for different values of hp/h, for a

particular E /E, as shown in Figure 5.1 Figure 5.5. The value of h, /R2 used in the

first five plots of each of the non-dimensional variables discussed in the section is 0.01.

Commercially available disc benders manufactured by APC International limited have

the Young's modulus ratio varying between 0.6 and 0.8. The Young's modulus of PZT

deposited in MEMS level device can be as low as 30 GPa. The Young's modulus of the

shim layer (Silicon) is around 150 GPa resulting in a Young's modulus ratio of 0.2.

Since, Silicon is a moderately anisotropic material with an anisotropic coefficient19 of

1.57 (which is close to unity. For an isotropic material, the value of anisotropic

coefficient is unity), the Silicon layer is treated as transversely isotropic in this analysis.

Hence values of 0.02, 0.2, 0.4, 0.6 and 0.8 were selected for the ratio of the Young's

Modulii. The Young's modulus ratio value was taken to as small as 0.02 to accommodate

PVDF Aluminum benders. In order to show that these non-dimensional variables do not

vary with change in aspect ratio R2 /h,, a plot of the non-dimensional variable with value

of Young's modulus ratio at 0.6 and the value of h, /R2 at 0.02. The 4th and 6th plots of

each of the non-dimensional variables discussed in the following section are exactly the

same even though the aspect ratios are different. This proves the non-dependent nature of

the non-dimensional variables on aspect ratio.

The denominator is multiplied with a factor of 0.25, to make its value equal to

that of the central deflection of a clamped classical circular plate acted upon by a pressure

load;










wO*(r) (5.18)

64DZ^)
The expression for non-dimensional vertical deflection is represented in terms of ratios

(mentioned in the beginning of this chapter) as


wo) W p(r)
PR24

16 Eh3
12(1-v)


(5.19)


which on simplification yields


WO (r)

wP*~ R2
W 1 '-
3. 2\ -Pf


(5.20)


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R1/R2
Figure 5.1: Non-dimensional plot of the center deflection of a piezoelectric unimorph
disc with E E, = 0.02 subjected only to a pressure load.





















0.6 0.9


0.5
1.2
0.4


0.3- Ep/E =0.2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R1/R2
Figure 5.2: Non-dimensional plot of the center deflection of a piezoelectric unimorph
disc with E IE, = 0.2 subjected only to a pressure load.


0.6

0.5

0.4

0.3

0.2 Ep/Es = 0.4

0.1 0.2 0.3 0.4 0.5 0.6
RI/R2
Figure 5.3: Non-dimensional plot of the center deflection
disc with E, E, = 0.4 subjected only to a pressure load.


0.7 0.8 0.9


of a piezoelectric unimorph































0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R1/R2
Figure 5.4: Non-dimensional plot of the center deflection of a piezoelectric unimorph
disc with Ep/E = 0.6 subjected only to a pressure load.


0.5

0.4

0.3

0.2
Ep/Es = 0.8

0.1 0.2 0.3 0.4 0.5 0.6
R1/R2
Figure 5.5: Non-dimensional plot of the center deflection
disc with E /E = 0.8 subjected only to a pressure load.


0.7 0.8 0.9


of a piezoelectric unimorph



















S 0.6\

0.5

0.4
4 0.9

0.3
1.2
0.2 EEs = 0.6

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Ri/R2
Figure 5.6: Non-dimensional plot of the center deflection of a piezoelectric unimorph
disc with Ep/E = 0.6 and h/R, = 0.02, subjected only to a pressure load.



Non-Dimensionalization for Voltage Loading

Equations (5.4) (5.11) are modified for the case when voltage alone is applied in

the following manner:


(5.21)



(5.22)



(5.23)



(5.24)


u, (r) = alr,)r



N(=r) = A ar,










(b(2)
U, (r) (2)r + ,
r


(5.25)









A(2)
8(2)(r) =a2)r+ 2, (5.26)
r


A (2)
N(2) (r) A(2) ( 1 + V )a(2)- A 2) V -) 1 (5.27)
^ ( YlUv (lV) ? (5.27)
r

and

(2) () = D)(2) 2) D., ,,2 ) (1-V) (5.28)
r 1- 2 DI l v). (5.28)
r

The expression for non-dimensional vertical deflection wo (r) is obtained from the

expression

R r
w (r) = O()dr +O(2)dr. (5.29)
0 R,
r
Substituting r in Eq. (5.29), we obtain
R2



