1. FACULTEIT INGENIEURSWETENSCHAPPEN Vakgroep Toegepaste Mechanica The design of a piezoelectric fan system for the flapping wing mircro-air-vehicle application Het ontwerp van een pizo fan systeem voor de flapping wing micro-air- vehicle toepassingEindwerk voorgelegd voor het behalen van de academische graad van Master in de Ingenieurswetenschappen door:Mohammad Ahmadi BidakhvidiAcademiejaar: 2008-2009 Promotor: Prof. Steve Vanlanduit

2. Dankwoord Ik wens iedereen die mij in de loop van dit academiejaar heeft geholpen om deze thesis tot een goed eind te brengen, te bedanken.Hierbij bedank ik vooral mijn promotor, Prof. Dr. Steve Vanlanduit, voor het scheppen van de mogelijkheid dit onderzoek te verrichten. Ook zou ik hem willen bedanken voor zijn vriendelijkheid, ondersteuning en advies. Mede door zijn opmerkingen en goede raad heeft hij me steeds op het goede spoor gezet.Prof. Dr. Ir. Dean Vucinic dank ik voor zijn toelichtingen en opmerkingen over de CFD simulaties. Ook Jean-Paul Schepens dank ik voor de technische ondersteuning die hij heeft gegeven bij de talrijke experimenten.Tot slot wil ik nog mijn familie bedanken om me gedurende heel mijn studie te steunen. i 3. Summary A micro aerial vehicle (MAV) is a semiautonomous airborne vehicle which measures less than 15 cm in any dimension. It can be used for video reconnaissance and surveillance. As demonstrated by birds and insects, flapping flight is advantageous for its superior maneuverability and much more aerodynamically efficient at small size than the conventional steady-state aerodynamics.Piezoelectric actuators are easy to control, have high power density and can produce high output force but typically the displacement is small. By using appropriate amplification mechanisms the piezoelectric actuators can generate enough thrust to be implemented in a MAV as a propulsion system. They can also be used to drive the flapping wings of MAVs.This research aims to develop a piezoelectric flapping wing system with 2 piezoelectric fans for MAVs. Various prototypes were made by attaching a flexible wing, formed by two spars and a flexible membrane, to two piezoelectric fans to make them coupled. The dynamic properties of the structures were characterized by using laser doppler vibrometer measurements.Theoretical models were used to analysis the performance of the piezoelectric fans at both quasi-static and dynamic operations, and the calculated results agreed well with the finite element analysis (FEA) modeling results. Several FEA models of the piezoelectric flapping wings were proposed and investigated. Selected factors such as geometric ratios, material selection, etc can affect the performances of the wing significantly. These influences have been investigated and optimization results were obtained using the FEA technique.Both numerical and experimental flow analyses are carried out on a piezoelectric fan. A 3D fluid- structure interaction computational fluid dynamics model was set up with commercial codes (CFX and ANSYS) to predict the velocity fields generated by the swinging movement of the piezoelectric fan. The flow measurements were carried out by using hot wire anemometry, particle image velocimetry and laser doppler anemometry. Thrust measurements were conducted to determine the feasibility of the use of piezoelectric flapping wing propulsion systemss in MAV applications.Two sinusoidal voltages with phase differences were then used to drive the coupled piezoelectric fans. High speed camera photography was used to characterize the two degrees of freedom motion of the wing. It has been observed that the phase delay between the driving voltages applied to the coupled piezoelectric fans play an important role in the control of the flapping and twisting motions (rotation) of the wing.ii 4. Samenvatting Een micro aerial vehicle (MAV) is een semiautonoom vliegtuig dat minder dan 15 cm in om het even welke afmeting meet. Het kan voor videoverkenning en toezicht worden gebruikt. Zoals aangetoond door vogels en insecten, is de klappende vlucht voordelig voor zijn superieure manoeuvreerbaarheid en veel meer aerodynamisch efficint bij kleine grootte dan de conventionele evenwichtstoestand aerodynamica.Pizo-elektrische actuatoren zijn gemakkelijk te regelen, hebben een hoge energiedichtheid en kunnen een hoge outputkracht veroorzaken, maar de verplaatsing is meestal klein. Door aangewezen versterkingsmechanismen te gebruiken kunnen pizo-elektrische actuatoren genoeg stuwkracht produceren die in een MAV als aandrijvingsysteem moet worden uitgevoerd. Zij kunnen ook worden gebruikt om de klappende vleugels van MAVs aan te drijven.Met dit onderzoek wordt getracht met 2 pizo-elektrische ventilatoren een pizo-elektrisch klappend vleugelsysteem te ontwikkelen voor MAVs. Diverse prototypen werden gemaakt door een flexibele vleugel, die door twee langsliggers en een flexibel membraan wordt gevormd, aan twee pizo- elektrische ventilators vast te maken om hen gekoppeld te maken. De dynamische eigenschappen van de structuren werden geanalyseerd door midden van laser doppler vibrometer metingen.De theoretische modellen werden gebruikt om de prestaties van de pizo-elektrische ventilators bij zowel quasi statische als dynamische werking te analyseren, en de berekende resultaten kwamen goed overeen met de resultaten van de eindige elementenanalyse (EEA). Verscheidene EEA modellen van de pizo-elektrische klappende vleugels werden voorgesteld en onderzocht. De geselecteerde factoren zoals geometrische verhoudingen, de materiaalkeuze, enz. kunnen de prestaties van de vleugel beduidend benvloeden. Deze invloeden zijn onderzocht en de optimalisatieresultaten werden verkregen door de EEA techniek te gebruiken.Zowel numerieke als experimentele stromingsanalyses werden uitgevoerd op een pizo-elektrische ventilator. Een 3D fluid-structure interaction computational fluid dynamics model is opgesteld met commercile codes (CFX en ANSYS) om de snelheidsvectorveld te voorspellen die door de harmonische beweging van de pizo-elektrische ventilator worden gegenereerd. De stromingsmetingen werden uitgevoerd door gebruik te maken van de hot wire anemometry, particle image velocimetry en laser doppler anemometry techniek te gebruiken. Metingen van de stuwkracht werden uitgevoerd om de haalbaarheid van het gebruik van pizo-elektrische klappende vleugelaandrijving systemen in MAV toepassingen te bepalen.Twee sinusodale voltages met faseverschillen werden gebruikt om de gekoppelde pizo-elektrische ventilators aan te drijven. Een hogesnelheidscamera werd gebruikt om de beweging van de vleugel met twee vrijheidsgraden te kenmerken. Er is vastgesteld dat het faseverschil tussen de elektrische voedingspanningen, die op de gekoppelde pizo-elektrisch ventilators worden toegepast, een belangrijke rol spelen in de controle van het klappen en verdraaien (rotatie) van de vleugel.iii 5. Resum Un micro aerial vehicle (MAV) est un aronef semi-autonome qui mesure moins de 15 cm dans n'importe quelle dimension. Il peut tre employ pour la reconnaissance et la surveillance visuelles. Comme dmontr par des oiseaux et des insectes, le battement du vol est avantageux pour sa manuvrabilit suprieure et est beaucoup plus arodynamiquement efficace de petite taille que l'arodynamique quilibre conventionnelle.Les dclencheurs pizolectriques sont faciles commander, ont la densit de puissance leve et peuvent produire une force haute production mais le dplacement typiquement est petit. En employant les mcanismes appropris d'amplification, les dclencheurs pizolectriques peuvent produire d'assez pousss pour tre mis en application dans un MAV comme systme de propulsion. Ils peuvent galement tre employs pour conduire les ailes de battement de MAVs.Ce recherche veut dvelopper un systme pizolectrique d'aile de battement avec 2 ventilateurs pizolectriques pour MAVs. Des prototypes divers ont t faits en attachant une aile flexible, constitue par deux longerons et une membrane flexible, deux ventilateurs pizolectriques pour les faire coupls. Les proprits dynamiques des structures ont t caractrises en employant des mesures de laser doppler vibrometer.Des modles thoriques ont t employs pour analyser l'excution des ventilateurs pizolectriques aux oprations quasi statiques et dynamiques, et les rsultats calculs taient conformes bien finite element analysis (FEA) modelant des rsultats. Plusieurs modles de FEA des ailes pizolectriques de battement ont t proposs et tudis. Les facteurs choisis comme des rapports gomtriques, le choix matriel, etc. peuvent affecter les excutions de l'aile de manire significative. Ces influences ont t tudies et des rsultats d'optimisation ont t obtenus utilisant la technique de FEA.Des analyses de flux numriques comme exprimentales sont effectues