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Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
Masters' Thesis Piezo Fan Propulsion
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Masters' Thesis Piezo Fan Propulsion

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  • 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 piëzo fan systeem voor de flapping wing micro-air- vehicle toepassing Eindwerk voorgelegd voor het behalen van de academische graad van Master in de Ingenieurswetenschappen door: Mohammad Ahmadi Bidakhvidi Academiejaar: 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 efficiënt bij kleine grootte dan de conventionele evenwichtstoestand aerodynamica. Piëzo-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 piëzo-elektrische actuatoren genoeg stuwkracht produceren die in een MAV als aandrijvingsysteem moet worden uitgevoerd. Zij kunnen ook worden gebruikt om de klappende vleugels van MAV’s aan te drijven. Met dit onderzoek wordt getracht met 2 piëzo-elektrische ventilatoren een piëzo-elektrisch klappend vleugelsysteem te ontwikkelen voor MAV’s. Diverse prototypen werden gemaakt door een flexibele vleugel, die door twee langsliggers en een flexibel membraan wordt gevormd, aan twee piëzo- 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 piëzo-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 piëzo-elektrische klappende vleugels werden voorgesteld en onderzocht. De geselecteerde factoren zoals geometrische verhoudingen, de materiaalkeuze, enz. kunnen de prestaties van de vleugel beduidend beïnvloeden. Deze invloeden zijn onderzocht en de optimalisatieresultaten werden verkregen door de EEA techniek te gebruiken. Zowel numerieke als experimentele stromingsanalyses werden uitgevoerd op een piëzo-elektrische ventilator. Een 3D fluid-structure interaction computational fluid dynamics model is opgesteld met commerciële codes (CFX en ANSYS) om de snelheidsvectorveld te voorspellen die door de harmonische beweging van de piëzo-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 piëzo-elektrische klappende vleugelaandrijving systemen in MAV toepassingen te bepalen. Twee sinusoïdale voltages met faseverschillen werden gebruikt om de gekoppelde piëzo-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 piëzo-elektrisch ventilators worden toegepast, een belangrijke rol spelen in de controle van het klappen en verdraaien (rotatie) van de vleugel. iii
  • 5. Resumé Un micro aerial vehicle (MAV) est un aéronef semi-autonome qui mesure moins de 15 cm dans n'importe quelle dimension. Il peut être employé pour la reconnaissance et la surveillance visuelles. Comme démontré par des oiseaux et des insectes, le battement du vol est avantageux pour sa manœuvrabilité supérieure et est beaucoup plus aérodynamiquement efficace à de petite taille que l'aérodynamique équilibrée conventionnelle. Les déclencheurs piézoélectriques sont faciles à commander, ont la densité de puissance élevée et peuvent produire une force à haute production mais le déplacement typiquement est petit. En employant les mécanismes appropriés d'amplification, les déclencheurs piézoélectriques peuvent produire d'assez poussés pour être mis en application dans un MAV comme système de propulsion. Ils peuvent également être employés pour conduire les ailes de battement de MAVs. Ce recherche veut développer un système piézoélectrique d'aile de battement avec 2 ventilateurs piézoélectriques pour MAVs. Des prototypes divers ont été faits en attachant une aile flexible, constituée par deux longerons et une membrane flexible, à deux ventilateurs piézoélectriques pour les faire couplés. Les propriétés dynamiques des structures ont été caractérisées en employant des mesures de laser doppler vibrometer. Des modèles théoriques ont été employés pour analyser l'exécution des ventilateurs piézoélectriques aux opérations quasi statiques et dynamiques, et les résultats calculés étaient conformes bien à finite element analysis (FEA) modelant des résultats. Plusieurs modèles de FEA des ailes piézoélectriques de battement ont été proposés et étudiés. Les facteurs choisis comme des rapports géométriques, le choix matériel, etc. peuvent affecter les exécutions de l'aile de manière significative. Ces influences ont été étudiées et des résultats d'optimisation ont été obtenus utilisant la technique de FEA. Des analyses de flux numériques comme expérimentales sont effectuées sur un ventilateur piézoélectrique. Un modèle informatique de dynamique des fluides d'interaction de la fluide-structure 3D a été installé avec des codes commerciaux (CFX et ANSYS) pour prévoir les champs de vitesse produits par le mouvement d'oscillation du ventilateur piézoélectrique. Les mesures d'écoulement ont été effectuées en employant hot wire anemometry, particle immage velocimetry et laser doppler anemometry. Des mesures de poussée ont été conduites pour déterminer la praticabilité de l'utilisation des systèmes piézoélectriques de propulsion d'aile de battement dans des applications de MAV. Deux tensions sinusoïdales avec des différences de phase ont été alors employées pour conduire les ventilateurs piézoélectriques couplés. La photographie de caméra à grande vitesse a été employée pour caractériser les deux degrés de mouvement de liberté de l'aile. On l'a observé que le retard de phase entre les tensions motrices s'est appliqué piézoélectrique couplé de ventilateurs a un rôle 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 1 Introduction and overview .............................................................................................................. 1 1.1 Motivation of research ............................................................................................................ 1 1.2 Micro Aerial Vehicles ............................................................................................................... 1 1.2.1 Fixed wing MAV ............................................................................................................... 3 1.2.2 Rotary wing MAV ............................................................................................................. 4 1.2.3 Flapping wing MAV .......................................................................................................... 5 1.3 Piezoelectric actuators ............................................................................................................ 6 1.3.1 Piezoelectricity ................................................................................................................ 6 1.3.2 Piezo fans....................................................................................................................... 11 1.4 Piezoelectric actuated flapping wings ................................................................................... 13 1.5 Research objectives ............................................................................................................... 15 2 Optimization of piezoelectric actuated wing structures ............................................................... 17 2.1 Introduction ........................................................................................................................... 17 2.2 Theoretical analysis of piezoelectric fans .............................................................................. 17 2.2.1 Introduction ................................................................................................................... 17 2.2.2 Analysis for bimorph actuators at quasi-static operation ............................................. 18 2.2.3 Analysis for unimorph actuators at quasi-static operation ........................................... 19 2.2.4 Analysis of the dynamic peak amplitude ....................................................................... 22 2.2.5 Electromechanical coupling factor (EMCF) ................................................................... 24 2.3 FEM analysis of piezoelectric fans ......................................................................................... 25 2.3.1 Piezoelectric FEM Equations ......................................................................................... 25 2.3.2 Finite element software: ANSYS .................................................................................... 27 2.4 Parametric optimization ........................................................................................................ 28 2.4.1 Validation of the finite element models ....................................................................... 28 2.4.2 Optimization Results and Discussion............................................................................. 30 3 CFD Simulations ............................................................................................................................. 57 3.1 Introduction ........................................................................................................................... 57 3.2 Fluid Structure Interaction: coupling of CFD and FE analysis ................................................ 57 3.2.1 Defining the problem .................................................................................................... 59 v
