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Modeling, Characterization and Optimization of Electrodynamic Zero-net Mass-flux (ZNMF) Actuators

Permanent Link: http://ufdc.ufl.edu/UFE0024236/00001

Material Information

Title: Modeling, Characterization and Optimization of Electrodynamic Zero-net Mass-flux (ZNMF) Actuators
Physical Description: 1 online resource (193 p.)
Language: english
Creator: Agashe, Janhavi
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: actuators, magnetic, znmf
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Zero-net mass-flux (ZNMF) actuators have been used for flow control applications such as separation control, thermal management. Electrodynamic transduction has been employed implementing these actuators in the past. However, a systematic design and modeling approach has been absent. This work presents a first-principles-based low order model for electrodynamic zero-net mass-flux (ZNMF) actuators. The lumped element modeling approach is used to model the multi-energy domain coupled system. The model developed was validated using prototype ZNMF actuators. A low cost, flexible, repeatable hybrid manufacturing technique was developed to realize these prototype actuators. Based on this model, the parameters that have the most impact on the actuator were identified. The model predicted the performance of several actuator configurations with less than 10 % error. Several design trends and tradeoffs were presented to enable intelligent design of these actuators. The model was also used to formulate a constrained optimization of the electrodynamic actuator. The optimized actuator was fabricated and characterized. The optimized actuator demon stared at least 50 % higher output velocities as compared to the baseline actuator. The optimized actuator achieved nearly 35 m/s maximum output velocity. The optimized actuator had volume of 15 cm3, overall mass of 90 g and maximum input power of 900 mW.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Janhavi Agashe.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Arnold, David.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024236:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024236/00001

Material Information

Title: Modeling, Characterization and Optimization of Electrodynamic Zero-net Mass-flux (ZNMF) Actuators
Physical Description: 1 online resource (193 p.)
Language: english
Creator: Agashe, Janhavi
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: actuators, magnetic, znmf
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Zero-net mass-flux (ZNMF) actuators have been used for flow control applications such as separation control, thermal management. Electrodynamic transduction has been employed implementing these actuators in the past. However, a systematic design and modeling approach has been absent. This work presents a first-principles-based low order model for electrodynamic zero-net mass-flux (ZNMF) actuators. The lumped element modeling approach is used to model the multi-energy domain coupled system. The model developed was validated using prototype ZNMF actuators. A low cost, flexible, repeatable hybrid manufacturing technique was developed to realize these prototype actuators. Based on this model, the parameters that have the most impact on the actuator were identified. The model predicted the performance of several actuator configurations with less than 10 % error. Several design trends and tradeoffs were presented to enable intelligent design of these actuators. The model was also used to formulate a constrained optimization of the electrodynamic actuator. The optimized actuator was fabricated and characterized. The optimized actuator demon stared at least 50 % higher output velocities as compared to the baseline actuator. The optimized actuator achieved nearly 35 m/s maximum output velocity. The optimized actuator had volume of 15 cm3, overall mass of 90 g and maximum input power of 900 mW.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Janhavi Agashe.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Arnold, David.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024236:00001


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1 MODELING, DESIGN AND OPTIMIZATION OF ELECTRODYNAMIC ZERO-NET MASSFLUX (ZNMF) ACTUATORS By JANHAVI S. AGASHE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2009

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2 2009 Janhavi Agashe

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3 To my parents Mr. S.H. Agashe and Mrs. Padmaja S. Agashe

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4 ACKNOWLEDGEMENTS This work was supported in part by NASA NRA # NNX07AD94A monitored by Dr. Brian Allan. This work would not have been possible w ithout constant guidan ce and input from my committee members Dr. David P. Arnold, Dr. Loui s N. Cattafesta, Dr. Mark Sheplak and Dr. Henry Zmuda. Their time and effort have not only made this work better but also made me an improved researcher. The lessons of patience, attention to detail and perseverance that I have learnt from Dr. Arnold will serve me for a long time to come. His encouragement and motivation have helped me survive this emotional roller-coaster. I woul d like to thank Dr. Cattafesta for making me a very careful and meticulous experimentalist. As much I have hated repeating every experiment several times, I have learnt th e value of cautiousness in taking and interpreting experimental data. Dr. Sheplak has taught me the importance of keeping the big picture always in sight without sacrificing the details. I have really enjoyed several non-academic discussions with him and I am going to miss them sorely. I also have to thank him for getting the espresso machine in the lab without which lot of long nigh ts would have been impossible. I would like to thank a ll my Interdisciplinary Microsyste ms Group (IMG) colleagues past and present, who made the three an d half years I spent here a lot of fun. I have learnt more from my interactions with my colleagues than any textbook or class could have taught me. I would like to extend my gratitude towards ex-IMG co lleagues Dr. Ryan Holman, Dr. David Martin, Dr. Karthik Kadirvel an d Dr. Steven Horowitz who were there to hold my hand when I was starting out. My colleagues Dr. Brian Home ijer, Matthew Williams, Benjamin Griffin, Brandon Bertolucci, Vijay Chandrasekharan, Chri stopher Bahr, Christop her Meyer, Matias Oyarzun, Justin Zito and Miguel Palavaccini who have been sounding board for all my

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5 research ideas. In addition, I would like to thank other colleagues who have become friends. Tailgating and talking football, summer barbeque s would not have been the same without Drew Wetzel, Jessica Meloy, Jessica Sockwell, Erin Patric k, Alex Phipps and Jeremy Sells. I am glad that I started out in IMG with a lot of colleagues and coworkers but I am leaving with so many friends. I would like to also thank my fr iends Dr. Madhura Bandhyopadhyay and Ms. Ankita Datta for the fun time I have spent with them. Lastly I would like to thank my family fo r their incessant and unconditional love and support for this long process. My parents Mr. Sh riniwas Agashe and Mrs. Padmaja Agashe have always been there for me. They have taught me everything I know and given me every possible freedom and opportunity. I would not have been here without them. I would like to thank my sister Ms. Ashwini Agashe for always reminding me what the important things in life were. My father-in-law Dr. O.P. Katyal ha s always been a source of inspiration and his dedication to his work will always motivate me. My mother-in-law Mrs. Manju Katyal and my sisters-in-law Dr. Vasuda Bhatia and Mrs. Bhavana Katyal-S arin have been extremely supportive and understanding through the years. I would also like to thank my c ousin and brother-in-law Dr. Jayashree Mahajan and Dr. Asoo Vakharia. They have been my family here and I will always cherish the time spent with them and my nephews Rohan and Ajit. Last, but not the least, I thank my husband Dr. Vipul Katyal who has been my rock. He has stood by me through some of my most vulnerable moments and ensured that I have emerged out of them a better stronger person. I love him dearly and am looking forw ard to many happy years with him.

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6 TABLE OF CONTENTS page ACKNOWLEDGEMENTS.............................................................................................................4 LIST OF TABLES................................................................................................................. ..........9 LIST OF FIGURES.......................................................................................................................11 ABSTRACT...................................................................................................................................16 CHAPTER 1 INTRODUCTION..................................................................................................................18 1.1 Background.......................................................................................................................19 1.2 Research Objectives..........................................................................................................25 1.3 Thesis Outline............................................................................................................. ......26 2 BACKGROUND.................................................................................................................... 27 2.1 Transduction Mechanisms................................................................................................27 2.2 Basics of Magnetic Transduction..................................................................................... 32 2.2.1 Magnetic Field Generated by Current Distributions..............................................33 2.2.2 Magnetic Field due to Intr insic Material Properties............................................... 35 2.2.3 Equivalence between Current Dist ributions and Magnetic Materials....................40 2.3 Magnetic Field Based Actuators....................................................................................... 41 2.3.1 Fixed-Field Actuators (Electrodynamic)................................................................42 2.3.2 Variable-Field Actuator s (Variable-Reluctance).................................................... 44 2.4 Scaling of Magnetic Actuators......................................................................................... 46 2.5 Magnetic Actuators...........................................................................................................49 2.5.1 Reported Zero-Net Mass-Flux Actuators............................................................... 50 2.5.2 Modeling of Zero-Net Mass-Flux Actuators.......................................................... 56 3 LUMPED ELEMENT MODELING...................................................................................... 58 3.1 Lumped Element Modeling Basics................................................................................... 58 3.1.1 Passive Lumped Elements...................................................................................... 59 3.1.2 Two-Port Elements................................................................................................. 61 3.2 Lumped Element Model of Electrodynamic ZNMF Actuator......................................... 64 3.3 Electrodynamic Transduction........................................................................................... 67 3.3.1 Calculation of the Transduction Coefficient using 1-D Magnetic Circuit Model.......................................................................................................................... .72 3.3.2 Calculation of the Transduction Coe fficient using Finite Element Analysis ......... 76 3.4 Diaphragm Model............................................................................................................ .79 3.4.1 Coupling between Mechani cal and Acoustic Doma ins.......................................... 79 3.4.2 Acoustic Model of the Diaphragm......................................................................... 81

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7 3.5 Models of the Acoustic/Fluidic Components................................................................... 84 3.5.1 Cavity Model..........................................................................................................84 3.5.2 Orifice Model.........................................................................................................85 3.5.3 Acoustic Radiation Impedance............................................................................... 87 3.6 Transfer Function for Electrodynamic ZNMF Actuator.................................................. 89 4 FABRICATION AND MODEL VALIDATION OF PROTOTYPE ELECTRODYNAMIC ZNMF ACTUATORS...................................................................... 93 4.1 Speaker-Driven Electrodynamic ZNMF Actuator............................................................ 93 4.1.1 Lumped Parameter Extraction................................................................................ 94 4.1.1.1 Electrical impedance(eC R and eCL ).............................................................95 4.1.1.2 Mechanical compliance (mDC ).....................................................................96 4.1.1.3 Electromechanical transduction coefficient (coil B L)....................................98 4.1.1.4 Diaphragm acoustic resistance (aD R ) and equivalent acoustic mass (aD M ).................................................................................................................100 4.1.2 Hotwire Measurements Overall Model Verification......................................... 105 4.2 Fabrication of Custom ZNMF Actuators........................................................................ 108 4.2.1 Magnetic Assembly.............................................................................................. 109 4.2.2 Diaphragm Assembly........................................................................................... 111 4.3 Model Validation........................................................................................................... .114 4.3.1 Electrical Impedance Measurements.................................................................... 115 4.3.2 Static Displacement Tests..................................................................................... 116 4.3.4 Hotwire Measurements......................................................................................... 121 4.3.5 Summary...............................................................................................................125 5 DESIGN AND OPTIMIZATION STRATEGI ES F OR ELECTRODYNAMIC ZNMF ACTUATORS......................................................................................................................127 5.1 Challenges in Design and Optimiza tion of Electrodynami c ZNMF Actuators.............. 127 5.2 Frequency Dependence of Actuator Performance.......................................................... 132 5.3 Design Tradeoffs for Composite Dia phragm Electrodynam ic ZNMF Actuators.......... 136 5.3.1 Magnetic Assembly Design Tradeoffs.................................................................136 5.3.2 Composite Diaphragm Design Tradeoffs............................................................. 144 5.3.2.1 Effective area ( S )........................................................................................145 5.3.2.2 Acoustic compliance ( CaD).........................................................................146 5.3.2.3 Acoustic mass ( MaD)..................................................................................148 5.3.2.4 Resonant frequency ( fres)............................................................................149 5.3.3 System-level Design Tradeoffs............................................................................150 5.3.4 Power Considerations...........................................................................................153 5.4 Optimization of Composite Diaphragm Electrodynam ic ZNMF Actuators................... 155 5.4.1 Choice of the Objective Function.........................................................................158 5.4.2 Parameter Variation..............................................................................................164 5.5 Characterization of the Optimized Electrodynamic Actuator........................................ 165 6 SUMMARY AND FUTURE WORK.................................................................................. 169

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8 6.1 Summary.........................................................................................................................169 6.2 Suggested Future Work.................................................................................................. 171 APPENDIX A MAGNETIC CIRCUIT MODEL......................................................................................... 173 A.1 Non-idealities in the Soft Magnet.................................................................................. 176 A.2 Magnetic Assembly for the Optimized Actuator........................................................... 178 A.3 Force Non-linearities.....................................................................................................180 B DERIVATION OF LUMPED MODEL PARAMETERS OF A CLAMPED ANNULAR PLATE.......................................................................................................................... ........182 C OPTIMIZATION DETAILS................................................................................................ 186 LIST OF REFERENCES.............................................................................................................188 BIOGRAPHICAL SKETCH.......................................................................................................193

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9 LIST OF TABLES Table page 2-1 Relative permeability of soft ma gnetic materials.................................................................. 39 2-2 Magnetic properties of hard magnet materials...................................................................... 40 2-3 Summary of ZNMF actuators reviewed................................................................................55 3-1 Effort and flow variab les in various energy doma ins............................................................ 60 3-2 Passive lumped element components in various energy domains......................................... 61 3-3 Dimensions of the magnetic assembly for 1-D circuit model and finite elem ent modeling comparisons.......................................................................................................77 3-4 Lumped parameters for electrodynamic ZNMF actuator...................................................... 89 4-1 Geometrical parameters fo r speaker-driven ZNMF actuator................................................ 94 4-2 Linear curve fit results for compliance measurements.......................................................... 97 4-3 Linear curve fit results fo r transduction factor measurem ents............................................ 100 4-4 Extracted parameters for the speaker-driven ZNMF actuator............................................. 104 4-5 Diaphragm material properties and geometry ..................................................................... 115 4-6 Model parameters for PDMS diaphragm corresponding to actuator A 1............................. 115 4-7 Electrical and magnetic parameters..................................................................................... 118 4-8 PDMS diaphragm parameters for actuator A1.................................................................... 120 4-9 Device configuration for hotwire tests................................................................................ 121 4-10 Various components of the damping.................................................................................124 5-1 Output velocity in di fferent frequency ranges and desi red actuator characteristics............ 135 5-2 Lower and upper bounds on the magnetic assembly design variables................................ 141 5-3 Constants used in the optimization formulation.................................................................. 141 5-4 Optimized designs for maximum blocke d force for various power constraints and no ma ss constraint (active cons traints indicated in bold)..................................................... 143 5-5 Diaphragm material properties and geometry ..................................................................... 148

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10 5-6 Parameters for the magnetic assembly................................................................................ 153 5-7 Lower and upper bounds on design variables..................................................................... 157 5-8 Baseline actuator and optimized designs for objective functions 13 (active constraints indicated in bold) ............................................................................................................. 160 5-9 Optimized designs for objective function 1 (maxi mum integrated velocity) for various power constraints (active cons traints indicated in bold).................................................. 160 C-1 Optimized magnetic assembly desi gns for ma ximum blocked force for power constraint and mass c onstraint of 150 g...........................................................................186 C-2 Optimized magnetic assembly desi gns for ma ximum blocked force for power constraint and mass c onstraint of 120 g...........................................................................186 C-3 Optimized designs for objective func tion 1 with various we ight constraints..................... 187

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11 LIST OF FIGURES Figure page 1-1 Actuation stress vs. actuation st ress for various transduction mechanism s.......................... 22 1-2 Frequency dependence of performa nce of various transduction mechanism s...................... 23 2-1 Two port model of a transducer........................................................................................... ..28 2-2 Free body diagram of the all the forces acting on an actuator............................................... 31 2-3 Operating characterist ics of a typi cal actuator...................................................................... 32 2-4 Magnetic field genera ted by a current element ..................................................................... 34 2-5 Magnetic field generated by a solenoidal coil....................................................................... 35 2-6 Soft magnetic material................................................................................................... ........38 2-7 Hard magnetic material.........................................................................................................39 2-8 Hard magnet representation as an equivalent current density............................................... 41 2-9 Electrodynamic transduction.............................................................................................. ...43 2-10 Typical mesoscale voice-coil actuator................................................................................. 44 2-11 Typical configuration of me soscale solenoid actuator........................................................ 45 2-12 Scaling of various ma gnetic interactions............................................................................. 46 2-13 Cross-section of a typical moving coil speaker................................................................... 50 2-14 Integrated micromachined electrosta tic ZNMF actuator adapter from Coe et al................ 51 2-15 Mesoscale piezoelectric ZNMF act uator adapted from Sm ith et al.................................... 51 2-16 Fixed field (electrodynamic) ZN MF actuator adapted from McCorm ick........................... 52 2-17 Integrated micromachined electrostatic ZNMF actuator demonstrated by Parviz et al. .....53 2-18 FSMA based ZNMF actuato r reported by Liang et al......................................................... 53 2-19 On chip electrodynamic ZNMF actuator for cooling reported by Yong et al..................... 54 2-20 IPMC driven ZNMF actuator adapted from Lee et al......................................................... 54 3-1 Passive lumped element................................................................................................... ......60

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12 3-2 Ideal transformer and gyr ator two-port circuit elements ....................................................... 62 3-3 Impedance and source transformation using a transformer................................................... 63 3-4 Impedance and source tr ansformations using a gyrator........................................................ 64 3-5 Schematic of electrodynamic ZNMF actuator....................................................................... 64 3-6 Lumped element model of the electrodynamic ZNMF actuator........................................... 66 3-7 Lorentz force on a current ca rrying coil in a ma gnetic field................................................. 68 3-8 Voltage induced on a moving coil in a magnetic field.......................................................... 69 3-9 Two-port gyrator model of electrodynami c transduction...................................................... 71 3-10 1-D magnetic circuit approach........................................................................................... .72 3-11 Fringing fields in the air gap.......................................................................................... ......74 3-12 Axissymmetri c COMSOL model........................................................................................ 77 3-13 Results from the COMSOL model radial ma gnetic flux density (Tesla) and streamlines of total magnetic flux density......................................................................... 78 3-14 Comparison of the magnetic flux density in the air gap between the 1-D circuit model predictions and FEM results (a veraged over entire gap volume)...................................... 79 3-15 Equivalent circuit repr esentation of coupling between the mechanical and acoustic dom ains........................................................................................................................ ......80 3-16 Composite diaphragm model as an annular plate................................................................ 81 3-17 Short closed cavity and its lumped model ...........................................................................84 3-18 Narrow circular orifice and its lumped model..................................................................... 86 3-19 Lumped model of a piston in an infinite baffle................................................................... 88 3-20 Lumped element model of the electr odynami c ZNMF actuator referred to the acoustic domain......................................................................................................................... .......90 4-1 Schematic of the speaker-dri ven electrodynamic ZNMF actuator ........................................94 4-2 Photograph of the speaker-d river, the cavity and the slot..................................................... 94 4-3 Electrical impedance of the speaker-driver........................................................................... 95 4-4 Experimental setup for compliance measurement................................................................. 96

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13 4-5 Results from co mpl iance measurement................................................................................. 97 4-6 Experimental setup for tr ansduction factor measure ment..................................................... 99 4-7 Results from the transduction factor measurement............................................................... 99 4-8 Setup for the impulse response test..................................................................................... 101 4-9 Results from the impulse response test................................................................................ 101 4-10 Step response of the diaphragm and the curve fit..............................................................103 4-11 Equivalent circuit fo r the impulse response test ................................................................ 105 4-12 Typical hotwire calibration curve......................................................................................106 4-13 Model predicted and expe rime ntally measured output ve locity of the speaker-driven ZNMF actuator for 3V pp voltage input.......................................................................... 107 4-14 Schematic of Polydimethylsiloxane (PDMS) composite diaphragm actuator..................109 4-15 Magnetic assembly schematic and photograph................................................................. 110 4-16 Two-flange design for the rigid central boss.....................................................................111 4-17 Diaphragm components.................................................................................................... .112 4-18 Centering ring for the central boss.................................................................................... 113 4-19 Mold for the PDMS diaphragm fabrication and released PDMS diaphragm with the rigid center boss .............................................................................................................. .113 4-20 Fully assembled device.................................................................................................. ....114 4-21 Impedance measurements of 150-turn 40 AWG copper coil without the ma gnetic assembly....................................................................................................................... ....116 4-22 Setup for dc displacement test and the step response test................................................. 117 4-23 DC displacement test results............................................................................................ ..118 4-24 Displacement measurement under pressure load............................................................... 119 4-25 Displacement measurements under uniform pressure load............................................... 120 4-26 Hotwire results and model comparisons for device A1..................................................... 123 4-27 Hotwire results and model comparisons for device A2..................................................... 123

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14 4-28 Hotwire results and m odel predictions for device A3....................................................... 124 4-29 Relative magnitudes of the slot da mping and the total dam ping for device A1................ 125 5-1 Loaded electrodynamic actuator.......................................................................................... 128 5-2 Electrodynamic actuator circu it referred to the acoustic doma in........................................ 128 5-3 Typical velocity frequency response of a second-order system.......................................... 129 5-4 Comparison of the cavity and slot impedances for actuator A2 .......................................... 136 5-5 Magnetic assembly........................................................................................................ ......137 5-6 Coil configuration using a mu ltiple layers of bondable copper wire.................................. 138 5-7 Optimization results for maximizing blocked force for various power and ma ss constraints........................................................................................................................144 5-8 Normalized effective area as a function of the solidity ratio............................................... 146 5-9 Acoustic compliance variat ion with the solidity r atio ( 0.08 ha )....................................147 5-10 Acoustic mass variation with the solidity r a tio ( 0.08 ha )............................................149 5-11 Resonant frequency of the diaphragm ( 0.08 ha )..........................................................150 5-12 Variation of electroac oustic transduction coefficient G with solid ity ratio ( 0.08 ha ).. 151 5-13 Variation of predicted outpu t v elocity with solidity r atio ( 0.08 ha )............................153 5-14 Input and output as a func tion of frequency for actuato r A2 ............................................. 154 5-15 Efficiency of actuator A2..................................................................................................155 5-16 Frequency response of actuator A1 for different diaphragm damping.............................. 157 5-17 Comparison of the frequency response of the baseline device and optim ized designs for the three objective functions ...................................................................................... 159 5-18 Maximum output velocity as a func tion of m a ximum allowable input power for objective function 1.......................................................................................................... 161 5-19 Maximum output velocity as a func tion of m a ximum allowable input mass for objective function 1.......................................................................................................... 161 5-20 Comparison of the frequency response of the baseline device and optim ized designs for the objective function s with a resonant frequency constraint.................................... 163

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15 5-21 Variation of the average velocity from 0 300 Hz with the change in the design variab les...................................................................................................................... .....164 5-22 Impedance measurement results for the optimized actuator............................................. 166 5-23 Damping measurement results for the optimized actuator................................................167 5-24 Optimized and baseline actu ator ho twire and LEM comparisons .....................................167 A-1 Magnetic assembly and the 1-D circuit model................................................................... 173 A-2 Arrow plot of magnetic fl ux density in the m a gnetic assembly......................................... 174 A-3 Reluctance components of the magnetic assembly............................................................ 174 A-4 Relative values of the reluctan ces of the magnet, co re and air gap .................................... 176 A-5 B-H curve for an ideal a nd typ i cal soft magnetic material................................................. 177 A-6 Results from the COMSOL model to tal m agnetic flux density (Tesla).......................... 177 A-7 Radial magnetic flux density in the ma gnetic assem bly of the optim ized actuator............179 A-8 Total magnetic flux density.............................................................................................. ..180 A-9 Non-linearity due to spatial variation of the flux density ................................................... 181 B-1 Schematic for the mechanical driver an the sim plified annular p late model...................... 182 B-2 Transverse displaceme nt Model & FEM results ( ba = 0.5, ha = 0.08, a = 12.7 x 103, 2 E = 360 kPa, 2 = 0.33).............................................................................................. 184

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16 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MODELING, DESIGN AND OPTIMIZATI ON OF ELECTRODYNAMIC ZNMF ACTUATORS By Janhavi Agashe May 2009 Chair: David P.Arnold Major: Electrical and Computer Engineering Zero-net mass-flux (ZNMF) actuators have been used for flow control applications such as separation control, thermal management. El ectrodynamic transduction has been employed implementing these actuators in the past. Howeve r, a systematic design and modeling approach has been absent. This work presents a first-pr inciples-based low order model for electrodynamic zero-net mass-flux (ZNMF) actuators. The lumped element modeling approach is used to model the multi-energy domain coupled system. The model developed was validated using prot otype ZNMF actuators. A low cost, flexible, repeatable hybrid manufacturing technique was developed to real ize these prototype actuators. Based on this model, the parameters that have th e most impact on the actuator were identified. The model predicted the performa nce of several actuator confi gurations with less than 10 % error. Several design trends and tradeoffs were presented to enable intelligent design of these actuators. The model was also used to fo rmulate a constrained optimization of the electrodynamic actuator. The optimized actuato r was fabricated and characterized. The optimized actuator demon stared at least 50 % higher output velocities as compared to the

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17 baseline actuator. The optimized actuator achiev ed nearly 35 m/s maxi mum output velocity. The optimized actuator had volume of 15 cm3, overall mass of 90 g and maximum input power of 900 mW.

