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TWOPORT 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 NAG12261 and NAG12249 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 twoport 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 thoughtprovoking 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 twoport 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 OPORT NETW ORK M ODELING .............................................. .................. 7 Lumped Element M odeling ........................................................ 7 TwoPort Network Modeling An Introduction................................................. 7 TwoPort Model of Piezoelectric Transduction .....................................................9 Tw oPort M odel of a 1D Piezoelectric.................................................. ............... 9 TwoPort 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 StrainD 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 NONDIM EN SIONALIZATION ........................................ .......................... 45 Buckingham Theorem ............................ .......................... ..... ............... 45 NonDimensional Deflection for Pressure Loading ...............................................48 NonDimensionalization for Voltage Loading................................... ..... .......... 54 NonDimensionalization of Lumped Element and TwoPort 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 ShortCircuit 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 twoport parameters..................................... ..........................42 LIST OF FIGURES Figure Pagi 1.1: Crosssectional schematic of a clamped axisymmetric piezoelectric unimorph disk tran sdu cer .................................. ................. ........... ............................ . 2.1: Schematic of a piezoelectric plate that can be approximated to be 1D. ........................... 9 2.2: Equivalent twoport 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 twoport 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 twoport circuit representation of axisymmetric piezoelectric unimorph disk at frequencies comparable to that of the primary resonance ........... .............. 16 3.1: An axisymmetric multilayered transversely isotropic circular plate with pressure load, radial load and a moment. All loads shown are considered positive.......................19 3.2: Crosssectional 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: Nondimensional plot of the center deflection of a piezoelectric unimorph disc with Ep E, = 0.02 subjected only to a pressure load ......................................................51 5.2: Nondimensional plot of the center deflection of a piezoelectric unimorph disc with Ep /E = 0.2 subjected only to a pressure load .......... ............................................. 52 5.3: Nondimensional plot of the center deflection of a piezoelectric unimorph disc with Ep E, = 0.4 subjected only to a pressure load .......... ............................................. 52 5.4: Nondimensional 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: Nondimensional 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: Nondimensional 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: Nondimensional plot of the center deflection of a piezoelectric unimorph disc with Ep /E = 0.02 subjected only to a voltage load ......... ....... ......... ........ ............. 57 5.8 Nondimensional plot of the center deflection of a piezoelectric unimorph disc with Ep E, = 0.2 subjected only to a voltage load.................................... ....... ......... 58 5.9: Nondimensional 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: Nondimensional plot of the center deflection of a piezoelectric unimorph disc with E /E, = 0.6 subjected only to a voltage load.............................................................59 5.11: Nondimensional plot of the center deflection of a piezoelectric unimorph disc with Ep /E = 0.8 subjected only to a voltage load .................. ......... ................... 59 5.12: Nondimensional 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: Nondimensional shortcircuit acoustic compliance plots for E /E, = 0.02. ...................61 5.14: Nondimensional shortcircuit acoustic compliance plots for E p/E = 0.2 ...................62 5.15: Nondimensional shortcircuit acoustic compliance plots for E /E = 0.4 ...................62 5.16: Nondimensional shortcircuit acoustic compliance plots for E p/E = 0.6 ...................63 5.17: Nondimensional shortcircuit acoustic compliance plots for Ep/E, = 0.8 .....................63 5.18: Nondimensional shortcircuit acoustic compliance plots for E I/E = 0.6 and h,/R 2 = 0.02 ...........................................................................64 5.19: Nondimensional acoustic mass plots for EIE, = 0.02 (Aluminum/PVDF) ..................65 5.20: Nondimensional acoustic mass plots for E /E = 0.05 (Aluminum/PVDF). ..................66 5.21: Nondimensional acoustic mass plots for E /E = 0.2(Silcon/PZT). ............................66 5.22: Nondimensional acoustic mass plots for E IE = 0.4(Silcon/PZT). .............................67 5.23: Nondimensional acoustic mass plots for E IE, = 0.6(Silcon/PZT) ............................67 5.24: Nondimensional acoustic mass plots for E IE = 0.6 (Brass/PZT). ............................68 5.25: Nondimensional acoustic mass plots for E IE = 0.8 (Brass/PZT). .............................68 5.26: Nondimensional acoustic mass plots for E /E = 0.6 and h/R2 = 0.02 (Silcon/PZT)................................... ................. .............. ...........69 5.27: Nondimensional effective acoustic piezoelectric coefficient plots for E /E = 0 .02 .................................................................................... .................. 7 1 5.28: Nondimensional effective acoustic piezoelectric coefficient plots for E /E = 0.2..........71 5.29: Nondimensional effective acoustic piezoelectric coefficient plots for E I/E = 0.4..........72 5.30: Nondimensional effective acoustic piezoelectric coefficient plots for E I/E = 0.6..........72 5.31: Nondimensional effective acoustic piezoelectric coefficient plots for E /Es = 0.8..........73 5.32: Nondimensional effective acoustic piezoelectric coefficient plots for Ep/Es = 0.6 and h = 0 .02 ................................................................ ................................ 73 5.33: Nondimensional A, for E /E Nondimensional Nondimensional Nondimensional Nondimensional Nondimensional Nondimensional Nondimensional Nondimensional Nondimensional Nondimensional Nondimensional 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 TWOPORT 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 modeshape 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 nondimensionalization procedure, which could be used to extract nondimensional 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 modeshape. 