sur un ventilateur pizolectrique. Un modle informatique de dynamique des fluides d'interaction de la fluide-structure 3D a t install avec des codes commerciaux (CFX et ANSYS) pour prvoir les champs de vitesse produits par le mouvement d'oscillation du ventilateur pizolectrique. Les mesures d'coulement ont t effectues en employant hot wire anemometry, particle immage velocimetry et laser doppler anemometry. Des mesures de pousse ont t conduites pour dterminer la praticabilit de l'utilisation des systmes pizolectriques de propulsion d'aile de battement dans des applications de MAV.Deux tensions sinusodales avec des diffrences de phase ont t alors employes pour conduire les ventilateurs pizolectriques coupls. La photographie de camra grande vitesse a t employe pour caractriser les deux degrs de mouvement de libert de l'aile. On l'a observ que le retard de phase entre les tensions motrices s'est appliqu pizolectrique coupl de ventilateurs a un rle important dans la commande du battement et des mouvements de vrillage (rotation) de l'aile. iv 6. Contents List of Figures.......................................................................................................................................... vii List of Tables ............................................................................................................................................ xi List of symbols ........................................................................................................................................ xii Abbreviations ........................................................................................................................................ xiv 1Introduction and overview .............................................................................................................. 1 1.1Motivation of research ............................................................................................................ 1 1.2Micro Aerial Vehicles ............................................................................................................... 11.2.1 Fixed wing MAV ............................................................................................................... 31.2.2 Rotary wing MAV ............................................................................................................. 41.2.3 Flapping wing MAV .......................................................................................................... 5 1.3Piezoelectric actuators ............................................................................................................ 61.3.1 Piezoelectricity ................................................................................................................ 61.3.2 Piezo fans....................................................................................................................... 11 1.4Piezoelectric actuated flapping wings ................................................................................... 13 1.5Research objectives ............................................................................................................... 15 2Optimization of piezoelectric actuated wing structures ............................................................... 17 2.1Introduction ........................................................................................................................... 17 2.2Theoretical analysis of piezoelectric fans .............................................................................. 172.2.1 Introduction ................................................................................................................... 172.2.2 Analysis for bimorph actuators at quasi-static operation ............................................. 182.2.3 Analysis for unimorph actuators at quasi-static operation ........................................... 192.2.4 Analysis of the dynamic peak amplitude ....................................................................... 222.2.5 Electromechanical coupling factor (EMCF) ................................................................... 24 2.3FEM analysis of piezoelectric fans ......................................................................................... 252.3.1 Piezoelectric FEM Equations ......................................................................................... 252.3.2 Finite element software: ANSYS .................................................................................... 27 2.4Parametric optimization ........................................................................................................ 282.4.1 Validation of the finite element models ....................................................................... 282.4.2 Optimization Results and Discussion............................................................................. 30 3CFD Simulations ............................................................................................................................. 57 3.1Introduction ........................................................................................................................... 57 3.2Fluid Structure Interaction: coupling of CFD and FE analysis ................................................ 573.2.1 Defining the problem .................................................................................................... 59 v 7. 3.2.2Modeling........................................................................................................................ 603.3Numerical model ................................................................................................................... 623.3.1Analysis settings ............................................................................................................ 633.3.2Sensitivity analysis and convergence test ..................................................................... 653.3.3Results and discussion ................................................................................................... 683.3.4Conclusion ..................................................................................................................... 73 4Experiments ................................................................................................................................... 744.1Introduction ........................................................................................................................... 744.2Prototype design ................................................................................................................... 744.3Laser Doppler Vibrometer measurements ............................................................................ 754.3.1Measurements .............................................................................................................. 774.4Propulsion and energy consumption measurements ........................................................... 854.5Flow experiments .................................................................................................................. 874.5.1Introduction ................................................................................................................... 874.5.2Hot wire Anemometry measurements.......................................................................... 874.5.3Laser Doppler Anemometry measurements ................................................................. 934.5.4Particle Image Velocimetry measurements .................................................................. 994.6High speed camera visualization ......................................................................................... 108 5Conclusions.................................................................................................................................. 1125.1Conclusions and final remarks............................................................................................. 1125.2Recommendations for future work ..................................................................................... 113 A. Time-history solution of the velocity of the flow (CFD) .............................................................. 116 B. Time-history solution of the velocity of the flow (LDA) .............................................................. 134 C. Contents of the DVD .................................................................................................................... 137 Bibliography......................................................................................................................................... 