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18 CHAPTER 1 INTRODUCTION Small, light and low power electromechanical actuators are desired for various consumer, biomedical, and industrial applications. Zero -net mass-flux (ZNMF) actuators (also called synthetic jets) have been studied recently for a pplications in flow control, thermal management, fluidic mixing, etc [1],[2],[3],[4]. This dissertation focuses on mesoscale magnetic ZNMF actuators for flow control applications. Flow controlparticularly separation controlhas been studied for automotive and aerodynamic applications to reduce drag and improve lift and fuel efficiency. The typical structure of a ZN MF actuator consists of an electromechanical transducer e.g. piezoelectric, electrostatic magnetic, etc.that drives an oscillatory flow through a small orifice or slot in an attached cavity. The peri odic expulsion and ingestio n of fluid constitutes the synthetic jet, so named because it requires only the ambient fluid to produce a jet like structure rather than hoses, tubes, pumps, etc. Since no additional fluid mass is added to the flow during the operation, these actuators are also called zer o-net mass-flux actuators. The device affects the flow field by addition of moment um to the flow. The device geom etry and material properties decide the device performance. However, the act uation scheme also has a large impact on the device performance, and different approaches offer certain advantages and disadvantages. For almost all applications of ZNMF actuato rs, large centerline ve locities over a large range of frequencies are desire d. To obtain high centerline ve locities, large actuator volume displacements are required, implying large area, large stroke, or both. Of all possible electromechanical actuation schemes, magnetic-field based actuators are well-suited due to their high energy densities, large st roke and low operating frequencies. Magnetic actuators are commonplace in many macroscale systems and ar e well known for their performance and

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19 reliability. Moreover, scaling analyses of these actuators indicate their applicability at smaller scales. Thus compact, efficient magnetic ZNMF actuators should be designable to meet performance requirements for emerging ZNMF applications [5],[6]. This thesis presents detailed modeling, desi gn, characterization, a nd optimization approach for magnetic ZNMF actuators. Lumped element and finite element modeling techniques are used for the modeling of these actuators. Detailed characterization of various electrodynamic ZNMF actuators is also presented in order to valid ate the developed models and design guidelines. Various tradeoffs involved in the design of the actuators to meet targ et specifications are discussed, such as mass and power consumption vs. actuation force. Based on these tradeoffs, a design strategy is presented to obtain the desi red actuator performance while meeting the necessary constraints. A preliminary constrai ned optimization was also performed and the optimized actuator was fabricated and characterized. 1.1 Background A transducer is defined as any device that converts energy from one form to another[7]. Thus a transducer is a two port device th at couples two different energy domains. Electromechanical transducers, which couple electrical and mechanical energy domains, are divided into two typessensors and actuators. Sensors produce an elect rical output proportional to some mechanical quantity, while actuators co nvert electrical energy into mechanical energy and thus produce mechanical motion. There are various transduction mechanisms by which this energy conversion can occur: magnetic, electrostatic, piezoelectric, etc. Actuators are used in a wide variety of app lications to achieve so me mechanical function in response to an electrical driv e or control signal. The perfor mance requirements of an actuator force, displacement, bandwidth, response time, power consumption, etc.differ greatly depending on the specific application, and differe nt tradeoffs in performance can be achieved by

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20 careful design of the transduction mechanism, actuator geometry and materials. Device and system models are necessary to correlat e these design parameters with performance specifications. Magnetic-field-based actuator s rely on the interaction be tween electrical input and magnetic fields to produce a mechanical force le ading to rotational or linear motion. Rotary actuators such as electri c motors have complete 360 range of motion for conti nuous rotation. On the other hand, linear actuators ha ve restricted range of motion a nd generate finite in-plane or out-of-plane displacements. Typical applications of linear actuators include valves, fuel injectors, head positioners for disk driv es, switches, relays, etc. The remainder of this section discusses variou s performance characteristics of actuators. A brief comparison of magnetic-field-based actuat ors and other actuation schemes is presented. This comparison shows that the transduction sche me is an important factor that affects the performance of an actuator. Thus for a given application, the selec tion of a particular transduction scheme is crucial. The various performance parameters for actua tors are now discussed. Two of the most commonly specified parameters are blocked force and free deflection. Blocked force, Fb, refers to the force exerted at a given input excitation level when the actuator is not allowed to move. Free deflection, Xf, refers to the maximum displacem ent at a given input excitation without the actuator working agains t any external load. A detailed discussion of these parameters is included in section 2.1 where the two-port model for electromechancial transducers is presented. Huber et. al. [8] discuss various performance indices of mechanical actuators that are important in choosing an actuator for a particular application, in cluding actuator stress, actuator

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21 strain, power density and bandwidth. Actuation stress is defined as the force per unit area produced by the actuator and thus is a measure of the force available from the actuator. The actuation strain is defined as the nominal change in length produced by the actuator and is a measure of displacement/stroke. Th e power density of the actuator is defined as the output mechanical power per unit volume of the actuato r. The maximum power density can thus be defined as 2bfFX Vol where is the frequency of operation and Vol is the volume of the actuator. Figure 1-1 shows the variation of actuation s t ress with actuation strain for various actuation schemes [8]. The solid lines show the maximum limits for each type of the actuator. Note two different types of magnetic-field-based actuators are listed: moving-coil and solenoid. In this thesis, these will be define d as fixed-field-actuato rs and variable-fieldactuators, respectively, and the differences will be fully described in Section 2.3. It can be seen that although both moving-coil and solenoid type actuators have re latively small actuation stress (force); they produce large strain (displacements ), which is a desirable for ZNMF actuators. Moreover, in the figure, the diagonal lines repr esent curves of maximum available work per unit volume or energy density. It can be seen th at the magnetic actuation schemes offer higher available work output per unit volume as compar ed to most piezoelectric devices. It should be noted however that these plots only consider direct actuation and do not account for more complex flexure-based or kinematic amplification schemes. Figure 1-2 shows the output power of various types of actuators as a function of frequency [8]. For ZNMF actuators, the frequency of operati on is usually fairly low (~100 Hz) in order to affect the flow condition s. Although m agnetostrictiv e actuators have larger power density, they have very small stroke and are not suitable for ZNMF applications. Piezoel ectric actuators have

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22 higher power density, attributed to their higher operating frequencies (recall piezoelectrics had lower energy densities). Figure 1-1. Actuation stress vs actuation stress for various tr ansduction mechanisms [8] (used with permission) Moreover, the total bandwidth for moving-coil type actuators is also larger as compared to most piezoelectric actuators. This is important when broadband type actuators are desired where a large output displacement is desired over a large range of frequencies. The comparisons discussed above incorporat e only mesoscale devices. In most cases (particularly for ZNMF actuators), the weight, po wer consumption and even the overall size of actuators are important criteria in design and se lection of actuators. Usually these parameters scale with the size, and it is im portant to know the scaling behavior of various actuators in order to determine their effectiv eness at smaller scales. Magneticfield-based actuators

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23 Figure 1-2. Frequency dependen ce of performance of various transduction mechanisms [8] (used with permission) The favorable scaling of magnetic-field-based actuators has been di scussed by [9], [10], [11]. However, the focus has been only on the scaling of the transduction scheme itself. The overall scaling of a functional actuator depends not only on the scaling of the transduction scheme but also on the scaling of the structures (beams, springs, coils, etc.) of the complete actuation system. Moreover, the manufacturing ch allenges presented at smaller scales may require different material selections, fabrication methods, geometries, etc. This work presents a detailed investigation of overall scaling of magnetic ZNMF actuators. In order to design ZNMF actuato rs best-suited for a given application, accurate models of the transduction scheme and overall actuators are necessary. Both computational techniques (such as finite element modeling) and low-orde r analytical models ha ve been developed for ZNMF actuators. Analytical models provide be tter physical insight in to device operation and Magnetic -fieldbased actuators

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24 hence provide excellent design tools. However, analytical models usually are first-order approximations, and finite element models are n ecessary in order to st udy the higher order and nonlinear effects present in the device. ZNMF actuators are coupled systems with va rious energy domainselectrical, mechanical and fluidic/acousticinteracting with each othe r. Lumped element modeling (LEM) is one of the low-order analytical approaches used to model such systems since the 1920s, particularly in the field of electroacoustics [7, 12, 13]. Mc Cormick [3] developed a coupled model of a magnetic speaker-driven ZNMF actuator. A transfer function for the output velocity as a function of input voltage was presented. Although an e quivalent circuit was presented for the overall actuator, no details were described for the i ndividual model parameters for the magnetic assembly and the mechanical driver. Gallas [14] presented coupled equivalent circuit based model for a piezoelectric ZNMF actor using the lumped element modeling technique. The model was successfully used to study the impact of various geometry parameters and material properties on the device performance. In additi on to LEM, various numerical techniques have also been used to model ZNMF actuators However, these methods are time-and computationally-intensive and provide little insi ght into the operation of the device [15-17]. Thus the goal of this work is to use th e lumped element modeling technique to develop and experimentally validate an equivalent circuit model for magnetic ZNMF actuators, paralleling the work Gallas [14] performed for the piezoelectric ZNMF devices. Finite element modeling will also be used to validate some as pects of the lumped model. The validated model will provide various physical insights into th e actuator physics, and design and optimization strategies will be develope d based on these insights.

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25 1.2 Research Objectives The primary goal of this research is a multi-faceted investigation of the modeling, design, scalability and optimization of magnetic-field-bas ed ZNMF actuators. The models developed are validated by a detailed characte rization experiments. These models are then used to design ZNMF actuators that meet necessary target spec ifications of centerline velocity, frequency of operation, power consumption, etc. The various tradeoffs that aff ect the actuator performance are investigated, and a systematic approach is pres ented for design of magnetic ZNMF actuators to meet performance metrics. The thesis also presents a systematic scali ng analysis of magnetic-f ield-based actuators using a lumped element modeling framework to develop coupled, multi-domain models of magnetic actuators that account for the magne tic, mechanical as well as acoustic/fluidic components of the device[7, 12]. These models w ill be used for scaling analysis and for design and optimization of magnetic-fie ld-based ZNMF actua tors. Although, several scaling analyses just for the magnetic transduction scheme are found in the literature, scaling analysis of the entire actuator has been lacking. Overall scaling analys is of the devices provides important physical insight into the biggest f actors that influence the pe rformance of the device. The research contributions of this work are as follows Development and validation of coupled, multienergy domain low-order (lumped-element) system models for magnetic ZNMF actuators. The models are validated experimentally and using finite element analysis. Development of fabrication methods for mesoscale ZNMF actuators using hybrid manufacturing techniques. The fabrications me thods, challenges, limitations etc. for all the actuator component s are detailed. Development of an overall design approach to ach ieve desired target specifications from a magnetic ZNMF actuator. A systematic discus sion of tradeoffs and trends involved in the design of magnetic ZNMF actuators is presented. Demonstration and characterization of an optimized magnetic ZNMF actuator.

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26 1.3 Thesis Outline This dissertation is divided into 5 chapters and 3 appendices. Chapter 1 detailed the motivation for this work and the proposed research goals as well the contri butions of this work. Chapter 2 provides detailed background information and literature review relevant to this work. Chapter 2 introduces the basics of magnetics and magnetic-fie ld-based actuators. Scaling analysis of magnetic-field-based actuators is also included. The literature review is divided into two different sections: a review of previously reported ZNMF act uators and a brief review of modeling techniques used to analyze ZNMF actuators. Chapter 3 discusses modeling of electrodynamic ZNMF actuators. Ch apter 4 discusses the verification of the lumped element model for a speaker-driven electrodynamic ZNMF actuator. The fabrication and characterization of custom-built mesoscale electrodynamic ZNMF actuators is also presented Chapter 4. Chapter 5 discusses optimization strategy and various tr adeoffs encountered in the design of magnetic ZNMF actuators. The characteriza tion of optimized magnetic ZNMF actuators is also presented in Chapter 5. Chapter 6 discusses the conclusions and the some future directions for this work. The feasibility of scaling these actuators at small scales is investigated. The fabrication, material choices and the geometry that are viable for smaller sizes are discussed.

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27 CHAPTER 2 BACKGROUND A brief overview of the basics of electromechan ical transduction is presented in Chapter 2 with an emphasis on magnetic actuators. For the purpose of this dissertation, magnetic actuators or more explicitly magnetic-field-bas ed actuators are those devices that use quasistatic magnetic fields as the primary fields for energy conversion and whose operation are governed by the magnetostatic Ma xwell equations [18]. There are numerous configurations using coils and magnets that can be employed for magnetic actuators, but these can be generally divided into two distinct types, defined in this thesis as: fixed-field actuators and variable-field actuators. In fixed-field actuato rs (FFAs), the magnetic field di stribution does not significantly change during the device operation, whereas in variable-field actuators (VFAs) the device operation relies on a large change in the magnetic field distribution. Ma gnetostrictive actuators, which use materials that develop strain in the presence of an external magnetic field, are not discussed in this work. In Chapter 2, the origin of magnetic fields from both material magnetization and current carrying conductors is first disc ussed. Then, the func tional operation of the magnetic-field-based actuators is described, and scal ing analysis for the two types of magnetic actuation schemes is presented. A detailed summary of various ZNMF act uators reported in the literature is presented, and an overview of some modeling approaches used for ZNMF actuators is also discussed. 2.1 Transduction Mechanisms A transducer is any device that converts energy from one domain to another. Transducers can be also classified into various categor ies depending on their energy conversion behavior. Energy-conserving transducers convert all energy in one domain to the other. Typical examples of this type are magnetic tran sducers (both FFA and VFA), electrostatic transducers, and

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28 piezoelectric transducers. Conve rsely, piezoresistive and ther moelastic transducers are nonenergy conserving. In non-energy -conserving transducers, some en ergy is dissipated in various losses inherent to the transduction mechanism. Fo r example in piezoresistive transducers, some energy is needed to maintain a current through th e resistors in order to measure the change in resistance due to the incident mechanical energy. Any energy-conserving transducer can be repr esented as a generic two-port network as shown in Figure 2-1 [13]. Here E1 and F1 represent the generalized effo rt and flow variables in the first energy domain. The product of the effort and flow variables is power. Figure 2-1. Two port model of a transducer The relationship between the effort and flow variables can be expressed in matrix from as 11 22EF AB EF CD (2.1) Note that this form is called the impedan ce representation as the effort variables ( E1 and E2) are expressed as the functions of the flow variables (F1 and F2). B and C are the coupling terms between the two energy domains. Thus for electromechanical tran sducers, the two-port model is shown below mme emeZT FU TZ VI (2.2)

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29 where F and V are the force and voltage, respectively. Tme and Tem are transduction coefficients [7]. The term Ze is the blocked electrical impedance, i.e. the input electrical impedance when the mechanical motion is blocked such that the velocity is zero. Similarly, Zm is the open-circuit mechanical impedance, i.e. the mechanical im pedance when the electrical terminals are opencircuited and no current is allowed to flow. Energy-conserving transducers can be classified further into various categories. A reciprocal transducer is the one where the tr ansduction coefficients are the same in both directions i.e. Tme = Tem = T Reciprocal transducers thus have the same transduction coefficients operating in either direction (elect rical to mechanical or mechanical to electrical). Non-reciprocal transducers, on the other hand, have different transduction coeffici ents dependent on the direction [7]. For example, piezoelectric and el ectrostatic transduction schemes are reciprocal, whereas piezoresistive sensing and electroth ermal actuation are not. When the transduction coefficients between the two domains is equal in magnitude but opposite in sign ( Tme = Tem), the transduction mechanism is called anti-reciprocal This usually arises due to the choice of the sign convention [7]. Fixed-fi eld or electrodynamic transducti on scheme is anti-reciprocal. When the transduction coefficients ( Tme and Tem) between the two energy domains are constant, i.e. the effort variable in the second energy domain is directly proportional to the effort (or flow) variable in the first domain, the transduc er is referred to as a linear transducer. In nonlinear transducers, the coupling between the ener gy domains is non-linear. Fixed-field magnetic transducers are linear, ener gy-conserving and anti-reciprocal. Variable-field magnetic transducers are also energy-conserving and anti-reci procal, but they are in general, non-linear. Based on the two-port network model above, the blocked force produced by actuator is defined as

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30 bmeFTI (2.3) The internal impedance or the open-circui t mechanical impedance of the actuator Zm is defined as 0 m IF Z U (2.4) Thus if the actuator is not work ing against any other external for ce, the blocked force is balanced by the restoring force due to the internal stiffness. In other words, for a given current, a force will be generated, resulting in a disp lacement that is limited by the inte rnal stiffness of the actuator. This resulting displacement is cal led the free displacement, where f ree indicates the actuator is not working against any external mechanical lo ad. The relationship between the blocked force and the free displacement can be given by b f mF X j Z. (2.5) Here, harmonic motion of the actuato r is assumed. Note that the ne gative sign indicates that the restoring force is created by the actuator intern al impedance, which opposes the useful output force generated by the actuator. Moreover, the bl ocked force and the free displacement represent the maximum force and displacement bounds, respec tively, that the actua tor can achieve. The open-circuit mechanical impedance Zm of the actuator is usually a function of the frequency and thus the free displacement is also a function of the frequency. The total available output force, F (the force available to perform useful work) can be expressed as[19] bmFFjZx (2.6) where x is the displacement. Figure 2-2 shows all the forces acti ng within an actuator pictorially in the form of a free body diagram It can be se en that the total useful force produced by the actuator is limited by its blocked force as well as internal stiffness.

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31 Figure 2-2. Free body diagram of the all the forces acting on an actuator Figure 2-3 shows typical operating characteristic s of a general actu ator w here the available force F is shown as a functi on of the displacement x. For a linear actuator, the operating characteristics of the actuator will be a straight line connect ing the free displacement and the blocked force points. The blocked force (and hen ce the free displacement) is a function of the electrical input to the actuator For example, in the case of a fixed-field magnetic actuator, the blocked force is the Lorent z force produced given by Fb = BLI where B is the magnetic flux density, L is the length of the coil and I is the input current. In th e case of piezoelectric actuators, the blocked force is given by Fb = dV where d is the piezoelectric charge modulus and V is the applied input voltage. Thus as the input to the actuator is increased, the blocked force and the free displacement both increase. This is indicated by the parallel lines in Figure 2-3. The slope of the oper atin g characteristics always remains the sa me. If the actuator is operating at frequency the maximum power it can supply is given by 2bfFX .It should also be noted that the free displacement of an actuator is al so the function of the frequency. This is because the moving structure of the actuator has a fi nite mass in addition to the co mpliance and hence behaves as a standard second-order system. For ZNMF type applications, large displacements are required. Thus actuators with large blocked force and small internal stiffness are idea l for ZNMF applications as can be seen from

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32 (2.6). However, the resonant frequency and band width are directly propor tional to th e internal stiffness. Thus there is an inheren t tradeo ff between maximum achievable free displacement and maximum bandwidth. Other considerations such as power consumption, quality factor, etc. may also be important. Actuator Force Figure 2-3. Operating characte ristics of a typical actuator 2.2 Basics of Magnetic Transduction Magnetic and electric fields ar e usually described in terms of the force they produce on charges. A charge q moving with a velocity v experiences a force F given by Electric force Magnetic forceFqEqvB (2.7) where E is the electric field intensity and B is the magnetic flux density The magnetic flux density is the most important parameter in the analysis of magnetic systems. The unit of magnetic flux density is thus Newton/(Ampere-me ter). However, the most commonly used unit is the tesla (T) or equivalently weber/m2 (Wb/ m2). Magnetic flux arises from two fundamental

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33 sources: electric currents (sometimes called free currents ) and intrinsic atomic moments in materials due to orbital motion of electrons and spin moments (sometimes called bound currents ) [20]. The total magnetic flux density at any point in space is given by the constitutive equation 0 B HM (2.8) where 0 is the permeability of free space (-7410H/m), and the two summed terms indicate the two origins of the magnetic flux density. The first term H is the magnetic field intensity or simply magnetic field and represents the contribution from free electric currents. The SI unit of magnetic field intensity is Ampere/meter (A/m). The second term M is the magnetization and represents the contribution from the atomistic magnetic moments induced inside the material under consideration. Magnetization has the same un its as magnetic field intensity (A/m) and may be a function of the applied magnetic field. In free space there is no induced magnetization; thus M = 0, and the relationship between B and H simplifies to0 B H In contrast, in any real material, the material magnetization M 0 and the full expression must be used. Moreover, for ferromagnetic materials, the material magnetizati on is usually a strong nonlinear function of the magnetic field, leading to complex magnetic field re lationships as will be described later. The following sections describe the magnetic fields produced by electric currents and induced magnetic fields inside various magnetic materials. 2.2.1 Magnetic Field Generated by Current Distributions The incremental magnetic field dH generated at a point P by a differential current element of length dL carrying current I is given by the Biot-Savart law [21], 2 4 I dLr dH r (2.9) Here, r is the distance between point P the current element, and r is the unit vector along the line connecting P and the current element as shown in Figure 2-4. The magnetic field is inversely

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34 proportional to the square of the distance and dir ectly proportional to curren t. The total magnetic field is calculated by integrating the contribution of the individual elements over the length of the current carrying filament, 2 4L I dLr H r (2.10) The differential current element I dL can also be expressed in terms of a surface current density K (units A/m) or a volume current density J (units A/m2). Thus the magnetic field can be calculated as 2 4sKr HdS r (2.11) 2 4VJr HdV r (2.12) dL r Figure 2-4. Magnetic field ge nerated by a current element An electromagnet refers to a setup that uses an electric current to generate a desired magnetic field. One of the most commonly used electromagnet configurations is a solenoidal coil. Figure 2-5 shows a long sole noidal coil of length L that carries cu rren t I The magnetic flux density H established inside the coil is fairly un iform everywhere and approximated by [21]

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35 NI H L (2.13) where N is the number of turns in the coil. The dir ection of the magnetic field is given by the right-hand rule. The magnitude and directi on of the magnetic field can be modified by changing the current in the coil. Mo reover, if the current is sinusoi dal, sinusoidal fields can be established. Figure 2-5. Magnetic field ge nerated by a solenoidal coil 2.2.2 Magnetic Field due to Intrinsic Material Properties The previous section desc ribed the magnetic field H arising from free currents. This section describes the origin of material magnetization M Every atom consists of a central nucleus and electrons revolving around it. These moving charges act like current loops that generate tiny magnetic moments, similar to the solenoidal coil described above. In addition to the electron orbital magnetic moments, electron spin and nuclear spin magnetic moments arise due to quantum effects [21]. The net internal moment is thus the sum of the contributions from the electron orbital moments and the electron and nuclear spin moments. All materials can be magnetically classified based on the distribution of these atomic moments and their response in the presence of an externally applied field. While details of the origins of material magnetization

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36 is beyond the scope of this thesis, a brief summar y of different classes of magnetic materials is included below [22]. In diamagnetic materials the magnetic moments due to orb ital motion of the electrons and the spin moments effectively cancel each othe r, and hence the net magnetic moment of each atom is zero. In the presence of an external field, however, the magne tic force on the orbital electrons changes the orbital velocity (the orbita l radius cannot change due to quantization) in such a way that the total internal moment is decreased. As a result, diamagnetic materials are repelled from a magnet, although this force is very weak at moderate fields. Water, bismuth and noble gases (helium, neon, etc.) are ex amples of diamagnetic materials. In paramagnetic materials, the orbital moment and spin moments do not cancel each other completely. Each atom exhibits a small moment, but the orientation of these atomic moments is randomly distributed, and as a result the material as a whole does not exhibit a net magnetic moment. When an external magnetic field is app lied, a torque on the in dividual atomic moments tends to align them and increase the net magnetic moment inside the material. When the external field is removed, the atoms return to their randomly aligned distribut ion and the net magnetic moment is again zero. Typical examples of paramagnetic magnetic materials are oxygen, tungsten, potassium, etc. Ferromagnetic materials are the most important type in the context of magnetic actuators. These are materials that have large net moments ev en in the absence of an external field. The atomic moments in large areas of the material are aligned in the sa me direction due to interatomic forces. These regions are called dom ains. Although individual domains have large moments, they are initially random ly distributed inside the materi al. Thus, the material has near zero net magnetic moment. When an external magne tic field is applied, however, the domains

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37 aligned in the direction of the field grow in size while the other domains grow smaller through a process called domain wall motion. When the field is removed, all the domains do not return to their initial random state. This results in a non-zero magnetic moment even when the external field is zero. This phenomenon is called magnetic hysteresis. The magnetization M of a magnetic material is the ne t magnetic moment per unit volume of the material. The magnetization of an ideal ferromagnetic material Mint is given by [21] int extr M HM (2.14) where Hext is the external applied field is the susceptibility of the ferromagnetic material and Mr is the remnant magnetization. Note that this equation does not describe the hysteresis and saturation behavior present in all ferromagnetic materials. The impact of these effects is discussed in later sections. The susceptibility is the measure of induced magnetization in the material, and Mr is the magnetization that remains when the external field is zero. Another parameter, the intrinsic coercivity Hci, is the field required to reduce the internal magnetization to zero. From (2.4), the coercivity can be defined as r ci M H (2.15) Ferromagnetic materials can be further classified into soft and hard on the basis of their intrinsic coercivity and remnant magnetization. Soft magnetic materials have small values of remnant magnetization Mr and intrinsic coercivity Hci. Figure 2-6 shows the magnetization induced inside a soft ma gne tic material and the M-H relationship for the same. The M-H loop of a soft magnetic material shows some hysteresis behavior where the internal magnetization is not zero when the external field is removed. Ideally, however, the remn ant magnetization is zero and the hysteresis is small, i.e. the coercivity is zero.

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38 Figure 2-6. Soft magnetic material Under this assumption, the relations hip between the magnetic flux density B and applied field for an ideal soft magnetic material can be described as 0i n t00()(1)ext extrext B HMHH (2.16) Here 1r is the relative permeability of the soft magnetic material. Thus the magnetic flux density created in a soft magnetic material is directly proportiona l to the external applied field and the relative permeability. This can also be seen from the blue curve in Figure 2-6 for the ideal soft magnet. At som e point, however, all of the domains in the soft magnetic material are aligned to the external magnetic field, and any in crease in the external field does not lead to an increase in the magnetization. This maximum magnetization achievable in the soft magnetic material is called the saturation magnetization Ms. Examples of soft magnetic materials are iron, nickel, permalloy (nickel-iron alloy) and other alloys of these metals that develop strong internal magnetiza tion only in the presence of an external magnetic field. In many applications, soft magnets are us ed to concentrate and guide the magnetic field in certain regions, and usually high relative permeability is desired. Typical values of relative permeability of commonly used soft magnetic materials are shown in Table 21.

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39 Table 2-1. Relative permeability of soft magnetic materials [23] Material Relative permeability r Mu metal (Nickel ~ 75 % Iron ~ 25 %Alloy, Copper and Molybdenum) ~25000 Carbon Steel ~700 Permalloy (Nickel-Iron) ~8000 Soft Ferrites (Iron oxide alloys) ~5000 In contrast to soft ferromagnetic materials, ha rd ferromagnetic materials have a substantial remnant magnetization Mr, and their coercivity is also large. When an external field is applied, the total magnetic flux density is the sum of the remnant magnetization and the external field and is given by 0i n t0000()(1)ext ext rrext r B HMHMHM (2.17) N S H M HciMr MsIdeal Hard Magnet Real Hard Magnet Figure 2-7. Hard magnetic material The magnetic field produced by a hard magne t and the typical M-H loop is shown in Figure 2-7. Ideal hard magnetic materials have a square M-H loop as sh own. This means that the rem n ant magnetization Mr and the saturation magnetization (Ms) are equal. In other words, the magnetization of the material changes very little in the presence of the applied field. An important figure of merit for a hard magnetic mate rial is the squareness ratio, defined as the

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40 ratio of the remnant magnetization to the satu ration magnetization. The squareness ratio of an ideal hard magnetic material is unity. Some of the most commonly used hard magnetic materials are neodymium-iron-boron (Nd-Fe-B), samarium-cobalt (Sm-Co) and alnico (Al-Ni-Co) metal a lloys as well as ferrites. For all applications, a large remnance and squareness ratio of 1 are desired. Typical values of remnance for commonly used hard magnet materials are shown in Table 2-2. Table 2-2. Magnetic properties of hard m a gnet materials [22] Material Remnance (T) Neodymium-Iron-Boron (Nd-Fe-B) 1.2 1.3 Samarium-Cobalt (Sm-Co) 0.87 1 Aluminum-Nickel-Cobalt (Al-Ni-Co) 0.74 1.2 Hard Ferrites (Barium or Strontium Ferrite) ~ 0.1 0.35 2.2.3 Equivalence between Current Distributions and Magnetic Materials Two very distinct origins of magnetic fields ar e discussed in the prev ious sections namely electric currents and intrinsic magnetizations in magnetic materials. However, the magnetization of a material can be represented as a current distribution and vice ve rsa. The relationship between equivalent current dens ities and the magnetization are JM KMn (2.18) Here M is the magnetization, J is the volume current density, K is the surface current density and nis the unit normal to the surface of the materi al. Thus for a hard magnetic material, where the magnetization can be assumed uniform and c onstant, the volume current density is zero and there exists a constant surface current density equa l to the remnant magnetiz ation of the magnet. This is shown schematically in Figure 2-8 [20]. In the case of a sof t magnetic material, there is no magnetization present without an external magnetic field, but (2.18) still can be applied by letting th e m agnetization be a f unction of the spatially-varying externally applied field.

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41 nK Figure 2-8. Hard magnet representati on as an equivalent current density 2.3 Magnetic Field Based Actuators A steady magnetic field is capable of producing a force on moving charges. The force Fon a charge q moving with a velocity v in the presence of external magnetic flux density B is given by the Lorentz force equation [21], FqvB (2.19) It can be seen that the force is in a direct ion perpendicular to bot h the velocity and the magnetic field. Thus the force on a current dist ribution, composed of a large number of moving electrons, can be calculated by integrating the for ce all the electrons that constitute the current distribution. Thus, the force, dF acting on a differential surface element dS with surface current density of Kis given by dFKBdS (2.20) The total force is calculated by integrating the differential force over the entire surface. The force acting on a volume element dV due to a volume current density J is also calculated in a similar fashion.