0 () R20(1)dr + 0(2)dr. (5.30)
0 11
The non-dimensional form of the slope when subjected to voltage loading alone is

obtained by dividing the slope with an equivalent of the forcing term found in Eq. (5.24)

to yield




D(2) D(2)
0(1 0(1) and 6(2)v* 2-- (5.31)



The letter V in the superscript is used to represent non-dimensional parameters that are

obtained when the piezoelectric unimorph is subjected to voltage loading alone. Even

though the slope by itself is non-dimensional, the above form is more useful because it








does not vary with change in loading, piezoelectric constant, relative permitivity of the

piezoelectric and the overall dimensions.

Substituting Eq. (5.31) in Eq. (5.30), we obtain


w (r) 1 d= : 'drO +f2)Vdr. (5.32)


Equation (5.32) implies that the non-dimensional vertical deflection is given by


(r)- w (r) (5.33)
Sp0 2I"
Mr R2
ID1 )
The non-dimensional vertical displacement obtained in Eq. (5.33) is also independent of

aspect ratio and is only a function of the radius ratio, thickness ratio and Young's

modulus ratio, which is much simpler to represent than the form indicated in Eq. (5.1);


w r = f (5.34)

The best way to represent the whole set of deflections is to plot it in the same

manner as in the case of pressure loading. The denominator was multiplied by a factor of

1
Sto make its value equal to that of the deflection obtained for a classical circular
1+v

plate acted on by a moment equal to piezoelectric couple shown in Eq. (3.27). The

revised expression is given by
wv (-\ woo(r)
wo = (r) (5.35)
MPR2
Dl?> (1+v)
The expression for non-dimensional vertical deflection is represented in terms of

ratios (mentioned in the beginning of this chapter) in the following manner:










w<(o W0rV) E, (5.36)

1-v Eh +Eh~

E,h,3
(1+v)
12(1-v2)

(i.e.)


Eh
r (5.37)
2 12 E h+1
E, h h, R,
The variation of the center deflection of a unimorph subjected only to a voltage

load, with variation in non-dimensional ratios is delineated in Figure 5.7 Figure 5.12.



0.25 I t t I --i
/EIEs=0.02 h /hs=0.00001

0.9
0.2 0.3 1.2



0.15

>o

0.1



0.05



0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R,/R2

Figure 5.7: Non-dimensional plot of the center deflection of a piezoelectric unimorph
disc with E IEE = 0.02 subjected only to a voltage load.











0.25 t I t
E IE =0.2
pS
h /h =0.00001





















0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

RI/R2

Figure 5.8 Non-dimensional plot of the center deflection of a piezoelectric unimorph disc
with E /E = 0.2 subjected only to a voltage load.





Ep /Es=0.4
0 S S
h /hs=0.00001
0.2 -0

















0.3


0.15
0.


0.1 -0.





















0.05




0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R,/R2
Ri/R 2


Figure 5.9: Non-dimensional plot of the center deflection
disc with E,/E, = 0.4 subjected only to a voltage load.


of a piezoelectric unimorph





















>/ 6
0.6
0.1

0.9

0.05- 1.2



0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
RI/R2

Figure 5.10: Non-dimensional plot of the center deflection of a piezoelectric unimorph
disc with E,/E, = 0.6 subjected only to a voltage load.


0 0.1 0.2 0.3 0.4 0.5
R IR2
Figure 5.11: Non-dimensional plot of the center d
disc with E /E =0.8 subjected only to a voltage load.


0.6 0.7 0.8 0.9 1

election of a piezoelectric unimorph



















>/ 6
0.6
0.1

0.9

0.05 1.2



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
RI/R2
Figure 5.12: Non-dimensional plot of the center deflection of a piezoelectric unimorph
disc with E /E = 0.6 and h/R2 = 0.02, subjected only to a voltage load.



To further enable design and optimization of a circular piezoelectric unimorph,

non-dimensional lumped element and two-port network parameters are needed.