138vi 8. List of Figures Figure 1.1: Basic concept of a MAV flight for surveillance applications. ................................................ 2 Figure 1.2: Black widow MAV .................................................................................................................. 4 Figure 1.3: Micro Flying Robot ................................................................................................................ 4 Figure 1.4: Microbat MAV ....................................................................................................................... 6 Figure 1.5: Piezoelectric material deformation depending upon the electric field and the polarization direction of the piezo material. ............................................................................................................... 6 Figure 1.6: Force-Deflection characteristics of piezoelectric actuators. ................................................. 8 Figure 1.7: Structure of bimorph piezo patches for (a) Bimorph in series connection (b) Bimorph in parallel connection. ................................................................................................................................. 9 Figure 1.8: Comparison between (a) bimorph bending actuator and (b) shear actuator. ..................... 9 Figure 1.9: A commercial bimorph piezo fan from [15]. ....................................................................... 11 Figure 1.10: Set-up and basic principle of an operating piezo fan. ....................................................... 11 Figure 1.11: Four-bar mechanism from [25] ......................................................................................... 13 Figure 1.12: Schematic of the coupled piezoelectric fans for MAV applications. ................................. 14 Figure 1.13: Piezoelectric flapping wing propulsion system. ................................................................ 14 Figure 1.14: Methodology used in this research to obtain the parameters for an optimal piezoelectric flapping wing prototype design............................................................................................................. 16 Figure 2.1: The effect of the thickness ratio on , fr, 0 and Fbl ............................................................. 21 Figure 2.2: Flow chart to determine the deflection at resonance. ....................................................... 23 Figure 2.3: Definition of short circuited and open circuited configuration for piezoelectric bending actuators................................................................................................................................................ 24 Figure 2.4: First three normalized bending modes for ANSYS and the analytical calculation. ............. 29 Figure 2.5: Amplitude of the tip of the piezo fan in function of the frequency. Comparison between the analytical and FEM (ANSYS) results................................................................................................. 30 Figure 2.6: Definition of the various models used in the FEA simulations............................................ 31 Figure 2.7: Element size h-convergence test on FEM models of the piezoelectric flapping wings. ..... 32 Figure 2.8: Basic piezo fan model with the definition of the length and width parameters. ............... 32 Figure 2.9: Optimization results for the piezoelectric fans with rectangular PZT-5H patches. First row: dynamic tip deflection in meters; Second row: EMCF (%); Third row: fA in m/s; Fourth row: first resonance frequency in Hz. ................................................................................................................... 35 Figure 2.10: Optimization results for the piezoelectric fans with triangular PZT-5H patches. First row: dynamic tip deflection in meters; Second row: EMCF (%); Third row: fA in m/s; Fourth row: first resonance frequency in Hz. ................................................................................................................... 36 Figure 2.11: The influence of the distance between the piezo patch and the clamping of the piezo fan on the Tip deflection, EMCF, fA and first resonance frequency of the piezo fan. ................................ 37 Figure 2.12: The influence of the elastic plate material on the Tip deflection, EMCF, fA and first resonance frequency of the piezo fan................................................................................................... 39 Figure 2.13: The influence of the thickness of the boundary layer, between the piezoelectric patch and passive plate, on the tip deflection of the piezo fan. ..................................................................... 40 Figure 2.14: The amplitude of the dynamic tip deflection in function of the frequency for different widths (under 120V). ............................................................................................................................. 40 Figure 2.15: A meshed FEA model of a piezoelectric flapping wing structure (Model 6a, bimorph). The mesh size is 0.5mm obtained after a convergence analysis. ................................................................ 41 vii 9. Figure 2.16: The amplitude of the dynamic tip deflection in function of the frequency for the diverse models under 120V. .............................................................................................................................. 41 Figure 2.17: The amplitude of the dynamic tip deflection in function of the frequency for different damping ratios under 120V. .................................................................................................................. 42 Figure 2.18: The influence of the voltage on the tip deflection and fA. ............................................... 42 Figure 2.19: The influence of the length ratio on the different optimization parameters for the different models. ................................................................................................................................... 44 Figure 2.20: Optimization results for Model 2a and Model 6a. First row: dynamic tip deflection in meters; Second row: EMCF (%); Third row: fA in m/s; Fourth row: first resonance frequency in Hz. . 45 Figure 2.21: Optimization results for the basic piezo fan model and model0a. First row: dynamic tip deflection in meters; Second row: EMCF (%); Third row: fA in m/s; Fourth row: first resonance frequency in Hz. ..................................................................................................................................... 47 Figure 2.22: Optimization of the length ratio for the proposed models (bimorph configuration) with different wing materials. ....................................................................................................................... 48 Figure 2.23: Piezoelectric flapping wing models with diverse plate materials. .................................... 50 Figure 2.24: The influence of the wing length on the performance of the piezoelectric flapping wing structure. ............................................................................................................................................... 52 Figure 2.25: Optimization results for models 4 and 6 with different width ratios. .............................. 53 Figure 2.26: The influence of the width of the piezo fans on the performance of the flapping wing structure. ............................................................................................................................................... 55 Figure 2.27: The influence of the thickness ratio and length ratio of the spar on the optimization quantities............................................................................................................................................... 56 Figure 3.1: The three dimensions of fluid dynamics. ............................................................................ 57 Figure 3.2: A time dependent pressure function can be applied on the plate to let it move similar to the first bending mode. ......................................................................................................................... 59 Figure 3.3: Total Mesh Displacement of the tip of the wing in CFX Solver. .......................................... 62 Figure 3.4: Velocity vectors colored by velocity magnitude (m/s) (Time=1.5280s) [40]. ..................... 63 Figure 3.5:The solution of the FSI model with the piezo fan oscillating at 60Hz with a tip deflection of 2cm. It can be observed that the solution converges to a certain result when smaller time steps are applied for the simulation. .................................................................................................................... 