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42 dFJBdV (2.21) Thus forces due to interactions between current distributions and magnetic fields are known. The forces on magnetic materials due to external magnetic fields can also be calculated by representing the magnetic materials as equiva lent surface and volume current densities as described in section 2.2.3. Magnetic actuators, where a specified force is produced in response to an electrical current or voltage, can be created in many different configurations. Most generally, these different configurations can be classifi ed into two different categor ies based on the magnetic field distributions in the actuators. Fixed-field actuators (FFAs) are actuators where the mechanical motion of the device does not resu lt in substantial change in th e magnetic field distribution. The most common configuration used for FFA is the interaction between a current coil and a magnetic field generated by a hard magnet. Th ese actuators are called voice-coil actuators, moving-coil actuators or electr odynamic actuators in the literature [7, 12]. In contrast, in variable-field actuators (VFAs), the magnetic field distribu tion changes when the components of the device move with respect to each other. These actuators usually include a combination of current carrying coils, soft magnets and/or hard magnets. In the literature, VFAs are typically called variable-reluctance actuators or sometimes just electromagnetic actuators. 2.3.1 Fixed-Field Actuators (Electrodynamic) These actuators involve interaction between ma gnetic fields, such th at the overall field distribution does not change substantially during the device operat ion. In most cases, a hard magnet creates a fixed magnetic field on a current carrying coil. Soft magnetic materials may be included in the magnetic circuit in order to incr ease the magnetic flux density acting on the coil. The force F produced on a current carrying wire as shown in Figure 2-9 is given by

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43 FILB (2.22) where B is the magnetic flux density, L is the vector along the length of the coil and I is the current through the coil. Note that force generated on the current carrying coil is just a 1-D solution of the Lorentz force on a moving charge, e.g. see (2.19). The force is directly proportional to the input current a nd is in the direction perpendicu lar to both the current and the m a gnetic field. The direction of the force is given by right-hand-rule [21]. Note that the direction of the current indicated here is the conventional direction of current and hence the electron velocity (v ) is actually opposite to th at of the current direction. However, as the electron has negative charge the force expressions (2.19) and (2.22) are equivalent. The electrodynam i c transduction scheme will be further discussed in Chapter 3. Figure 2-9. Electrodynamic transduction A typical configuration of an axisymmetric mesoscale voicecoil actuator is shown in Figure 2-10. The magnetic circuit consists of a hard m a gnet and soft magnetic core used to direct the field in the radial direction. A solenoid coil is attached to a movable piston that is mounted to a rigid support via a spring. The hard magnet is magnetized in the z-direction as shown, but the soft magnet directs the field so that it is in the ra dial (r) direction in the ai r gap. The current is in the -direction (in and out of the page). The current entering into the coil is indicated by a cross, and current leaving the coil is s hown by a dot. Thus the force produ ced in the z-direction. This NS F B I 1 2 dL

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44 results in transverse motion of the piston. Th e dynamics of the actuator are determined by the mechanical components (the piston and the spring) of the actuator. Magnet Soft magnetic core Coils Piston z r x x x x x x x x x x Figure 2-10. Typical mesoscale voice-coil actuator 2.3.2 Variable-Field Actuators (Variable-Reluctance) Variable-field actuators generate a mechan ical force using the interaction between a magnetic field produced by an electromagnet and th e induced field in a so ft magnetic material. Reluctance of a magnetic circuit is analogous to resistance in an electrical circuit and depends on the geometry and material properties of the magnetic path. The magnetic interactions in these devices always tend to reduce the reluctance seen by the magnetic field, and hence they are sometimes also called variab le-reluctance actuators. The typical configur ation of these actuators is shown in Figure 2-11 [13], commonly used in m a gnetic relays, magnetic plungers, etc. The flux in the gap is set up by the electromagnet (current-carrying coil). The soft magnetic core is used to direct and concentrate the flux. The equilibrium position of the moving soft iron slab is achieved when the magnetic force balances the mechanical restoring force exerted by the spri ng. The magnetic force is given by [24], [13] magmagFW (2.23) where magWis the magnetic energy,

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45 2 211 22magNI MMF W (2.24) where MMF is the magnetomotive force (the product of the current I through the coil and N is the number of turns on the coil) and is the reluctance of the circ uit. The reluctance of the magnetic circuit is directly proportional to the length of the magnetic circuit and inversely proportional to both the cross-sect ional area and the magnetic perm eability of the magnetic path. The reluctance of a magnetic path of length l and cross-sectional area A is given by l A (2.25) where is the magnetic permeability of the material. Figure 2-11. Typical configuration of mesoscale solenoid actuator It can be seen from (2.23) and (2.24) that force is not linear with respe ct to the input current. Moreover, the force is unidirectional a nd always tries to clos e the gap and reduce the gap. The specific expressions for the force can be calculated based on the geometry of the device. The total magnetic force for the topology shown in Figure 2-11 is given by [13] 2 0 0 02m gNI A F L x x (2.26)

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46 where x0 is the fraction of the gap area affected by the moveable portion, xg is the length of the air gap and Lm is the length of the magnetic path and is the magnetic permeability of the core material. 2.4 Scaling of Magnetic Actuators The previous section describes the differences in operation and geomet ry of the two types of magnetic-field based actuators. These are impo rtant considerations when choosing an actuator for a given application. In additi on to these, the scaling of these two actuators is also significant particularly when compact actuators are desired. Several studies have compared scaling of magnetic actuators with other types of actuation schemes [10], [9], [25]. Cugat et. al. [11] di scuss the scaling of the two types of magnetic actuators. The effect of improved cu rrent density in microcoils due to better heat dissipation is discussed. As the coils are scaled down by a factor k, the volume and the Joule heating reduce as k3, while the surface area and h eat dissipation scale as k2. Thus current density up to 10,000 A/mm2 can be tolerated in microscale coils, whil e mesoscale coils can tolerate only up to 10 A/mm2. The scaling of magnetic actuators, taking into consideration the increased current densities, is shown in Figure 2-12. Figure 2-12. Scaling of various magnetic interactions (used with permission) [5] FFAs VFAs

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47 Here k is the scale factor by whic h the device is reduced and ki is the factor by which the current density is increased at smaller scales. It can be seen that at smaller scales, electrodynamic transduction is advantageous. To illustrate these effects, consider if all the dimensions of a fixed-field actuator are scaled by some factor s. The length scales by s. (Note k, described in Figure 2-12 is the scale down factor i.e. the device di mensions are divided by k, 1s k ). The current through the coil is the product of the volume current density (J) and the cross-section area (A). Thus if the current density is held constant, the total current scales as s2. As the gap and magnetic components both scale by the same factor, the flux density remains constant. Thus the force per unit volume of the device will stay constant as the device is scaled as shown below 23() ,constant ,, constant FBLIBLJA BJ LsAsVs F V (2.27) If the current density is assu med to increase by a factor of ki (ki > 1) as the device is scaled down, the force scaling change s slightly as shown below 23() constant ,,i iFBLIBLJA B Jk LsAsVs F k V (2.28) Thus force per unit volume increases by the same factor as the increase in the current density. The force generated by a variab le-field actuator is given by (2.23). Scaling analysis by assum i ng constant volume current density (J), is shown below

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48 22 4,constant 1magNINJA FW NJ l As Fs F s V (2.29) It is seen that the when variable-field actuators are scaled down (s < 1), the force per unit volume reduces. Thus, the scaling of these actuators with constant current density does not provide any performance improvement. If the cu rrent density is assumed to be increased, an improvement in force per unit volume can be expected as shown 22 4constant 1 *c mag i iNINJA FW RR N Jk l R As Fs F sk V (2.30) Thus, it may be possible to achieve improved force per unit volume in (s*ki >1). In summary, qualitative scaling laws in the literature indicate that fixed-field actuators (interactions between a magnet and a coil) scale the best amongst all the magnetic interactions, and furthermore magnetic actuators may offer advantages over other actuation schemes. Variable-field actuators using the interaction be tween soft magnet and current do not scale well as the devices are made smaller. These scaling analyses only account for the sc aling of the transduction scheme itself. The overall scaling of the actuator including the mechanical comp onents is also necessary to understand the physics of the entire device. In orde r to study the effects of scaling of the entire

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49 actuator, accurate models of various component s are required. Chapter 3 develops the lumped element models for all the components of an electrodynamic ZNMF actuator and some design trends and tradeoffs are discussed in Chapter 5. 2.5 Magnetic Actuators This section presents an overview of magnetic actuators in general fo llowed by a review of several ZNMF actuators reported in the literature so far. Section 2.5.2 discusses several modeling efforts related to ZNMF actuators including low-or der analytical techniques as well as numerical approaches. Magnetic actuators have been around for over a hundred years. The most common applications of magnetic actuators are loudspeak ers, magnetic relays, switches, and solenoids. A typical design for an electrodynamic ZNMF actuato r is essentially a loud speaker with a cavity on the front and a slot (or orifi ce) through which the jet is formed. The purpose of an acoustic loudspeaker is to effectively ra diate sound over the audio-range of frequencies (20 Hz 20 kHz). In contrast, the purpose of a ZNMF actuator is to achieve large volume velocity displacements to form a strong jet through the slot (or orifice). The frequency of interest is also usually much lower, from 10 Hz 500 Hz. Although these are slightly different requirements, both the actuators need to produce large displ acement of the mechanical component. A basic moving-coil (fixed-field) loudspeaker is shown in Figure 2-13. The major com ponents of the loudspeaker are m agnet assembly and voice coil, speaker cone and the spider. The soft magnet and the annular hard magnet ge nerate a radial field in the air gap. The interaction between the current in the voice-coil and the magnetic field produces a force. The coil is attached to the speaker cone by means of th e corrugated membrane and the spider. These two components determine the mechanical compliance of the device.

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50 The magnetic assembly and the voice-coil that form the driving mechanism of the loudspeaker shown here, use an annular hard magnet. However, other magnetic circuit topologies can be used to achieve the radial fiel d in the gap. An altern ative magnetic circuit is shown in Figure 2-10 where a cylindrical hard magnet is used in the cen tra l leg of the magnetic circuit. An annular soft magnet forms the retu rn path. The choice of the magnetic topology is based on various considerations magnet performan ce, weight, ease of manufacturing, cost, etc. Figure 2-13. Cross-section of a typical moving coil speaker 2.5.1 Reported Zero-Net Mass-Flux Actuators ZNMF actuators have been stud ied for various applications su ch has flow control, mixing enhancement and thermal management. Va rious transduction schemespiezoelectric, electrostatic and magnetichave been used to implement ZNMF actuators over a wide size ranges. Most efforts in the fluids community have focused on the modeling of the jets and studying their impacts on various flow situati ons. There are numerous studies that have concentrated on the visualization, and/or measurem ents of ZNMF actuators as well. This section discusses some of the mesoscale as well as micr oscale ZNMF actuators reported in the literature. Coe et al. [26] reported a micromachined electrostatic ZNMF actuator array ( Figure 2-14). The m e mbrane consisted of metalized polyimide released using anisotropic potassium hydroxide

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51 etch and sacrificial nickel layer. The membrane was 4 mm wide while the orifice opening was 50 m. Voltage could be applied to individual membranes to actuate them. Typical velocity of 20 m/s was obtained from these jets. Figure 2-14. Integrated micromachined electrosta tic ZNMF actuator adapter from Coe et al. [26] Smith et al. [2] tested a mesoscale piezoelectric ZNMF actuator resulting in a two dimensional jet through a slot 75 mm long and 0.5 mm wide slot ( Figure 2-15). Typical velocities up to 20 m / s were obtained. The frequency range of operation of this particular ZNMF actuator was 10 Hz 100 Hz. Figure 2-15. Mesoscale piezoelectric ZNMF actuator adapted from Smith et al. [2] There have been very few magnetic-field-based ZNMF actuators reported in the literature so far. McCormick [3] demonstrated a mesos cale fixed-field ZNMF act uator and developed a lumped element model of the devi ce to predict its performance ( Figure 2-16). The model

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52 predicted the cavity pressure as well as the out put velocity accurately. The peak performance of the actuator was observed at about 100 Hz and the maximum frequency of operation was 400 Hz. Figure 2-16. Fixed field (e lectrodynamic) ZNMF actuator adapted from McCormick [3] Gallas et al. [14] reported a mesoscale piezoel ectric ZNMF actuator. The structure of the device is similar to the one shown in Figure 2-15. The major focus of the work was to develop detailed coupled equivalent circuit model fo r the actuator based on lum p ed element modeling technique. The overall diameter of the actuators was less than 50 mm an d maximum velocities of about 60 ms/ was obtained at resonant frequency of 800 Hz. Mossi et al. [27] investigated three differe nt piezoelectric ZNMF actuators a unimorph disc, a bimorph disc and a radial field disk. Th e overall diameter of the actuator in case was about 10 mm. Maximum velocities of 30 m/s we re achieved for bimorph actuators. Typical operating frequency was about 500 Hz. Parviz et al. [28] demonstrated the use of micromachined el ectrostatic ZNMF actuators as a propulsion mechanism fo r micro air vehicles ( Figure 2-17). The overall width of the diaphragm was 1.2 mm and the predicted output velocity was 16 m/s at about 60 kHz.

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53 Liang et al. [29] used ferromagnetic shape memory alloy composite (FSMA) diaphragm and an electromagnet to implement a ZNMF actua tor for active flow control applications ( Figure 2-18). A m a ximum velocity of 190 m/s was achieved at 220 Hz. Figure 2-17. Integrated micromachined electros tatic ZNMF actuator dem onstrated by Parviz et al. [28] (used with permission) Yong et al. [4] used a microscale electrodyna mic ZNMF actuator on printed circuit boards to achieve better cooling ( Figure 2-19). The device is fabricated in the b ac k side of the substrate using a lamination process for the fluidic components. The diaphragm is made out a fluroelastomer. A bulk samarium-cobalt magnet a nd a copper coil form the actuation mechanism. A peak jet velocity of 14 m/s was achieved for input power of 60 mW. Figure 2-18. FSMA based ZNMF act uator reported by Liang et al. [30] (used with permission)

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54 Figure 2-19. On chip electrodynamic ZNMF act uator for cooling reported by Yong et al.[4] (used with permission) Lee et al. [31] reported an ionic-polymer metal composite driven ZNMF pump for a MAV application. The device was 10 mm diameter diaphragm and achieved 0.96 mm center displacement and 8 12 m/s output velocity for 2V input. The resonant frequency of the actuator was about 100 Hz. Figure 2-20. IPMC driven ZNMF actuator adapted from Lee et al. [31] A summary of various ZNMF actuators discussed here is provided in Table 2-3. Several other ZNMF actuators ha ve been reported on the litera ture for various applications. Gordon et al. [32] studied the effects use of an electrodynam i c ZNMF actuator on the surrounding flow. The actuator was a modified co mmercial electrodynamic speaker. Campbell et al. [33] used electrodynamic ZNMF actuators to achieve cooling of electronic components. They report lowering of the temperature of processor ch ip 22 % for a test case. However, no details of the actuator are provided.

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55 Hong et al. [34] used a piezoceramic based ZNMF actuator for laminar flow separation. A similar device was also used by Lee et al. [35] for boundary layer flow control. Several other piezoelectric ZNMF actuators have been reported for full flight applications as well. [36, 37]. Tian et al. [38] used piezoelect ric ZNMF actuators in conjuncti on with pressure transducers to implement closed-loop control for sepa ration control over an air-foil. Table 2-3. Summary of ZNMF actuators reviewed Year Author Transduction Mechanism Notes 1995 Coe et al.[26] Electrostatic Microm achined device, typical velocity ~ 20 m/s 1998 Smith et al. [2] Piezoelectric Mesosc ale device, typical velocity ~ 20 m/s, Bandwidth 10 Hz 100 Hz 2000 McCormick et al. [3] Fixed-field actuator (Electrodynamic) Up to 30 m/s velocity for 5 W input power, Bandwidth 100 Hz 400 Hz 2003 Gallas et al. [14] Piezoelectric Me soscale device (~ 50 mm), typical velocity 60 m/s at 800 Hz 2004 Mossi et al. [27] Piezoelectric Diffe rent types of piezoelectric drivers studied. Maximum velocity ~ 30 m/s 2005 Parviz et al. [28] Electrostatic Micromachined device, typical velocity ~ 16 m/s @ 60 kHz 2005 Liang et al. [29] Variable-field actuator Mesoscale device, typical velocity 190 m/s @ 220 Hz 2005 Yong et al. [4] Fixed-field actuator On chip cooling device, velocity of 14 m/s, Resonant frequency 100 Hz 2006 Lee et al. [31] IPMC driven Micr o-air vehicle application, ~ 12 m/s velocity, Resonant frequency ~ 100 Hz In spite of several efforts on applicati ons and effects of ZNMF actuators on the surrounding flow detailed discussion of design and fabrication of the actuators themselves has been missing. This work presents a systematic approach to design an electrodynamic ZNMF actuators to meet the necessary target specifications of the app lication such as output velocity, bandwidth, maximum power consumption.

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56 2.5.2 Modeling of Zero-Net Mass-Flux Actuators In order to design and optimize ZNMF actuators, accurate models are necessary. Surprisingly, very few analytical models for ZNMF actuators have been reported. McCormick et al. [3] modeled the coupling betw een the electrical and mechanical domain as an ideal gyrator. The diaphragm is modeled as a mass-spring-dam per system. The cavity a nd orifice are modeled using the lumped element modeling approach as will be discussed in Chapter 3. The model was verified by comparing the predicted cavity pressure with experimental results at several different input levels and frequencie s. There was very good agreement between the model and experimental results. Gallas et al. [14] also used a similar appro ach to model piezoelectr ic ZNMF actuator. The coupling between the electrical and the mechanical domain was modeled as an ideal transformer. The piezoelectric disc was modeled as a clamped co mposite circular plate. The cavity and orifice models used were similar to the ones used by Mc Cormick et al. [3]. The model for the composite plate and the acoustic cavity were verified indi vidually. The output veloc ity generated by the jet was measured using hotwire anemometer and co mpared with the model predictions. The model predicted the output velocity ve ry well over the entire range of frequencies of interest. Most other modeling approach es have used numerical techniques to model the flow physics of a ZNMF actuator. Seve ral reduced-order models have been developed to reduce the computational requirements. Kral et al. [15] simulated a single ZNMF actuator for a twodimensional, incompressible case. Instead of simulating the actuator and the cavity, they used a sinusoidal velocity profile as a boundary condition at the entrance of the orifice. The results of the simulations indicated that some of the expe rimentally observed phenomena were not entirely captured by using just the sinusoidal velocity bou ndary condition. Rizetta et al. [16] used a decoupling approach to solve the compressible Na vier-Stokes equations inside and outside the

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57 cavity. For the cavity simulations, oscillating boundary conditions was used on one of the cavity surfaces. The external flow filed was calculated us ing recorded exit velocity profiles from the jet as the boundary. Thus simultaneous simulations of the flow field inside and outside the cavity were not necessary. Several other modeling efforts have also fo cused on detailed mode ling of the cavity and the slot that form important components of the actuator. This work will focus on lumped element modeling based approach similar to that used by McCormick et al. [3]. Detailed models of individual components electrodynamic coupling, diaphragm, cavity and the orifice are developed in Chapter 3.

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58 CHAPTER 3 LUMPED ELEMENT MODELING Chapter 3 discusses the low-order, lumped element modeling of a fixed-field (electrodynamic) zero-net mass-flux (ZNMF) actuato r. This modeling approach is useful for design and optimization of the device. The basi c assumptions involved in the lumped element approach are described in section 3.1, followed by an overview of conjugate power variables. The overall structure of the ZNMF actuator a nd the lumped element model is discussed in section 3.2. The subsequent sections present each component of the lumped element model in detail. Section 3.3 develops the model for th e electrodynamic transduction scheme. The model for the moving coil/diaphragm assembly is presen ted in section 3.4. Finally, the acoustic models for the cavity and the slot (or orifice) are discussed in section 3.5. The transfer function relating the input electrical excitation and output average velocity is derived in section 3.6. 3.1 Lumped Element Modeling Basics The inherent assumption of lumped element mode ling (LEM) is that the length scale of the device (d) is much smaller than wavelength () of the physical phenomenon governing the device [12], [13]. In such a case, th e governing partial diffe rential equations of the system may be simplified to ordinary differential equations. For example, the wave equation governs various physical phenomenon such as acoustic, electroma gnetic (EM), and mechanical membrane/string vibration phenomena. Consid er the wave equation with h as the independent variable, 2 2 221 h h ct (0.31) where c is the speed of propagation of the wa ve. For harmonic motion at frequency the left hand side of (0.31) scales as 2 22 22212 hh h ctc (3.32) If the largest length scale of interest is d the spatial derivative on th e right hand side scales as

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59 2 2h h d (3.33) The spatial variation and the time va riations can be decoupled if 2 2 221 h h ct (3.34) Substituting by the relevant scaling parameters, (3.4) can be represented 2 22 h h d (3.35) Thus, when the wavelength of the governing physical phenomenon is mu ch greater than the length scale of interest ( >> d ), the spatial variations can be analyzed independent of the time harmonic temporal variations. The distributed spatial quantities of interest can then be lumped at a reference location, and the time-depe ndence of these quantities can be studied independently. In the case of an electrodynami c ZNMF actuator, the maximum frequency of interest is about 500 Hz, and the acoustic waveleng th at this frequency is about 0.7 m. Thus, if the size of devices considered is much smaller th an 0.1 m, the lumped assumption is valid. Note that the electrical wavelength is much larger th an the acoustic wavelengt h and hence the acoustic wavelength is the more limiting factor. The lump ed assumption is also valid for the diffusion equation and the bending wave equation. 3.1.1 Passive Lumped Elements Lumped element modeling is used to capture the energy behavior of various components of any system and represent it as electrical circuit. The overall behavior of the system can then be predicted by using standard circ uit analysis techniques. Within such a circuit model, a passive lumped element is used to represent each individual energy storage/dissipation mechanism in the device. Every lumped element has an effort ( e ) and a flow ( f ) variable associated with it ( Figure 3-1) [13]. The product of the effort and fl ow variables in each of the dom ains is power, w ith units of Watts. The effort and flow variables are hence ca lled conjugate pow er variables.

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60 Note the sign convention used is such that the terminal through which the flow variable f enters is defined as the positive terminal and vice vers a. Thus by convention, the power absorbed by the lumped element (power going into the element) is defined to be positive. The effort and flow variables in various energy doma ins of interest are shown in Table 3-1. Figure 3-1. Passive lumped element Table 3-1. Effort and flow va riables in various energy domains Domain Effort E Flow f Electrical Voltage ( V ) Current ( I ) Mechanical Force (F ) Velocity ( U ) Acoustic/Fluidic Pressure ( P ) Volume velocity ( Q ) To build a lumped element model for a dynami c system, the energy storage and dissipation mechanisms of the system must be identified an d modeled using three different passive lumped elements. A generalized resistance (resistor) represents dissipation of energy in the form of heat. Kinetic energy storage is repr esented as a generalized iner tance (inductor), while potential energy storage is represented as a generalized compliance (capacitor). The impedance Z of a passive element is defined as e Z f (3.36) Thus electrical impedance is the ratio of voltage a nd current with units of V / A or Mechanical impedance is defined as the ratio of force to velocity and has units of (N-s)/m Acoustic

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61 impedance, the ratio of pressure to volume velocity, has the units of N-s/m5. Table 3-2 summarizes the lump ed parameters in different energy domains. Table 3-2. Passive lumped element components in various energy domains Domain Generalized resistance Generalized inertance Generalized compliance Energy behavior Dissipation Kinetic energy Storage Potential energy storage Electrical Resistance ( ) Inductance (H) Capacitance (F) Mechanical Damping Factor (Ns/m) Mass (kg) Compliance (m/N) Acoustic Acoustic resistance (N-s/m5) Acoustic mass (kg/m4) Acoustic compliance (m5/N) 3.1.2 Two-Port Elements In addition to the one-port lumped elements, two-port elements may be used to represent the energy coupling between different energy domain s or to represent impedance transformations within a single domain. The two most common tw o-port circuit elements used for transducer modeling are transformers and gyrators. Ideal transformers and gyrators do not store or dissipate energy; they just convert energy from one energy domain to another. A transformer is used when the effort variables in the two energy domains are proportional to each other, while a gyrator is used when the effort variable in one domain is proportional to the flow variable in the other. Figure 3-2 shows the two circuit elem ents with standard con ven tions. The transformer equations are given as 21 210 1 0 T EE FF T (3.37) Similarly gyrator equations are given as 21 210 1 0 G EE FF G (3.38)

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62 where T and G are transduction coefficients that repr esent the coupling between the two energy domains. It can be seen that the power in both the domains is equal, that is1122EFEF Figure 3-2. Ideal transformer and gyrator two-port circuit elements For circuit analysis, it is often convenient to simplify a circuit by eliminating the transformer or gyrator; this is accomplished by reflecting circuit el ements from one side of the transformer or gyrator to the other, as describe d below. Consider the transformer with impedance Z2 connected to the second energy domain as shown in Figure 3-3. The effort variable E2 in the second domain is given by 2122ETEZF (3.39) The relationship between the effort and flow va riable in the first domain is thus given by 2 11 2Z EF T (3.40) Thus 2 2Z T is the equivalent or referred impedance in the first domain as shown in the figure. The impedance transformation for a transformer is reciprocal. In other words, if an impedance Z1 is connected in the first domain the equi valent impedance in the second domain is 1 2Z T. Moreover, an effort source V2 connected in the second energy domain, can be replaced by an equivalent effort source 2V in the first energy domain given by

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63 2 2V V T (3.41) Note that the source transformation is also reciprocal. Thus an effort source V1 in the first energy domain, is equivalent to an effort source 21VTV in the second energy domain. A flow source transformation can be performed in a similar manner. Figure 3-3. Impedance and source transformation using a transformer Impedance transformation from one energy domain to the other when a gyrator is used is now discussed. Consider impedance Z2 connected in the second energy as shown in Figure 3-4. The effort variable E2 in the second energy domain is given by 2221EFZGF (3.42) The effort variable E1 in the first domain is given by 2 1 121 22GF G EGFGF ZZ. (3.43) Thus it can be seen that the equivale nt impedance in the first domain is 2 2 2G Z Z Note that the original impedance is in the denominator; thus if a capacitive load is connected in the second energy domain, the equivalent impedance seen by the first energy domain is inductive, and vice versa. As gyrator represents relationship between an effort variable in one domain to a flow variable in the other domain, when an effort s ource is connected on one si de establishes a flow

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64 on the other side or vice vers a. Thus, an effort source V2 connected in the second energy domain is equivalent to a flow source given by 2 1V I G as shown in Figure 3-4. Note that, as in the case of a transformer, the im pedance and source transf ormations due to a gyrator are also reciprocal. Figure 3-4. Impedance and source transformations using a gyrator 3.2 Lumped Element Model of Elec trodynamic ZNMF Actuator A schematic drawing of an electrody namic ZNMF actuator is shown in Figure 3-5. A circul ar diaphragm of radius a is attached to a current carryin g coil. The circular diaphragm forms the bottom portion of a cavity. The top of the cavity is bound by a rigid plate containing a slot or a circular orifice (an axisymmetric orifice is shown in the schematic). Figure 3-5. Schematic of el ectrodynamic ZNMF actuator.