Non-Dimensionalization of Lumped Element and Two-Port Network Parameters

Some of the lumped element and two-port network parameters are obtained from

the transverse deflection by expressions described in Chapter 2 of this thesis. The non-

dimensional parameters of these variables, has the same dependence on the

corresponding dimensional variables, as the transverse deflection. In other words, the

equivalent non-dimensional parameter is obtained from the corresponding dimensional

parameter by dividing it with the value of the parameter for a classical circular plate.

The non-dimensional short-circuit acoustic compliance is given by










C
CAS AS (5.38)


where (CAS)Shm represents the short-circuit acoustic compliance of the shim alone;


Shm 16E h 3 (5.39)

Above expression for (CAS) Shm in terms of non-dimensional ratios is given by


(CAS)h 1-(i 2)2)K4 + R2j2 (5.40)

The non-dimensional acoustic compliance is also a function of the Young's

modulus ratio, thickness ratio and radius ratio. The best way to represent these variables

will be similar to that of the deflections.

Figure 5.13 Figure 5.18 delineates such plots of short-circuit acoustic

compliance of the piezoelectric unimorph.

h /hs = 0.00001



0.6
0.95-

.0.9
0.9


0 1.2
0.85-



0.8-



0.75
75 E /E = 0.02
p s
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
RI/R2
Figure 5.13: Non-dimensional short-circuit acoustic compliance plots for E/E, = 0.02.






















00.9
0.6


0.5- 1.2


0.4


0.3 Es = 0.2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
RI/R2
Figure 5.14: Non-dimensional short-circuit acoustic compliance plots for E /E,



h /h = 0.0000
P s
0.9
0.3
0.8


0.7
00.6

0 < 0.6-

0.5- 0.9


0.4-


0.3


0.2 Ep/E = 0.4

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
RI/R2
Figure 5.15: Non-dimensional short-circuit acoustic compliance plots for Ep/Es


0.9


0.2.


0.9


0.4.




































0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R IR
R1/R2
Figure 5.16: Non-dimensional short-circuit acoustic compliance plots for E /E = 0.6.




h /h = 0.00001
PS
0.9-

0.8-
0.3
0.7-


uo 0.6-

0.6
0.5-

0.4-
0.9

0.3-
1.2

0.2-
E /E = 0.8
ps

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R1/R2
Figure 5.17: Non-dimensional short-circuit acoustic compliance plots for E /E = 0. 8.



















0< 0.6 0.6

0.5
0.9
0.4

0.3- .2

0.2 E /E = 0.6
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R1/R2
Figure 5.18: Non-dimensional short-circuit acoustic compliance plots for Ep/E, =0.6
and h, R2 = 0.02.



The non-dimensional acoustic mass is given by


MA MA (5.41)
((M A)Sh/m

where (MA )Shm represents the acoustic mass of the shim alone;


(M (PSA )Shm (5.42)



where (PA,)Sh, represents the areal density of the shim layer.

The non-dimensional mass like the deflections are a function of the usual three

ratios. In addition it is also a function of density ratio. Density ratio, like Young's

Modulus ratio, is unique for a certain set of material. Aluminum/PVDF benders have










Young's modulus ratio of 0.02 0.05. Therefore the density ratio of Aluminum/PVDF

benders (i.e. 1760/2700) is chosen for the plots shown in Figure 5.19 Figure 5.20. The

Silicon/PZT benders have their Young's Modulus ratio varying from 0.2 0.6. Therefore

the density ratio of Silicon/PZT benders (i.e. 7500/2300) is chosen for the plots shown in

Figure 5.21 Figure 5.23 and Figure 5.26. The Brass/PZT benders have their Young's

Modulus ratio varying from 0.6 0.8. Therefore the density ratio of Brass/PZT (i.e.

7500/8700) benders produced by APC International Ltd. is chosen for the plots shown in

Figure 5.24 Figure 5.25. Figure 5.26, like the plots of previous non-dimensional ratios,

has all ratios similar to Figure 5.23 except the aspect ratio. The plots appear the same

indicating that the non-dimensional acoustic mass does not vary with change in aspect

ratio.