  • 7. 3.2.2 Modeling........................................................................................................................ 60 3.3 Numerical model ................................................................................................................... 62 3.3.1 Analysis settings ............................................................................................................ 63 3.3.2 Sensitivity analysis and convergence test ..................................................................... 65 3.3.3 Results and discussion ................................................................................................... 68 3.3.4 Conclusion ..................................................................................................................... 73 4 Experiments ................................................................................................................................... 74 4.1 Introduction ........................................................................................................................... 74 4.2 Prototype design ................................................................................................................... 74 4.3 Laser Doppler Vibrometer measurements ............................................................................ 75 4.3.1 Measurements .............................................................................................................. 77 4.4 Propulsion and energy consumption measurements ........................................................... 85 4.5 Flow experiments .................................................................................................................. 87 4.5.1 Introduction ................................................................................................................... 87 4.5.2 Hot wire Anemometry measurements.......................................................................... 87 4.5.3 Laser Doppler Anemometry measurements ................................................................. 93 4.5.4 Particle Image Velocimetry measurements .................................................................. 99 4.6 High speed camera visualization ......................................................................................... 108 5 Conclusions.................................................................................................................................. 112 5.1 Conclusions and final remarks............................................................................................. 112 5.2 Recommendations 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......................................................................................................................................... 138 vi
  • 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 λ, fr, δ0 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.................................................................................................................................. 72 viii
  • 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 50µs (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. ............................................................ 113 x
  • 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 20°C 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. .......................................................................... 90 xi
  • 13. List of symbols ROMAN Area 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 ,ℎ Components of the stress ⁄ Thickness/height of layer k , , Internal energy density Displacements relative to , , respectively Voltage across electrodes , , Volume Cartesian coordinates GREEK Dielectric 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 frequency xii
  • 14. ⁄ ⁄ Frequency ⁄ Natural frequency of mode k − Density Damping ratio MATRICES Dielectric permittivity matrix Structural damping matrix Mechanical stiffness matrix Piezoelectric coupling matrix (electrical field/strain) Dielectric permittivity matrix Piezoelectric coupling matrix (electrical field/stress) ℎ Piezoelectric coupling matrix (electric flux density/strain) Piezoelectric coupling matrix (electric flux density/stress) ⋆ Structural stiffness matrix Equivalent stiffness matrix ( ∅ ∅ Piezoelectric stiffness matrix ∅∅ Dielectric stiffness matrix Structural mass matrix Mechanical compliance matrix VECTORS Electric flux density vector Electric field vector Force vector ⋆ Vector with nodal structural forces Equivalent structural forces Vector with nodal charges Part of with prescribed charge boundary distribution ∅ Part of with prescribed voltage boundary distribution ∅ Part of ∅ with prescribed charge boundary distribution Vector with nodal charges ∅ Part of ∅ with described voltage boundary distribution Polarization vector Strain vector Stress vector Vector with nodal structural displacements MISCELLANEOUS Matrix Matrix transposed Vector , Vector transposed First and second time derivatives of , First and second spatial derivatives of Spatial derivation operator xiii
  • 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 AC Alternating current CCD Charge-coupled device CFD Computational fluid dynamics CFRP Carbon fiber reinforced polymer CTA Constant Temperature Anemometry DARPA Defence Advanced Research Projects Agency DC Direct current DOF Degree of freedom EAP Electroactive Polymers EMCF Elektromechanical coupling factor FE Finite 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 OC Open circuited PIV Particle image velocimetry PTV Particle tracking velocimetry PVDF Polyvinylidene Fluoride PZT Lead zirconate titanate RMS Root mean square SC Short circuited SLDV Scanning laser doppler vibrometer UAV Unmanned aerial vehicle VAC Volts Alternating Current xiv
  • 16. 1 Introduction and overview 1.1 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 low 1
  • 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 number due 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 their 2
  • 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 MAV 1.2.2 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 Robot The 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 DARPA’s 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 MAV Researchers 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. Delfly’s 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]. Wood’s 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 material’s 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 properties PZT PVDF d33 (10-12 m/V) 300 -25 d31 (10-12 m/V) -150 15 Relative permittivity ⁄ d32 (10-12 m/V) -150 3 1800 12 (ε0 = 8.854 x 10-12 F/m) Young modulus 50 5 Maximum operating 140 90 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 the 7