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65 The magnetic assembly generates a uniform radial magnetic flux density B in the annular air gap. A solenoida l coil of total length Lcoil resides in the magnetic gap and is rigidly attached to the diaphragm. The diaphragm is driven into motion due to the Lorentz force acting on the coil when a current is passed through it. The oscillat ory motion of the diaphragm causes the fluid in the cavity to be expelled or injected, forming th e synthetic jet. Thus it can be seen that, large displacement of the diaphragm will result in larg e volume of fluid being moved and large output velocity (ignoring compressibility effects in the cavity). The electrodynamic ZNMF actuator has two different energy do mains electrical (magnet, coil) and acoustic (diaphragm, cavity and slot/orifice). Note that the magnetic energy domain is not represented as a separate domain explicitly within the system-level LEM. The magnetic domain is modeled separately to predict the magnetic fields necessary for the electrodynamic coupling between the electrical and acoustic domain. Figure 3-6 shows the complete lumped elemen t m odel of the ZNMF actuator, which will be developed in detail throughout Chapter 3, but introduced briefly here first. In the circuit model nomenclature, the first subscript denotes the domain ( a for acoustic and e for electric), and the second subscript describes the component (C for coil, D for diaphragm, Cav for cavity, and OS for the orifice or slot). Beginning on the left side of the circuit, the electrical impedance of the coil is represented as a resistor ReC and inductor LeC in series ( eCeCeC Z RjL ). A gyrator is used to represent the electrodynamic coupling between the electrical and acoustic domains. Everything to the right of the gyrator is in the acoustic domain, where the effort variable is pressure, and the fl ow variable is volume ve locity. The potential and kinetic energy storage in the diaphragm is represented as an acoustic compliance CaD, and an acoustic mass MaD respectively. All the en ergy losses in the diaphrag m support losses, viscous

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66 losses due to the diaphragm motion, structural losses etc. are represented as an acoustic damping resistance RaD The compression or expansion or fluid in the cavity resu lts in storage of potential energy and this is denoted as an acoustic compliance CaCav. Note that the back cavity formed by the housing, annular gap in the magnetic assembly also adds an additional compliance. However, for all the devices considered in this work, the back cavity volume is much larger and hence the corresponding impedance is much smaller than the impedance of the front cavity. This assumption will be validated in Chapter 4 where the characterization of these actuators is discussed. The losses in the slot (or the orifice) are modeled as a combination of linear (RaOS and RaRad) and nonlinear (RaO) resistors. RaOS and RaO represent the viscous losses along the walls and the corners of the slot (or orifice) and RaRad represents the losses due to radiation of sound from the slot (or orifice) into the semi-infinite medium. The kinetic energy storage in the slot (or orifice) due the physical motion of the fluid is represented by acoustic masses MaOS and MaRad. G=BLcoil/S P MaD Q RaOS+RaRadMaOSMaRadRaOQCavQout CaCavVac Electrical domain Acoustic/fluidic domain RaD LeC ReC V I CaD Figure 3-6. Lumped element model of the electrodynamic ZNMF actuator The interconnections (series/parallel) of th e lumped elements are determined by their energy sharing behavior. Elements that share co mmon flow through their terminals (current or volume velocity) are connected in series, whereas elements that share common effort across their terminals (voltage or pressure) are connected in parallel. In operation, the equivalent acoustic

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67 pressure acts on the diaphragm and causes its motion. The motion of the diaphragm results in compression (or expansion) of the fluid in the ca vity and expulsion (or suction) of fluid through the slot or the orifice. Thus the cavity and slot share the same effort variable (pressure) and are connected in parallel as shown. Once the equivalent circuit model is es tablished many relevant system performance parameters can be determined. The following sections will present the models for each individual components of the lumped circuit model. 3.3 Electrodynamic Transduction Electrodynamic or fixed-field transduction us ually involves the interaction between a current carrying coil and the field from a hard magnet as discussed in section 2.3.1. One primary assumption is that the overall magnetic fiel d distribution in the device does not change significantly during the device opera tion. Specifically, this implies (i) the magnetic fields arising from current flow in the coil are small compared to the static magnetic field from the magnetic assembly and (ii) the motion of the coil/diaphragm assembly does not alter the magnetic field. Note, this does not imply the magnetic field is uniform, but this assumption is often required to ensure a constant transduction factor and thus li near operation. In the most general case, the differential Lorentz force dF on a small volume element dV of a conductor is given by dFJB (3.44) where J is the current density in A/m2 and B is the magnetic flux density. The total force is calculated by integrating dF over the volume of the conductor. The model for electrodynamic tr ansduction scheme developed here is an idealized case where it is assumed that the current density J and the magnetic flux density B are constant and that the coil is perpendicular to the direction of the magnetic field. The cross-secti onal area of the

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68 coil is Acoil and overall length of the coil is Lcoil. Thus Lorentz force F acting on the coil, subjected to constant magnetic flux density B is [21] coilcoilcoil VFJBdVJBALBLI (3.45) The force produced is in the di rection normal to both the current and the field according to the right-hand rule as shown in Figure 3-7. The electromechani cal transduction coefficient BLcoil is a measure of the coupling between the two energy domains. For an actuator, it should be as large as possible to achieve maximum force for a given input current. B F I Figure 3-7. Lorentz force on a curren t carrying coil in a magnetic field The electrodynamic coupling is anti-reciprocal. In other words, a conductor moving in the presence of a magnetic field will have a voltage induced across it [21]. But the direction of voltage induced is negative as per the standard sign conventions This can be deduced from the Faradays law given by B A V tt (3.46) where V is the induced voltage and is the magnetic flux given by the product of the magnetic flux density and the area of coil l oop (note not the crosssection area of the co il). Consider a coil

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69 loop of length Lcoil, moving with velocity U in a homogenous magnetic flux density B as shown in Figure 3-8. The free electrons in the co il as given by the L o rentz force equation (FqUB ), where q is the charge of the electron. This magnetic force can be represented as that exerted by an electric field EUB (3.47) This electric field is called motional electric field [39] and the direction of this field is perpendicular to both the veloc ity and the magnetic flux density. This electric field induces a voltage called the motional emf across the two te rminals of the wire given by the line integral of the electric field as shown below 1 2VUBdl (3.48) where 1 and 2 indicate the two terminals of th e wire. For the case shown in the figure, the velocity and the magnetic flux density are perpendi cular to each other, thus the induced voltage is coilVBLU (3.49) x z y B U 2 1 V Figure 3-8. Voltage induced on a moving coil in a magnetic field

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70 This combined interaction between the electrical and mechanical domains is represented in the matrix from as shown below (section 2.1) 0 0coil coilBL FU BL VI (3.50) Note that the induced voltage is indicated as positive here. The implications of choosing this sign convention is discussed shortly. In the overall lumped element mode l for the ZNMF actuator shown in Figure 3-6, the diaphragm cavity and the slot (or orifice) are modeled in the acoustic domain. The Lorentz force acting on the coil can be converted to an equivalent pressure given by P = F/S, where S is the effective area of the diaphragm to which the coil is attached. Note that the effective area is used instead of the overall area of the diaphragm because the diaphragm displacement is not necessarily constant over the enti re diaphragm. If the diaphragm displacement is constant over the entire diaphragm, then S is the overall area of the diaphr agm. The combined interaction between the electrical and ac oustic domains is shown below 0 0coil coilBL PQ S VBLI S (3.51) As discussed in section 2.1, it can be seen that the diagonal elements of the matrix are zero, and the electrodynamic transduction is a direct transduction scheme. However, it should be noted that the coil is always attached to a mech anical component (e.g. a diaphragm in the case of ZNMF actuator). Thus free displacement of the act uator as defined by (2.4) in Chapter 2 is not infinite. The free displacement is bounded by the im pedance of the mechanical structure, which is completely independent of the electrodynamic transduction.

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71 The two-port model for electr odynamic transduction using the gyrator representation is shown in Figure 3-9. The force produced in the mech anical dom ain (or the pressu re in the acoustic domain) is directly proportional to the input current in the elec trical domain, and the voltage produced in the electri cal domain is directly proporti onal to the velocity in the mechanical domain (or the volume velocity in the acoustic domain). Thus electrodynamic transduction is linear (under the assumption that the transduction coefficient is constant with respect to displacement). Note that the gyrator re presentation is chosen throughout this thesis in order to use the impedance analogy [7]. A transfor mer representation can also be used. However, this will require the use of admittanc e analogy in one of the energy domains. I V U coil B L F Electrical domain Mechanical domain I V coil B L SPElectrical domain Acoustic domainQ Figure 3-9. Two-port gyrator mode l of electrodynamic transduction As discussed before, electrodynamic trans duction scheme is anti-reciprocal or antisymmetric by definition, where the transduction coefficients are equal in magnitude but opposite in sign (3.48). This is a consequence of the sign conventions used in defining the various quantities for electrodynam i c transduction [7]. From equations (3.44) and (3.46) the two transduction coefficients can be defined as coil coil me em B LB L GG SS (3.52) However, by choosing appropriate sign convention, electrodynamic transduction can be represented as a reciprocal transduction scheme.

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72 It can be seen that the assumptions of the constant current density and constant magnetic flux density are crucial to the electrodynamic transduction model de veloped here. For most cases (particularly for low frequencies) the assumption of constant current density through the coil is reasonable. However, the magnetic flux density B may vary along the length of the coil, and for actual wire conductors with non-zero diameters, the flux density may vary over the wire crosssection area. Moreover, as the coil and the magn et move with respect to each other during the device operation, the flux dens ity changes with position. Hence, for more complex configurations, the magnetic field generated can be analyzed using fi nite element methods (FEM) to calculate the effective electromechanic al transduction coefficient. The calculation of transduction coefficient using the 1-D circuit mode l and finite element simulations is discussed in the following sections. 3.3.1 Calculation of the Transduction Coeffi cient using 1-D Magnetic Circuit Model The magnetic assembly of the electrodynamic ZN MF actuator consists of a hard magnet, a soft magnetic core and the air gap in which the coil resides. Figure 3-10 shows the schematic of the m agnetic assem bly and the equivalent 1-D circuit model. The 1-D magnetic circuit is used to predict the magnetic flux density in the air gap, which, combined with the length (and position) of the coil determines the transduction coefficient. magnet core g ap Figure 3-10. 1-D magne tic circuit approach

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73 The 1-D magnetic circuit model is based on th e integral form magnetostatic Maxwells equations [24]. The key assumption involved in this approach is that the ma gnetic flux is entirely confined to the magnetic path. The entire magnetic path is represented as a magnetic circuit analogous to an electrical circuit as shown in Figure 3-10. The hard magnet acts as the source of the m agnetic field. The soft m agnetic core serves to guide the flux across the air-gap. The soft magnetic core and the air-gaps are repr esented in the magnetic model by their magnetic reluctance, which is the resistance offered by the magnetic path to the flow of magnetic flux. Magnetic reluctance is a function of the geometry and the material and is given by 0 rl A (4.53) where l is the length of the magnetic path, A is the cross-sectional area through which magnetic flux flows, 0 is the magnetic permeability of free space and is given by 1.25 x 10-6 H/m and r is the relative permeability of the material. For example, the reluctance of the magnet for the magnetic assembly in Figure 3-10 will be 2 0 0 mm magnet magnetmagnet magnetmhh A r (3.54) where hm is the height of the magnet ,rm is the radius of the magnet and for neodymium-ironboron magnet = 1.05 [23]. Similar expressions can be obt ained for the reluctance of the air gap. However, the magnetic flux in the air gap the flux is not restricted just to the cross-sectional area of the gap ( Figure 3-11). This phenomenon is called f ringing. The flux sp reads out across a larger cro ss-section, and correc tions are used to calculate th e effective area and hence the reluctance of the air gap. The effective area for the annular air ga p is given by [39] 2effgg ggAhlrl. (3.55) The reluctance of the air gap is thus given by

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74 0 g gap effl A (3.56) Core Core Fringing field Flux lines Figure 3-11. Fringing fi elds in the air gap Note that the fringing fi eld correction shown in (3.55) is only valid for cases where the area of the cross-section of the core on both sides of the air gap is the sam e and hence the fringing fields are symmetrical. In the case of the magnetic topology illustrated in Figure 3-10, the air gap is bounded by different core areas on the two sides, and hence the fringi ng fields will be asym m etric. The error introduced due this asymmetry is small as shall be shown using finite element analysis. The reluctance of the core is composed of three different components the bottom, the top and the annular portion. Th e reluctances of these componen ts, along with the assumptions involved are detailed in Appendi x A. The reluctance expressions for the three components are given below 22g top coremh r (3.57) 22 cg annular coremgcmgwh rlwrl (3.58) and

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75 2 2t bottom c coremg gr w rlh (3.59) where core is the permeability of the soft magnetic ma terial used for the core. The overall core reluctance is the sum of these three components (coretopannularbottom ). Note that the relative permeability of the soft magnetic core is usually extremely high ( core ~ 100000) and hence the reluctance of the core is negligible as compared to that of the air gap and the magnet. This is discussed in detail in Appendix A. The magnetomotive force (MMF) generated by the hard magnet is given bycm M MFHh Here Hc is the coercivity of the hard magnet and hm is the height of the magnet. The coercivity Hc and the remnant magnetic flux density Br of an ideal magnet (square M-H magnetic hysteresis loop) are related by 0 r c magnetB H (3.60) Thus the total flux (in Weber) flowing throug h the gap is given by cm cm magnetcoregapmagnetgapHh Hh (3.61) The magnetic flux density in the air gap is thus calculated by dividing the flux with the crosssection area of the gap. The magnetic flux density in the air gap B is thus given by 2 0 02cm eff mgggg g magnetmHh B A hhlrl l r (3.62) The transduction coefficient for the axisymmetric magnetic assembly and coil configuration shown in Figure 3-10 is thus given by

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76 2 0 02coil cm coil mgggg g magnetm B LH hL G SS hhlrl l r (3.63) Note that the 1-D magnetic circuit model is used only to predict the magnetic flux density in the air gap and is not a part of the lumped element modeling paradigm where the power interactions between various components are represented in terms of conjugate power variables. 3.3.2 Calculation of the Transduction Coe fficient using Finite Element Analysis The 1-D magnetic circuit model may not be a pplicable for all magnetic assemblies. For example, if the length of the gap is much larger than the height of the ma gnet, the effect of the fringing fields cannot be captured by the simple corrections given by (3.55). In these cases, finite elem ent analysis is u sed to determine the field pattern produced by the magnetic assembly. The overall transduction coefficient is calculated by computing the average radial flux density over the coil volume. COMSOL Multiphysics finite element software is used in this work. Figure 3-12 shows the axisym m etric model used. Both the hard and soft magnets are assumed to be ideal (constant magnetization for the hard magnet). The hard magnetic material is modeled using the constitutive relationship given by 0magnetr B HB (3.64) where B and H are the magnetic flux density (in T Tesla) and magnetic field intensity (in A / m ). The soft magnetic material is modeled using the constitutive relationship given by 0core B H (3.65) The various material properties used are shown below [23], [40] Hard magnet remnant magnetic flux density ( Br) = 1.2 T Hard magnet relative permeability ( magnet) = 1.05 Core relative permeability ( core) = 100,000 (Hydrogen annealed). However, here coldworked EFI-50 was used and the relative permeability of 100 was used.

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77 Figure 3-13 shows the results from the finite elem ent sim ulations. The plot shows the color surface plot of the radial magnetic flux density in the air gap (in Tesla). The streamlines of magnetic flux density are also shown in the same plot. It can be seen that the magnetic flux density in the air gap is primarily directed in the radial direction as expected (all the streamlines are nearly horizontal). The fringing fiel ds can also be seen in the figure. Figure 3-12. Axissymmetric COMSOL model Table 3-3. Dimensions of th e magnetic assembly for 1-D circuit model and finite element modeling comparisons Parameter Value (mm) Magnet radius ( rm) 4.7 Magnet height ( hm) 6.35 Height of the air gap ( hg) 3.2 Width of the annular position ( wc) 3.2 Height of the bottom ( hb) 3.2 Length of the gap ( lg) Variable

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78 Figure 3-14 shows the comparison between th e m a gnetic flux density calculated using finite element analysis and the simp le 1-D model predictions as given by (3.62). The magnetic assem b ly geometry is detailed in Table 3-3. The magnetic circuit ap proach slightly over-predicts the f l ux density in the air gap, particularly fo r smaller air gaps. Thus the corrections for the fringing fields are adequate, but they do not account for the fringing completely. The general trend, however, is captured reasonably well (with in 10%) by the 1-D analytical model for various gap lengths. Hence the 1-D circuit model will be used in this work to calculate the magnetic flux density in the air gap, as it much simplifies th e design and optimization of the final structure. Figure 3-13. Results from the COMSOL mode l radial magnetic flux density (Tesla) and streamlines of total magnetic flux density Bottom Gap Magnet Top Annular ring Fringing fields

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79 0 1 2 3 4 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Length of the gap (mm)Magnetic flux density (T) 1-D circuit model FEM Figure 3-14. Comparison of the magnetic flux de nsity in the air gap between the 1-D circuit model predictions and FEM results (a veraged over entire gap volume) 3.4 Diaphragm Model In order to develop the complete model of the electrodynamic ZNMF actuator, models for the cavity and the slot or orifice need to be developed. It is most convenient to model these components in the acoustic or the fluidic domain. Hence the overall lumped element model uses acoustic domain models for the diaphragm as well. The next section hence presents the coupling between the mechanical domain and acoustic domain. The following section presents the acoustic model of the diaphragm. 3.4.1 Coupling between Mechanical and Acoustic Domains The conjugate power variables in th e acoustic domain are the pressure P and volume velocity Q The conjugate power variables in the mechanical domain are force F and velocity u. If the effective area of the diaphragm is S, the relationship between these quantities can be expressed as Typical gaps

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80 1 0 0 p F S Qu S (3.66) The pressure is the force divided by the area, whereas the volume velocity is the velocity multiplied by the area. These equations can be re presented using an equivalent transformer with the turns ratio 1 S as shown in Figure 3-15. Figure 3-15. Equivalent circu it representation of coupling betw een the mechanical and acoustic domains As discussed before, impedance is defined as th e ratio of the effort variable and the flow variable. The acoustic impedance is thus defined as the ratio of the pressure p and the volume velocity Q, ac p Z Q (3.67) Basic transformer relationships then can be used to relate the acoustic impedance Zac and mechanical impedance Zmech as shown below 22 mech acF Z pF S Z QSuSuS (3.68) The following sections develop the acoustic model for the diaphragm.

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81 3.4.2 Acoustic Model of the Diaphragm Various diaphragm structures can be used for the electrodynamic ZNMF actuators. The most common configurations are a uniform isot ropic diaphragm or a composite transversely isotropic diaphragm. Some co mmercial devices, such as co mmon loudspeakers, use rigid diaphragms (implemented via a paper cone) with co rrugated edges. However, it is much easier to fabricate and analyze the composite diaphragms, and hence they are used for this work. The schematic of the compos ite diaphragm is shown in Figure 3-16. The diaphragm consists of a relatively rigid central boss of radius b, Youngs modulus E1, Poissons ratio 1 and density 1. The outer annular compliant region extents from r = a to r = b. The solidity ratio of the composite diaphragm s is defined as b s a The homogenous diaphragm is thus a special case of the composite diaphragm when b = s = 0. In this work, the central boss is assumed to be perfectly rigid. Thus the diaphragm is modeled as an annular plate with a clamped outer edge [41]. Figure 3-16. Composite diaphrag m model as an annular plate The transverse displace ment of the diaphragm w(r) under a uniform pressure load p can be obtained by applying appropriate boundary and matching conditions. The central boss region

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82 moves as a piston with displacement w(0). The transverse displacement of the annular compliant region (w(r)) is a function of the radial distance. These are given by 24 24 2 3 231 (0)() 14ln 16 pa bbb wwb Ehaaa (3.69) and 2 2224 22 2 34 231 () 14ln222 16brap brrbbrr wr Ehaaaaaa .(3.70) The assumptions and limitations of this model are described in Appendix B. The acoustic compliance accounts for the change in volume ( ) for a unit of applied pressure, when the electrical voltage to the coil is zero (V = 0). The change in volume is calculated by integrating the displacement of th e diaphragm over the entire diaphragm. Thus the lumped acoustic compliance CaD is given by 120 0 0()2()2 ()2ba a b aD Vwrrdrwrrdr wrrdr C PPP (3.71) As the central region is assumed to be perfectly rigid, the compliance can be simplified to 26 246 2 3 21 133 16aDa bbb C Ehaaa (3.72) and the acoustic compliance is only a function of the solidity ratio and the material properties of the annular region. The detailed derivation of the acoustic compliance is included in Appendix B If there is no rigid central boss, where s = b = 0, the diaphragm is just a homogenous, compliant plate, and the compliance reduces to 62 2 3 0 21 16aD sa C Eh (3.73) The effective area of the diaphragm is also determined by calculating the total change in volume and dividing it by the center displacement w(0). Thus the effective area is given by

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83 (0)S w (3.74) If the central boss is assumed perfectly rigid, the effective area can be expressed as 246 2 624246 24 4224133 3 33 312log3 14ln abbb aaa abaabb S b bbb aabb a aaa (3.75) In the limit when the solidity ratio is 0 (homoge nous compliant diaphragm), the effective area is 23 a. The effective mass of the diaphragm is calc ulated by computing the total kinetic energy of the diaphragm and lumping it to the center velocity. Harmonic motion of the diaphragm is assumed. Since the coil is attached to the cen tral boss of the diaphragm assembly, it also contributes to the total mass of the diaphragm. The acoustic mass of the diaphragm is given by [42] 2 2 0() 2a coil aDM wr Mhrdr S (3.76) where Mcoil is the mass of the coil and S is the effective area of the diaphragm. Since the inner central boss is assumed perfect ly rigid, the lumped acoustic mass can be simplified to 2 22 2 2 12 2 2 2(0) 2 2a coil aD b a rigid coil bwr M w Mbhhrdr S M wr M hrdr S S (3.77) where 1 and 2 are the densities of regions 1 and 2 as indicated in Figure 3-16 and Mrigid is the physical mass of the central boss. The detailed derivation is included in Appendix B. In the limit when the solidity ratio is 0 ( homogenous compliant diaphragm), the lumped acoustic mass approaches the acoustic mass of a uniform, isotropic diaphragm given by

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84 2 2 2 0 29 9 5coil aD sM h M a a (4.78) where the second term represents the c ontribution from the mass of the coil. 3.5 Models of the Acoustic/Fluidic Components The acoustic impedance was defined in the previo us section. However, in some cases it is convenient to define another ac oustic quantity called the specific acoustic impedance as the ratio of the pressure p and the particle velocity u, s pac acpp Z SZ Q u S (3.79) The specific acoustic impedance is useful in deve loping the models for the cavity and the orifice. 3.5.1 Cavity Model A short closed cavity of volume V and length LC driven by a piston is shown in Figure 3-17 [43]. The area of cross-section of the piston is S, and thus the volume isCVSL The input specific acoustic impedance is given by [43] 00 00 002 tantanin CCcc Z Lf L jj cc (3.80) Here0 is the density and 0c is the speed of sound for the fluid in the cavity. The driving frequency is f. p Figure 3-17. Short closed cavity and its lumped model

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85 The cavity is considered short when the length LC is small compared to the wavelength of the drive signal, 02 1CCLL c (3.81) The specific acoustic impedance can be simplified using the Maclaurin expansion for the tangent function. The simplified specific acoustic impedance is obtained by ignoring the higher order terms and is given by 2 000000 3 0 001 tan ... 3spCav C C CCccc Z j L L LL j j c cc (3.82) The higher order terms can be ignored for 00.3CLc The acoustic impedance is thus given by 22 0000 aCav Ccc Z j LSjV (3.83) This behavior is analogous to a capacitor in the electrical doma in. Thus, comparing to capacitive impedance 1CZ j C the acoustic compliance is defined as 2 00 aCavV C c (3.84) 3.5.2 Orifice Model This section discusses the acous tic model of a circular pipe. The circular pipe has radius aOS and length LOS as shown in Figure 3-18. This narrow orifice can be modeled as an open cavity. Thus the spec ific acousti c impedance is given by [43] 00 00 002 tan tanOS OS spOSLf L Zjcjc cc (3.85) The low-order lumped element model for a 2D-slot is developed by assuming a channel flow in the slot.

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86 OSa OS L Figure 3-18. Narrow circular orifice and its lumped model Again assuming a short t ube per the condition of (3.81), the specific acoustic impedance can be sim p lified to 00 0 0tanOS s pOS OSL Z jc jL c (3.86) Thus the corresponding acoustic impedance is given by 0OS aOS j L Z S (3.87) This behavior is same as that of a mass in the mechanical domain or an i nductor in the electrical domain. Thus, comparing to inductive impedanceM Z jM the acoustic mass of the orifice is defined as 0OS aOSL M S (3.88) Note that the length LOS can include end corrections for one or both sides of the tube, depending on whether the end is open or flanged [43]. Al so this model assumes uniform flow through the pipe. If viscous effects are considered and fully developed Poiseuille flow is assumed through the pipe, the acoustic mass is slight ly different than that given by (3.88). The fully developed Poiseuille flow assum e s that the velocity profile through the slo t/orifice is the same irrespective of the where the velocity profile is measured.

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87 A complete orifice model also requires the incl usion of the viscous di ssipation in the pipe. The acoustic resistance is given by [43] 48aOSOS OSL R a (3.89) where is the dynamic viscosity of the fluid medium. In a ZNMF actuator, it is necessary to m odel the non-linear dump loss in addition to the linear orifice/slot resistance. The non-linear resistance term RaO, accounts for the losses around the corners of the neck. This lo ss is derived by modeling the orif ice as a Bernoulli flow meter and is given by [44],[3] 0 241 2 D out aO OSKQ R a (3.90) where, KD is the nondimensional loss coefficient (usually assumed to be unity), 0 is the density of the fluid and Qout is the output volume velocity. The loss coefficient, KD is usually a function of the orifice geometry, frequency and the Reynolds number of the flow. Although a circular orifice is considered here, similar approach is used to derive the model parameters of a rectangular slot. 3.5.3 Acoustic Radiation Impedance In addition to all the fluidic effects that are associated with the orifice discussed in the previous section, some acoustic effects are also important. The orifice (or the slot) is modeled as a piston in an infinite baffle to determine its sound radiation behavior. The radiation impedance associated with it is calculated by computing th e average pressure amplitude on the face of the piston in reaction to a prescrib ed piston velocity. The radia tion impedance is given by[43] 11 002222 1 22aRad aRadOS OS P OS OS RXJkaKka Zc j ka ka (3.91)

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88 Here 02 f k c is the wavenumber at the drive frequency and J1 and K1 are the Bessel and Struve functions of order one respectively. RaRad and XaRad are the real and imaginary parts of the piston impedance. These can be simplified at low frequencies (21OSka ), to 2462 2221212312342OS OS OS OS aRadkakaka ka R (3.92) 35 22222 28 4 3353573OS OS OS OS aRadkaka ka ka X (3.93) The reactive behavior ( XaRad) is analogous to an inductor in th e electrical domain (kinetic energy storage). Thus, comparing toM Z jM the acoustic radiation mass is defined as 2 08 3aRad OSM ca. (3.94) Thus for low frequencies, a piston is repres ented as a frequency dependent resistance (RaRad) and an acoustic radiation mass (MaRad) as shown in Figure 3-19. Figure 3-19. Lumped model of a piston in an infinite baffle All the lumped parameters for the elec trodynamic ZNMF actuator are summarized in Table 3-4 below. The sim p lified expressions for each para meter are included here as a quick reference. Note that the basic scaling information of the individual parameters can be easily deduced from these equations. For example, if all the dimensi ons of the diaphragm are scaled by a factor of k, the compliance scales as 3k.