E IE =0.02 1.2
1.7


1.6
0.9

1.5


<1.4 0.6


1.3


1.2 0.3


1.1

h /h =0.00001
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R1/R2

Figure 5.19: Non-dimensional acoustic mass plots for E pE = 0.02 (Aluminum/PVDF).














1.7


1.6


1.5


S<1.4


1.3


1.2


1.1



0.1 0.2 0.3 0.4 0.5 0.6
R1/R2

Figure 5.20: Non-dimensional acoustic mass plots for E IE,


4.5


4


3.5


S< 3


2.5


2


1.5


0.7 0.8 0.9


0.05 (Aluminum/PVDF).


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R1/R2

Figure 5.21: Non-dimensional acoustic mass plots for Ep/Es = 0.2(Silcon/PZT).













4.5 1.2


4

0.9
3.5


< 3
0.6

2.5


2- /0.3


1.5

h /h =0.00001

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R,/R2

Figure 5.22: Non-dimensional acoustic mass plots for E /E, = 0.4 (Silcon/PZT).


0.1 0.2 0.3 0.4 0.5 0.6 0.7
R1/R2

Figure 5.23: Non-dimensional acoustic mass plots for Ep/E


0.8 0.9


0.6 (Silcon/PZT).















1.9

1.8 1.2

1.7
0.9
1.6

H 1.5
0.6
1.4

1.3
0.3
1.2

1.1
h /h =0.00001
PS
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R1/R2

Figure 5.24: Non-dimensional acoustic mass plots for EP/E, = 0.6 (Brass/PZT).





2
E IE =0.8
p s
1.9

1.8 1.2

1.7

1.6 0.9

H 1.5

1.4- 0

1.3

1.2 0.3

1.1
h /h =0.00001

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R1/R2

Figure 5.25: Non-dimensional acoustic mass plots for ElE = 0.8 (Brass/PZT).





























0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R1/R2

Figure 5.26: Non-dimensional acoustic mass plots for Ep/E, =0.6 and hJR2 =0.02
(Silcon/PZT).



Unlike the parameters that were non-dimensionalized by quantities possessing a

physical significance, some of the two-port network parameters do not have quantities for

the shim alone described. We can obtain the non-dimensional form of the two-port

network parameters by finding the equivalent parameters for the shim alone from first

principles.

The effective acoustic piezoelectric coefficient as defined in Chapter 2 is obtained

by computing the volume displaced by a unit application of voltage due to the vertical

deflection field, analytical expression for which is obtained in Chapter 3. Therefore, the

non-dimensional form is derived by evaluating the volume displaced due to a

displacement field obtained by the application of a moment equal to the piezoelectric

couple generated on the shim of radius R ;









d d (5.43)
(d)L

where (dA, ) represents the effective acoustic piezoelectric coefficient of the

piezoelectric with dimensions of the shim that is obtained from the following expression:

2rd2)
DJ(2) (1+ V)
(d, )shim 0v (5.44)


Substituting for the terms appearing in the above expression, we obtain

--d 31Vh ,(R-r2)
SEAh
(l+v)
12(l-v)
(d,)s, Shim 1(5.45)


Pulling the constants out of the integral in the above expression, we obtain


(d,) =m 24r f R (R- rZ)rdr. (5.46)


The above expression is simplified as


(dA)shim = 6r r d31 (5.47)

which can be written as


(d,)Shm= 6 I3 R (5.48)
""" )E)h V
























0.03


0.02


0.01






Figure 5.27:
E,/E, = 0.02.




E
0.045 p

0.04

0.035

0.03

0.025
-D
0.02

0.015

0.01

0.005





Figure 5.28:
Ep/E, = 0.2.


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
RIR,
1 /R2
Non-dimensional effective acoustic piezoelectric coefficient plots for


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R1/R2
Non-dimensional effective acoustic piezoelectric coefficient plots for











0.04 -

0.035

0.03

0.025

'o 0.02


0.015

0.01

0.005





Figure 5.29:
EIE, = 0.4.


0.04


0.035


0.03


0.025


0.02


0.015


0.01


0.005


Figure 5.30:
EP/E = 0.6.