66 Figure 3.6: Time step size convergence plots for the velocity. (A) Relative error for velocity v. (B) Relative error for velocity vu. (C) Relative error for velocity vv.............................................................. 67 Figure 3.7: Comparison between the velocity vector field obtained by using a fine (left) and coarse (right) mesh. .......................................................................................................................................... 67 Figure 3.8: Geometrical properties of the enclosed space and the definition of the several locations in the fluid domain. The Cartesian coordinate system is placed in the base of the piezo fan. ................ 68 Figure 3.9: The velocity in function of the time in point 3. ................................................................... 68 Figure 3.10: The velocity in function of the time for the different positions........................................ 69 Figure 3.11: Velocity vector plot of the induced flow by the harmonic movement of the piezo fan at time=0.1s. .............................................................................................................................................. 70 Figure 3.12: Velocity vector field induced by a piezo fan [40] .............................................................. 70 Figure 3.13: Streamline for the flow pattern that has developed after time=0.1s. .............................. 71 Figure 3.14: The influence of the tip deflection on the velocity of the flow in point 3. ....................... 71 Figure 3.15: The influence of the frequency on the induced velocity by a piezo fan with a tip deflection of 2 cm.................................................................................................................................. 72viii 10. Figure 3.16: Influence of the frequency (averaged). ............................................................................. 72 Figure 3.17: 3D velocity vector field plot for a piezoelectric fan at time=0.12s. .................................. 73 Figure 4.1: First prototype of a piezoelectric flapping wing built for this work, using two coupled piezo fans. ....................................................................................................................................................... 74 Figure 4.2: The prototype with balsa wood operating at resonance. ................................................... 75 Figure 4.3: Basic set-up principle for the LDV measurement applied on the piezoelectric flapping wing. ............................................................................................................................................................... 75 Figure 4.4: First two mode shapes of a commercial piezoelectric fan with the frequencies for the open and short circuited configuration................................................................................................. 77 Figure 4.5: Results from the SLDV measurements. The tip deflection is obtained by driving the piezoelectric flapping wing at resonance and 130 VAC. ....................................................................... 83 Figure 4.6: Determination of the quality factor for Prototype 2. ......................................................... 84 Figure 4.7: Calculated damping ratios for the different piezoelectric flapping wing prototypes (by using the LDV measurement data). ....................................................................................................... 84 Figure 4.8: The electrical current and tip deflection in function of the applied voltage for the prototype in Figure 4.2. ......................................................................................................................... 85 Figure 4.9: The experiment set-up for the thrust measurements. ....................................................... 85 Figure 4.10: Definition of the geometrical variables for the thrust measurements. ............................ 86 Figure 4.11: Results of the thrust measurements. ................................................................................ 86 Figure 4.12: Hot wire probe measurements in a piezo fan flow. .......................................................... 88 Figure 4.13: Scheme of the CTA principle. ............................................................................................ 89 Figure 4.14: Set-up for the calibration of the hot wire anemometer. .................................................. 90 Figure 4.15: Calibration of the hot wire anemometer. ......................................................................... 91 Figure 4.16: Hot wire probe placed in the flow of the piezo fan. ......................................................... 92 Figure 4.17: Location of different measurement position for the hot wire experiment. ..................... 92 Figure 4.18: Results of the hot wire measurements. ............................................................................ 93 Figure 4.19: The LDA principles [44]...................................................................................................... 94 Figure 4.20: Location of the measurement grid for the LDA experiment. ............................................ 96 Figure 4.21: Results of the flow velocity in the different positions obtained using LDA measurements. ............................................................................................................................................................... 96 Figure 4.22: Results of the flow velocity using CFD simulation (the point is located in position 1 defined for the LDA measurements). .................................................................................................... 97 Figure 4.23: LDA velocity vector field of the flow generated by the piezo fan. The piezo fan is placed horizontally pointing in the positive y-direction. .................................................................................. 98 Figure 4.24: The time-averaged velocity vector field of the generated flow by the piezo fans over 1 oscillation (LDA measurement). The piezo fan is placed vertically pointing in the positive y-direction. ............................................................................................................................................................... 98 Figure 4.25: The basic set-up principle of particle image velocimetry. .............................................. 100 Figure 4.26: Basic working principle of PIV. ........................................................................................ 100 Figure 4.27: The employed experiment set-up for the PIV measurements. ....................................... 103 Figure 4.28: Residues on the glass enclosure due to the generated smoke during measurements. The bending of the piezo fan can clearly be observed. ............................................................................. 103 Figure 4.29: The piezoelectric flapping wing prototype in the Plexiglas enclosure. ........................... 104 Figure 4.30: A recorded image pair with a separation time of 50s (PIV measurement). The velocity vector field near the tip is obtained. ................................................................................................... 105 ix 11. Figure 4.31: Post processing of the results of the PIV measurements. .............................................. 106 Figure 4.32: The velocity vector field of the flow generated by the harmonic motion of a piezo fan with a tip deflection of 3cm, simulated with CFX (see 3.3.3). ........................................................... 107 Figure 4.33: Velocity vector field of the flow induced by a flapping piezo fan obtained with PIV measurements..................................................................................................................................... 107 Figure 4.34: Two piezoelectric fans with a phase delay of 180 degrees. ............................................ 108 Figure 4.35: Set-up for the high-speed camera recordings................................................................. 109 Figure 4.36: The tip deflection and used electric current in function of the phase delay for the prototype in Figure 4.2. ....................................................................................................................... 109 Figure 4.37: High-speed camera recordings of a prototype moving at the first bending mode. ....... 110 Figure 5.1: Stacking of multiple piezoelectric flapping wings. ............................................................ 113x 12. List of Tables Table 1.1: Comparison of PZT and PVDF material properties. ................................................................ 