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89 Table 3-4. Lumped parameters for electrodynami c ZNMF actuator Parameter Symbol Value Reference Lumped acoustic mass of the diaphragm MaD 22 2 22a rigid coil bM wr M hrdr S S (3.77) (Appendix B) Lumped mechanical compliance of the diaphragm CaD 26 246 2 3 21 133 16a bbb Ehaaa (3.72) Lum p ed acoustic compliance of the cavity CaCav 2 00V c (3.84) Orifice Slot Lumped acoustic mass of the circular orifice or 2D slot MaOS 0 2OS OSL a 06 5OS OSOSL wd (3.88) Orifice Slot Lumped acoustic resistance of a circular orifice or 2D slot RaOS 48OS OS L a 312OS OSOSL wd*[45] (3.89) Non-linear resistance of the orifice/slot RaO 0 21 2out D OSQ K S* (3.90) Orifice Slot Lumped acoustic radiation resistance of the circular orifice RaRad 2 002OScka 22 2 001 1 23 6OSkd ck [46] (3.92) Orifice Slot Lumped acoustic radiation mass of the circular orifice MaRad 0 28 3OSa 0 211 1 ln(2) 1 6OS OSOS OSOSw dw k wd (3.94) wOS and dOS are the width and the depth of the 2D slot and SOS is the area of the sl ot or the orifice. 3.6 Transfer Function for Electrodynamic ZNMF Actuator The overall lumped element model for the electrodynamic ZNMF actuator was developed in section 3.2 and all the indivi dual components were detailed in following sections. This section uses the overall lumped model to obtain a relatio nship between the input el ectrical excitation and the output jet velocity. For a given electrica l excitation, the output volume velocity Qout is first

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90 calculated from the nonlinear lumped element model as a function of frequency using an iterative technique to account for the nonlin ear neck resistance term. Once the output volume velocity is known, the average jet velocity and centerlin e velocity can be calculated by assuming a Poiseuille flow in the slot or the orifice. To facilitate a simpler electrical circuit, the gyrator can be eliminated from the circuit by referring the electrical components to the ac oustic domain or vice versa (section 3.1). As drawn in Figure 3-20, the electrical impedance eC Z can be referred to an effective acoustic impedance a Z via 2 2 coil coil eC a coil B L Q BL Q VG S Z S I SPZ P BL (3.95) Figure 3-20. Lumped element model of the electrodynamic ZNMF actuator referred to the acoustic domain The equivalent circuit referred entirely to the acoustic domain is shown in Figure 3-20. The electrical vo ltage source Vac in is represented as a pressure source given by ac eCGV P Z (3.96) Using the current divider relation, 21 1out D aCavaOSaCavaOSaO aRadQ Q CMMsCRRs (3.97)

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91 The volume velocity QD is given by 2 ac eC D eC effGVZ Q Z GZ (3.98) where ||aD aCavaO effZZZZ The overall transfer f unction is thus given by 2 21 1out eC ac aDaD aCavaOaCavaOSaO aOSaOaO aD eCGZ Q V G RsMCMsCRRsRRsM ZsC (3.99) where aOaOS aRadMMM is the effective acoustic mass of the orifice that includes the radiation mass of the orifice. Using the transfer function, a few limiting cases can be investigated to obtain some insight into the device operation. Here ||aD aCavaO effZZZZ is the effective acoustic impedance of the mechanical and acoustic components. Also QD is defined as 2 ac eC D eC effGVZ Q Z GZ CASE 1. aOaCavZZ ,that is the orifice impedance is much larger than the cavity impedance, then 0outQ There is no output volume velocity as all the fluid displaced by the diaphragm is compressed or expanded with in the cavity. This case will arise for a highly compliant cavity (i.e., very large volume or compliant walls). CASE 2. aCavaO Z Z that is the orifice impedance is much larger than the cavity impedance, then 0CQ In this case, all the fluid displaced by the diaphragm is pushed out of the orifice and there is no compression of the fluid in the within the cavity. This makes sense for a very stiff cavity (i.e., very small volume). CASE 3. aDZ that is the diaphragm impe dance is very high, then 0outQ Physically, this corresponds to case when the diaphragm does not provide significant displacement. This can happen for a diaphrag m with large resistance and mass and a low compliance (i.e., high stiffness). CASE 4. 2 eC effGRZ that is the referred electrical impedance is larger than any other impedance in the model, then ac DV Q G Here the electrical components of the actuator dominate the overall behavi or of the actuator.

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92 CASE 5. 2 eC effGRZ then ac D eC effQGVRZ. Physically, this corresponds to the case where the effective acoustic impedance of the components dominates and limits the diaphragm volume velocity. The overall lumped element model for the el ectrodynamic ZNMF actua tor is developed in Chapter 3. This model can be used as a predic tive tool for determining the performance of any device. The model can also be used as a de sign and optimization tool to meet target specifications for various applications.

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93 CHAPTER 4 FABRICATION AND MODEL VALIDATION OF PROTOTYPE ELECTRODYNAMIC ZNMF ACTUATORS The overall lumped element model for the electrodynamic ZNMF actuator was developed in Chapter 3. Chapter 4 presents fabrication and experimental validation of two electrodynamic ZNMF actuators. Section 4.1 describes a prot otype ZNMF actuator bu ilt around a conventional loudspeaker. The parameter extraction and the m odel validation of this actuator are also presented in this section. Sec tion 4.2 discusses the fabrication details for a completely custommade, composite-diaphragm ZNMF actuator. The model validation for this custom-built actuator is discussed in section 4.3. 4.1 Speaker-Driven Electrod ynamic ZNMF Actuator From the schematic of a generic electrodynamic ZNMF actuator shown in Figure 3-3, it can be seen that the structure is essentially an oscillating di aphragm with a cavity and a slot through which the jet is established. For initia l validation of the lumped model, a prototype device with a commercial speaker as the mechanical driver and a custombuilt cavity and slot (or orifice) was built. A 1 W, 8 speaker (CUI, Inc. Model number GF0771) was used as the mechanical driver [47]. The cavity and the slot we re rapid prototyped using Acryl onitrile butadiene styrene (ABS) plastic. A schematic and the photog raphs of the device are shown in Figure 4-1 and Figure 4-2 respectively. The dim e nsions of the speaker driv er, the cavity, and the slots are summarized in Table 4-1. The volume of the cavity is one of the param eters need to ca lculate its lumped compliance. The volume was calculated using the manufacturer specified dimensions of the speaker cone. The profile of th e speaker cone was also veri fied using the Keyence LKG 32 (Model Number LKGD 500) laser displacement sensor.

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94 Figure 4-1. Schematic of the speak er-driven electrodynamic ZNMF actuator Figure 4-2. Photograph of the speak er-driver, the cavity and the slot Table 4-1. Geometrical parameters for speaker-driven ZNMF actuator 4.1.1 Lumped Parameter Extraction Since the speaker was purchase d and not designed in-house, the lumped element model parameters for the speaker-driver such as the acoustic compliance, acoustic mass, diaphragm damping, and transduction coefficient were not known beforehand. In order to model the actuator performance, it is esse ntial to know these pa rameters. This section discusses generic parameter extraction procedures to dete rmine the relevant model parameters. Parameter Value Piston radius (a) 38.5 mm Cavity volume (V) 2.7x10-4 m3 Slot height (LOS) 1 mm Slot width (dOS) 1 mm Slot length (wO S ) 60 mm

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95 4.1.1.1 Electrical impedance(eC R and eCL) The speaker driver had a rated resistance of 8 The dc resistance was measured to be 7.8 using Fluke digital multimeter. The impedance of the speaker versus frequencyboth the resistance and inductancewas measured using the Agilent 4294A precision impedance analyzer over a frequency range of 40 Hz 1000 Hz The diaphragm was free to move; thus this actually was a measure of the series combination of the electrical impe dance and the reflected mechanical impedance 2 coil aDBL Z, where( 1aDaD aD aDZRjM C ). The measured free impedance of the speaker -driver is shown in Figure 4-3. As this test wa s perform ed in air, there is an additional acoustic load on the diaphrag m (acoustic radiation resistance and acoustic radiation mass). Figure 4-3. Electrical impe dance of the speaker-driver

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96 The peak seen in the data at about 165 Hz is due to the electromechan ical resonance. Since the electrical coil parameters (Rec and Lec) could not be isolated from this measurement, the dc resistance of the coil was used, and the induc tance was ignored as at low frequencies the contribution of the inductiv e reactance is small. 4.1.1.2 Mechanical compliance (mDC) The speaker-driver has corrugated edges that determine the mechanical compliance. The maximum displacement and the resonant frequency of the actuator are both controlled by the compliance. The mechanical compliance (mDC) of the speaker diaphragm was measured using static mechanical pull tests, as illustrated in Figure 4-4. The speaker-d river (without the cavity assemb ly) was mounted on a rigid platform a nd loaded with known weights. The resulting center displacement of the speaker-driver was measured using a laser displacement sensor (LDS) (Keyence LKG-32 Model Number LKGD 500). Figure 4-4. Experimental se tup for compliance measurement To extract a suitable complianc e parameter, the displacement range mimicked that during standard operation. The maximum displacement observed for the speaker was about 0.3 mm (peak). Hence the displacement range used for the compliance and transduction factor

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97 measurements was 0.2 mm. The results from a static displacement test are shown in Figure 45. It can be seen that for the measured range of applied forces the speaker-driver displacement is very linear. The extracted mechanical complia nce of this speaker-driver was 0.46 mm/N, as indicated in Table 4-2. Figure 4-5. Results from compliance measurement Table 4-2. Linear curve fit results for compliance measurements Parameter Value Extracted mechanical compliance 0.46 mm/N 95 % confidence intervals 0.4469 0.4739 mm/N R2 0.9953 The overall lumped model for the ZNMF act uator developed in Chapter 3 requires the acoustic complianceaDC. Thus it is necessary to convert th e measured mechanical compliance to an acoustic compliance. The relationship betwee n the mechanical and acoustic impedances was presented in section 3.41. Using this relation, the acoustic comp liance can be converted to the equivalent acoustic compliance as follows 2 aDmDCCS (4.100)

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98 where Sis the equivalent area of the mechanical st ructure. Here it is assumed that the entire speaker-driver moves as a rigid piston, since the corrugated edge dia phragm accounts for only a small portion of the surface area. Thus the equivale nt area of the speaker-driver is assumed to be the physical area. The acoustic co mpliance of the speaker-driver wa s calculated to be 9.97 x 10-9 m5/N. 4.1.1.3 Electromechanical transduction coefficient (coil B L) The transduction factor from the electrical to the mechanical domain is given bycoil B L as discussed in section 3.2. As th e details regarding magnetic assemb ly and the coil structure of the speaker driver were not known, th e transduction factor was determ ined indirectly by exciting the piston electrically with a dc cu rrent and measuring th e displacement with a LDS, as shown in Figure 4-6. Using the force balance, coil mDFBLIxC the transduction factor was determined since the mechanical comp liance was known, and the displacement x and current I were directly measured. The displacement was measured using th e Keyence LKG-32 LDS (Model Number LKGD 500), and the current was measured using a Tektronix TCPA300 current probe. Like the compliance test, the displacement range mimicked the operational range of the actuator. A least squares method was used to fit a line to the data, as shown in Figure 4-7. The displacem ent at each drive current w as measur ed 31 times. The uncertainty estimates in the measurement are also indicated in the plot. Note that this measurement was carried out at both positive and negative currents, and the device was shown to exhibit some hysteresis behavior. This could be due to magnetic or mechanical nonlinear effects, especially since the speaker was being driven beyond its normal operational limits. Also, since the detailed design and operation of the speaker was outside the scope of this work, the source of this hysteresis was not fully

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99 investigated. The curve fit results for the tran sduction factor measurem ents are indicated in Table 4-3, and the extracted transduction factor was 1.35 N/A. The electromechanical transduction coefficient was converted to equivalent electroacoustic trans duction coefficient (G) by using the effective area of the speaker-dri ver. The electroacoustic transduc tion coefficient of the speakerdriver is 289.9 Pa/A. Figure 4-6. Experimental setup fo r transduction factor measurement Figure 4-7. Results from the transduction factor measurement

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100 Table 4-3. Linear curve fit results for transduction factor measurements Parameter Value Extracted slope 0.621 mm/A 95 % confidence intervals 0.5907 0.6513 mm/N R2 0.9809 4.1.1.4 Diaphragm acoustic resistance (aD R ) and equivalent acoustic mass (aD M ) The diaphragm and voice coil are all permanently attached to one another, and hence the effective mechanical mass of the piston was measur ed indirectly by performing an impulse test. The damping ratio and consequent ly the acoustic resistance (aD R ) corresponding to the diaphragm damping is also obtained from this test. In order to measure the diaphragm parameters, it is essential to isolate the electrical components of the device (to avoid any unwante d electromechanical coupling) but also to provide consistent mechanical excitation. A know n force was applied by exciting the coil with a dc current to achieve a static displacement of the diaphragm. An electrical switch was then opened to open-circuit the coil such that curre nt could no longer flow, thereby removing the mechanical force. Any electrical transients were assumed to be much faster than the resulting mechanical motion, so this procedure simula ted an impulse response with a known initial condition. The resulting diaphrag m oscillation was measured using the laser displacement sensor (LDS) (Keyence LKG-32 Model Number LKGD 500). Th e setup of the experiment is shown in Figure 4-8. The tim e-series data of the displacem ent is shown in Figure 4-9. The initial displacement of -0.3 mm i s seen when the coil is excited wi th a dc current. When the switch is opened, the displacement decays to zero. The displacement is essentially the free response of the diaphragm with the dc displacement as the initial condition (initial value problem). The characteristic underdamped system response is observed. The dampin g ratio and the damped natural frequency are

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101 extracted from this data using the log decr ement method described be low[48]. A non-linear curve fit may also be used to extract th ese parameters as shown in the figure. Figure 4-8. Setup for the impulse response test Figure 4-9. Results from the impulse response test d1 d2 t

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102 If 1dand 2dare the values of displacements measured at two consecutive peaks as shown, the damping ratio is given by 1 22 2ln, 4d d (4.101) The damped natural frequency d f can be extracted from the tim e difference between the two peaks as follows 1df t 4.(102) The natural frequency is then given as 212 1 2nf t (4.103) The natural frequency is related to the mechanical mass and compliance 1 2n mDmDf M C (4.104) Thus the effective mechanical mass can be obtai ned and it is converted to equivalent acoustic mass (aD M ) as follows 2mD aDM M S (4.105) The extracted values of the damping and natural frequency were 0.015 and 193 Hz respectively. Thus the effective mechanical mass calculated using (4.104) is 1.5 g. The effective acoustic mass calcu lated u sing (4.105) is 70.3 kg/m4. The damping ratio is converted to an equivalent acoustic resistance as follows 2aD aD aDM R C. (4.106) Thus the extracted acoustic resistance is 2.52 x 103 N-s/m5. Another experiment was also used to extract the diaphrag m damping. The experimental setup is similar to the one shown in Figure 4-6. The only difference was that the speaker-driver

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103 was excited with a 5 V p-p, 1 Hz, 50 % duty cycle square wave using an Agilent 33120A function generator. The electrical stimulus was never disconnected. In this configuration, the total observed damping originates from mechan ical and electrical losses (due to the electromagnetic coupling). This system exhibited a characteristic underdamped second-order displacement step response, as shown in Figure 4-10. The displacement step response for a standard second order system is given by 21cos1nt n x te t (4.107) where x(t) is the displacement of the diaphragm, n is the natural frequency in rad/sec and is the damping ratio of the system. A non-linear curve fit was used to fit the standard second order system response to the measured displacemen t data. The curve fit is also shown in Figure 4-10. The extracted values of dam p ing ratio and the to tal damping from this experiment were 0.08 and 1.23 x 104 N-s/m5 respectively. Figure 4-10. Step response of the diaphragm and the curve fit

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104 As mentioned above, the damping measured here is a combination of the diaphragm damping and an effective electrical damping. The equivalent electrical circuit is shown in Figure 4-11. Note that the im pe dance of the source drivi ng the speaker is usually small and is ignored in this analysis. Thus the total damping term measured is given by 2measured aD eCG R R R. (4.108) Based on other previous measurements, the equivalent electrical damping 2eCGR is calculated to be 9.55 x 103 N-s/m5. Thus the diaphragm damping RaD is 2.72 x 103 N-s/m5. Note, the electrical damping component is nearly four times higher than the mechanical damping. Comparing this closed-circuit result to the other open-circuit damping experiment, fairly good agreement is seen in the extracted diaphrag m damping. However, the open-circuit damping measurement is used for this work, since it is a more direct measurement of the damping. Table 4-4 sum m arizes all the extracted parameters of this actuator. These parameters are used in the lumped element model to predict the output volume velocity produced by the actuator. Table 4-4. Extracted parameters for the speaker-driven ZNMF actuator Extracted parameter Value Electrical resistance ( ) (ReC) 7.8 Mechanical transductio n coefficient [49] (BLcoil) 1.35 Acoustic transduction co efficient [Pa/A] (BLcoil/S) 289.9 Mechanical compliance [mm/N] (CmD) 0.46 Acoustic compliance [m5/N] (CaD) 9.97 x 10-9 Mechanical mass [gm] (MmD) 1.5 Acoustic mass [kg/m4] (MaD) 70.34 Damping ratio () (Diaphragm only) 0.015 Damping ratio () (Diaphragm + Electrical) 0.083 Natural frequency [Hz] 193 Equivalent diaphragm damping [N-s/m5] (RaD) (Diaphragm only) 2.52 x 103 Equivalent diaphragm damping [N-s/m5] (RaD) (Diaphragm + Electrical) 2.72 x 103

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105 Figure 4-11. Equivalent circuit for the impulse response test 4.1.2 Hotwire Measurements Ov erall Model Verification The validation of the lumped model for speaker-driven actuator is presented in this section. The output velocity from the actuator was m easured using a constant temperature hotwire anemometer with a standard 5 m diameter, 1.25 mm long wire (Dantec 55P11). The Dantec Streamline 90N10 Frame and 90N02 flow unit were used for calibration of the hot wire. The hotwire calibration provides th e relationship between flow velo city and hotwire voltage. The empirical relationship between hotwire voltage E and the flow velocity U is given by Kings law as 2 n E ABU (4.109) where A, B and n are constants[50]. A least square curve fit is used to extract the constants. Thus the velocity for any measured hotwire voltage can be calculated. A t ypical hotwire calibration curve is shown in Figure 4-12. In this case, th e co nstants A, B and n are 0.0069, 0.0036 and 0.4537 respectively. After calibration, the hotwire was moved to the center of the actuator slot. The speaker actuator was then excited using sinusoidal voltage The voltage signal sh own on the oscilloscope was used to adjust the depth of the wire in the slot such that the unrectified and rectified voltage signals were equal in amplitude; this ensured that the hotwire was exactly in the center of the slot, though the length of the slot Thus the peak velocity produ ced by the jet is measured using

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106 the hotwire. The lumped element model of the ZNMF actuator developed in Chapter 3 predicts the total output volume velocity Qout. This is converted to the peak velocity by assuming Poiseuille flow in the orifice. For Poiseuille flow through an orifice, the peak (centerline) velocity is 1.5 times the average velocity [51]. For the flow through the 2D slot, the channel flow model is used [45]. The peak (cen terline) velocity in this case is twice the average velocity. Note that the velocity profile in the slot/orifice is dependent on the Stokes number of the flow defined as 2Std Here is the radian frequency of operation, d is the depth of the slot and is the kinematic viscosity of the fluid. The Poiseu ille flow assumption is only valid for Stokes number of zero. For non-zero Stokes number a frequency dependent correction factor is necessary to convert from average velocity to cen terline velocity[45]. For example, at 100 Hz the Stokes number corresponding to the 1 mm radius orifice is about 1. Thus the ratio of the centerline velocity and the average velocity is lesser than 1.5. Figure 4-12. Typical hotwire calibration curve

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107 The excitation frequency and amplitude was varied over the range of interest for each device, with the maximum voltage limited by the maximum allowable current through the coils. Figure 4-13 shows the comparison b etween the model-pred icted and experimentally measured output velocity. 10 blocks of data were taken and the 95 % confidence intervals B for the measured velocity were calculated using th e standard t-distribution as follows [52] 1.812 10B (4.110) where is the standard deviation of the velocity. It can be seen th at the model captures the first resonant peak (both the frequency and amp litude) of the output velocity well. The LEM predictions deviate from the expe rimental results particularly fo r higher frequencies. This may be due to the non-linearities in the speaker compliance or the transduction coefficient. In addition, there is significa nt uncertainty in the volume of the cavity. The slot model also needs to be refined to be applicable at higher frequencies. Figure 4-13. Model predicted a nd experimentally measured out put velocity of the speakerdriven ZNMF actuator for 3V pp voltage input

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108 It can thus been concluded that the LEM de veloped in Chapter 3 pr ovides a powerful tool for analysis and prediction of the performan ce of ZNMF actuator. However, the LEM was validated only for an actuator where many im portant lumped parameters were extracted experimentally. The next secti on presents the fabrication met hod and validation of a composite diaphragm ZNMF actuator. The models for both the magnetic assembly and mechanical structure developed in Chapter 3 are used to predict the device performance. 4.2 Fabrication of Custom ZNMF Actuators This section describes the fabrication of a custom-built, ZNMF actuator, as shown in Figure 4-14. The actuator consists of three m ajor com ponentsa magnetic assembly for creating a static magnetic field, a movable composite di aphragm/coil assembly for creating fluid motion, and a cavity with an orifice (or slot) through which the fluid is expelled or injected. The magnetic assembly comprises permanent and soft magnets to establish a fixed, uniform magnetic field across the coil windings. A multi-turn, multi-layer solenoidal coil of total length coilLresides in the annular air gap. The mo veable portion of the ZNMF actuator consists of an axisymmetric composite diaphrag m clamped at its outer edges. The rigid central boss has a radius band the outer annular compliant region has an outer radius of a. The thickness of the diaphragm ish. The solenoidal coil is rigidly a ffixed to the central boss. The diaphragm forms the bottom portion of a sealed cavity of lengthCL. The top of the cavity is bounded by a rigid plate c ontaining an orifice (or slot). A circular orifice with radiusosaand lengthosLis shown in the schematic. Many of the sub-components in the actuatorthe housing, diaphragm central boss, cavity, and orifice (or slot) structurewere rapid pr ototyped using a ZCorp Z415 3D printer. The resulting material is a composite powder that is held together by a binder. The material is

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109 extremely light and sturdy. Thus the housing pieces do not contribute sign ificantly to the overall mass of the actuator; in fact, most of the total mass of the actuator arises from the magnetic assembly. The overall density of the binder material was measured to be 1190 kg/m3. The manufacturing tolerances in the 3D printing pr ocess decide several key dimensions of the actuator. Thus the lower limits on the cavity length, the orifice (or slot) dimensions and the rigid center boss thickness are set by the tolerances of th e 3D printing. Both the vertical and lateral tolerances on the Z415 are +/0.05 mm. In add ition to the 3D printer tolerances, the minimum dimensions were restricted to ensure repeatability of the manu facturing process and maintain structural strength of all the pa rts built. Based on these limits, the smallest diaphragm/central boss thickness was limited to 1 mm, the mini mum cavity length was set to 3 mm, and the minimum slot/orifice dimensions were 0.5 mm. Figure 4-14. Schematic of Polydimethylsil oxane (PDMS) composite diaphragm actuator 4.2.1 Magnetic Assembly The primary objective of the magnetic assembly is to produce a large, constant magnetic field in the annular air gap. The two most commonly used high-performance permanent magnet materials are neodymium-iron-boron (NdFeB) and samarium cobalt (SmCo). From the 1-D magnetic circuit model developed in section 3.3.1, it was seen that material with high coercivity

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110 cH and remnant magnetic flux density r B should be used to obtain large magnetic flux density in the air gap. Hence NdFeB magnets (from K J Ma gnetics) were used, as they have the highest remnant magnetic flux density among commercially available magnets (1.3 T). The soft magnetic material in the magnetic assembly provides a low reluctance path for the magnetic flux. Hence soft magnetic material with very large relative permeability should be selected. Here, EFI50 (nickel-iron alloy) from Ed Fagan Industries with relative pe rmeability of 100,000 was used [40]. In addition, EFI 50 also has relati vely high saturation magne tization of 1.45 T. The hard magnet was purchased at a specific si ze, and the soft magnetic core pieces (top, bottom and annular piece) were conventionally machined to the desired dimensions. All the pieces of the magnetic assembly were held toge ther using a two-part epoxy. The geometrical dimensions of the magnetic assembly were chosen to keep the final actuat or device confined to a size of approximately a 25 mm on a side. The fi nal dimensions of the magnetic assembly are shown in Figure 4-15, along with a photograph of th e m a gnetic assembly sitting within the bottom housing of the actuator. The magnetic asse mbly is attached to the bottom housing piece using a two-part epoxy. Figure 4-15. Magnetic assembly schematic and photograph

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111 4.2.2 Diaphragm Assembly The diaphragm assembly is the most complex component of the actua tor. Fabrication was achieved using a multi-step processthe rigid center boss and the coil assembly were fabricated first, followed by assembly with the PDMS to form the composite diaphragm. In order to ensure that the PDMS adheres we ll to the center boss, a two-flange design was employed in the center boss, as detailed in Figure 4-16. During assembly of the diaphragm, the liquid PDMS would flow between the flanges, providing a reliable connection. The overall diam eter of both the flanges was set to 2 bas decided by the desired solidity ratio. The two discs were connected together by a centr al region with a diameter of b. Six additional smaller outer posts around the periphery provided additional su pporting structures to which the PDMS could attach. Note that the dimensions of the central post and the outer posts are not critical as long as the overall structural integrity and rigidity of the central boss is en sured. The entire structure was rapid prototyped using the ZCorp Z415 3D printer. Figure 4-16. Two-flange desi gn for the rigid central boss The coil winding was formed using bondable co pper wire from MWS Wire Industries [53]. The required number of coil turns were wound on a cylindrical post of desi red coil diameter. The

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112 copper wire has a thin coating of Polyvinyl Butryl, which is a thermoplastic adhesive. When the adhesive is heated to 130C, it binds the coil windings toge ther, and the self -supporting coil is then slid off the winding post. The coil was then attached to the diaphragm central boss using a two-part epoxy. Several different coil configur ations were manufactured, as will be described later. Figure 4-17 shows photographs of the coil and the rigid center boss. The annular com p liant portion of the compos ite diaphragm was fabricated using PDMS. PDMS is a silicon-based organic polymer elastomer material. It is very suitable as a structural material because it is very compliant (Young s Modulus in the 100 kP a 760 kPa range [54]) and thus can enable in large displacements for a small forces. PDMS is a viscoelastic siliconbased organic polymer material th at conformally reflows when pour ed into a mold. After curing, it behaves as an elastic solid. Dow Corning Sylg ard 184 Elastomer Kit was used in this work. The kit consists of an elastomer base and a curi ng agent, which are mixed in 10:1 weight ratio. The mixture was allowed to degas for 30 minutes before pouring into mold s (fabricated using the 3D printer) to form the diaphragms. This prev ented the formation of bubbles in the PDMS during the curing process. During the curing process, the polymer chains in the PDMS cross-link making the material stiffer. The final material properties are a function of the curing temperature and curing time [54]. A 2 hour cure at 50C on a hotplate was used for this work. Figure 4-17. Diaphragm components

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113 To form the diaphragm assembly, the PDMS mixture was poured into to a shallow mold. The central boss was placed, cen tered into the uncured PDMS and centered using the centering mechanism as shown in Figure 4-18. The PDMS was then cured at 50 C for 2 hours. Note the PDMS m o ld has eight posts to create holes in the PDMS layer to accommodate the screws for the final assembly. After the curing process is completed, the diaphragm was removed from the mold, with the central boss/coil attached as show n. The photographs of the mold and the released diaphragm-coil assembly are shown in Figure 4-19. Finally, the di aphragm was placed between the two housing pieces. The bo ttom housing holds the m agnetic assembly, and the top housing forms the cavity/slot. Bo th these pieces have th rough holes on the edges th at align with the holes in the diaphragm. The photograph of the fully assembled device with a 2D-slot is shown in Figure 4-20. Figure 4-18. Centering ring for the central boss Figure 4-19. Mold for the PDMS diaphragm fabrication and releas ed PDMS diaphragm with the rigid center boss

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114 Figure 4-20. Fully assembled device 4.3 Model Validation Before validating the overal l model of the electrodynami c ZNMF actuators, several experiments were carried out in order to verify the models of the individual components of the ZNMF actuator. The coil electrical impedance was measured to verify the coil resistance and inductance (eC R and eCL). Similarly, experimental character ization of the diaphragm assemblies is very important in order to verify the di aphragm model. The acoustic compliance and the acoustic mass of the diaphragm are discussed in section 3.4.2. Two different characterization experiments were performed to experimentally veri fy the diaphragm model (w ith attached coils). First, the static center deflection was measured using a dc loading test to measure the center displacement. The center displacement of the PD MS diaphragm under a uniform pressure load was also measured. The resonant frequency and the diaphragm damping were then extracted using the impulse response measurements. The comparison between measured and model predicted resonant frequency will provide an independent but indirect validation of diaphragm model.