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R1/R2
Non-dimensional effective acoustic piezoelectric coefficient plots for


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R /R2
Non-dimensional effective acoustic piezoelectric coefficient plots for










0.04
E

0.035


0.03


0.025


S< 0.02
-0

0.015


0.01


0.005





Figure 5.31:
Ep E, = 0.8.


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R /R2
Non-dimensional effective acoustic piezoelectric coefficient plots for


0.04
E /E = 0.6
P 0.3
0.035
0.6
0.03

h 0.025/h = 0.00001 0.9
0.025


S< 0.02


0.015


0.01


0.005


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R1/R2
Figure 5.32: Non-dimensional effective acoustic piezoelectric coefficient plots for
E p/E = 0.6 and hJ/R, =0.02.









The best way to represent this non-dimensional parameter is the same as that of

the previously obtained parameters (see Figure 5.27 Figure 5.32). It is found that dA is

directly proportional to the square of the aspect ratio.

Non-dimensional electrical free capacitance was defined in the same manner as

the ratio of the electrical free capacitance of the piezoelectric composite circular late to

capacitance of a piezoelectric disc with the dimensions of the shim. (i.e.)

C
C*F EF (5.49)
(Cf )Shim

where

(CEF)Shm R2 (5.50)
h


(i.e.)

eCrR2 ] 2?
hP R 2
C*F 2 (5.51)
he h2


Now that the non-dimensional form of all the independent two-port parameters

and lumped parameters are known, non-dimensional representation of other parameters

can be found directly by replacing the form of variables visible in the expressions with

the corresponding non-dimensional variables.

Hence the non-dimensional representation for A is given by


dA
OA' A (5.52)
CAS


Eq. (5.52) implies that










S= A C)shim (5.53)
(dA shim
Substituting Eqs. (5.40) and (5.47) in Eq.(5.53), we obtain


1 (Iv2 P R2 4 R22h,

S= A (5.54)
E1 P
6 V Jl R2 ) R3 hs
R2 E, h V

which on simplification, gives




'PA ='PA 1.2 (5.55)
P '2 96(1 v2)CV
Es h)

The plots of non-dimensional 'A are shown in Figure 5.33 Figure 5.38.



E E = 0.02 1.2
P s
0.08 0.9

0.07 -
0.6
0.06 -
/0.3
0.05- /
h /h = 0.00001
0.04 P s

0.03-

0.02 -

0.01


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R1/R2
Figure 5.33: Non-dimensional 'A for Ep E, = 0.02.
































0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
RF/R2
Figure 5.34: Non-dimensional A for E /E, = 0.2.


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
RIR,
1 N 2
Figure 5.35: Non-dimensional 0A for Ep/E, = 0.4.


0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

































0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R1/R2
Figure 5.36: Non-dimensional 0A for E E, = 0.6.


0.07


0.06


0.05


< 0.04


0.03 h / = 0.00


0.02


0.01


0.1 0.2 0.3 0.4 0.5 0.6 0.7
R /R2
Figure 5.37: Non-dimensional 0A for E /E, = 0.8.


0.8 0.9





























0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R1/R2
Figure 5.38: Non-dimensional 0A for E /E = 0.6 and h,/R2 = 0.02.


The non-dimensional representation for is obtained in the same manner as that


of A and is given by


A* d
CFT


(5.56)


Substituting Eqs. (5.43) and (5.49) in Eq. (5.56), we obtain


which on simplification, gives


-A
S (d )shim
S CEF

(CEF ),hm



A -A (d ,)h
A (cI


Substituting Eqs. (5.48) and (5.50) in Eq.(5.58), we obtain


(5.57)


(5.58)












6'A = d ) "s-p {'2 ) (5.59)

R2 E3l h, V

Multiplying both sides of the above equation with the denominator of right hand

side of Eq. (5.59), we obtain



0= 6 2 E 3 ,J (5.60)

which on simplification, gives

E p R2 P R2 6d312E, V (5.61)
_'X = AEJc (5.61)
ofE s h, ) E s d31V e P

Figure 5.39 Figure 5.44 depicts the variation of non-dimensional OA with

variation in non-dimensional ratios.



E IE =0.02
0.3

1.2
0.25


0.2- 0.9

-e- ^^^\
0.15
0.6

0.1

0.3
0.05


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R,/R2

Figure 5.39: Non-dimensional 0 for E /E = 0.02.