7 Table 1.2: Compressed Matrix Notation. .............................................................................................. 10 Table 2.1: Material properties of PZT-5H for analytic and FEA calculations. ........................................ 28 Table 2.2: Validation of the FEM model: comparison between the analytical and ANSYS solution..... 29 Table 2.3: Validation of the FEM model. ............................................................................................... 30 Table 2.4: Material properties of the elastic plate, wing and spars used in the FEA of piezoelectric flapping wings........................................................................................................................................ 41 Table 3.1: Material properties of air at 20C used in the numerical flow simulations. ........................ 60 Table 4.1: Definition of the geometrical variables of the models for the thrust measurements. The first resonance frequency is also specified. .......................................................................................... 86 Table 4.2: Parameters for the hot wire anemometer. .......................................................................... 90xi 13. List of symbols ROMANArea of cross section A Amplitude of the tip of the piezoelectric actuated structure,Width of beam and piezoelectric actuator Components of the mechanical stiffness Components of the electric flux density vector Components of the piezoelectric coupling (electrical field/stress) Components of the electric field vector Youngs modulus Components of the piezoelectric coupling (electrical field/strain) Frequency Components of body force vector Resonance frequency Blocking force Shear modulus h Enthalpy Moment of inertia = 1 Current Imaginary unit Length, Electromechanical coupling factor, Electromechanical coupling factors of piezo material Electromechanical coupling factors of a bimorph/unimorph Dynamic/effective electromechanical coupling factor Power, Pressure Surface change Quality factor Components of the strain Components of the mechanical compliance tensor , Components of the stress Thickness/height of layer k, , Internal energy density Displacements relative to , , respectively Voltage across electrodes, , Volume Cartesian coordinates GREEKDielectric permittivity in vacuum (= 8.85 10 ) Components of the dielectric permittivity tensor Variation (of length), distance Tip deflection Electromechanical coupling factor Dynamic viscosity Poisson ratio Normalized frequencyxii 14. FrequencyNatural frequency of mode kDensityDamping ratio MATRICES Dielectric permittivity matrixStructural damping matrixMechanical stiffness matrixPiezoelectric coupling matrix (electrical field/strain)Dielectric permittivity matrixPiezoelectric coupling matrix (electrical field/stress)Piezoelectric coupling matrix (electric flux density/strain)Piezoelectric coupling matrix (electric flux density/stress) Structural stiffness matrixEquivalent stiffness matrix ( Piezoelectric stiffness matrixDielectric stiffness matrixStructural mass matrixMechanical compliance matrix VECTORS Electric flux density vectorElectric field vectorForce vector Vector with nodal structural forcesEquivalent structural forcesVector with nodal chargesPart ofwith prescribed charge boundary distributionPart ofwith prescribed voltage boundary distributionPart of with prescribed charge boundary distributionVector with nodal chargesPart of with described voltage boundary distributionPolarization vectorStrain vectorStress vectorVector with nodal structural displacements MISCELLANEOUS MatrixMatrix transposedVector ,Vector transposedFirst and second time derivatives of, First and second spatial derivatives ofSpatial derivation operatorxiii 15. Spatial derivative of with respect to , Short notation for the spatial derivative of with respect to Complex conjugate of Abbreviations ABS Acrylonitril butadieen styreen ACAlternating current CCD Charge-coupled device CFD Computational fluid dynamics CFRPCarbon fiber reinforced polymer CTA Constant Temperature Anemometry DARPA Defence Advanced Research Projects Agency DCDirect current DOF Degree of freedom EAPElectroactive Polymers EMCFElektromechanical coupling factor FEFinite element FEA Finite element analysis FEM Finite element method FFT Fast fourrier transformation FRF Frequency response function FSI Fluid-structure interaction LDA Laser doppler anemometry LDV Laser doppler vibrometry MFI Micro flying insect MAV Micro aerial vehicle OCOpen circuited PIV Particle image velocimetry PTV Particle tracking velocimetry PVDFPolyvinylidene Fluoride PZT Lead zirconate titanate RMS Root mean square SCShort circuited SLDVScanning laser doppler vibrometer UAV Unmanned aerial vehicle VAC Volts Alternating Currentxiv 16. 1 Introduction and overview1.1 Motivation of research The recent advances of small CCD cameras, infrared sensors, etc have led to significant interest in small flying vehicles called Micro Air Vehicles (MAVs), which can perform as highly portable platforms for the tiny cameras and sensors. These aerial vehicles were originally proposed as extremely portable observation platforms for military applications. The potential of these flapping wing MAVs has resulted in extensive work in recent years.It is demonstrated by flying birds and insects that flapping flight and thus flapping wing MAV is advantageous for its greater maneuverability and lifting capability at low flight speeds in indoor environments. Insects can commence complex maneuvers like taking off backwards, flying sideways and landing upside down. Small flapping wing MAVs would not only move like insects, but with typical dimensions in only the millimeter range can also function almost unnoticed. The aerodynamic mechanisms allowing the high lift forces and maneuverability of insect flight are complex. Dickenson et al. [1] addressed and modeled three separate aerodynamic lift mechanisms in fruit flies. These mechanisms have been named delayed stall, rotational lift and wake capture. Delayed stall is a leading edge vortex on the wing due to a high angle of attack that would eventually cause the wing to stall. However, before stall occurs, a large increase in lift force is observed. Since the wing soon reverses direction, the leading edge vortex does not separate (stall). Rotational lift occurs when the wing is simultaneously translating and rotating. Finally, wake capture occurs when the wing reverses direction; since it has rotated, when the wing now meets the vortex that was attached to the wing during the previous stroke, a significant inertial lift spike is observed.The main goal of this present work is to obtain a propulsion system that can let a MAV achieve autonomous flight; specifically designing the power plant in such a way that the maximum thrust to power ratio is obtained.This work commenced with two difficult constraints. The first constraint was that the flying construction must be a MAV. By definition, MAV must have a total wingspan less than 15 cm. In our case this was an essential constraint for the flapping wing. The second constraint was that the flying object must fly by flapping wings or using flapping wings to maintain flight. The aerodynamics of flapping-wing flight, especially MAV size, is still not a fully-explored subject. There have been studies of insect flights, but unlike fixed-wing aerodynamics there have not yet been any available design rules for flapping-wing aerodynamics for MAV size.1.2 Micro Aerial Vehicles DARPA (Defense Advanced Research Projects Agency), the research and development organization for the Ministry of Defense of America, introduced the concept of an insect like miniature vehicle. The purpose of such flying objects was originally for military applications [2]. They defined a MAV to be sized less than 15 cm in length or width or height, weight less than 50 grams and capable of staying in flight for 20 to 60 minutes for a distance of 10 km. These size and weight restrictions put MAVs in a size class which is at least an order of magnitude smaller than other Unmanned Air Vehicles (UAVs). The MAVs could be applied to enter environments which are too risky for direct human intervention, for instance, searching for disaster survivors or detecting explosive devices planted in buildings. Other applications are communications, traffic monitoring, inspections, etc (Figure 1.1). This would necessitate a highly maneuverable capability of evading obstacles to access the targets. So next to the military applications a large number of commercial applications exist for this technology. The low1 17. detectability, low noise production, the ability to broadcast real-time data