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115 The material properties and the geometrical dimensions of the composite diaphragm are tabulated in Table 4-5. Based on these parameters, the lumped acoustic compliance, acoustic m a ss and resonant frequency of the diaphragm were calculated (section 3.4), as shown in Table 4-6. N o te that this particular PDMS diaphragm and coil assembly is used in actuator A1. The volume of the front cavity for the prototype actuators is 1.5 cm3 and the total volume of the back cavity is nearly 12 cm3. Moreover, the bottom housing se ction has six large openings as can be seen from Figure 4-20. Thus it can be seen that the back cavity is at leas t 8 tim es larger than the front cavity and its effect can be neglected. Table 4-5. Diaphragm materi al properties and geometry Parameter Value Outer radius (a) 12.7 mm Inner radius (b) 9.5 mm Height (h) 1 mm Youngs modulus of the annular portion (E2) 360 kPa [55] Poissons ratio of the annular portion (2) 0.33 Density of the annular portion (2) 970 kg/m3 Youngs modulus of the central portion (E1) 20 GPa[56] Poissons ratio of th e central portion (1) 0.33 Density of the central portion (1) 1190 kg/m3 Mass of the coil (Mcoil) 0.31 g Table 4-6. Model parameters for PDMS diaphragm corresponding to actuator A1 Parameter 1 inch diameter Effective area (S) 3.89 x 10-4 m2 Acoustic compliance (CaD) 2.70 x 10-10 m5/N Acoustic mass (MaD) 5.08 x 103 kg/m4 Mass contribution of the coil ( 2coil M S) 1.85 x 103 kg/m4 Predicted resonant frequency (Hz) 151.7 Hz 4.3.1 Electrical Impedance Measurements The electrical resistance of the coil can be calculated based on the number of turns, the average diameter of each turn and the gauge of the wire used. Thus, for 150 turn, 40 AWG turn with mean diameter of 10.6 mm is calculated to be 23 The coil resistance and inductance was measured using the Agilent 4294A precision impe dance analyzer from 40 Hz 1000 Hz. The

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116 results of the resistan ce and inductance measurements are shown in Figure 4-21. At each frequency, 30 averages were used. The re sistance of the coil is shown to be 25.2 and the inductance was measured to be 0.48 mH. Note th at at the maximum frequency of interest here (500 Hz ), the inductive portion of the coil impedance is 1.5 Thus it can be seen that the inductive impedance of the coil even at the maxi mum frequency of interest is nearly 20 times smaller than the resistance of the coil. The magn etic assembly and the diaphragm were not used during this experiment. Figure 4-21. Impedance measurements of 150turn 40 AWG copper coil without the magnetic assembly 4.3.2 Static Displacement Tests The Lorentz force on the coil F = BLI. Thus if the coil is excite d by a dc current a static force is applied on the diaphragm. The center deflection w(0) of the diaphragm under this condition can be predicted from the model de veloped in Chapter 3 and is given below

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117 114 22 2 12 24 4 4 11 (1) (2) 11(0) ln 4ln 264264 Pb bb abPb wb ba aba Da D a ,(4.111) where the constants are detailed in the Appendix B. Note that this deflection assumes a uniform pressure load. The Lorentz force load on the coil is converted to an equivalent pressure as follows coil B LI F P SS (4.112) Thus if the magnetic field B and the coil length Lcoil are known, the applied force F is known for a given current I. Note that S is the effective area of the diaphragm. The radial magnetic flux density in the gap was measured using a Lakeshore 475 DSP Gaussmeter with a Lakeshore hall probe HMNT-4E04-VR. The average measured magnetic flux density in the air gap (B) was 0.33 T. This measured value matched well with finite element analysis of th e magnetic assembly. The coil consists of 150 turns of 40 AWG copper wire. The average diameter of the coil was 10.6 mm, making the total coil length 6.7 m. All the electrical and magnetic parameters of the magnetic assembly are tabulated in Table 4-7. Figure 4-22. Setup for dc displacemen t test and the step response test

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118 Table 4-7. Electrical and magnetic parameters Number of coil turns 150 Coil resistance (ReC) 25.2 Coil Inductance (LeC) 0.48 mH Coil length (Lcoil) 6.7 m Magnetic field in the gap (B) 0.33 T Maximum current 110 mA For the testing, the coil was excited using a DC current source, and the current through the coil (I) was measured. The displacement of the diaphragm was measured using the Keyence LKG 32 laser displacement sensor. The test was performed for a range of currents up to the maximum allowable current through the coil (110 mA for 40 AWG copper wire). The force was calculated using the Lorentz force equation (F = BLcoilI), and the equivalent acoustic pressure corresponding to each case was calculated using (4.112). Figure 4-23. DC displacement test results Figure 4-23 shows the measured and predicted values of the center di splacement as a function of the applied equivalent pressure. The displacement was measured 31 times at each

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119 applied pressure and the uncertainty estimates calculated are also indicated on the plot. The displacement is fairly linear with respect to a pplied pressure and matche s very well to the model predicted displacement. The center displacement was also measured by loading the diaphragm by a uniform pressure. The experimental setup for this measurement is shown in Figure 4-24. The setup consists of a vacuum cha mber whose one wall was formed by the diaphragm under test. A vacuum was generated in the chamber by means of a small electric motor. The pressure inside the chamber was measured using a Heise ST-2H pressure gauge. The diaphragm which forms one wall of the chamber was clamped in be tween two annular magne ts as shown. The displacement of the diaphragm was measured us ing the Keyence laser displacement sensor as described in the previous experiment. Figure 4-24. Displacement measurement under pressure load Figure 4-25 shows the displacement as a functio n of the press ure. The comparisons with the model predictions are also shown in the same plot. It can be seen that the model matches the e xperimental data well. Note that due to the limitations of the experimental setup, the displa cement of the diaphragm could not be measured

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120 over the range of displacements that were obs erved during the device operation. A better experimental setup with a better control of the pr essure inside the chamber will be necessary to characterize the static displacement profiles more co mpletely. In addition, a setup with means to pressurize as will as to establish a vacuum in the chamber can be used to study any hysteresis behavior present. Figure 4-25. Displacement measurements under uniform pressure load Table 4-8. PDMS diaphragm parameters for actuator A1 Parameter Value Extracted resonant frequency 143.7 Hz 95 % Confidence intervals (frequency) 142.8 144.6 Hz Model predicted resonant frequency 151 Hz Extracted damping ratio 0.015 95 % Confidence intervals (Damping ratio) 0.0146 0.154 R2 0.9476 Diaphragm damping (aD R ) 1.3 x 105 N-s/m5 It can be seen from the dc displacement test ( Figure 4-23) and the pr es sur e loading test ( Figure 4-25) that the composite diaphragm mode l pred icts the center dis placement accurately. Moreover, the resonant frequency measured using the step response test is within 5 % of that

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121 predicted by the model ( Table 4-8). This indicates that the di aphragm model is reliable and can be used to predict the overall response of the ZNMF actuator. 4.3.4 Hotwire Measurements The purpose of the lumped element model of the electrodynamic ZN MF actuator is to predict output velocity obtained from the jet. The previous sections discussed verification of the diaphragm model and extracted the diaphragm damp ing. This section discusses the validation of the overall model by measuring th e output velocity from the je t using hotwire anemometer measurements. The general experimental set up, calibration and procedure for the hotwire measurements were described in section 4.1.2. Table 4-9. Device confi guration for hotwire tests Actuator A1 A2 A3 Magnetic flux density in the gap (B) T 0.4 0.4 0.4 Length of the coil (coilL) m 6.6 (150 turns, 40 AWG) 4.5 (90 turns, 34 AWG) 4.5 (90 turns, 34 AWG) Solidity ratio ( ba) 0.75 0.75 0.75 Effective area ( D A ) m2 3.89 x 10-4 3.89 x 10-4 3.89 x 10-4 Transduction factor (G) Pa/A 6.8 x 103 4.62 x 103 4.62 x 103 Resistance of the coil (eCR) 23.3 3.8 3.8 Equivalent electrical resistance ( 2 eCGR) N-s/m5 1.28 x 106 5.62 x 106 5.62 x 106 Mass of the coil (coilM) g 0.56 0.31 0.31 Predicted resonant frequency (res f ) Hz 151 177 177 Extracted resonant frequency (res f ) Hz 144 167 167 Extracted damping 0.015 0.012 0.013 Slot/orifice geometry 24 mm x 1 mm x 1mm slot 24 mm x 1 mm x 1mm slot 2 mm diameter x 1 mm deep orifice Maximum current (rms) (mA) 110 330 330 Maximum input power (mW) 200 400 400 Hotwire results for three different actuators are discussed here. The only difference in the three devices was the coil and slot configurations; the diaphragm, cavity, and magnetic assembly dimensions, were identical. The geometries are summarized in Table 4-9. Figure 4-26 Figure 4-

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122 28 show the hotwire results for th e three devices. 100 blocks of data were used to obtain the error estimates and the bounds on the data based on (4.113) and (4.114). The error estimates for each frequency were obtained by analyzin g the time data at each frequency. The normalized random error r for the measurement is given by [52] r (4.113) where is the standard deviation of the measurement and is the mean of the measurement. The 95 % confidence interval for the mean of the measurement is given by 2B N (4.114) where N is the number of independent observations. Th is is obtained from the t-distribution and assuming that number of independent observations N is greater than 31. Thus the mean of the measurement is bounded by B B (Here, N = 100). The overall rms and mean errors for each of the actuators were cal culated. The maximum rms error was calculated to be 11.3 %, and maximum mean error was 9.8 %. The model predictions for both the devices are also shown in the plots. The hotwire results indicate that the lumped element model predicts the resonant frequency of the device well. Note that device 2 has a much broader peak around the resonant frequency. This is because of the difference in equivalent electrical damping in the two devices as shown in Table 4-9. The resis t ance of the coil for device A1 and device 2 is 23 and 4.5 respectively. Thus when this resistance is referred to the acoustic domain, the equivalent electrical damping for device 2 is much higher than fo r device A1. Device A3 shows even flatter frequency response because in additional to extra el ectrical damping, this device also has higher fluidic resistance (RaOS and RaO). In all three cases, the model predicts th e behavior of the device up to the resonant frequency extremely well. At hi gher frequencies the deviation from the model predictions is

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123 higher. This may be due to nonlinear effects in the composite diaphragm, magnetic assembly, fluidic components etc. Figure 4-26. Hotwire results and model comparisons for device A1 Figure 4-27. Hotwire results and model comparisons for device A2

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124 Figure 4-28. Hotwire results and model predictions for device A3 In almost all actuators, the overall damping is constant. ZNMF actuators are unique in this respect because the damping is frequency depe ndent that due to the non-linear slot/orifice resistance (aO R ), the damping is frequency dependent. Figure 4-29 shows the relative ma gnitudes of the slot/orifice damping and the total damping for device A1. Table 4-10 shows the relative ma gnitudes of various damping components in the three actuators. Table 4-10. Various components of the damping Damping term Actuator A1 Actuator A2 Actuator A3 Equivalent electrical damping ( 2 eCGR) 1.28 x 106 5.62 x 106 5.62 x 106 Diaphragm damping (aD R ) 1.3 x 105 0.9 x 105 1.1 x 105 Linear slot/orifice damping (aOS R ) 0.4 x 105 0.4 x 105 0.85 x 105 Non-linear slot/orifice damping (aO R ) at resonance 5.1 x 105 2.7 x 106 2.9 x 106 Total damping 1.86 x 106 8.3 x 106 8.6 x 106 It can be seen that the diaphragm damping is very small portion of the overall damping in each of the actuators. For actuator A1, the equivalent electrical damping is constant with

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125 frequency ( 2eCGR) and forms the bulk of the damping (1.28 x 106 N-s/m5 ). However, near the resonant frequency of the device, the slot damping term is at least 50 % of the equivalent electrical damping. Thus the total damping in th e ZNMF actuator particularly near the resonant frequency is frequency dependent. Similar trends are also seen for actuators A1 and A2. This presents several challenges in design and optimi zation of these actuators as will be discussed in Chapter 5. Figure 4-29. Relative magnitudes of the slot damping and the total damping for device A1 4.3.5 Summary The model has been validated for three differe nt electrodynamic ZNMF actuators. It can be seen that lumped element model developed is a very powerful predictive tool for analysis and design of electrodynamic ZNMF actuators. The only parameter extracted e xperimentally was the equivalent diaphragm damping. Se veral trends and tradeoffs can be inferred based on the models developed in Chapter 3 and the experimental results presented in Chapter 4. The parameters that

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126 have the most effect on the device performance can also be identified. The electrical resistance of the coil determines the breadth of the velocity peak. The resonant freq uency of the actuator is predominantly determined by the diaphragm dimensions. A more detailed discussion of various design trends and tradeoffs will be presented in Chapter 5. The custom-built device is very compact, with a volume of ~ 15 cm3 and weight of about 80 g. The magnetic assembly is the heaviest com ponent of the actuator w ith weight of about 50 g. The maximum power consumption of all three actuators was under 500 mW. Note that both the input and output power are a function of the drive frequency and the maximum occurs at the resonance. A fabrication methodology for PDMS based co mposite diaphragms was presented. The fabrication methodology is ve ry robust and cost effective. Mo reover, the fabrication is modular where each of the components can be designed and then fabricated to meet the desired specifications of the actuator. In addition, the fabrication can be adapted to a wide range of sizes and thus much smaller actuators than the ones described in this work can be fabricated using this fabrication methodology.

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127 CHAPTER 5 DESIGN AND OPTIMIZATION STRATEGIES FOR ELECTRODYNAMIC ZNMF ACTUATORS Chapter 5 presents several design and optimizat ion strategies for de signing electrodynamic ZNMF actuators based on the lumped element model for these actu ators developed in Chapter 3 and validated in Chapter 4. Section 5.1 outlines some of the challenges for design of ZNMF actuators. Section 5.2 presents detailed discussion of the performan ce of the electrodynamic ZNMF actuator in various frequency ranges. The design tradeoffs and trends for various performance parameters of a composite diaphrag m ZNMF actuator developed in Chapter 4 are discussed in section 5.3. This section is divided into three major sections the first section discusses the design methodology for the magnetic assembly, the next section discusses some trends, and the final section disc usses tradeoffs involved in the PDMS diaphragm. Some systemlevel design tradeoffs are then presented. Section 5.4 presents a complete constrained optimization formulation for the composite diaphragm electrodynamic ZNMF actuator, including the choice of the objective functi on and sensitivity analysis of the optimized design to design variables. 5.1 Challenges in Design and Optimizati on of Electrodynamic ZNMF Actuators As discussed in Chapter 2, two most importa nt performance metrics of an actuator are blocked force and free displacement. Most actuator performance requirements such as maximum achievable force, maximum displacement, resona nt frequency are dependent on application. Depending on the type of load that the act uator is driving, the desired performance characteristics are different. In order to optimize any given actuator perfor mance, it is necessary to understand the load ing on the actuator. Figure 5-1 shows the equivalent lumped-element circuit m odel for an electrodynam ic actuator without a load impedance load Z The equivalent

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128 acoustic domain representation of the electrodynamic actuator is shown in Figure 5-2. The details of cir cuit analysis to eliminate th e gyrator were discusse d in section 3.1.2. Figure 5-1. Loaded electrodynamic actuator Figure 5-2. Electrodynamic actuator circ uit referred to the acoustic domain In many cases, the load consists of series -connected resistive and reactive components where the total load impedance can be expressed loadload load load j ZRjM C If so, the output volume velocity Q generated by the actuator is 2() 1eCaDload aT aTP Q GRRRjMjC (5.115) where is the frequency of operation and aTC and aT M are the overall acous tic compliance and acoustic mass respectively. The total acoustic compliance and acoustic mass are

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129 loadaD aT loadaDCC C CC (5.116) and aTaDloadMMM (5.117) Note that in the case of a ZNMF actuator, there is the additional branch due to the compliance of the cavity (aCavC). However, for low frequency operation and compact cavities, the impedance of the cavity is very high and is ignored for simplified analysis. A typical velocity (or volume velocity) vs. fr equency curve for a second-order system is shown in Figure 5-3. The low-frequency response of the actuator is dom inated by the acoustic com pliance, and the high frequency response is determined by the total acoustic mass as indicated by (5.115). Figure 5-3. Typical velocity freque ncy response of a second-order system At dc, the volume velocity (or velocity) gene rated by the actuator goes to zero. Thus at dc the change in volume or volume displacement achieved by the actuato r is an important Compliance dominated Mass dominated

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130 performance metric. The dc volume displacement is determined by the total acoustic compliance of the overall system, and the resonant frequency is determined by the overall system dynamics as follows 0aT fVCP, (5.118) and 1 2res aTaTf M C, (5.119) where V is the volume displacement, Pis the applied pressure and aTC and aT M are the total acoustic compliance and acoustic mass, respectively. The product of dc volume displacement and the resonant frequency for a second-orde r systemthe gain-ba ndwidth productis an important performance metric[ 57]. Here, this is given by 1 2aT res aTC Vf M. (5.120) Thus a common objective function used for actua tor optimization is the maximization of the volume displacement-resonant frequency product. However, depending on the load requirements, other objective functions may be chosen. In contrast to the simple formalism described above, the design of ZN MF actuators is more complicated for several reasons. First, the load impedance is not represented by a simple RLC network. The load consists of the parallel combination of the cavity compliance and the slot/orifice impedance as discussed in Chapte r 3 (section 3.6, Figure 3-20). Thus, a ZNMF actuator does not, in general, reduce to a second -order system. For this work, however, compact actuators are considered, and the low frequency performance of the ZNMF actuator is of interest. Hence the cavity compliance can be ignored (ass umed infinitely stiff), and the ZNMF actuator can be reduced to a nominal second-order system.

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131 Second, the load impedance is nonlinear, as di scussed in section 3.5.2; specifically, the dump loss resistance aO R is dependent on the output volume velocity. Thus a simple gainbandwidth type metric cannot be easily defined for these actuators. As the output volume velocity increases, the non-linear dump-loss re sistance increases driving the output volume velocity lower. Thus the output volume velocity is calculated iteratively and a linear input-output transfer function cannot be defi ned. In other words, the slop e of the volume velocity vs. frequency in the compliance dominated region (especi ally for frequencies n ear the resonance) is dependent on the input signal. Another key challenge for ZNMF actuators is identifying an appropriate objective function that maximizes the fluidic impact of these actuat ors. Since the applicatio n of ZNMF actuators in flow control is still an ongoing research topic, the specific needs are not well documented. Gallas et al. [58] presented optimiza tion strategies for improving th e performance of piezoelectric ZNMF actuators for flow control applications. The authors investigated two different objective functions(i) maximizing the total velocity over a range of frequencies and (ii) maximizing the velocity at a specific frequency. They also di vided the optimization problem into two separate problems(i) optimization of the orifice and cav ity for a given piezoelectric driver and (ii) optimization of the piezoelectric driver indepe ndent of the slot and cavity configuration. Chapter 5 first investigates the actuator pe rformance in various frequency regimes based on the lumped element model developed. In ZNMF actuators, the magnetic assembly and mechanical structure can be designed almost completely independently. These sub-components impact different performance parameters of the actuator. Hence these sub-components are investigated individually to identify various trends and limita tions. The overall system-level tradeoffs are then presented. To ensure maxi mum fluidic impact from these actuators three

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132 different objective functions are investigated be fore choosing the most appropriate one. The magnetic assembly and the diaphragm are then individually investig ated. Although the design tradeoffs and limits presented here are specific to the topology of the actuator chosen, the overall strategy is generic and can be adapte d to any actuator configuration. 5.2 Frequency Dependence of Actuator Performance The overall lumped element model for the electrodynamic ZNMF actuator was developed in section 3.6. The output res ponse function of the ZNMF actuato r was given by (3.61) and is included here again 2 211aD out eC ac aDaDaDaD aD aCavaOaCavaOSaO aOSaOaO eCsGCZ Q V G s CRsMsCCMsCRRsRRsMsC Z .(5.121) For compact actuators, the cavity volume V is chosen to be small, and hence the cavity compliance is also small. Thus for operating fr equencies below the Helmholtz frequency of the cavity, the cavity impedance aCavZ is very high. The validity of ignoring the cavity impedance essentially depends on the relative ma gnitudes of the cavity impedance (aCavZ) and the total slot/orifice impedance (aOS Z ). It should be noted that the sl ot/orifice impedan ce is not only frequency dependent but also am plitude dependent. In this work compact cavities are chosen. Hence for all cases, aCavaOS Z Z and thus ignoring the cavity impedance is reasonable. It should be noted that at frequencies closer to the Helmholtz freque ncy of the cavity and slot/orifice, the cavity ca n no longer be ignored. The equivalent circuit thus reduces to a single loop and the output response function in this case can then be simplified to

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133 2 2 2 21 1eC out ac aDaOS eC eC aDaD aOSaRadaOaO eC aD eCaD aDaOaDaD aDaosaRadaO eCGR Q V G ZZ R GR G RsMRRRsM Rs C GRsC G sMMCsCRRRR R (5.122) The resonant frequency of th e ZNMF actuator is given by 1 2res aPaPaOf CMM (5.123) A larger compliance and total acoustic mass will result in a lower resonant frequency. The output response function of the ZNMF actua tor can be further simplified for various frequency regimes. In the low frequency region, when the operating frequency is lower than the resonant frequency of the actuator (res f f ), the contribution of th e acoustic compliances is dominant, and the acoustic masses can be neglected. In this case, the outp ut response function of the actuator can be simplified to resout eCaD ac ffQ GRsC V (5.124) This can be further simplified by substituting for the transduction coefficient Gand the coil resistance eC R 1/res eCout coilwire wireaD aD aD eC ac wire wire coil ff G RBL C AB A Q GRsC sCs VS LS (5.125) where B is the flux density in the air gap, S is the effective area of the diaphragm, coilL is the total coil length and wireA and wire are the cross-sectional area and resistivity, respectively, of the wire. It can be seen that the output vol ume velocity obtained from the actuator at low

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134 frequencies increases linearly with the frequency. Moreover, in order to obtain large volume velocity at low frequencies, larg e values of magnetic flux density B and large aDC S are desired. Note however, if a large acoustic compliance is chosen to maximize the low frequency performance, the resonant frequency of the actuator will be lower. Also, the largest coil crosssectional area should be chosen to improve the low frequency response. At the resonant frequency of the device (res f f ), the reactive component of the overall impedance is 0. Note that the equivalent circui t of the ZNMF actuator is actually a two-loop circuit. Thus the reactive com ponent may not be exactly zero but very small. Thus the output response is given by 2reseC out ac ff aPaosaRadaO eCGR Q G V R RRR R (5.126) For most ZNMF actuators, the equivalent elec trical damping and the non-linear slot/orifice damping are the most dominant. Thus the out put response can be further simplified to 22 2 0 21 2reseC eC eC out out ac D ff aos aP aO aO aRad eC eC eC OSGR GRGR Q KQ V GG G RRRRR RR R S ,(5.127) where the expression for aO R presented in Chapter 3 is substituted. It can be seen that the output volume velocity at resonance is a comple x function of the actuator parameters. For frequencies greater than the resonant frequency but lesser than the Helmholtz frequency of the cavity and the slot/orifice ( H res f ff), the acoustic masses dominate the overall impedance of the actuator. The output re sponse function in this case is given by 2reseC eC out ac aDaO ff aPaDaosaRadaOaO eCGR GR Q G Vs M M RsMRRRsM R .(5.128)

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135 It can be seen that at high frequencies the output volume velocity decrea ses inversely with the frequency. It can be seen from (5.122)(5.128) that, to maximize the output of the actuator, the ratio eCGR is an important parameter. Moreover, the low frequency performance of the actuator is dominated by the acoustic compliance of the diap hragm. The performance in various frequency regimes and the desired characteristics of the actuator are summarized in Table 5-1. Table 5-1. Output velocity in different frequency ranges and desired actu ator ch aracteristics Frequency range Output volume velocity ( Q ) Desired device characteristics Low frequency ( f << fres) aD eCGsC R High G High CaD Low ReC At resonance ( f = fres) 2 eC aO eCGR G R R High G Low ReC High frequency ( f >> fres) eC aDaOG R sMM High G Low ReC Note that this analysis has been performed by assuming that the cavity impedance is much larger than the slot/orifice impedance. This assu mption is not valid for higher frequencies, where the capacitive impedance of th e cavity is not very high. Figure 5-4 shows the magnitude of the cavity im pedance and th e total slot impedance for the actuator A2 described in Chapter 4. The cavity impedance monotonically decreases with frequency. The predominant component of the slot impedance is the non-linear dump loss resistance, which is proportional to the output volume velocity. Thus the slot impedance shows a peak at near the resonance where the output volume velocity is high. It can be seen that the cavity impedance is at least 10 times higher than the slot impedance for frequencies up to 1000 Hz. Thus the fluid in the cavity behaves almost as an incompressible fluid. At higher fr equencies, the compressibility effects will be significant and

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136 thus the cavity cannot be ignored. Thus the assu mption that the cavity impedance can be ignored is valid. (Note that the slot and cavity impedan ces are in parallel). Similar trends are seen for actuator A1 and A3 as well. Figure 5-4. Comparison of the cavity and slot impedances for actuator A2 5.3 Design Tradeoffs for Composite Dia phragm Electrodynamic ZNMF Actuators Chapter 4 presented the fabric ation and model validation of a composite diaphragm ZNMF actuator. This section presents the general trends and tradeoffs affecting the overall performance of these actuators. The design tradeoffs for the ma gnetic assembly are discussed in section 5.3.1. In section 5.3.2, the variation of the lumped acoustic parameters for the composite diaphragms are first discussed, followed by discussion of the overall actuator performance. 5.3.1 Magnetic Assembly Design Tradeoffs The magnetic assembly determines the maximum force (blocked force) produced by the actuator, as discussed in Chapter 2. The bl ocked force produced by the electrodynamic actuation scheme is given by bcoilFBLI (5.129)

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137 where B is the magnetic flux density in the coil, Lcoil is length of the coil and I is the current through the coil. The cavity and slot impedance can be seen as the load against which the actuator has to act as shown in Figure 5-1. Thus in order to produce m a ximum output volume velocity, large blocked force is desired. The 1-D magnetic circuit model to predict the magnetic flux density in the air gap was presented in section 3.3.1. The magnetic asse mbly with all the dimensions is shown in Figure 55. The m a gnetic flux density in the air gap is given by 00 cm meffg magnetmHh B hAl A (5.130) where Hc is the coercivity of the hard magnet, and Am and Aeff are the area of the magnet and the effective area of the air gap. Since neodymium-iron-boron magnets are used in this work, 1magnet Thus the magnetic flux density can be further simplified to 0 0 cmm c effg effmmg mmHAh H B Al AhAl Ah (5.131) lg hg rg hm 2rm 2rt wcGap hb Figure 5-5. Magnetic assembly

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138 Some interesting observations can be made based on (5.131). It can be seen that if it is desired to m a ximize the magnetic flux density in the air gap, the ratio of the areas of the air gap and the magnet should be minimized. In addition, the ratio of the length of the air gap to the height of the magnet should be minimized. Thus, in order to maximize the flux density, a magnet with the largest volume (mmAh) should be chosen. By substituting (5.131) into (5.129), the blocked force is given by 0 cmmcoil b effmmgHAhLI F AhAl (5.132) Thus it can be seen that the blocked force is determined entirely by the magnetic assembly geometry and the length of the coil coilL. The coil resides in the air gap of the magnetic assembly. Thus the length of the coil is not co mpletely independent of the magnetic assembly geometry. The relationship between the length of the co il and the gap geometry can be investigated by considering a multilayer coil winding as described in Chapter 4. The schematic of the coil configuration is shown in Figure 5-6. The diameter of the copper wire is dwire, and the overall diameter of the wire with the insulation is dins. hg hm Figure 5-6. Coil configuration using a multiple layers of bondable copper wire

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139 The length of coil is a function of the magnet a ssembly geometry and the number of turns. The average length of each turn Lturn is given by 220.5turng mgLrrl. (5.133) Thus the total length of the coil with Nl layers and Nt turns per layer is given by 20.5coilltturn ltmgLNNLNNrl (5.134) The number of layers and the number of turns are also dependent on the magnetic assembly geometry. The maximum thickness and height of the coil are limited by the length of the gap lg and height of the gap hg, respectively. Thus the maximum limits on Nl and Nt are given by g l insl N d (5.135) and g t insh N d (5.136) The actual values of Nl and Nt are chosen to be the closest integer values smaller than the limits indicated above. Thus it can be seen that the gap size determin es the coil configuration. The interaction between the coil configuration and th e magnetic assembly design can thus be seen. The wire cross-section diameter corresponds to its wire gauge ( AWG ) and can be chosen as one of the design variables. The empirical relationship between th e wire gauge and the diameter of the wire ( dwire in mm) is given by [59] 36 390.127*92AWG wired (in mm). (5.137) The insulation thickness is usuall y dependent on the gauge of the wi re chose. For this work, the coil wire gauge is restricted to be between 34 AWG and 40 AWG. For this range, the insulation thickness is 0.0076 mm [53]. Thus the overall diameter of the insulated wire is given by 0.015inscoildd (in mm). (5.138) The maximum length of the coil Lcoil is given by

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140 220 5gg coiltlg mg insinslh LNNrrl dd (5.139) Thus from (5.132) and (5.139), it can be seen that maxi m i zing blocked force involves a coupled optimization of coil design and magnetic assembly design. If a very small gap is chosen, higher flux density can be achieved in the gap, but the le ngth of the coil will be smaller and vice versa. 5.3.1.1: Optimization of the magnetic assembly to maximize the blocked force As discussed in the previous section, there are conflicting design variables for maximizing the blocked force. This section presents the fo rmulation of the optimization problem to maximize the blocked force. All the dimensi ons of the magnetic assembly and the wire gauge are chosen to be design variables as listed below Magnet radius ( rm) Magnet height ( hm) Height of the gap (hg) Length of the gap (lg) Wire gauge ( AWG ) The design variable vector is thus mathematically represented as ,,,,mmgg X rhhlAWG (5.140) The constraints used are Upper bounds on all geometrical dimensions to limit the size of the device to 30 mm diameter, 30 mm tall cylinder. Lower bounds on all geometrical dimensions to ensure that manufacturing limits are not violated. The length of the gap is constrained to be 50 % of the magnet height to ensure that the 1D circuit model is valid. The power consumed in the coil is constrained. The overall mass of the magnetic assembly is also constrained.