80




0.11

0.1 1.2

0.09 0.9

0.08

0.07

<0.06
-e-
0.05 0

0.04

0.03

0.02

0.01
h /hs=0.00001

0.1 0.2 0.3 0.4 0.5 0.6 0.7
R1/R2

Figure 5.40: Non-dimensional for E E = 0.2.





0.9


0.06


0.05 0.6


0.04 0.3


0.03


0.02



0.01

h /h =0.00001
Sps
0.1 0.2 0.3 0.4 0.5 0.6 0.7
R1/R2

Figure 5.41: Non-dimensional for E /E, = 0.4.


0.8 0.9


0.8 0.9








81



0.05 1.2 0.9

0.045

0.04 0.6

0.035
0.3
0.03

<0.025

0.02

0.015

0.01

0.005
h /hs=0.00001

0.1 0.2 0.3 0.4 0.5 0.6 0.7
R1/R2

Figure 5.42: Non-dimensional for E E = 0.6.





0.04
0.9
1.2
0.035


0.03
0.3

0.025


0.02


0.015


0.01


0.005

h /h =0.00001
p p s I II
0.1 0.2 0.3 0.4 0.5 0.6 0.7
R1/R2

Figure 5.43: Non-dimensional for E IE, = 0.8 .


0.8 0.9


0.8 0.9










0.05
1.2 0.9 EPIES=0.6
0.045

0.04 0.6

0.035
0.3
0.03

0.025

0.02

0.015

0.01

0.005
h /hs=0.00001

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R1/R2
Figure 5.44: Non-dimensional 0 for Ep/E = 0.6 and h/R2 = 0.02.




The coupling coefficient represents the ideal fraction of energy transduced to the

other domain and by definition is non-dimensional. The coefficient provides an indication

of the electroacoustic energy conversion for the unimorph, but does not yield the actual

value because it does not account for electrical and mechanical losses as well as electrical

and mechanical loads.1 Therefore the equivalent representation for K will be referred to

as the universal representation for the coupling coefficient, since this representation is

independent of the electric and dielectric properties of the piezoelectric layer. It is


denoted by K* and is given by




K*= d)2 (5.62)
CASCEF


which on simplification yields










dK'
K =. (5.63)
\ c* .c* .

Substituting Eqs. (5.40), (5.48) and (5.50) in Eq. (5.63), we obtain



K = K d3 2E .E (5.64)
4 (l-v 2)eE

It is noted that K is directly proportional to d31/ Even though Young's

modulus iatio appears in the relationship, nothing can be said about the dependency of K

on Young's modulus ratio.

Since these parameters depend on the fore-mentioned parameters, the best

possible representation of these non-dimensional parameters is in no way different from

the representation of those basic parameters. Figure 5.45 Figure 5.50 depict the

variation of K* with variation in non-dimensional ratios.


0.14 t t I t t
E IE =0.02
PS
1.2
0.12

0.9
0.1


0.08 0.6


0.06

0.3
0.04


0.02

h /lh=0.00001
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Ri/R2

Figure 5.45: K* for Ep/E, = 0.02.











0.09 t I t ii
E /E=0.21.2
0.08
0.9
0.07

0.6

0.06

0.05-

0.3
0.04


0.03


0.02


0.01

h /lh=0.00001
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R,/R2

Figure 5.46: K* for E /E, =0.2.




0.07
E IE =0.4
P s 1.2

0.06 0.9


0.05


0.04
0.3


0.03


0.02


0.01

h /h =0.00001

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
RI/R2

Figure 5.47: K* for E/E, = 0.4.











0.06 -I tii
E IE =0.6
p s 1.2

0.9
0.05
0.6

0.04

0.3

S0.03



0.02



0.01


h /h =0.00001
ps

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
R,/R2

Figure 5.48: K* for E /E, =0.6.




0.05
E IE =0.8.2
p s
0.045 0.9

0.04 0.6

0.035
0.3
0.03

, 0.025

0.02

0.015

0.01

0.005
h /h =0.00001

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
RI/R2

Figure 5.49: K* for E/E, = 0.8.