from an area of observation and the ability to maneuver within confined spaces, makes MAVs perfect for those applications. In recent years the size and weight constraints set in the definition of MAVs by DARPA have become quite flexible, with MAVs ranging from less than 10 grams to more than 300 grams. Figure 1.1: Basic concept of a MAV flight for surveillance applications.The Reynolds number is a ratio of the inertial to viscous aerodynamic forces used to characterize flight regimes, and is defined as: == =where is the fluid density, is the characteristic length (in this case the chord), is the fluid(10 or below) as compared to the conventional aircrafts (over 10 for fast-flying commercial aircraft) viscosity, and is the dynamic viscosity of the fluid. MAVs fly at a extremely low Reynolds numberdue to their small dimensions and low speed.A number of aerodynamic challenges exist for designing a MAV to obtain enough lift and low drag. The MAV must have only small amounts of material. This could also give the possibility to manufacture them very economically, meaning a swarm of MAVs could be used to deal with the problem at hand without requiring an optimal performance of each vehicle.Latest studies in the understanding of aerodynamics of flapping wing flight have led to new methods to realize the flapping wing flight. A vital challenge in creation of bird/insect-mimicking flapping machine is to select an actuator which could produce sufficient wing deflections.Existing MAVs can be classified into three broad categories based on the aerodynamic mechanisms used to generate lift: fixed wing, rotary wing and flapping wing. In the development of MAVs, an analogy can be drawn with the development of their larger, manned counterparts during the last century. Fixed wing technology was always a step ahead of rotary wing technology because of the added complexities involved in rotary wing flight. Likewise, among the existing MAVs, fixed-wing MAVs perform better than both rotary and flapping wing MAVs. Flapping wings, with their unsteady wing beating, establish an extra level of complexity above and beyond rotary wings, and hence their2 18. development seems to be the slowest. Fixed-wing MAVs have a better endurance than rotary and flapping wing MAVs. However, their major shortcoming is the lack of hover capability, which allows an MAV to maneuver in much smaller spaces.Propulsion mechanisms continue to stay an important limitation of MAV advancement. Most recent MAVs are electric-powered. Electric-powered systems efficiently convert stored energy into usable energy, but present battery equipment has a low energy density. Gasoline has a very high energy density, but combustion engines are very inefficient at the MAVs scale and produce a lot of noise. Existing propulsion systems obtainable for MAVs are not appropriate for long endurance, allowing less than 10 minutes of flight in many cases. MAV endurance is limited primarily by the efficiency of the system and by the propulsive efficiency. Rotary-wing MAVs are especially limited in endurance as they consume great amounts of power in order to hover. Thus, scientists have begun to investigate rotary and flapping-wing MAV designs, which can securely operate at low speeds and offer the possibility to hover. However, unlike fixed wings, these MAVs function in a more complex aerodynamic environment.1.2.1 Fixed wing MAV Most of the MAV have fixed wings. While the small size of MAVs is attractive, there are associated technology barriers. The most obvious are the complexity linked with miniaturization and our imperfect understanding of the complex low Reynolds number aerodynamic regime where MAVs operate. The MAV in [3] could cruise at the speed of 65 km/h at Reynolds number of about 130,000. However, as the size of a MAV reduces, the Reynolds number of the flow surrounding the MAV also decreases. The challenge to the fixed wing aircraft is that at low Reynolds number, the aircraft lacks of maneuverability and needs a large turning radius to navigate and avoid obstacles in a confined space. The fixed-wing designs are based on the conventional scaled-down aerodynamics and flight control approaches, and despite the ongoing research, these MAVs are not suitable for operations in constrained areas because of their relatively high stall speeds.A well-known fixed wing MAVs is Black Widow MAV [4], which has a wing span of 15.2 cm and can achieve a fly speed to 48.2 km/h, the maximum fly range to 1.8 km and the maximum fly altitude to 769 feet, and the endurance to 30 minutes. Multidisciplinary design optimization was employed to determine the battery, motor, gearbox, power requirements, propeller diameter, wingtip chord, and cruise velocity combination that would result in the best configuration. Black Widow can be used for missions such as target tracking and video monitoring. It could deliver live images in real-time via a custom-made color camera and transmitter. The model is based on a disc shape structure as shown in Figure 1.2, and many other MAVs have been developed with slight modification using a similar shape and concept.3 19. Figure 1.2: Black widow MAV1.2.2 Rotary wing MAV The main purpose to develop MAVs is for the military surveillance applications; hence an agile MAV would be more beneficial. If the MAV could hover or fly slowly, then it could explore and relay clear images back to the control centre. Helicopter MAVs are interesting because of their capability to hover. They have been constructed and studied by many researchers. Except carefully designed, rotors can suffer from performance degradation at low Reynolds numbers since their airfoils operate in a more challenging environment.The Micro Flying Robot (see Figure 1.3) is a rotary wing MAV developed by Seiko Epson in Japan, which weighs 8.9 gram. It has the ability to transmit images to a control centre via Bluetooth technology. Figure 1.3: Micro Flying RobotThe Pixel and Proxyflyer Micron are other types of rotary wing MAVs which can hover and fly in every direction. The Pixel weighs 6.9 grams and is a fully functional helicopter controlled by an infrared signal. The Proxyflyer has the identical weight as the Pixel, but designed under a different technical concept.The MAVs mentioned above are scaled down versions of helicopters. The smallest scaled down version of the helicopter is the Small Flying Helicopter, developed by Microtechnology in Germany with the dimensions of 24 x 8 x 0.4 mm and weight of 0.4 grams. It is powered by a 5mm long motor with a diameter of 2.4 mm. This tiny flying helicopter could take off at 40,000 revolutions per minute, but did not include remote control capability.4 20. 1.2.3 Flapping wing MAV For the miniature scale of MAVs, flapping wing vehicles could be the favored approach because of their presence in nature, and their capability to harness low Reynolds number unsteady vortex lift effects. That is why scientists are trying to mimic the wing motions of birds and insects to build flapping wing MAVs. There is a great amount of biological inspiration offered by nature. With the introduction of a continuously accelerating and decelerating wing, the aerodynamics of such vehicles is highly unsteady. Because they operate at low Reynolds numbers where high viscous effects dominate, they need high flapping frequencies and consume large amounts of power. Their tiny size also restricts their payload capacity. Additionally, the highly evolved motions involved with insect flight renders mechanical replication difficult and costly in terms of weight. Flapping is much more aerodynamically efficient than the conventional steady-state aerodynamics at small size [5].The wing kinematics of insects and birds are both based on flapping wings, but there is a fundamental difference between both [6]. Birds primarily utilize wing flapping for propulsion, while lift is generated by a combination of forward speed and wing flapping, causing the lack of hover capability. Most birds flap their wings in a vertical plane with small changes in the pitch of the wings during a flapping cycle. Since birds are much larger than insects, incorporating muscles, feathers and other moving parts into the wings is easier. Birds can control the shape and even the span of their wings to adapt to different flight modes. However, without large changes in pitch, this kind of flapping cannot produce sufficient vertical force to support the weight in the absence of any forward velocity. As a result most birds cannot hover. There are a great number of insects that can hover. These insects flap their wings in a nearly horizontal plane, accompanied by large changes in wing pitch angle to produce lift even in the absence of any forward velocity. Birds like the