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141 Each of the design variables is bounded by upper and lower bounds. This can be mathematically stated as LXU (5.141) These bounds are defined in Table 5-2. Table 5-2. Lower and upper bounds on the magnetic assembly design variables Design variable Lower bound ( L ) Upper bound ( U ) Magnet radius ( rm) (mm) 0.47 9.4 Magnet height ( hm) (mm) 0.635 12.7 Height of the gap (hg) (mm) 1.6 9.6 Length of the gap (lg) (mm) 1 8.5 AWG (gauge of the coil wire) 34 40 The magnetic permeability of the soft magnetic core material is very high and is ignored in the optimization as discussed in Chapter 3. The current density in the coil is assumed to constant and is limited by the maximum allowabl e Joule heating in the coil. The width of the annular soft magnetic piece ( wc) and the height of the bottom piece ( hb) are held constant at 5 mm. These dimensions mainly affect the we ight of the device. However, making these dimensions too small may result in the satura tion of the soft magnetic core. Finite element simulations of some typical magnet assembly geom etry show that there is no saturation for these dimensions of the soft magnetic core. The detail s of the finite element simulations are included in Appendix A. All the constants used are summarized in Table 5-3. Table 5-3. Constants used in the optim i zation formulation Parameter Value Remnant magnetization of the hard magnet 1.2 T Permeability of free space 1.25 x 10-6 H/m Resistivity of copper 17.2 x 10-9m Current density in the coil 4 x 107 A/m2 Density of the hard magnetic material 7500 kg/m3 Density of the soft magnetic material [40] 8180 kg/m3 Height of the bottom soft magnetic piece 3.2 mm Width of the annular soft magnetic piece 3.2 mm

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142 The objective function is to maximize the blocked force. The fmincon function from the MATLAB optimization toolbox is used to implement this problem. As this optimization routine is set up to minimize a given objec tive function, the objective function is stated mathematically as follows Minimize: ()obj b f XF (5.142) The optimization was first carried out with no po wer or mass constraints. In this case, the optimal design required the largest possible ma gnet and largest gap height to maximize the number of coil turns. The length of the gap was limited to 50 % of the magnet height as per the modeling constraint described a bove. The optimal design also re quired the largest wire crosssection (smallest AWG ). The maximum blocked force obtai ned in this case was 5.84 N with power consumption of 20.8 W and mass of 208 g. Clearly, the power and mass limits in a practical actuator would likely be much smalle r than these. Hence the optimal designs for various power and mass rest rictions were studied. Figure 5-7 shows the maximum achievable bloc ked force for various pow er constraints. The blue curve indicates m aximum achievable bl ocked force when only the power is constrained and the magnetic assembly is not constrained. Table 5-4 shows the optimized designs for this case. The param eters that are at the upper or lower bound ar e indicated in bold. It was seen that as the m aximum power consumption in the coil was reduced, the optimal design resulted in a magnetic assembly that had the largest allowable magnet (mr and mh ). The length and the height of the gap ( g l and g h ) reduced with reduced power c onsumption. Thus for lower power consumption, the optimal design had a smaller coil. However, when the maximum allowable power was set to very small value (< 0.15 W), the resulting optimal design was different from these trends. It can be seen from Table 5-4 that for very low power constraints (0.08 W 0.1 W

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143 0.15 W) the optimal design required the smallest possible gap dimensions ( g l and g h ). Consequently small wire cross-section is necessary in order to fit sufficient coil in the gap. The relationship between the maximum achieva ble blocked force and maximum allowable power can be deduced from the slope of th e line in the log-log scale as follows 1 3bFP. (5.143) The details of the optimized designs with mass constraints are included in Appendix C. Table 5-4. Optimized designs for maximum bloc ked force for various power constraints and no mass constraint (active cons traints indicated in bold) rm (mm) hm (mm) hg (mm) lg (mm) AWG P (W) Fb (N) Mass (g) 4.7 12.7 1.6 1 40 0.05 0.095 52 6 12.7 1.6 1 40 0.1 0.15 65 9.2 12.7 1.6 1 40 0.15 0.36 105 9.4 12.7 1.6 1 34 0.2 0.48 108 9.4 12.7 1.6 1.2 34 0.3 0.65 110 9.4 12.7 1.6 1.6 34 0.5 0.9 113 9.4 12.7 2 1.9 34 0.8 1.18 117 9.4 12.7 2.2 2.1 34 1 1.33 120 9.4 12.7 2.7 2.4 34 1.5 1.66 125 9.4 12.7 3.2 2.6 34 2 1.93 130 9.4 12.7 4.5 3.4 34 4 2.74 144 9.4 12.7 5.4 3.9 34 6 3.34 156 9.4 12.7 6.3 4.4 34 8 3.83 165 9.4 12.7 7 4.7 34 10 4.24 174 9.4 12.7 7.6 5.1 34 12 4.62 181 9.4 12.7 8.5 5.5 34 15 5.1 192 9.4 12.7 9.3 5.9 34 18 5.54 202 9.4 12.7 9.6 6.3 34 21 5.85 209 When mass constraints were included in a ddition to the power constraints, maximum achievable blocked force was reduced. Figure 5-7 shows the maximum achievable blocked force for various power constraints and three different m ass constraints. Two distinct regions can be observed in the optim al designs. When the maxi mum allowable power is small, the optimal blocked force is limited by the geometry parameters of the actuator. This is can be considered the geometry limited regime of the actuators. Similarly, the flatter regions in plot are mass

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144 limited. As the maximum allowable mass is increased, the mass limited region becomes smaller. For extremely small maximum allowable power (few cases shown here 0.08 W, 0.1 W, 0.15 W), the optimal design is limited only by the input maximum power allowed. Figure 5-7. Optimization results for maximizing blocked force for various power and mass constraints The optimal designs of the magnetic assembly in order to achieve maximum blocked force indicate that for a gi ven power and mass constraint, the la rgest magnet should be chosen. The geometry, power and mass constraints have a very large impact on the maximum blocked force. Depending on which of the two constraints is more restrictive, the device may be in a geometry limited or mass limited regime as indicated before. Note that in each of these regions, the power constraint is also active. However, for extremely small power constraints, the optimal design performance is limited onl y by the power as indicated. 5.3.2 Composite Diaphragm Design Tradeoffs The magnetic assembly and the coil configuration determine the blocked force achievable by the ZNMF actuator. In contrast, the composite diaphragm largely contro ls the dynamics of the Geometry limited Weight limited

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145 actuator. For compact actuators, where the cavity is small, the Helmholtz frequency of the cavity and the slot/orifice is much higher than the di aphragm resonant frequency. Thus the overall actuator resonant frequency is determined prim arily by the diaphragm resonance. The acoustic model for this composite diaphragm was intr oduced in section 3.4. 2 and all the detailed derivations are included in A ppendix B. This section will study the variation of diaphragm lumped parameters as a function of the geometry of the diaphragm. 5.3.2.1 Effective area ( S ) The concept of effective area was first introduced in section 3.4.2. The effective area of the PDMS diaphragm can be calculated by equating the to tal volume it displaces to that displaced by a rigid piston. This effective area is used to calculate the electro acoustic gyrator ratio G. Thus the diaphragm effective area is determined by calculating the total change in volume V and dividing it by the center displacement (0)w. Thus the effective area is (0)V S w (5.144) As the central boss is assumed to be ri gid, the effective area is given by 246 2 624246 24 4224133 3 33 312ln3 14ln abbb aaa abaabb S b bbb aabb a aaa (5.145) Note that when 0ba the effective area is 1/3 of the phys ical area of the diaphragm. When 1ba the effective area approaches the physical ar ea of the diaphragm, since the diaphragm is assumed to be rigid. The variation of the effective area of the diaphragm is shown in Figure 5-8. The effective area is normalized by th e physical area of the diaphragm (2a ). It can be seen that the effective area of the diaphragm increases mo notonically as a function of the solidity ratio. The two limiting cases ( 0baand 1ba ) can also be seen clearly in the figure.

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146 Figure 5-8. Normalized effective area as a function of the solidity ratio 5.3.2.2 Acoustic compliance (CaD) The acoustic compliance of the diaphragm is defined as the volume displaced by the diaphragm under unit pressure loadin g, when the electrical excita tion is zero. The expression for the acoustic compliance of the composite diaphr agm was presented in section 3.4.2 and is repeated here 022 6 246 22 624246 33 2211 33133 16 16aDaD Ca bbb Cab aa b b aaa Eh Eh .(5.146) The acoustic compliance is only a function of the solidity ratio ba, the diaphragm thickness and the material prope rties of the annular region fo r a rigid central region. The compliance of the diaphragm is the larg est when the solidity ratio is zero (0aDC i.e., when there is no inner circular boss a nd the diaphragm is homogenous, 0b ). As the solidity ratio increases, the diaphragm becomes stiffer, and the complian ce reduces. For the case wh en the solidity ratio is 1, (ba ), the acoustic compliance is zero, because the central boss is assumed to be rigid and the boundary condition at ra specifies zero displacement.

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147 Figure 5-9 shows the variation of the normali zed acoustic co m pliance as a function of the solidity ratio. The compliance is normalized by the correspond ing maximum compliance at a solidity ratio of 0 (0aDC). Note that this normalization produ ces a single curve that is solely a function of ba The plot indicates that the compliance becomes smaller as the solidity ratio increases in accordance with (5.146). The values of typical geom etrical parameters and m aterial properties are provided in Table 5-5. Figure 5-9. Acoustic compliance vari ation with the solidity ratio ( 0.08 ha ) It was seen from (5.125) that the low frequency perf orm a nce of the electrodynamic ZNMF actuator depends on the ratio aDCS This ratio can be derived from (5.145) and (5.146) and is given by 24 24 2 3 231 14ln 16aDa C bbb SEhaaa (5.147) It was discussed before that in order to maximize the low frequency output volume velocity of the ZNMF actuator, the ratio aDCS needs to be maximized. It can be seen from

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148 (5.147) that this can be achieved by choosing a diaphragm with large radius (a). However, it can also be seen that the ratio aDCS reduces monotonically as the solidity ratio ba increases. Thus the smallest possible solidity ratio should be chosen. However, it wi ll be shown further in this section that the resonant fre quency and hence the bandwidth of the ZNMF actuator increases with the solidity ratio. Thus th ere is inherent tradeoff between the low frequency performance of the ZNMF actuator and its bandwidth. Table 5-5. Diaphragm materi al properties and geometry Parameter Value Outer radius a 12.7 mm Inner radius b Variable Thickness h Variable Youngs modulus of the annular portion 2E 360 kPa Poissons ratio of the annular portion 2 0.33 Density of the annular portion 2 970 kg/m3 Youngs modulus of the central portion 1E 20 GPa Poissons ratio of the central portion 1 0.33 Density of the central portion 1 1190 kg/m3 Mass of the coil coilM 0.31 g 5.3.2.3 Acoustic mass ( MaD) The effective mass of the diaphragm is calcul ated by computing the total kinetic energy of the diaphragm and lumping it to that of an e quivalent mass moving with the center velocity, 00 wjw as discussed in section 3.4. 2. The attached coil also contributes to the total mass of the diaphragm assembly. The overall acous tic mass of the diaphragm is given by 22 2 22aDa rigid coil bM wr M Mhr d r SS (5.148) where Mrigid and Mcoil are the mass of the rigi d central boss and the mass of the coil respectively. Figure 5-10 shows the normalized acoustic ma ss of the diaphragm for various solidity ratios. The total acoustic m ass is normalized by the corresponding value of the total acoustic

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149 mass with 0 ba Again, this normalization procedure collapses the parametric dependence on ba to a single curve. As baincreases, the effective mass of the diaphragm decreases. Figure 5-10. Acoustic mass variat ion with the solidity ratio ( 0.08ha ) 5.3.2.4 Resonant frequency ( fres) The resonant frequency of the diaphragm can be calculated once the diaphragms total acoustic mass and acoustic compliance are known. The resonant frequency is given by 1 2res aDaDf M C (5.149) Figure 5-11 shows the variation of the resonant frequency with ba The diaphragm geometry and material propert ies used are indicated in Table 5-5. The thickness of the diaphragm is fixed at 1 mm It can be seen that the resonant frequency increases rapi dly as the solidity ratio becomes larger. This is due to the fact that both the acoustic compliance and the total acoustic mass reduce with the solidity ratio. The comparis on with finite element re sults is also shown on the same plot. Note that at solidity ratio of 1, the resonant frequency is infinite as the compliance is zero for the solidity ratio of 1.

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150 Figure 5-11. Resonant fre quency of the diaphragm ( 0.08ha ) 5.3.3 System-level Design Tradeoffs The two previous sections discussed th e design tradeoffs involved in two major components magnetic assembly and composite diaphragm of the electrodynamic ZNMF actuator. The system-level design tradeoffs are di scussed in this section. The electroacoustic transduction coefficient G is one of the key parameters of the actuator. The transduction coefficient G is defined as coil B L G S (5.150) Thus, for a given magnetic assemb ly and coil configuration if coilBL is assumed to be constant, the transduction coefficient is only a function of the diaphragm effective area S From (5.145) it can be seen that the effectiv e area is a f unction of the solidity ratio. Figure 5-12 shows

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151 the variation of the normalized transduction coeffi cient as a function of the solidity ratio. The transduction coefficient in each case is normaliz ed by the maximum transduction coefficient (at solidity ratio 0). As the solidity ratio increases, the effective area becomes larger. As the solidity ratio approaches 1, the annular compliant regi on of the diaphragm becomes smaller, and the effective area approaches the physical area of the diaphragm. Hence the normalized transduction coefficient approaches 1/3. It should also be noted that the transduction coefficient is independent of the thickness of the diaphragm. Thus in order to ach ieve large transduction coefficient, smaller solid ity ratios are preferred. Figure 5-12. Variation of electro acoustic transduction coefficient G with solidity ratio ( 0.08 ha ) As indicated by (5.124) (5.128), in addition to the tran sduction coefficient, the ratio eCGR is also an important factor impacting the ac tuator performance. The ratio is given by coilcoilwirewire eCeCwirecoilwire B LBLABA G RSRSLS (5.151)

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152 where Awire is the cross-sectional area of the wire and wire is the resistivity of the coil material in -m. Thus in order to maximize eCGR, a coil with small resistivity wire and large area Awire should be chosen. The magnetic flux density in the gap Bshould also be maximized. Thus a long magnet (large mh ) with large coercivity cH should be chosen. Note that a larger magnet will result in a heavier actuator. Thus there is a tradeoff between device performance and weight of the device. The resonant frequency of the actuator is dete rmined primarily by the resonant frequency of the diaphragm assembly, because the cavity is de signed to have a small volume. The resonant frequency of the diaphragm increases rapidly with the solidity ratio as can be seen from Figure 511. Thus high resonant frequency and high bandwidth require larger solidity r a tio. However, as discussed before, a large volume ve locity requires a small solidity ratio. Thus there is a tradeoff involved between large output volume velocity and high bandwidth. The tradeoff between the output volume velocity and the bandwidth can be studied by investigating the overall performance of the ZNMF actuator for various solidity ratios. Figure 5-13 shows the predicted output velocity as a function of the frequency for four different solidity ratios ( 0.50.8 ba ). The magnetic assembly parameters are detailed in Table 5-6 and the diaphragm geom etry and material properties are indicated in Table 5-5. The diaphragm dam ping ratio is assumed to be constant at 0.15. As the solidity ratio increases, the diaphragm becomes stiffer, thus increasing the resonant frequency and lowering the output velocity. Thus there is an inherent tradeoff between maximum velocity and bandwidth. It can be seen that the maxresvf (maxv is the maximum rms velocity) product increases slightly with the solidity ratio, as the resonant frequency increas es faster than the dr op in the maximum output velocity.

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153 Figure 5-13. Variation of predicted ou tput velocity with solidity ratio ( 0.08ha) Table 5-6. Parameters for the magnetic assembly Parameter Value Magnet radius mr (mm) 4.7 Magnet height mh (mm) 6.35 Height of the air gap g h (mm) 3.2 Width of the annular position cw (mm) 3.2 Height of the bottom bh (mm) 3.2 Length of the gap g l (mm) 2.8 Total radius tr (mm) 10 Remnant magnetic flux density r B (T) 1.3 Relative permeability of the magnet magnet 1.05 Predicted magnetic flux density in the gap B (T) 0.4 5.3.4 Power Considerations The input and output power of the actuator can also be calculated using the lumped element model. Both input and output powers are a function of th e frequency of operation. The input power is the total powe r dissipated and is defined from the lumped-model as 2 2inD aPaOaos eCG PQRRR R (5.152)

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154 Similarly the output power is defined as the power lost as the dump losses or the power delivered to the non-linear resistance aO R 2 outoutaOPQR (5.153) where both QD and Qout are defined in section 3.6. Figure 5-14. Input and ou tput as a function of fr equency for actuator A2 Figure 5-14 shows both the input and output power as a functi on of frequency of operation for actuator A2 discussed in Chapter 4. It can be seen that the m a ximum of both the input and output power occur at the resonant frequency. It can be seen that the maximum efficiency of the actuator, defined as the ratio of output power to th e input power, is also a function of the frequency as seen in Figure 5-15. The maximum efficiency o ccurs at the resonance of the actuator.

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155 Figure 5-15. Efficiency of actuator A2 5.4 Optimization of Composite Diaphr agm Electrodynamic ZNMF Actuators This section details the coupled optimizati on problem that optimizes the overall ZNMF actuator including the magnetic a ssembly and the diaphragm. Fi rst various constraints for various components are discussed. Based on the give n application, it may be desired to establish the synthetic jet through an axisy mmetric orifice or a 2D slot. In Chapter 5, the optimization of an actuator with a 2D-slot is considered. The optimization routine uses the entire lumped model of the ZNMF actuator in cluding the cavity compliance, although it was shown that the effect of the including the cavity compliance is extremely small. The schematic of the electrodynamic ZNMF actua tor with all the important dimensions is shown in Figure 4-13. The design variables chos en for the optimization problem are all the dimensions of the magnetic assembly, dimensi ons of the diaphragm, the cavity length and the gauge of the coil used. All the de sign variables are listed below

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156 Magnet Radius ( rm) Magnet Height ( hm) Height of the gap (hg) Length of the gap (lg) Diaphragm outer radius ( a ) Diaphragm inner radius ( b ) Diaphragm height ( h ) Height of the cavity ( hC) AWG (gauge of the coil wire) The design vector is thus mathematically represented as ,,,,,,,,mmggC X rhhlabhhAWG (5.154) The constraints considered are the same as those chosen for the optimization of the magnetic assembly except the power constraint The constraints are listed below Upper bounds on all geometrical dimensions to limit the size of the device to 30 mm diameter, 30 mm tall cylinder. Note that thes e size constraints were chosen for a fair comparison with the previously investigated unoptimized prototype devices. (In general, the maximum size usually will depend entirely on the target application. The size used here is suitable for typical automotive flow control application.) Lower bounds on all geometrical dimensions to ensure that manufacturing limits are not violated. The length of the gap is constrained to be 50 % of the magnet height to ensure that the 1D circuit model is valid. The maximum diaphragm deflectio n is restricted to be 30 % of the diaphragm thickness. The total input power given by (5.152) is constrained. The overall mass of the magnetic assembly is also constrained. The upper and lower bounds for the magnetic asse mbly, diaphragm and the cavity are indicated in Table 5-7. The m echanical d amping of the diaphragm structure is not known beforehand and is usually extracted experimentally. The change in the damping ratio will affect the performance of the actuator and affect the optimal design. It was shown in Chapter 4 that the diaphragm damping is the smallest contribution to the overall damping in the actuator. The equivalent

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157 electrical damping and the fluidi c damping in the slot/orifice dominate the total damping in the actuator. However it is important to study the impact of change in mechanical damping on the actuator performance. Figure 5-16 shows the frequency response of actuator A1for several different damping ra tios (0.01-0.2). Figure 5-16. Frequency response of actuator A1 for di fferent diaphragm damping Table 5-7. Lower and uppe r bounds on design variables Design variable Lower bound ( L ) Upper bound ( U ) Magnet radius ( rm) (mm) 0.47 9.4 Magnet height ( hm) (mm) 0.635 12.7 Height of the gap (hg) (mm) 1.6 9.6 Length of the gap (lg) (mm) 1 8.5 AWG (gauge of the coil wire) 34 40 Diaphragm outer radius ( a ) (mm) 10 15 Diaphragm inner radius ( b ) (mm) 0.1 a Height of the diaphragm ( h ) (mm) 1 3 Height of the cavity ( hC) (mm) 1 3 It can be seen that as the damping ratio incr eases, the frequency response of the device is flatter. However, even if the damping ratio increases by a factor of 20, the maximum output

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158 velocity reduces only by about 30 %. For all the prototype actuators characterized in Chapter 4, the diaphragm damping was measured to be be tween 0.015 and 0.07. Thus as a conservative estimate, the diaphragm damping ratio is assumed to be 0.1 for optimization setup. 5.4.1 Choice of the Objective Function The choice of objective function in the case of electrodynamic ZNMF actuators depend on the application based on the type of fluidic impact these actuators need to produce on the flow field. Gallas et. al [58] used the total velocity over a predefined range of fre quencies as the objective function, which can be stated mathematically as Minimize:lim1 0()()f objrms f Xvfdf (5.155) Here rmsvis the rms centerline velocity. The frequenc y range chosen for this work was 0 300 Hz as indicated. However, several other objec tive functions may be used, and the resulting design may be significantly different in each case. A gain-bandwidth type objective function stated mathematically as Minimize:2m a x()obj res f Xvf (5.156) was also investigated. Here maxv is the maximum rms centerline velocity, and res f is the resonant frequency of the actuator. A third objective func tion investigated here was simply maximizing the rms centerline velocity at any given frequenc y. This is stated mathematically as follows Minimize:3m a x()obj f Xv (5.157) The baseline actuator considered for the opt imization is the actuator A2 described in Chapter 4. This baseline actuator was optimized with the two objectiv e functions described above. The power and mass was not constrained in both the cases. Figure 5-17 shows the output velocity f o r the baseline actuato r and the three optimized actuators. The first objective function (integrated frequency over a range of frequencie s) shows nearly 50 % improvement in the peak

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159 velocity as compared to the baseline device. However, the resonant frequency of the optimized device is lower (~ 85 Hz). The performance of the optimized device is slightly worse than that of the baseline device above 150 Hz. It can be seen that the when the sec ond objective function is used, the resonant frequency of the actuator is pushed very high while there is no significant improvement in the peak velocity obtained and the low frequency performance of the actuator is also poor. The use of the third objective function (m aximum velocity), results in an actuator with very high peak velocity (nearly twice the baseline actuator case). However, the actuator has a very low resonant frequency. Similar trends are observed for all three ob jective functions when the geometry constraints are changed and/or power and mass constraints are included. Figure 5-17. Comparison of the frequency resp onse of the baseline device and optimized designs for the three objective functions The optimized designs and some key perfor mance parameters for each of the objective functions are tabulated in Table 5-8. It can be seen that each optimal design uses the maximum diaphragm radius, m aximum wire size (smallest AWG ), largest cavity height and smallest diaphragm thickness. Moreover, fo r objective function 3 (maximum max resvf ), the diaphragm inner radius is the largest, making a very stiff device. Conversely for objective function 2