  • 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 plates to increase the reliability and mechanical strength. Unlike the PZT stack, bimorphs are operated in the d31 mode. • Shear actuator: This mechanism is another way to deflect the beam and create the well known fan moves. Piezo fans with this actuation mechanism are more difficult to make. The proposed 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 a transverse shear strain. The sandwich plate exists of the following components: the top and bottom layers are for example aluminum and the core is a shear actuated piezolayer. In case the patch of this material does not cover the whole length of the sandwich plate, the core should be filled with a rigid foam material. The core should be softer than the faces and thick enough 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 ε 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 1.2 Ij 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 0 0 0 0 = 0 0 0 0 0 0 0 0 where = , = . 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 (εT) 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: = ∙ − ∙ = ∙ + ∙ = ∙ + ∙ =− ∙ + ∙ = ∙ − ℎ ∙ =− ℎ ∙ + ∙ Where the matrices (cE) and (εS) represent the short-circuited stiffness and clamped (i.e., {S} = {0}) dielectric permittivity, respectively. The matrices (sD), (cD), (βT) and (βS) 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 (<100Hz). Although it is possible to run the piezoelectric fans at higher modes of resonance, this is not preferred since the frequencies corresponding to these mode shapes fall in the audible range for relatively small-sized fans (1-5 cm). Piezoelectric fans were first discussed in the seventies [16]. The surge of portable electronics devices in the past decades has generated renewed interest in the use of piezoelectric fans as a very compact, low power, noiseless air cooling technology for applications as varied as mobile phones, laptop computers, DVD player and automobile multimedia boxes etc. A number of researches on piezo fans have been reported in the literature; almost of them are focused on the thermal performance for the cooling application. [16] found by placing a piezoelectric fan on the side of a power transistor panel of a television receiver it could decrease the temperature by 17 °C on the panel surface. Schmidt studied the local and average transfer coefficients on a vertical surface cooled by two piezoelectric fans resonating out of phase and found changing the distance between the fans and the surface, or the distance between these two fans would significantly change the transfer coefficients [17]. Ihara et al. [18] investigated the flows around the tips of an oscillating piezoelectric fan, and the discrete vortex method was used to numerically simulate the flow field. Yoo et al. [19] developed and tested several types of piezoelectric fans at 60 Hz, and at two AC voltage levels, 110 and 220 V. Different vibrating metal plates were tested and analyzed with PZT used for actuation. An optimization was also performed for the material of the vibrating plate for a piezoelectric fan to have a resonance frequency of 60 Hz, with the size of the piezoelectric patch unchanged. It was found that the most effective fan was the one made from a phosphor bronze shim and with PZT in a bimorph configuration. This phosphor bronze fan had a patch length of 33 mm, total length of 65 mm, width of 26 mm and thickness of 0.10 mm; under an input of 110 VAC at 60 Hz, this fan resulted in a tip deflection of 35.5 mm. The velocity of the air 1mm away from the fan was measured to be 3.1 m/s. The structure optimization of piezoelectric fans was studied by Bürmann et al. [20], and Basak et al. [21] performed an optimization study of a piezoelectric fan with two symmetrically placed piezoelectric patches. An analytical Bernoulli-Euler model, as well as a finite element (FE) model of the composite piezo-beam, were used in the modeling of the piezoelectric fan. A closed-form analytical solution was developed for the piezoelectric fan, and optimal patch-to-blade length and piezoceramic-to-blade thickness ratios were calculated for maximizing the electromechanical coupling factor (EMCF), tip deflection and rotation, and stroke volume rate. Simple design guidelines can be developed for low-power high-stroke piezoelectric fans based on such optimization studies. Using piezo fans in small-scale electronics for cooling applications were investigated by Wait et al. [22].The thermal performance of piezoelectric fans was investigated by experiments on the cooling of mobile phones and laptops. The thermal management of low-power electronics components was investigated experimentally and numerically for the piezoelectric fans for the cooling applications. Different parameters including the vibrational amplitude, the distance between the fan and the heat source, the fan length, resonant frequencies, and the fan offset from the centre of the heat source are considered in the investigation on the effects of the heat transfer from a small heat source. Although piezoelectric fans as a cooling application are a novel technology, the basic design rules for applying these fans into practical thermal solutions are not yet established. Complicated thermal phenomenon coupling with the three-dimensional, unsteady temperature and flow fields involved in such a small- 12
  • 28. scale device would be a significant challenge to the integration of piezo fans into portable commercial electronic devices. 1.4 Piezoelectric actuated flapping wings As we have seen earlier flapping wing systems are inspired by insect flight and usually involve the wing completing pitching, yawing and sweeping components of motion over one flapping cycle. Various mechanisms such as motor-driven actuators have been utilized to imitate this difficult flapping motion. Yet weight and mechanical system complexity are frequently experienced using these mechanisms. One of the most important constraints to the development of flapping wing MAV is the lack of a compact, high energy density propulsion mechanism. This is a great challenge, since at small MAV scales the mechanical components (bearings, conventional joints, etc) are too heavy and give inefficient actuation. They also require a complex control system. Piezoelectric ceramics offer the possibility for a flapping wing propulsion system with integrated electronics and simplified control systems. The piezoelectric actuators are also light weight and easy to integrate into the MAV platform. Some kind of motion amplification mechanism is required to achieve large displacements with these actuators. Piezoelectrically actuated four-bar mechanisms for MFI thorax were developed by Yan et al. [23]. A system with one piezoelectric unimorph actuator and three flexural-based mechanisms to transform the linear output of the actuator into single-degree-of freedom flapping motion was developed to imitate the flapping flight [24]. A four-bar linkage system (Figure 1.11) driven by lightweight piezo- composite actuator was developed by Park et al. [25] to mimicking the flapping wing system of insects. The University of California has done considerable work on a piezoelectrically actuated flapping wing MAV and used a pair of piezoelectric unimorph actuators and four bar mechanisms [26]. The four-bar mechanism had two flexible links. PZT-5H and PZN-PT based unimorph actuators were utilized at the input link of the four-bar. The kinematics and dynamics of the proposed wing structure with two parallel four-bar mechanisms were analyzed, and DC forces generated at the wing were computed for checking the feasibility of the design. They constructed a four-bar prototype using laser micromachining and folding techniques. Figure 1.11: Four-bar mechanism from [25] In comparison with the four-bar and other mechanism, the simplest motion amplification system is using a piezo fan at resonance [27]. The motion pattern of an insect flapping is quite complex as mentioned earlier and has three DOF's. To realize those three DOF, multiple piezo fans need to be used, since one piezo fan can only produce one DOF motion. By using two piezo fans in parallel connected to a wing structure, the ability is created to provide flapping and rotation motion by controlling the phase of the actuator input and the amplitude difference between the two piezo fans. 13
  • 29. Therefore for obtaining complex wing motions a control system has to be designed to drive multi piezo actuators with signals of different amplitudes and phases. This can also allow the tuning and control of the phase between the flapping and rotational motion, which is a key for flapping flight control. Commercial piezo fans are used to make flapping wing prototypes to obtain two degrees of freedom (DOF) motion for the wing of a flapping MAV. A new optimization criterion × has been introduced for piezo fans where is the fundamental frequency and is the vibration amplitude at the resonant frequency [28]. can also be described as the free tip deflection at the quasi-static operation. Optimizations for several piezo fan configurations have been calculated through analytical method and finite element modeling and then have been the criterion × is that it is measurable. It was concluded that this approach and this criterion compared to experimental results. Good agreements have been found between them. The advantage of provide a promising method to optimize piezo fans and piezo fan constructions for flapping wing MAV applications. Figure 1.12 shows two piezoelectric fans in anti-phase which will enable the wing to rotate. The wing will be flapping when the two fans are in-phase. To obtain complex wing motions it should be possible to drive multi piezoelectric actuators with signals of different amplitudes and phases. This design will also enable the adjustment and control of the phase between the flapping and twisting motions which is a key for flapping flight control. Figure 1.12: Schematic of the coupled piezoelectric fans for MAV applications. Another working principle is placing the piezoelectric flapping wings on a steady wing of a MAV (Figure 1.13), thus replacing the conventional propulsion system by a piezoelectric flapping wing propulsion system. If enough thrust can be generated using these flapping wing structures, the MAV would be able to sustain flight. This work is mainly focused on designing and optimizing such wings, however also attention is given to the first working principle (Figure 1.13). Figure 1.13: Piezoelectric flapping wing propulsion system. 14