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 twoport 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. Microfluidic 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 multilayered isotropic composite circular plate. As the problem consists of various domains, a twoport 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 twoport network model of the device. This twoport 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: Crosssectional schematic of a clamped axisymmetric piezoelectric unimorph disk transducer Figure 1.1 shows a crosssection 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 bendingextension 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 microelectromechanical 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 multilayered 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 nondimensional r groups using the Buckingham theorem. They used a higherorder 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 flexuralextensional vibratory response of metalpiezoceramic unimorphs and PZT5H 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 electroelastic theory assuming free boundary conditions, which is nonphysical 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 nonlinear 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 firstorder 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 twoport, lumpedelement 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 twoport 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 twoport 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 TWOPORT NETWORK MODELING The piezoelectric composite plate actuator represents a coupled electro mechanicalacoustic system with frequency dependent properties determined by device dimensions and material properties. The analysis and design of such a coupleddomain 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 twoport modeling. TwoPort 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. TwoPort 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). TwoPort Model of a 1D Piezoelectric plate of area A 3dir 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 1D. In case of a 1D 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 shortcircuit mechanical compliance and effective mechanical piezoelectric coefficient (that is responsible for a strain in 3 direction due to application of electric field in the 3direction) 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 1D piezoelectric that is given by CEF (2.9) hP where k is the dielectric constant of the piezoelectric in the 3direction due to application of an electric field in the 3direction and eo is permitivity of free space. The transduction in the case of a 1D 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, TwoPort 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 1D piezoelectric described above. Also the focus in this thesis is oriented towards an electroacoustic model rather than an electromechanical 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 1D 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 shortcircuit 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 twoport 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 shortcircuit 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 energycoupling 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 VEF 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 CEFK2). (2.20) Q 'Ai 1:'A EF 1 SEF P C A AP V  AO Figure 2.5: Another equivalent twoport 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 acoustoelectric 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 SA 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 energycoupling 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 (IK2). (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 twoport 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 zeroradius 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 shortcircuit natural frequency f, in Hz of the system is given by 1 f, = (2.27) while the opencircuit 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 (lIK2)MA (K2) which on simplification yields f ( K2) = (2.30) The deflection field of the piezoelectric composite plate, due to unitapplied pressure and due to unitapplied 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 twoport 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 zaxis. In other words, the normal stress oz and the shear stress zr, are approximately equal to zero. Figure 3.1: An axisymmetric multilayered 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: Crosssectional 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 multilayered 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 multilayered 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 r0 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 multilayered 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 + (MOr, (3.12) dr r and d + +P= 0. (3.13) dr r StrainDisplacement 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)OrO (3._u = (3.18) rO r and ldw 0 KC = (3.19) r dr r Constitutive Equations For an elastic plate, the stressstrain 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][cN(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 flexuralextensional 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 straindisplacement relationships and curvaturedisplacement 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) (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 ^34aA ~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)+ (1v) + (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= 1i {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 lv2 (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 1v2 3 4 which is simplified further to yield 1 E h3 2 2 (4z2h+4zh2 +h3 1v2 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) 1v2 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 shortcircuit 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 8noded 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 0500 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 1direction due to application of field in the 3direction (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.) x107 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 107 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 twoport parameters Parameter Theory Experiment Error % 0 142 Pa/V CAS 1.40e013 m4S2 kg MA 13800 kg/m4 dA 1.98e011 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 shortcircuit and opencircuit resonant frequencies, obtained from the theory (see extracted lumped and twoport 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 shortcircuit resonant frequency and a piezoelectric in static case (zero current) would correspond to the opencircuit 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 opencircuit and the shortcircuit value. Table 4.3 shows the value of the lumped element and twoport 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 twolayered 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 NONDIMENSIONALIZATION 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 nondimensionalization procedure, the effect of Poisson's ratio is ignored. Morris and Foster4 have also neglected the effect of Poisson's ratio. Nondimensionalization 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 (113 = 8) independent nondimensional 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 nondimensionalize variables with units of pressure and leaves h, or R] to be the basic dimension to nondimensionalize 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 nondimensional variables. R2 R2 The basic field variables can be expressed in nondimensional 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 nondimensional 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 nondimensional representation of these field variables, are conventionally obtained by nondimensionalizing the plate equations. However, nondimensionalizing the plate equations is also a complex task. Therefore, governing equations that provide physical insight are used for the purpose of nondimensionalization. 