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160 (maximum maxv ), the optimal design has the smallest i nner diaphragm radius resulting in a very compliant device. Thus the resonant freque ncy in this is the smallest (~ 20 Hz) Table 5-8. Baseline actuator and optimized designs for objective functions 1-3 (active constraints indicated in bold) Case rm (mm) hm (mm) hg (mm) lg (mm) a (mm) b (mm) LC (mm) AWG h (mm) Power (W) Mass (g) Baseline actuator 4.77 9.5 3.2 2.4 12.7 9.5 3 34 1 0.8 57 Objective function 1 5.6 12.7 7.2 2.1 12.7 8 3 34 1 1.3 88 Objective function 2 3.6 12.7 9.6 4.1 12.7 1.4 3 34 1 2.716 86 Objective function 3 44.1 8.8 5.6 3.5 12.7 10.8 3 34 1 0.536 64 Table 5-9. Optimized designs for objective function 1 (maximum inte grated velocity) for various power constraints (active constraints indicated in bold) rm (mm) hm (mm) hg (mm) lg (mm) a (mm) b (mm) LC (mm) AWG h (mm) 5.6 12.7 7.2 2.1 12.7 8 3 34 1 5.6 12.7 6.6 2.1 12.7 8.1 3 34 1 5.7 12.7 5.5 2 12.7 8.3 3 34 1 5.8 12.7 4.4 1.9 12.7 8.3 3 34 1 6.3 10.6 4.9 1.4 12.7 8 3 34 1 5.3 8.8 2 2 12.7 7.7 3 34 1 5.4 11.9 2.9 1.1 12.7 7.3 3 36 1 For this work, objective functi on 1 (integrated velocity over 0 300 Hz) is used. The slot height, width and length are usually specified for a given application and are held constant here. The constraints upper and lowe r bounds set by manuf acturing limits, overall size specifications and power and mass limits have already been de scribed. If the maximum input power to the device defined by (5.152) is now constrained, the resu lting optim ized devices are different. Figure 5-18 shows the variation of the peak cen terline velo city as a function of the maximum allowable input power. The peak centerline velocity reduces (almost linearly) as the maximum input power is lowered, as exp ected. Similar behavior is also observed when the mass of the actuator is constrained. The optimized designs fo r the various power constr aints are tabulated in

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161 Table 5-9. The active constraints are indicated in bold. The other perform ance param eters of the optimized actuators are de tailed in Appendix C. 0 5 10 15 20 25 30 35 40 00.20.40.60.811.21.4 Maximum power (W)Peak centerline velocity (m/s) Figure 5-18. Maximum output ve locity as a function of maxi mum allowable input power for objective function 1 0 5 10 15 20 25 30 35 40 02 04 06 08 01 0 0 Maximum mass (g)Peak centerline velocity (m/s) Figure 5-19. Maximum output ve locity as a function of maxi mum allowable input mass for objective function 1

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162 When the maximum mass of the actuator was co nstrained, (without a ny power constraint) the maximum output velocity is initially mass limited, as seen from the linearly increasing region of the curve in Figure 5-19. As the maximum allowable mass is increased, the maximum output velocity obtained from the actuator starts tapering off as th e actuato r performance is now limited by the geometry constraints. It was seen that for the objective function 1, for any power constraint, the resulting optimized actuator had a lower resonant frequency than the baseline actuator. This can be seen from Figure 5-17 that the resonant frequency of the baseline actuator was close to 150 Hz and that of the o p timized device was about 80 Hz. In mo st cases, it is desired that the actuator have a certain resonant frequency and bandwidth. Thus optimization wa s performed again by including an resonant frequency in equality constraint (120r f Hz ). The frequency response of the optimized device without power and mass cons trains but including a resonant frequency constraint was shown in Figure 5-20. It can be seen including the resonant frequency resulted in an actuator that has bette r perform a nce over all frequencies as compared to the baseline actuator. The low frequency performance is slightly lower than the case where the resonant frequency was not included. It was seen that the using obj ective function 3 (maximum velocity) pegged the resonant frequency to 120 Hz and maximum veloci ty obtained was lower than the previous case. It was seen that the resonant frequency constraint did not impact the optimized design for objective function 2 (maximum max rvf ). For this work the optimized actuator for obj ective function 2 with a resonant frequency constraint is fabricated. The op timized design and the key performance parameters are shown in Table 5-10. The corresponding parameters for the baseline actuator are also shown. The active

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163 constraints are indicated in bold. The value of the objective f unction for the optimized device was 5640 m/s2. This can be converted to an average velocity of 18.8 m/s. Figure 5-20. Comparison of the frequency resp onse of the baseline device and optimized designs for the objective functions with a resonant frequency constraint Table 5-10. Optimized and ba seline actuator parameters fabricated for this work Design Variables Baseline Actuator Optimized Actuator Magnet radius ( rm) (mm) 4.7 5.5 Magnet height ( hm) (mm) 9.5 12.7 Gap height ( hg) (mm) 3.2 8.1 Gap length ( lg) (mm) 2.8 2.2 Diaphragm outer radius ( a ) (mm) 12.7 12.7 Diaphragm inner radius ( b ) (mm) 9.3 9.5 Cavity height (LC) (mm) 3 3 Diaphragm height ( h ) (mm) 1 1 Coil size (AWG ) 34 34 Other Parameters Number of coil layers 3 4 Number of turns per layer 30 30 Mass of the device (g) 80 95 Maximum input power (mW) 700 900 Predicted resonant frequency (Hz) 147 143 Predicted peak velocity (m/s) 21 35.2

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164 5.4.2 Parameter Variation One important consideration before fabric ating the optimized device is the study the sensitivity of the optimized actua tor performance to the various design variables. The linearized sensitivity of each of the design variables about the optimized point can be obtained from the optimization routine. This sensitivity is essent ially the slope of the objective function vs. the design variable curve at the optimal point. It can be seen that the optimal design is most sensitive to the diaphragm outer radius a. The sensitivity of the objective function to the gap height ( g h ) and the cavity height (CL ) is very small. However, the optimization problem is nonlinear and the linearized sensitivity cannot be extrapolated to predict the objective function for large variation in the design variables. Figure 5-21. Variation of the average velocity from 0 300 Hz with the change in the design variables Figure 5-21 shows the variation of the norma lized averag e v elocity generated by the actuators as the design variables are varied i ndividually. Only four de sign variables are shown

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165 here (magnet radius, diaphragm outer radius, inne r radius and wire gauge) The variation in the objective function when the other de sign variables are changed is relatively small as compared to these four variables. It can be seen that as the magnet radius (mr ) is reduced by 20 %, the actuator performance also reduces. However, the increase in the magnet radius (mr ) by 20 % does not produce similar increase in the obj ective functions. The objective function reduces significantly when the diaphragm radius (a) is reduced. It can also be seen that smaller wire gauge (larger coil diameter) is always desired. 5.5 Characterization of the Optimized Electrodynamic Actuator The actuator corresponding to the optimized de sign presented in the previous section ( Table 5-10) was fabricated using th e techniqu es described in Chap ter 4. This section presents the characterization of the optimized actuator. The fabrication and characterization procedure followed is the same as that described for the prototype actuators desc ribed in Chapter 4. The magnetic flux density for the optimized magnetic assembly predicted by the 1-D magnetic circuit model was 0.24 T. The average magnetic flux density in the gap predicted by finite element analysis was 0.27 T. The radi al magnetic flux density in the gap was also measured using a Lakeshore 475 DSP Gaussmeter with a Lakeshore hall probe HMNT-4E04VR. The average measured magnetic flux density in the air gap ( B ) was 0.25 T. The average coil diameter was measured to be 11.8 mm. The total length of the coil was calculated to be 5.1 m. The electrical impedance of the coil from 40 Hz 2000 Hz was also measured using the Agilent 4249A precision impeda nce analyzer. The resu lts of the impedance measurements are shown in Figure 5-22. The dc resistance of th e coil was m easured to be 4.7 The inductance of the coil was m easured to be 0.15 mH. The inducti ve reactance at 500 Hz (maximum frequency interest), was 0.47 Thus the coil inductance was ignored.

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166 The damping was measured using the impulse res ponse test as described in section 4.1.1 b. The results for the damping measurements are shown in Figure 5-23. The damping ratio and the resonant frequency were extracte d using the log-decrem ent method described in Chapter 4. The extracted damping ratio and resonant freque ncy were 0.053 and 154 Hz respectively. The corresponding diaphragm acoustic resistanceaD R is 1.26 x 105 N-s/m5. The model predicted resonant frequency of the diaphragm is 143 Hz. Figure 5-22. Impedance measurement results for the optimized actuator The output centerline velocity from the actua tor was measured using hotwire anemometry. The experimental setup and calibra tion procedure are detailed in Chapter 4. The hotwire results for the optimized actuator and the corres ponding model predictions are shown in Figure 5-24. The baseline actuator results are also shown on th e same plot. It can b e seen that the maximum velocity from the optimized actua tor nearly 50 % more than that from the baseline actuator. Moreover, the optimized actuator generates high er output velocities from 100 Hz 200 Hz.

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167 Figure 5-23. Damping measurement results for the optimized actuator Figure 5-24. Optimized and baseline actuator hotwire and LEM comparisons

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168 Thus it can be seen that the opt imization strategies developed in Chapter 5 can be used to achieve better performance from these actuators Even before trying to optimize the overall actuators, the inherent limitations based on size, power weight c onstraints can be deduced from the model.

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169 CHAPTER 6 SUMMARY AND FUTURE WORK The overall research goals of this work and the key contributions are summarized in Chapter 6. Several suggestions for improving a nd extending this work a nd possible avenues for further research are also identified. 6.1 Summary Active flow control is an expanding resear ch area with applications ranging from separation control in automotiv e and aerospace industry to ther mal management and fluidic mixing. One of the most important components of any active flow control scheme is the actuator itself. ZNMF actuators have been used for both open and closed loop flow control schemes. Thus understanding the behavior of the actuators and intelligently designing th em forms a significant portion of the flow control efforts. Most ZNMF act uators available in the literature have used piezoelectric transduction schemes. There ha ve been fewer efforts using electrodynamic actuation schemes. However, no systematic modeling and design tools have been reported. Thus this work aimed to present a detailed, first-principles-based, low-order model to predict the performance of electrodynamic ZNMF actuators and provide an effective tool for design and optimization of these actuators. An axisymmetric magnetic assembly with a cen tral cylindrical magnet was chosen for this work. The magnetic assembly was modeled using th e classical 1-D circuit approach to predict the magnetic flux density in the air gap. A circul ar clamped plate with a central rigid region and an annular region was chosen as the mechanical driver. This mech anical driver was modeled as an annular plate. The overall ener getic interactions in the system were modeled using the lumped element modeling approach. Note that the model developed here is specific to the actuator

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170 topology chosen. However, the general modeling approach can be adapted to any actuator configuration. In order to validate the model, several protot ype actuators were built. A unique, flexible hybrid fabrication method was developed to realize these actuators. The magnetic assembly was built using traditional machining techniques. The annular compliant portion of the mechanical driver was manufactured using an elastomer material (PDMS). The elastomer material provided the advantages of very large compliance, ease, flexibility and repeatability in manufacturing. The overall validity of lumped elemen t modeling approach was established by characterization of speaker-driven ZNMF actuators. Detailed characterization experiments on three different prototype PDMS diaphragm devices were pe rformed to validate the individual component models and the overall behavior of the actuator. The model predicted the output velocity of several ac tuator configurations with maximum error of 10 %. The validated model was then used to inve stigate several design trends and tradeoffs impacting the actuator performan ce. These tradeoffs can provide a first level design tool to achieve target actuator specifi cations without performing a full-blown constrained optimization. Based on the model, a complete constrained optim ization for three different objective functions was also performed. One optimized actuator was bu ilt and characterized. It was observed that the optimized device generated 50 % higher output veloci ty as compared to the baseline actuator. Thus the overall objective of the workto model, validate and optimize low-cost, compact, low power electrodynamic ZNMF actuatorswas reached. The optimized electrodynamic actuator had total volume of 15 cm3, total mass of 90 g, maximum input power consumption of 900 mW and generated a maxi mum output velocity of about 35 m/s.

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171 6.2 Suggested Future Work There are several different directions that ca n be pursued in order to improve and extend this work. Firstly, the model and design tradeoffs presented in this work are specific to the actuator topology used. However, the overall approach is generi c and can be extended to many different actuator topologies. So me possible actuator configurati ons may include rectangular or elliptical actuator configurations. Rectangular or elli ptical topologies may be able achieve comparable actuator performance for smaller volume as they conform more to the 2D slot configurations. The biggest ch allenge in this case would be to develop and validate the appropriate models for the mechanical driver. The mass of electrodynamic actuators is usually more than comparable-sized piezoelectric ZNMF actuators. The biggest contribution to wards the mass of the electrodynamic ZNMF actuator is the magnetic assembly. A detailed investigation of the magnetic assembly design particularly the soft magnetic core can provide several ways to reduce the mass. The other important improvement on this work w ill be the detailed investigation of scaling these actuators. Some preliminary model-based s caling analysis was presented in [60]. However, the fabrication methodology presented in this work will not be applicable for very small actuators. The most fabrication of the mechanical driver is probably th e most restrictive for realizing smaller actuators. Some possible alternative manufacturing techniques for the mechanical driver can include spin-coating of the elastomer material. The magnetic assembly fabrication for smaller size may include polymer or wax bonded hard magnets and electroplated soft magnetic materials. The fabrication of solenoidal coils will also present significant challenges at smaller size. Ribbon conductors may be used for very small gaps. Electroplating coils in preformed molds may be another possi ble option. However, beyond a certain size, conventional fabrication methods will be no longer applicable and the use of some surface and

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172 bulk micromachining techniques usi ng for the fabrication of MEMS (M icroE lectroM echanicalS ystems) will be necessary. These techniques can provide tremendous flexibility in the manufacturing and possibl e geometries of small electrodynamic ZNMF actuators. Finally, one of the most relevant extensions of this work will include the characterization of fluidic impact of these actua tors in actual flow conditions a nd identifying the key actuator requirements. This synergy between in actuator design and application requirements will be significant step towards taking these actuators fr om bench-top laboratory setting to real world application.

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173 APPENDIX A MAGNETIC CIRCUIT MODEL The 1-D circuit approach was used in Section 3.3.1 to model the magnetic assembly and predict the magnetic flux density in the air gap of the magnetic assembly. The inherent assumptions in this approach ar e that all the flux is flowing in the closed path defined by the magnetic circuit. The hard magnet acts as the source of the magnetic flux. Each of the components of the magnetic circuit has reluctance associated with it. The reluctance is based on the geometry and the material properties of the component and is a measure of the resistance the component offers to the flow the magnetic flux. The magnetic assembly and the 1-D circuit model are shown in Figure A-1. magnet core g ap Figure A-1. Magnetic assembly and the 1-D circuit model The reluctances of the magnet and air gap were de tailed in (3.24) and (3.26) respectively. The reluctance of the core was divided into three pa rts top, bottom and annular (3.27)-(3.29). The details of the core reluctance ca lculation are now detailed. The magnetic flux distribution in the magnetic assembly is shown in Figure A-2. It can be seen that the m a gnetic flux turns around the corner in the soft magnetic pieces. Thus to define the reluctance of th e core pieces, certain assumptions are required.

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174 Figure A-2. Arrow plot of magnetic flux density in the magnetic assembly Figure A-3. Reluctance compone nts of the magnetic assembly Magnet Bottom Annular Top

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175 Figure A-3 shows the reluctance components of the m a gnetic path. The green arrows indicate the reluctances of the magnet and the air gap discussed in Chapter 3. The red arrows represent the average flux path in the soft magnetic core and are used to define the reluctances of the top, bottom and the annular piece of the core. It was observed that the EFI 50 nickel-iron alloy used for the soft magnetic core had lower relative permeability than the 100,000 specified in the datasheet. Machining of this material and lack of high temperature anneal after machining result ed in relative permeability of about 100. However even with this lower permeability, the core reluctance is still much smaller than the air gap and the magnet reluctance and the average flux de nsity in the air gap is nearly the same. The core reluctance is thus ignored in the calculations. This can be justified by looking at the relative magnitudes of the reluctances of th e core, the air gap and the magnet. Figure A-4 shows the m a gnitudes of reluctances for the magnet, core and the air gap for various va lues of the air gap. The magnet and the core reluctance are independent of the air gap length. It can be seen that even for very small air gaps (~ 100 200 m) the air gap reluctance is the same order of magnitude as the core reluctance. However, due to manufacturing constraints the gaps considered will be at least a few millimeters and at these values of gap, the air gap reluctance is nearly 100 times the core reluctance. Thus it is justified to ignore the core reluctance in the magnetic flux density calculations.

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176 Figure A-4. Relative values of the relu ctances of the magnet, core and air gap A.1 Non-idealities in the Soft Magnet The magnetic flux density (B ) created in a soft magnetic materi al is directly proportional to the external applied field (H ) and the relative permeability (core) as shown by Figure A-1. At some applied field, however, all of the dom ains in the soft magnetic material are aligned to the external magnetic field, and any increase in the exte rnal field does not lead to an increase in the magnetic flux density (Figure A-5). This maximum magnetic fl ux density achievable in the soft m agnetic m aterial is called the saturation magnetic flux density Bs. The saturation flux density for EFI50 is specified to be 1.45 T [40]. It is always advisable to avoid saturation in the soft magnetic core in order to redu ce losses, non-linearities etc.

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177 Figure A-5. B-H curve for an ideal and typical soft magnetic material In the magnetic assembly chosen for this desi gn, the corners where the flux lines have to turn sharply are most susceptible to high flux densities and hence saturation of the soft magnetic core. Figure A-6. Results from the COMSOL m odel total magnetic flux density (Tesla) Gap Magnet Top Annular ring Bottom

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178 Figure A-6 shows the total magnetic flux density (in Tes l a) in the magnetic assembly. It can be seen that at one of the corners the magne tic flux density is high and can lead to magnetic saturation. If the thickness of the magnetic core material is chosen to be very small, the saturation problem will be more severe. In this case the thickness of the soft magnetic core (the thickness of top and bottom pieces and the thickne ss of the annular ring) is set to be 3.2 mm. Thicker core pieces will ensure no saturation, however the overall weight of the magnetic assembly will increase. The average magnetic flux density in the magnetic assembly is calculated to be 1.32 T. A.2 Magnetic Assembly for the Optimized Actuator The dimensions of the magnetic assembly for the optimized actuator are tabulated in Table 5-10. COMSOL Multiphysics was also used to veri fy the magnetic flux density in the air gap and ensure that the soft magnetic core does not have saturation. Figure A-7 shows the radial magnetic f l ux density in the magnetic assembly. The averag e radial magnetic flux density in the air gap was calculated to be 0.26 T. The 1-D magnetic ci rcuit model predicted the flux density in the air gap to be 0.24 T.

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179 Figure A-7. Radial magnetic fl ux density in the magnetic assembly of the optimized actuator The total magnetic flux density in magnetic assembly is shown in Figure A-8. It can be seen that near the corners of th e total flux density is high. However, the average flux density in the sof t magnetic core is 1.2 T which is lowe r than the saturation magnetic flux density of 1.45 T.

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180 Figure A-8. Total magnetic flux density A.3 Force Non-linearities The most generic expression for the Lo rentz force on the coil is given by r VVFJBdVJBdV (A.158) where Jis the current density through the coil and B is the magnetic flux density. For the axisymmetric magnetic assembly topology chosen he re, the current is in the angular direction ( ) and thus only the radial magnetic flux density (r B ) generates a force in the axial direction. The assumption of constant current density in the coil is reasonable particularly for low frequencies. If the radial flux de nsity linking with the coil is al so assumed to be constant, the classical Lorentz force equation (coil F BLI ) is obtained (section 3.3). However, the radial flux

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181 density is a function of both the axial and radial position. Moreover, when the coil moves during the device operation the flux density linking with the coil changes. This will result in an inherent directional non-linearity in the device operation. Figure A-9 shows the force calculated from COMSOL Multiphysics for various coil positions during the devi ce op eration for th e optimized actuator. The range of motion chosen here (0.3 mm) is the maximum range of motion in actual device. It can be seen that the maximum deviation from the cons tant force assumption is about 8 %. Figure A-9. Non-linearity due to sp atial variation of the flux density

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182 APPENDIX B DERIVATION OF LUMPED MODEL PARAMETERS OF A CLAMPED ANNULAR PLATE The mechanical driver of the electrodynamic ZNMF actuators consists of rigid center boss of radius band annular complaint re gion with outer radius a. The coil of mass coilMis attached to the rigid portion of the driver The schematic and the material properties of the two regions (Region 1 = rigid central bo ss, Region 2 = annular comp liant region) are shown in Figure B-1. The centr al boss is as sumed to be perfectly rigi d and the coil is assumed to add an additional mass. The mechanical driver is thus modeled as an annular plate. The as sumptions involved in the model are as follows The annular portion has uniform thickness and is isotropic. The maximum transverse deflection (w) is small (not more that 30 % of the thickness of the diaphragm i.e.()0.3 wrh ). Note that this constraint also depends on the solidity ratio ( ba ) [61]. A curve-fit may be used to obtain a solidity ratio dependent constraint on the maximum deflection. The detaile d investigation of this depe ndence is beyond the scope of this work. For moderate solidity ratios considered here (0.5 0.8), this assumption seems reasonable. All forces are normal to the plane of the diaphragm. The diaphragm thickness is small (0.1ha ). Figure B-1. Schematic for the mechanical driver an the simplified annular plate model

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183 Note that additional constraints to prevent buc kling of the annular diaphragm may also be included in the optimization formulation. Ba sed on these assumptions, the transverse displacements in the two regions are defined as 1 2()() for 0 () wrwrrb wrforbra (B.159) The outer edge r = a, is clamped and the boundary conditions can be stated mathematically as ()0, 0radw wa dr (B.160) The boundary conditions at the cente r require finite values of th e displacement and slope as given by 0(0), rdw w dr (B.161) In addition, matching conditions ar e necessary at the in terface between the inner circular portion and the annular outer region at r = b. The displacement and slope matching conditions at the interface are given by 12 12()(),rbrbdwdw wbwb drdr (B.162) From these boundary conditions and matching cond itions, the displacement of the diaphragm is calculated under a uniform pressure load p [62] as 24 24 2 3 231 (0)() 14ln 16pa bbb wwb Ehaaa (B.163) and 2 2224 22 2 34 231 () 14ln222 16brap brrbbrr wr Ehaaaaaa .(B.164) The typical displacement prof ile for 10 Pa uniform pressure load is shown in Figure B-2.

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184 Figure B-2. Transverse displa cement Model & FEM results ( ba= 0.5, ha= 0.08, a = 12.7 x 10-3, 2E= 360 kPa, 2 = 0.33). The acoustic compliance accounts for the change in volume ( ) for a unit of applied pressure, when the electrical voltage to the coil is zero ( V = 0 ). The change in volume is calculated by integrating the displacement of th e diaphragm over the entire diaphragm. Thus the lumped acoustic compliance CaD is given by 12 00 0()2()2()2aba b aD Vwrrdrwrrdrwrrdr C ppp (B.165) Using the displacement profile described prev iously, the acoustic compliance is given by 26 246 2 3 21 133 16aDa bbb C Ehaaa (B.166) The effective mass of the diaphragm is calc ulated by computing the total kinetic energy of the diaphragm and lumping it to the center velocity. Harmonic motion of the diaphragm is assumed. Since the coil is attached to the cen tral boss of the diaphragm assembly, it also contributes to the total mass of the diaphragm. The acoustic mass of the diaphragm is given by [42]

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185 2 2 0() 2a coil aDM wr Mhrdr S (B.167) where Sis the effective area of the diaphragm defined in Chapter 3. This can be simplified to 22 12 12 2 0() () 22ba coil aD brigid annularMMM wrwr Mhrdrhrdr S (B.168) where rigidM and annularMare the contributions of the rigid cen tral boss and the annular compliant region respectively. These are given by 2 64824 2 1 6810224 4918ln16ln8ln2 20156 1615rigidbbbbbbbb bh aaaaaaaa M bbbbbb a aaaaaa (B.169) and 4242 2 6810224 2 286120ln3236ln 5 20156 1615 2425110240annualrbbbbbb aaaaaa h M a bbbbbb aaaaaa bbb aaa 24 681 0224909 201561615 bb aa bbbbbb aaaaaa (B.170)

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186 APPENDIX C OPTIMIZATION DETAILS Chapter 5 detailed the optimization problem for the magnetic assembly as well as the overall electrodynamic ZNMF actuators. The optimization of magnetic assembly fo r maximum blocked force was presented in section 5.3.1. The optimized magnetic assembly designs for the mass constraint of 150 g and 120 g are tabulated in Table C-1 and Table C-2 respectively. Table C 1. Optimized magnetic assembly de signs for maximum blocked force for power constraint and mass c onstraint of 150 g rm (mm) hm (mm) hg (mm) lg (mm) AWGP (W)Fb (N) Mass (g) 6 12.7 1.6 1 40 0.1 0.15 65 9.4 12.7 1.6 1.2 34 0.3 0.65 110 9.4 12.7 1.6 1.6 34 0.5 0.9 113 9.4 12.7 2 1.9 34 0.8 1.18 117 9.4 12.7 2.2 2.1 34 1 1.33 120 9.4 12.7 2.7 2.4 34 1.5 1.66 125 9.4 12.7 3.2 2.6 34 2 1.93 130 9.4 12.7 4.5 3.4 34 4 2.74 144 9.4 11.8 5.4 3.9 34 6 3.32 150 9.4 10.3 6.2 4.4 34 8 3.74 150 9.4 8.7 7.9 4.3 34 10 4.05 150 Table C-2. Optimized magnetic assembly de signs for maximum blocked force for power constraint and mass c onstraint of 120 g rm (mm) hm (mm) hg (mm) lg (mm) AWGP (W)Fb (N) Mass (g) 6 12.7 1.6 1 40 0.1 0.15 65 9.4 12.7 1.6 1.2 34 0.3 0.65 110 9.4 12.7 1.6 1.6 34 0.5 0.9 113 9.4 12.7 2 1.9 34 0.8 1.18 117 9.4 12.7 2.2 2.1 34 1 1.33 120 9.4 11.8 2.7 2.4 34 1.5 1.65 120 9.4 11 3.2 2.6 34 2 1.9 120 9.4 8.6 4.5 3.3 34 4 2.6 120 9.4 6.9 6 3.5 34 6 2.98 120 9.4 6.9 6 3.5 34 8 2.98 120 It can be seen that the as the mass of the magnetic assembly becomes restrictive, the mass constraint is satisfied by making the magnet smaller.

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187 The optimized designs for the overall el ectrodynamic ZNMF actuator for maximum integrated velocity over 0 300 Hz (objective function 1) for various mass constraints are tabulated in Table C-3. Table C-3. Optim ized designs for o bjective function 1 with various weight constraints rm (mm) hm (mm)hg (mm) lg (mm) a (mm) b (mm) LC (mm) A W G h (mm) Mass (g) Average velocity (m/s) 5.5 12.7 8.1 2.2 12.7 9.7 3 34 1 90 13.6 5.5 9.9 7.7 2.2 12.7 9.6 3 34 1 80 17.9 5.5 7.2 7.4 2.2 12.7 9.5 3 34 1 70 18.7 5.5 4.9 688 2.2 12.7 9.3 3 34 1 60 18.6 5.5 4.5 4.2 2.2 12.7 9.3 3 34 1 50 18.1 4.9 4 3.4 2 12.7 9.3 3 34 1 40 16.7 It can be seen that as in the case of the optimization of the magne tic assembly alone, the mass constraint is met by making the magnet sma ller even for the optimized designs of the overall actuator.

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193 BIOGRAPHICAL SKETCH Ms. Janhavi Agashe received her bachelors degree in e lectrical engineering from Veermata Jijabai Technological Institute in August 2000. She got her Ma ster of Science in electrical e ngineering from Iowa State Univ ersity in August 2003. She started the Ph.D. program in the Department of El ectrical and Computer Engineering at th e University of Florida in August 2004. She received her Ph.D. from th e University of Florida in the spring of 2009