  • 30. It can be concluded that a piezoelectric flapping wing construction based on piezo fans show potential to be used as a flapping wing for a MAV. In this work the focus mainly went to the propulsion of such flapping wings, and not the full control of these flapping wings. It has been tried to find a way to generate enough thrust to fly forward using these propulsion systems in a most efficient way. The total power consumption must be lower or equal to the existing MAV systems that are driven by other propulsion systems like propellers. To be effective the piezo fan needs to be in the same range of creating thrust as to a propeller. Basically the available commercial piezo fans have not enough tip deflection and surface area to generate an appropriate thrust. Therefore a wing construction is fabricated and used. This wing prolongs the piezo fan and exists out of two stiff spars and a skin between these spars to generate the air displacement. 1.5 Research objectives The primary objective of this research is to provide general guidelines for the design of piezoelectric actuated wings for MAVs. To achieve this general goal, an extensive research investigation has been conducted with the following objectives: 1. Developing miniature piezoelectric flapping wings for MAVs. 2. Analytical and numerical prediction of dynamic performance for piezoelectric bimorph structures, especially the dynamic peak amplitude at resonances; 3. Investigation of the effects of geometrical parameters, physical parameters, boundary conditions, bonding layers, etc by finite element method; 4. Optimization of geometric design including thickness, shape, location of piezoelectric actuators and wing construction; optimal material selection. 5. Numerical and experimental study of flow and generated by the proposed wing constructions with different conditions. Developing fluid-structure interaction models for the flow induced by the piezoelectric flapping wings. Understanding the principles of operation of the piezoelectric flapping wings by experimentally visualizing the fluid-structure interaction of the piezoelectric flapping wings. 6. Determining the feasibility of the use of piezoelectric flapping wings in MAV applications by conducting thrust measurements. This thesis is divided according to the above goals. In Chapter 2, analytical prediction of dynamic performance for piezoelectric bimorph structure is presented. Effects of material type of the flexible plate and wing are analyzed. Finite element analysis (FEA) is used to on the design of piezoelectric resonating structures for generating flapping wings which may be used for MAVs. The vibration characteristics of different piezoelectric structures are simulated by the finite element method and validated with analytical approaches. In Chapter 3 a fluid-structure model is established to investigate the velocity field created in the flow of piezoelectric flapping wings. In Chapter 4, the piezo fans and prototype flapping wings are experimentally investigated. Different flapping wing structures are designed and experimentally investigated using the Laser Doppler Vibrometry, Hot Wire Anemometry, Particle Image Velocimetry and Laser Doppler Anemometry. Finally, Chapter Fout! Verwijzingsbron niet gevonden. presents conclusions from this research and gives direction for future research. The methodology used in this research is summarized as a flowchart in Figure 1.14. Images and plots contain much more than tabular data. Generally the preference is thus given to graphs instead of tables. However the programs and scripts (MATLAB, ANSYS, CFX, ...), measurement data, tabular data, photographs of the experiments, etc can be found on the DVD of this thesis. 15
  • 31. Figure 1.14: Methodology used in this research to obtain the parameters for an optimal piezoelectric flapping wing prototype design. 16
  • 32. 2 Optimization of piezoelectric actuated wing structures 2.1 Introduction In this chapter important optimization parameters are studied. The influence of the other physical and geometrical parameters on the optimization parameters is investigated for the design of future piezoelectric flapping wings. First the analytical models applied on simplified models are introduced. For more complex models the optimization is done with the finite element method, which is described in the second part of this chapter. The optimization of the piezo fan design is complicated because the optimization criterion is application dependent and can vary among optimal mode shape, flapping frequency, tip deflection, maximal electromechanical coupling factor (EMCF), etc. A strongly related topic is the optimization of piezoelectric unimorph/bimorph structures since these have been thoroughly studied for the static and dynamic operations. The effectiveness of piezoelectric bimorph actuator to perform mechanical work under varies constant loading conditions using constituent equations for quasi-static operation has been investigated by Hsien-Chung et al. [29].The electromechanical coupling and output efficiency of piezoelectric bimorph and unimorph actuators in terms of maximization of three actuator characteristic parameters, namely electromechanical coupling coefficient, energy transmission coefficient and mechanical output energy for quasi-static operation was investigated by Smits et al. [30]. Dynamic and topology optimization of the piezo fan structures without load for the cooling application based on the maximizing EMCF using analytical solution and finite element modeling (FEM) have been reported recently [21]. Optimal design of the piezo fan configuration in practical operation is difficult because it requires a precise knowledge of the fan damping model which is lacking at present. A type of a piezo fan mechanism has been proposed for flapping wing micro-aerial- vehicle (MAV) application by Wang et al. [27]. = ∙ / , which divides stroke The Strouhal Number is often used in the analyzing of oscillating, unsteady fluid flow dynamics. It is an important dimensionless number and can be expressed as frequency ( ) and amplitude ( ) by forward speed ( ). is known to govern a well-defined series of vortex growth and shedding regimes and propulsive efficiency is high over a narrow range of St and usually peaks within the interval 0.2 < < 0.4. Most swimming and flying animals when cruising animals as = ∙ / , and can also be used as design guide ∙ = × operate at 0.2 < < 0.4. This can be used for the prediction of cruising flight and swimming speed for morphology and kinematics in MAV application, where ≈ 0.3 [28]. So an optimization criterion for wing can be introduced for the design of the flapping wing actuators where is the vibration frequency amplitude is largest at this frequency, ∙ can also be used as an optimization criterion for piezo fan and the amplitude. Since piezofan is usually operated at its first resonant frequency and its vibration where is its fundamental resonant frequency and its vibration amplitude. In this work, analytical solution and finite element modeling (FEM) will be used to analyze the performance of piezoelectric unimorph and piezo fan structures at quasi-static and dynamic operations, and these theoretical results are compared with experimental measured ones. 2.2 Theoretical analysis of piezoelectric fans 2.2.1 Introduction Before one can even try to understand the performance or the behavior of a piezoelectric flapping structure, it is necessary to gain more insight in the working of the actuators themselves. The bimorph and unimorph configuration are investigated here. Bimorph, meaning two piezoelectric patches 17