hummingbird, which are capable of hovering, have wing motions very similar to hover capable insects. Thus, insect-based bio-inspired flight may present a hover-capable and highly maneuverable solution for MAVs. There exist large differences between the flight kinematics of diverse insect species.It has been reported that the mass of the system has a large influence on the performance of the flapping wing MAV. Singh et al. [7] reported that when the mass of the flapping wing MAV increases, the maximum frequency of the mechanism needs to increase as well due to high inertial power requirements. Also, wing tests showed a decrease in thrust at high frequencies.Nature had millions of years to optimize its designs through the process of natural selection, therefore the understanding of the fundamental physics of flapping flight is important. However, simply copying biological morphology, kinematics, or behaviors could not certainly lead to an optimum system [8], because even if such an optimum system could be achieved, it may not be practical (and economical) due to external constraints such as availability of suitable material or nonexistent manufacturing techniques. In flapping flight, a mechanism that can imitate insect wing kinematics is also a major obstacle which requires newer materials such as Electroactive Polymers (EAP) for artificial muscles [9].The latest flapping wing MAV projects have adopted a battery powered electrical motor as an actuator. The rotary motion of the electrical motor is converted to a linear or flapping motion by a mechanism. The most common mechanism that can convert rotary motion to linear or flapping motion is a slider crank type four-bar mechanism.The flapping wing MAVs fly like birds in which the wings function as static lifting surfaces similar to conventional airplanes. Flapping is used along with changing angle of attack to generate forward thrust. One of the first flapping MAVs with static lifting surface wings was the Caltech Microbat [10].5 21. The Microbat was a 23-cm span, electric-powered ornithopter, developed in response to DARPAs original MAV initative. Microbat was built primarily of carbon fiber and Mylar, weighted 12.5 g and an endurance of 22 min, and was remotely piloted. This MAV flew like a bird however several claims of wake capture have been made for the vehicle. A super capacitor powered electric motor was applied. Some enhanced models have been derived from the original Microbat (Figure 1.4). Figure 1.4: Microbat MAVResearchers at Delft University of Technology in the Netherlands have created a micro UAV they have termed DelFly [11]. The DelFly had two sets of flapping wings, which allowed it to fly both fast forward flight missions and very slow (almost hovering) missions. It had a 35 cm wingspan, flapping frequency of 6 Hz, and weight of 17 g. The DelFly carried a video camera payload, allowing it to identify targets. It had an endurance of 12 minutes at a cruise velocity of 1.8 m/s. The latest version, Delfly II is a 30cm device and can fly horizontally at 15m/s but can also hover. Delflys performance is certainly noteworthy but with such a large vehicle, constrained indoor flight is still difficult due to maneuverability issues. At Harvard Micro Flying Insect (MFI) technology is used to realize takeoff of a tethered 60 mg flapping vehicle [12]. Woods vehicle flapped at approximately 110Hz. Though the vehicle was tethered and uncontrolled, it is the first vehicle of its size to produce thrust greater than its weight.1.3 Piezoelectric actuators 1.3.1 Piezoelectricity Piezoelectricity is a coupling between a materials mechanical and electrical behaviors. When a piezoelectric material is squeezed, an electric charge collects on its surface (direct effect). Conversely, when a piezoelectric material is subjected to an electric field, it exhibits a mechanical deformation (inverse effect). A basic illustration of converse piezoelectricity is shown in Figure 1.5. Applying an electric voltage to the electrodes of piezoelectric material will induce a mechanical deformation according to the magnitude and sign of applied voltage [13]. Figure 1.5: Piezoelectric material deformation depending upon the electric field and the polarization direction of the piezo material.6 22. The piezoelectric effects can be seen as transfers between electrical and mechanical energy. Such transfers can only occur if the material is composed of charged particles and can be polarized. For a material to exhibit an anisotropic property such as piezoelectricity, its crystal structure must have no centre of symmetry. Most of the piezoelectric materials are crystalline solids. They can be single crystals, either formed naturally or by synthetic processes, or polycrystalline materials like ferroelectric ceramics. Certain polymers can also be made piezoelectric by stretching under an electrical field.Piezoelectric ceramics are formed by conventional ceramic processing techniques, such as dry pressing, casting or extrusion. The ceramic material is then sintered, machined into the desired dimensions and pasted on electrodes. Polarization of the ceramic element is the final step in processing which involves heating the ceramic above the Curie temperature and subsequently cooling the material in the presence of a strong DC electric field. This poling process aligns the molecular dipoles of the ceramic in the direction of the applied field and thus induces its piezoelectric properties. Piezoelectric ceramics are hard, chemically inert and completely insensitive to humidity or other atmospheric influences. Their mechanical properties resemble those of the better known ceramic insulators and they are manufactured by much the same processes. Furthermore they are extremely stiff. They are capable of exerting or sustaining great stresses. One of the principal advantages of Lead Zirconate Titanate (PZT) ceramics is that their properties can be optimized to suit specific applications by appropriate adjustment of the zirconate-titanate ratio. They can be tailored to suit specific applications.Piezoelectricity can also be obtained by orientating the molecular dipoles of polar polymers such as Polyvinylidene Fluoride (PVDF) in the same direction. The PVDF can be made piezoelectric because fluorine is much more electronegative than carbon. The fluorine atoms will attract electrons from the carbon atoms to which they are attached. A sequence of processes, including elongation, annealing, evaporation of electrodes and poling, has to be performed to make the material piezoelectric. PVDF differs in many ways from the conventional crystalline and polycrystalline materials. In particular, PVDF is characterized by such properties as flexibility, ruggedness, softness, lightweight, relatively low acoustic impedance and low mechanical quality factor. The material is also available in thin films and in large sheets and is inexpensive to produce.Material propertiesPZTPVDF d33 (10-12 m/V)300-25 d31 (10-12 m/V)-150 15Relative permittivity d32 (10-12 m/V)-150 31800 12 (0 = 8.854 x 10-12 F/m) Young modulus50 5 Maximum operating14090 1 10 500 10 temperature (C) Maximum electric field (V/m) Density (kg/m3)7600 1800 Table 1.1: Comparison of PZT and PVDF material properties.A comparison of some of the physical properties of PVDF with those of PZT is given in Table 1.1 It is clear from the table that the piezoelectric strain constant d31, which relates the induced in-plane strain due to the electric field in the thickness direction, of PVDF is considerably smaller than the constant of PZT. Also the maximum operating temperature of PVDF is much lower than that of PZT which makes it less useful working in high temperature environment. The advantage of PVDF over PZT is that the7 23. maximum electric field strength that can be applied to the polymer without danger of depolarization is much greater.Actuator is a device which transforms energy into controllable motion. The primary performance characteristics of any linear actuator are displacement, force, frequency, size, weight and electrical input power. Piezoelectric materials are known for their excellent operating bandwidth and can generate large forces in a compact size, but traditionally they have very small displacements. They cannot be used directly as actuators in their raw form. So amplification is required.Piezoelectric actuators are usually specified in terms of their free deflection and blocked force. Free deflection (Xf) refers to displacement obtained at the maximum recommended voltage level when the actuator is completely free to move. Blocked force (Fb) refers to the force exerted at the maximum recommended voltage level when the actuator is totally blocked and not allowed to move. Deflection is at a