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 (1v) 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. NonDimensional Deflection for Pressure Loading The nondimensional vertical deflection ,, (7) is obtained from the expression shown in Eq. (5.12). The symbol in the superscript indicates a nondimensional 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 nondimensional 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 nondimensional parameters that are obtained when the piezoelectric unimorph is subjected to pressure loading alone. Even though the slope by itself is nondimensional, 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( dr. (5.15) R2 PR2 0 16D(12) Eq. (5.15) implies that the nondimensionalized vertical deflection is given by w Pr) (r) (5.16) PR4 16D(12 The nondimensional 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 nondimensional 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 nondimensional 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 nondimensional 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 nondimensional variables do not vary with change in aspect ratio R2 /h,, a plot of the nondimensional 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 nondimensional variables discussed in the following section are exactly the same even though the aspect ratios are different. This proves the nondependent nature of the nondimensional 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 nondimensional vertical deflection is represented in terms of ratios (mentioned in the beginning of this chapter) as wo) W p(r) PR24 16 Eh3 12(1v) (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: Nondimensional 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: Nondimensional 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: Nondimensional 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: Nondimensional 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: Nondimensional 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: Nondimensional 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. NonDimensionalization 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 ) (1V) (5.28) r 1 2 DI l v). (5.28) r The expression for nondimensional 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 nondimensional 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 nondimensional parameters that are obtained when the piezoelectric unimorph is subjected to voltage loading alone. Even though the slope by itself is nondimensional, 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 nondimensional vertical deflection is given by (r) w (r) (5.33) Sp0 2I" Mr R2 ID1 ) The nondimensional 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 nondimensional 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) 1v Eh +Eh~ E,h,3 (1+v) 12(1v2) (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 nondimensional 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: Nondimensional 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 Nondimensional 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: Nondimensional 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: Nondimensional 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: Nondimensional 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: Nondimensional 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, nondimensional lumped element and twoport network parameters are needed. NonDimensionalization of Lumped Element and TwoPort Network Parameters Some of the lumped element and twoport 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 nondimensional parameter is obtained from the corresponding dimensional parameter by dividing it with the value of the parameter for a classical circular plate. The nondimensional shortcircuit acoustic compliance is given by C CAS AS (5.38) where (CAS)Shm represents the shortcircuit acoustic compliance of the shim alone; Shm 16E h 3 (5.39) Above expression for (CAS) Shm in terms of nondimensional ratios is given by (CAS)h 1(i 2)2)K4 + R2j2 (5.40) The nondimensional 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 shortcircuit 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: Nondimensional shortcircuit 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: Nondimensional shortcircuit 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: Nondimensional shortcircuit 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: Nondimensional shortcircuit 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: Nondimensional shortcircuit 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: Nondimensional shortcircuit acoustic compliance plots for Ep/E, =0.6 and h, R2 = 0.02. The nondimensional 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 nondimensional 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 nondimensional ratios, has all ratios similar to Figure 5.23 except the aspect ratio. The plots appear the same indicating that the nondimensional 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: Nondimensional 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: Nondimensional 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: Nondimensional 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: Nondimensional 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: Nondimensional 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: Nondimensional 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: Nondimensional 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: Nondimensional acoustic mass plots for Ep/E, =0.6 and hJR2 =0.02 (Silcon/PZT). Unlike the parameters that were nondimensionalized by quantities possessing a physical significance, some of the twoport network parameters do not have quantities for the shim alone described. We can obtain the nondimensional form of the twoport 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 nondimensional 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 ,(Rr2) SEAh (l+v) 12(lv) (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 Nondimensional 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 Nondimensional 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 Nondimensional 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 Nondimensional 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 Nondimensional 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: Nondimensional effective acoustic piezoelectric coefficient plots for E p/E = 0.6 and hJ/R, =0.02. The best way to represent this nondimensional 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. Nondimensional 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 nondimensional form of all the independent twoport parameters and lumped parameters are known, nondimensional representation of other parameters can be found directly by replacing the form of variables visible in the expressions with the corresponding nondimensional variables. Hence the nondimensional 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 nondimensional '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: Nondimensional '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: Nondimensional 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: Nondimensional 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: Nondimensional 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: Nondimensional 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: Nondimensional 0A for E /E = 0.6 and h,/R2 = 0.02. The nondimensional 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 ) "sp {'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 nondimensional OA with variation in nondimensional 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: Nondimensional 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: Nondimensional 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: Nondimensional 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: Nondimensional 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: Nondimensional 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: Nondimensional 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 nondimensional. 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 (lv 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 forementioned parameters, the best possible representation of these nondimensional 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 nondimensional 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. 