  • 33. attached to each other and unimorph, a composition of a piezoelectric patch and another non- piezoelectric elastic layer. After all, a piezoelectric fan can be a unimorph configuration (asymmetric fan) or an extension of a bimorph, namely a three-layer structure (symmetric fan). In §1.3.1 bimorph and unimorph configurations are already mentioned. In this chapter they will be examined in a more detailed way on how they both act in a static way. Concerning the dynamic behavior, a closer look is taken to the cantilever bimorph actuators. To end this chapter, the electromechanical coupling factor in general is introduced and a closer look is taken to both the bimorph and unimorph configurations, regarding the static case. Bimorph actuators consists of two thin ceramic plates bonded together and driven with opposite electrical field. Two types of connections are often used in bimorph fabrication. One is a series or antiparallel connection, in which two piezoelectric sheets with opposite polarization direction are bonded. The electrical voltage is applied across the total thickness. The electrical field E3 is the voltage divided by actuator total thickness 2h, with h being the thickness of a single piezoelectric layer. The other is the parallel connection, in which the two piezoelectric layers have the same polarization directions. The electric voltage is applied between the intermediate electrode and the top/bottom electrodes. Within the two piezoelectric layers, the polarity of driving voltage is opposite. The electric field E3 now is the voltage V divided by h. In both cases, one plate expands while the other contracts. The net result is a bending deflection. In parallel connection, the driving voltage can be reduced to half the value in the series case while keeping the same field strength. In some cases, a triple-layer structure is used in which a neutral elastic layer is sandwiched between two piezoelectric layers. In a unimorph actuator, one piezoelectric layer and one elastic layer are bonded together. When the piezoelectric layer is driven to expand or contract, the elastic layer resists this dimension change, leading to bending deformation. The use of an elastic layer can greatly increase the mechanical reliability of the actuator, which is an important issue in practical applications. One significant characteristic of bimorph and unimorph actuators is that they can generate the largest displacement among all piezoelectric actuators in the range of tens of microns to several millimeters, depending on the geometrical dimensions of the actuators and applied voltage. Which makes these devices such useful tools as convertors of electrical energy to mechanical energy and vice versa. 2.2.2 Analysis for bimorph actuators at quasi-static operation L, w and h are respectively the length of the materials, the width and the thickness of each strip. The length direction of the bimorph is chosen along the x axis, the width direction along the y axis and the height direction along the z axis. The deflection of the bimorph as it moves under the effect of a voltage is then measured along the z axis. The deflection of the tip of the bimorph is called , while the slope of the bimorph at its tip is called , this can also be seen as the tip rotation. The voltage across the electrodes is indicated with V and the charge on the electrodes is Q. An external moment at the tip is indicated with M, a force at the tip is represented with F and a body force, acting uniformly over the entire length of a bimorph, is indicated with p. The applied electrical voltage V across the total thickness of the actuator, results in an electric field E3, which is the voltage divided by the actuator total thickness 2h. The deflection of the elastic curve is written as z. Because of the choice of the orientation of the bimorph and the coordinate system, the constitutive equations reduce to = + 18
  • 34. = + with the occurrence of the elastic constant at constant electric field , the piezoelectric constant and the dielectric constant at constant stress . To obtain the constituent equations of the bimorph bender, one has to define the elements of the following matrix E: = The elements are obtained by differentiating the total internal energy of the bender with respect to the extensive parameters (V,M, F, p, which are the inputs of the system) [31]. For the piezoelectric fan concept only the output parameters tip deflection and tip rotation, with respect ot the applied voltage are valuable. So only elements and are important, which means the equation can be written as: = Which finally gives the used equation to calculate the tip rotation and static deflection: 6 = 2 2ℎ 3 = 2 2ℎ 2.2.3 Analysis for unimorph actuators at quasi-static operation An unimorph bender is a composite cantilever beam made up of an upper piezoelectric element, as always sandwiched between a pair of metal electrodes (this time made visible in the scheme), and a lower passive elastic element. The material for this element can be any elastic material, for example silicon. The same conventions are used for the axes of the piezoelectric as in the bimorph situation. When an external voltage, , is applied across the electrodes such that the external electric field, , is antiparallel to the polarization of the piezoelectric element, the piezoelectric material will expand in the plane perpendicular to and contract in the direction of if < 0 and > 0 as is commonly the case. Since the piezoelectric element is tightly joined to the lower elastic element, there will be reaction forces at the interface from the elastic element opposing the expansion of the piezoelectric element. This motional restriction will result in the downward bending of the structure. By changing the polarity of the external voltage, the beam will bend in the opposite direction. We have the following equations for the tip rotation and deflection: 6 = 3 = 19
  • 35. with the following definitions: = ℎ + ℎ ℎ ℎ +ℎ = ℎ + ℎ = ℎ +4 ℎ ℎ +6 ℎ ℎ +4 ℎ ℎ + ℎ When both the heights of the elastic ( ℎ ) and piezoelectric element ( ℎ ) and the respectively compliances and are equal, the equation for the bimorph configuration follows. Wang et al. [32] derived analytical expressions relating the bending resonance frequency, tip deflection, blocking force, equivalent moment with the geometrical dimensions, Young’s modulus, densities and piezoelectric coefficients of the cantilever When a piezoelectric unimorph cantilever is used as an actuator, part of the input electric energy is stored as electric energy due to the capacitive nature of the piezo-layer, and the other part stored as mechanical energy .Under a fixed external voltage , the freetip deflection is the maximum tip deflection and the blocking force is the maximum generative force (opposite in direction). For the first bending resonance of a unimorph cantilever we have: 3.52 +2 2 +3 +2 +1 = 4 3 1+ +1 1+ where = ; = ; = ; = + : Fundamental bending resonant frequency. : Young's modulus of passive elastic layer (metal). : Young's modulus of piezoelectric layer. : Thickness of passive elastic layer (metal). : Thickness of piezoelectric layer. : Total thickness of the unimorph cantilever. : Density of passive elastic layer (metal). : Density of piezoelectric layer. : Length of the unimorph actuator. : Young's modulus ratio of passive elastic layer and piezoelectric layer. : Thickness ratio of passive elastic layer and piezoelectric layer. : Density ratio of passive elastic layer and piezoelectric layer. For an unimorph cantilever, the tip deflection is expressed as equation: 3 2 1+ = 2 +2 2+3 +2 +1 is the electric voltage in Z-axis over the piezoelectric patches and is the transverse piezoelectric coefficient for the piezoelectric layer. This equation is under the condition of a fixed actuator thickness 20
  • 36. t, which is the piezoelectric layer thickness plus the elastic layer thickness. Sometimes, for obtaining transforming = ⁄ 1 + , so the equation becomes: the displacement for the fixed piezoelectric layer thickness with various elastic layer thicknesses, then 3 2 1+ = 2 +2 2+3 +2 +1 For blocking force, the unimorph cantilever has the formula: 3 2 = 8 +1 +1 actuator displaced at an external load , then the work carried out is − . This work reaches Where is the width of the cantilever, and the Young's modulus for the piezoelectric layer. If the maximum when the load is half of the maximum generative force −0.5 . The ratio between this maximum mechanical output energy and the input electric energy is defined as the energy transmission coefficient [33]: 8 +2 2 +3 +2 +1− 1+ 1+ = = −1 + 9 1+ Here is the piezoelectric coupling coefficient. Wang et al. used as a criterion for actuator design. 0,045 700 0 0 0,04 600 0 10 20 -0,01 0,035 -0,005 500 -0,02 0,03 Fbl fr 400 -0,01 -0,03 0,025 λ δ0 0,02 λ δ0 300 -0,015 -0,04 0,015 fr Fbl 200 -0,05 0,01 -0,02 0,005 100 -0,06 0 0 -0,025 -0,07 0 10 20 Thickness ratio Thickness ratio Figure 2.1: The effect of the thickness ratio on λ, fr, δ0 and Fbl The best actuators are those that have the largest energy transmission coefficient. The optimal thickness ratio obtained using this optimization criterion agrees with the results obtained based on the optimization of the EMCF ( [34], see later). However, is not a directly measurable quantity therefore solutions for , , is difficult to be compared with measurement results. Figure 2.1 shows the calculated analytical and as a function of the thickness ratio b for the PZT (PSI-5H4E)/Stainless- steel unimorph cantilevers. Considering the unimorph cantilever actuators with 43 mm in length and 10 mm in width, the calculation of the parameters was done as functions of the thickness ratio (telastic/tpiezo) for cantilevers under the applied voltage of 120V. It can be seen from this figure, that increases initially with the increasing thickness ratio, and reach a maximum, depending on the exact elastic layer. After the peak, it decreases with the increasing thickness ratio. So an optimal thickness ratio can be found. 21