maximum when the force is zero, and force is at a maximum when the deflection is zero. All other values of simultaneous displacement and force are determined by a line drawn between these two points on a force versus deflection line, as shown in Figure 1.6. For the piezoelectric actuators, the focus of research has been on an attempt to amplify the deflection of the material.Figure 1.6: Force-Deflection characteristics of piezoelectric actuators.There are different types of piezoelectric actuators: Stack actuator: a large number of piezo layers can be stacked to linearly increase their overall deflection while maintaining a low voltage requirement. The displacement and force of a stack actuator are directly proportional to the actuator length and cross-sectional area, respectively. Unimorph actuator: a composite beam is formed by attaching a plate with one active layer and one inactive layer, or substrate. Bimorph actuator: two thin ceramic plates bonded together and driven with opposite electric field. One plate expands while the other contracts. The net result is a lateral deflection of the plates. Piezoelectric bimorph is a bending element that generates horizontal displacement at the drive of electric field using the converse piezoelectric effect. There are two different electrical connections which are usually used in bimorph fabrication: one is series connection in which two piezoelectric plates have opposite polarization directions and the actuator is driven by applying electrical field between the top and bottom electrodes (see Figure 1.7(a)); the other is parallel connection in which two piezoelectric plates are of the same polarization directions and the actuator is driven by applying electrical field between surface electrodes and the bonding layer (see Figure 1.7(b)). In the latter case, two ceramic plates are electrically connected in parallel and driven voltage is applied across half the actuator thickness, thus enabling half driving voltage to achieve the same electrical field as in the series case.8 24. Figure 1.7: Structure of bimorph piezo patches for (a) Bimorph in series connection (b) Bimorph in parallel connection. Usually a metallic sheet or middle shim is sandwiched between the two piezoelectric platesto increase the reliability and mechanical strength. Unlike the PZT stack, bimorphs areoperated in the d31 mode.Shear actuator: This mechanism is another way to deflect the beam and create the wellknown fan moves. Piezo fans with this actuation mechanism are more difficult to make. Theproposed configuration is such that, this time, the d15 coupling coefficient dictates the design.In this situation the electric field is applied perpendicularly to the poling direction, inducing atransverse shear strain. The sandwich plate exists of the following components: the top andbottom layers are for example aluminum and the core is a shear actuated piezolayer. In casethe patch of this material does not cover the whole length of the sandwich plate, the coreshould be filled with a rigid foam material. The core should be softer than the faces and thickenough to produce shear stresses.Figure 1.8: Comparison between (a) bimorph bending actuator and (b) shear actuator.The use of stack actuators as bending actuators and shear actuators have been investigated in [14] and led to smaller tip deflections, therefore in this work only bending actuators were used.1.3.1.1 Constitutive equations An important characteristic of piezoelectric materials compared to other smart materials is its linear behavior within a certain range. The constitutive relations are based on the assumption that the total strain in the actuator is the sum of the mechanical strain induced by the stress and the controllable actuation strain caused by the electric voltages. The strain (S) - stress (T) - electric field (E) - electric 9 25. displacement (D) relationships of the piezoelectric materials can be approximated to have linear behavior:= += +where s and represent short-circuited elastic compliance and free electric permittivity of the materials, respectively. Note that IEEE compressed matrix notations (IEEE 1978) are used to denote the tensor variable. This consists of replacing subscripts ij and kl by p and q according to Table 1.2Ij or kl p or q 11 1 22 2 33 3 23 of 32 4 31 of 13 5 12 of 21 6 Table 1.2: Compressed Matrix Notation.The symbol dip represents the electro-mechanical coupling and is called the piezoelectric strain constant. Here, the first subscript i refers to the direction of applied electric field and the second subscript p refers to the direction of resulting strain. The piezoelectric strain constants of PZT and PVDF are (IEEE 1978):0 00 00=0 000 000 0where= ,=. It is clear that PZT and PVDF have three normal strains (S1, S2 and S3) when an electric field is applied in the thickness direction (subscript 3). When an electric field is applied in the inplane direction (subscript 1), the materials can also have shear strain (S5). In linear piezoelectricity, the constitutive relationships are often expressed with matrix notations as= += +where {S}, {T}, {E} and {D} represent strain, stress, electric field and electric displacement vector, respectively.=, = ,=, = Here, the matrices (SE), (d) and (T) represent the short-circuited (i.e., {E}={0}) elastic compliance, piezoelectric strain constants and free (i.e., {T}={0}) dielectric permittivity, respectively. 10 26. The alternate forms using alternative choices of independent variables for the above representation are: ==+ = + = + = = + Where the matrices (cE) and (S) represent the short-circuited stiffness and clamped (i.e., {S} = {0}) dielectric permittivity, respectively. The matrices (sD), (cD), (T) and (S) represent the open-circuited (i.e., ({D} = {0}) elastic compliance and stiffness, free and clamped dielectric impermittivity, respectively. The matrix (e) is the piezoelectric stress constants. The element dij of (d) represents the coupling between the electric field in the direction i (if a poling occurred, its direction is taken as direction 3) and the strain in the j direction. Sj = dijEi.1.3.2 Piezo fans The two most remarkable characteristics of the piezoelectric fans are their low noise levels and their low power consumption. These qualities make the piezoelectric fans well-suited for applications in the thermal management of portable electronic devices (see Figure 1.9). Figure 1.9: A commercial bimorph piezo fan from [15].A piezoelectric fan is fabricated by bonding a piezoelectric patch or several patches to a shim stock. If only one patch is used, this patch may be in any orientation on any side of the fan. However, in a two- patch configuration (one on each side) the patches are oriented such that when one expands the other contracts. A two-patch configuration is shown in Figure 1.10. An alternating voltage is applied to the piezoelectric patch in the fan. This causes the patch to alternately expand and contract. As this happens, the blade bonded to the piezoelectric patch flaps back and forth like a Japanese fan, but much faster to create a fluid flow. Figure 1.10: Set-up and basic principle of an operating piezo fan.11 27. The applied alternating voltage is at the frequency of the first resonance mode of the piezoelectric fan, driving the fan in resonance and letting it move in the first bending mode. Therefore the power consumption is minimized for the maximum tip deflection. Since the piezoelectric fans are driven at resonance, they are designed such that their fist mode of resonance will be outside the audible range (2.5). The thickness ratio must be as small as possible to obtain the maximal tip deflection. The EMCF gets maximal for a certain length ratio (between 0.6 and 1) and a thickness ratio (between 0.67 and 2.5). This optimal ratios are not coincident with the optimal ratios for the tip deflection and frequency. A compromise has to be made and one needs to determine what quantity has the greatest priority in the application. A greater oscillating frequency of the piezo fan creates a greater flow when the same dimensions are used. Multiplication of the resonance frequency with the dynamic tip amplitude gives an important quantity ( ) that needs to be maximized as well. From the obtained surf plots it can be seen that depending on the configurationgets maximized for a certain length ratio (0.36-1) and thickness ratio (0-1.17). The maximumfor the bimorph design is about twice the size of the unimorph configuration.It was found for the application of flapping wing actuation application, the best thicknesses of the materials available were the 127m thick PZT-5H and a stainless steel plate with a thickness greater than 110m. This thickness combination can produce the largest value with f