  • 37. 2.2.4 Analysis of the dynamic peak amplitude Smits and Ballato [35] have extensively analyzed the dynamic properties of bimorphs. Their model shows itself to be efficient in locating the resonance and antiresonance frequencies of symmetrical bimorphs. However, the dynamic tip deflection at resonances cannot be determined by their expressions when an electric field is applied on the bimorph structure. Hence, there is a need for studies determining the peak amplitude at resonances, which is very important in many practical applications of bimorphs. This is what Wu and Ro [36] do in their exploration and they proposed a new method to calculate the tip deflection at resonances, based on the damping ratio of the structures. 2.2.4.1 Determination of the resonances and the amplitude at resonance The dynamic behavior of piezoelectric bimorphs under excitation of harmonically varying voltage. is given by: 3 sin sinh = 4ℎ 1 + cos L cosh L 3 , = − cos − cosh cos − cosh 4ℎ 1 + cos cosh + − sin + sinh sin − sinh known that resonance occurs for the bimorph cantilever when the equation 1 + cos cos =0 is the normalized frequency and plays a role in the vibrating system as a natural frequency. It is is satisfied [36].In order to find the right eigenfrequencies, the following equation must be satisfied: 1 0 1 0 0 0 0 0 = − cos − sin cosh sin 0 sin − cos sinh cosh 0 or = 0. For this to be true either must be equal to zero or must be singular. If were equal to = 0 . Simplifying this equation, one finds again zero there would be no vibration, therefore, the matrix must be singular; in other words, its 1 + cos cos = 0, which may be solved numerically for the characteristic values L, and determinant must be zero hence the natural frequencies may be found. The numerator and denominator do not reach zero at the same value of in equation. This indicates, when the frequency approaches resonance, the tip deflection goes to infinity, which cannot represent the real system. So some correction should be made for the tip deflection at resonance. The quality factor is defined as 1⁄2 , where is the damping ratio of the mechanical system. By making an assumption of the damping ratio of the system, the amplitude at the resonance can be obtained by the flow chart (see Figure 2.2). 22
  • 38. Figure 2.2: Flow chart to determine the deflection at resonance. In the flow chart, is the amplitude at frequency , is the tip amplitude at the resonance and the resonance frequency. By using this method together with Smits’s dynamic expression, the finite tip deflection can be fully determined. Starting from the characteristic values , the eigenfrequencies can be calculated through the parameters a [35]: = 2 with ∙ = ∙ = 2ℎ ∙ 2ℎ ∙ = 12 where , , 2ℎ , , and are respectively the area of cross section, Young's modulus of the piezoelectric actuator, the thickness, the moment of inertia of the piezoelectric actuator, length and width of the bimorph. It is assumed that the bimorph under consideration is long, narrow and thin, this thickness 2ℎ. means that the length is much greater than the width , which in its own is much greater than the In order to calculate the mode shape analytically, one needs to calculate , . All parameters are fixed or determined, only varies (representing the considered point on the beam), namely from zero untill L, with small step size. 23
  • 39. 2.2.5 Electromechanical coupling factor (EMCF) The EMCF is based on a quasi-static cycle of deformation is defined as the square root of the ratio of the convertible to the total internal energy of the structure [34]: − = = Indices (OC) and (SC) refer to open-circuited and short-circuited electrodes, respectively. The convertible energy is the difference of the energy for open-circuited electrodes and short- circuited electrodes for a given strain field . The EMCF is a measure for the relative amount of energy that can be converted from the mechanical to the electrical ports of the system and vice versa, in a quasi-static deformation cycle. For open or short-circuited electrodes the internal energies would be different for the same displacement field. In the case of forced vibration of a conservative system excited by a piezoelectric actuator, resonance occurs when the frequency of excitation matches the resonance frequency , which is the eigenfrequency of the same system with the electrodes of the actuator short-circuited. In this case, the electrical impedance of the system is zero. The eigenfrequencies of the system with free electrodes are the so-called antiresonance requencies . Being excited at these frequencies, the system vibrates in the corresponding antiresonance modes and the electric impedance of the system is infinite, i.e. the amplitude of the electric current is zero. Both, the resonance and the antiresonance mode are eigenmodes of the system, for short-circuited or open electrodes, respectively. Now, for the particular (hypothetical) case where the displacement fields of resonance and antiresonance are identical, the respective internal energies are computed for the same deformation but different electrical boundary conditions. It follows for the case of identical resonance and antiresonance modes of deformation [34]: − − = = This frequency relation is also referred to as the effective or dynamic electromechanical coupling factor of a piezoelectric resonator. Figure 2.3: Definition of short circuited and open circuited configuration for piezoelectric bending actuators. Although bending actuators can generate the largest displacement among all piezoelectric actuators; they have low electromechanical coupling and small generative force, because flexural strength is lower than tensile or compressive strength in general. Also, internal stresses are built up in bending actuators when transverse motion is converted into bending motion, which lessens the output mechanical energy. For a single end clamped actuator two parameters, tip deflection and generative force, usually are used to characterize actuator performance. These properties have been shown to strongly depend on actuator dimensions and piezoelectric and elastic properties of each component. The electromechanical coupling is another important concern in actuator design. 24
  • 40. In the case of bimorph actuator, if the effect of the bonding layer is neglected, these effective coefficients are directly related to the transverse material coupling factor . In the case of unimorph actuator, these parameters also depend on the Young’s modulus and thickness of the elastic layer. Maximum values for these parameters can be obtained by choosing an appropriate thickness ratio and Young’s modulus ratio of elastic and piezoelectric layers. 2.3 FEM analysis of piezoelectric fans The study of physical systems frequently results in partial differential equations which either cannot be solved analytically or lack an exact analytic solution due to the complexity of the boundary conditions or domain. For a realistic and detailed study, a numerical method must be used to solve the problem. The finite element (FE) method is often found the most adequate. Over the years, with the development of modern computers, the finite element method has become one of the most important analysis tool in engineering. The advantage of finite element analysis over analytical solutions is that stress and electrical field measurements of complex geometries, and their variations throughout the device, are more readily calculated. FE allows calculation of the stress and electric field distributions under static loads and under any applied electrical frequency, and so the effect of device geometry can be assessed and optimized without the need to manufacture and test numerous devices. Basically, the finite element method consists in a piecewise application of classical variational methods to smaller and simpler subdomains called finite elements connected to each other in a finite number of points called nodes. The fundamental principles of the finite element method, concerning structural analysis, are: • The continuum is divided in a finite number of elements of geometrically simple shape. • These elements are connected in a finite number of nodes. • The unknowns are the displacements of these nodes. • Polynomial interpolation functions are chosen to describe the unknown displacement field at each point of the elements related to the corresponding field values at the nodes. • The forces applied to the structure are replaced by an equivalent system of forces applied to the nodes. A finite element formulation accounting for the coupling between the equations of electrostatics and elastodynamics becomes necessary when the piezoelectric material represents a non negligible fraction of the entire structure. 2.3.1 Piezoelectric FEM Equations The finite element modeling of plates and shells with surface bonded or embedded piezoelectric patches has received considerable attention in the past recent years. As a result, a large number of finite element formulations have been developed and published. Nowadays commercial finite element codes are equipped with elements with piezoelectric capabilities. Nevertheless, it stays a multiphysics problem, which is not easy to solve. The finite element equations of motion for a structure exhibiting linear piezoelectric behavior are given by [37]: 0 0 + + = 0 0 0 0 25
  • 41. with: : the structural mass matrix : the structural damping matrix = : the structural stiffness matrix and : the piezoelectric stiffness matrices : the dielectric stiffness matrix : the vector with nodal structural displacements : the vector with nodal voltages : the vector with nodal forces : the vector with nodal charges Hereby must be remembered that contains all the external loads that are applied on the structure and that the piezoelectric coupling is represented by the piezoelectric stiffness matrices. In this equation no distinction is made between the structure and the piezoelectric material. Obviously, the piezoelectric stiffness matrix and dielectric stiffness matrix are zero in the structure. The basic equations for the derivation of the piezoelectric FEM formulation were introduced in Section §1.3.1.1. Quasi-static behavior of the electric field was assumed there, which explains why the mass and damping matrices in the above equation do not contain contributions related to the electric field. Piezoelectric material can be used either to excite a structure, i.e. as a voltage or charge driven actuator, or to measure vibration, i.e. as a voltage or charge sensor. A distinction is made between two types of electrodes: • Electrodes with a prescribed voltage, which applies for voltage driven actuators and charge sensors (short circuit). • Electrodes with a prescribed charge, which applies for charge driven actuators and voltage sensors (open circuit). The vector with nodal voltages and the vector with nodal charges are divided according to this subdivision of electrical boundary conditions: = = where the superscript refers to prescribed voltage and the superscript refers to prescribed charge. Note that and are given inputs to the system, whereas and are outputs = 0. With the distinction between prescribed voltage and charge of the system. The electrical DOF of nodes which are not on an electrode surface are handled as prescribed charge DOF, with DOF, the equation of motion becomes: 0 0 0 0 0 0 0 + 0 0 0 + 0 0 0 0 0 0 = In order to perform an eigenvalue analysis, the nodal voltages are condensed from the system. 26
  • 42. The third row in the above equation can also be written as: = − − + Substitution of this equation into the first row in equation gives an equation of motion in terms of the structural displacement vector : + + ⋆ = ⋆ ⋆ ⋆ where the equivalent stiffness matrix and equivalent force is defined as, respectively: ⋆ = − ⋆ = − ⋆ − where ⋆ = − . This equation shows that the electrical inputs (prescribed voltages) and (prescribed charges) are written as equivalent structural loads. Once the structural displacement vector has been solved, the voltages in the nodes with a prescribed charge boundary condition, , can be calculated. The charges associated with the set of prescribed voltages, , can be calculated as follows: = ⋆ + ⋆ + where ⋆ = ⋆ and ⋆ = − , which is obtained by substitution. 2.3.2 Finite element software: ANSYS ANSYS is applied to investigate the multiphysics problem. ANSYS is a general purpose finite element modeling package for numerically solving a wide variety of mechanical problems. These problems include: static/dynamic structural analysis (both linear and non-linear), heat transfer and fluid problems, as well as acoustic, piezoelectric and electromagnetic problems. The number of variables acting at each nodal site is called the degree of freedom (DOF). For piezoelectrics there are four DOF at each node; UX, UY and UZ (3D displacement) and voltage V. To each DOF there is a reaction force FX, FY , FZ to the displacement and charge Q to the voltage. In this case, it is necessary to use coupled-field analysis to couple the interaction between applied stress and electric field. ANSYS has multiple specific coupled field elements for piezoelectric analysis. The following 3D elements are mainly used in this work: • SOLID226 Element (3D 20-Node Coupled-Field Solid) • SOLID227 Element (3D 10-Node Coupled-Field Solid) If the total structure contains as well non-piezoelectric elements, the normal meshing elements (e.g. SOLID92, a 3D 10-Node Tetrahedral Structural Solid) can be used. The choice between the coupled field elements depends upon the geometry being modeled. For example, a tetrahedral coupled-field solid element is more suited to dividing a 3D spherical body into elements, whereas SOLID5 constructs with cuboid elements and hence is used to discretise 3D cuboid bodies. A FEA modeling of the unimorph and bimorph cantilever type actuators by ANSYS 11.0 software have been developed in this work. When developing the FEA modeling, at first the bonding between the piezoelectric material and the metal shim is assumed to be perfect, and the thickness of the 27
  • 43. bonding layer is negligible as in the analytical models. Also the clamping is assumed to be infinitely stiff. Three-dimensional linear solid brick SOLID95 element is used for the piezo beam modeling in all of these FEA models because it has large deflection and stress when used in the structural and piezoelectric analyses. A regular three-dimensional mesh was generated in the piezoelectric actuators using 3D brick piezoelectric 20-node element SOLID226. Note that the planar isotropy of the piezoelectric material requires the prescription of more material constants than simply the three constants employed in the one-dimensional analytical beam model. The model is constructed by first defining keypoints, creating areas and volumes, and assigning the boundary conditions and material properties to the areas and respective volumes, respectively. As a boundary condition, the degrees of freedom are of the area of the beam which represents the clamped end, are fully constrained. All nodes underneath and above the piezoelectric patches, representing the electrodes, are assigned a voltage coupling corresponding to the SC/OC electrical boundary conditions. For the SC condition, all the electrodes (the 4 surface area’s of the patches) are assigned zero voltage. For the OC case, the voltages at all nodes on the top electrode are forced to be identical. Material properties used for the analytical model are presented in Table 2.1, while the material properties used in the 3D finite element analysis are presented in matrix form: The correct mesh size is determined by conducting a convergence test, where the mesh was refined until convergence of the results was achieved (§2.4.2.1). In Table 2.1 the material parameters used in the FEA modeling are summarized. 62 Property Value Young modulus Y1, Y2 7800 (GPa) 320 ∙ 10 Density (kg/m3) 700 ∙ 10 d31, d32 (m/V) 3130 d15, d24 (m/V) 3130 ε11 3400 ε22 −12 ε33 22.22 ε31 19.39 ε33 126 ∙ 10 ε15 79.5 ∙ 10 c11 (N/m2) 84.1 ∙ 10 c12 (N/m2) 117 ∙ 10 c13 (N/m2) 23 ∙ 10 c33 (N/m2) c44 (N/m2) Table 2.1: Material properties of PZT-5H for analytic and FEA calculations. Once the FEA modeling of the unimorph and bimorph actuators have been developed, the FEA modeling for piezoelectric fans was developed by changing parameters. 2.4 Parametric optimization 2.4.1 Validation of the finite element models In §2.2.2, piezoelectric unimorph cantilever actuators were investigated. As has been discussed before, this type of piezoelectric actuator can produce a relatively small displacement only. Next, another 28
  • 44. configuration will be investigated, namely the fan configuration where the passive elastic layer is longer than the active PZT layer. These piezoelectric fan actuators have been proved capable of producing much larger displacement. Unlike the unimorph cantilever structure, for the piezoelectric flapping wing structure based on piezo fans there is no simple analytical equation to calculate its performance parameters such as the tip displacement, the blocking force, etc. That is why we want to develop FEM models to investigate the performance of the complex piezoelectric flapping wing structures. In order to validate the used finite element models, comparisons have been made between the finite element models and the analytical formulas. The parameters of interest for the explored bimorph configuration are static and dynamic deflection; also the results of modal analysis are compared. The models have a length of 50mm, thickness of 200µm and are based on a series connection of the two piezoelectric layers from PVDF. A voltage of 120V is applied in the analytic and FEM models. The simulations are performed in 2D. To compare the static tip deflection, the equation for the tip deflection in §2.2.2 was applied, with a voltage equal to 120V. It can be seen that ANSYS and the analytical method show excellent agreement. Modal analysis is used to determine the first five bending eigenfrequencies and to check if the normalized mode shapes correspond with each other. Concerning the analytical calculation of the resonance frequencies, tip deflection and mode shapes, the same method is followed as §2.2.4. Method Static 1st 2nd 3rd 4th 5th deflection resonance resonance resonance resonance resonance frequency frequency frequency frequency frequency Analytical 6.1875e-4 27.24 170.74 478.07 936.83 1548.65 ANSYS 6.1701e-4 27.28 170.89 478.28 936.56 1546.83 Table 2.2: Validation of the FEM model: comparison between the analytical and ANSYS solution. In Figure 2.4 the three bending modes are plotted for ANSYS and the analytical calculation. The three mode shapes show excellent agreement and can be considered identical. 1 First mode shape (ANSYS) 0,5 Second mode shape (ANSYS) Amplitude Third mode shape (ANSYS) 0 First mode shape (analytic) 0 10 20 30 40 50 -0,5 Second mode shape (analytic) Third mode shape (analytic) -1 Location (mm) Figure 2.4: First three normalized bending modes for ANSYS and the analytical calculation. A harmonic analysis is performed to explore the dynamic properties of the bimorph configuration. Harmonic response analysis is a technique used to determine the steady-state response of a linear structure to loads that vary harmonically with time. The idea is to calculate the structure's response at several frequencies and obtain a graph of some response quantity (usually displacements) versus frequency. "Peak" responses can then be identified on the graph at the peak frequencies. 29
  • 45. To start, the analysis is performed analytically without taking damping into count. The results from ANSYS are obtained by using damping parameters straight away. The applied damping ratio is chosen to be 2%, which is the most common value for physical systems. In Figure 2.5, one would expect that, for the analytical solution, an infinite amplitude is achieved, when the bimorph is excited at its eigenfrequencies. This is not the case because the exact solution is not reached because of the frequency step size that is used (0.1 Hz). Evidently, the smaller the time step is, the higher the peak will be. The aim of the plot is only to show a global comparison between the two different methods. The step size in the finite element programs is 0.5 Hz. It can be seen that ANSYS shows excellent agreement with the determined analytical solution for a varied range of frequencies. The slightly different amplitudes at the peaks are also caused by the higher step size by ANSYS. 1,E-02 0 200 400 600 Amplitude (m) 1,E-04 Analytic 1,E-06 ANSYS 1,E-08 1,E-10 Frequency (Hz) Figure 2.5: Amplitude of the tip of the piezo fan in function of the frequency. Comparison between the analytical and FEM (ANSYS) results. In order to determine the dynamic peak amplitude analytically, the method exposed in paragraph §2.2.4 is followed. By chosing a damping ratio the quality factor can be defined. A small difference with the algorithm proposed by Wu and Ro [36], is that starting from the known eigenfrequency and amplitudes and thus as well the amplitude of the resonance frequency (√2 ∙ quality factor, the matching frequencies and are determined. Knowing this frequencies, their ) can be calculated. The analytical results, together with the FE ones (where a smaller step (0.01Hz) has been used this time) can be seen in Table 2.3. It can be noticed that ANSYS shows good agreement with the analytical method. Method Dynamic deflection at Dynamic deflection at Dynamic deflection at 1st Freq. 2nd Freq. 3rd Freq. Analytical 1.56e-3 1.40e-4 2.93e-5 FEM (ANSYS) 1.48e-3 1.32e-4 2.75e-5 Table 2.3: Validation of the FEM model. 2.4.2 Optimization Results and Discussion The dynamic response of a piezoelectric fan configuration with a substrate layer and a piezoelectric material layer was derived, after obtaining the solution for the dynamic response of a piezoelectric unimorph and bimorph actuator at first. Afterwards, configurations resembling the actual structure of the piezoelectric flapping wing which includes three different sections: a section for clamping, a piezoelectric active section and a passive blade section (or wings) are discussed. In Figure 2.6 various piezoelectric flapping wing models are proposed. Model 0 is a basic piezo fan with an extra wing structure on the flexible end. This wing structure consists of two stiff spars or ribs and a thin flexible sheet. The basic thickness of the thin sheet for all the models is 50µm. Model 1 consists of two discrete piezoelectric patches. This gives the possibility to apply a phase difference between the 30
  • 46. different patches and obtain a rotation in the bending movement. In Model 2 the piezoelectric patches are not connected to the same flexible passive plate. This model can be obtained by using two basic piezo fans. Like the earlier model it is expected that the phase difference between the two piezo fans will play an important role. Model 3 has a wing with a larger width. This width is expected to be important to create more flow and thrust. Model 4 can also be constructed using two piezo fans. Model 5 and Model 6 have also a larger wing area than Model 2. These models exist in two types of variations: rectangular(a)/triangular(b) piezo patches and unimorph/bimorph (e.g. Model 6b is Model 6 with triangular patches). Figure 2.6: Definition of the various models used in the FEA simulations. The area of the tip of the wing is expected to play an important role in the flow generation and thus is an important parameter together with the tip deflection, tip rotation and frequency for the thrust generation. The main objective for these flapping wing mechanisms is the generation of enough thrust, so they can be used as a propulsion system in MAVs. 2.4.2.1 Convergence test The solution of a finite element model converges to the analytical solution if the mesh element size is chosen small enough. However the smaller the element size is, the more calculation time is required to obtain the results. Therefore a convergence test is applied on the finite element model to determine the required element size, to reduce the calculation time of the simulation to its minimum. The h-convergence analysis, where the element size is reduced iteratively, is used to solve this problem. The convergence test was applied on all models and the results for the basic piezo fan model, model 2, model 4 and model 6 can be seen on Figure 2.7. For the basic piezo fan model the results convergence when a element mesh size of 0.7 mm or smaller is chosen, for the other models this becomes 0.5mm. This is due to the fact that the other models have more complex geometrical properties. 31
  • 47. Basic piezo fan Model 2 70 Mode 1 60 Mode 1 60 Mode 2 Mode 2 50 50 Mode 3 Mode 3 Mode 4 40 Mode 4 Error (Hz) Error (Hz) 40 Mode 5 30 Mode 5 30 Mode 6 Mode 6 20 20 Mode 7 Mode 7 10 10 Mode 8 Mode 8 0 Mode 9 0 Mode 9 0 0,5 1 0 0,5 1 Mode 10 Mode 10 Mesh element size (mm) Mesh Element size (mm) Model 4 Model 6 10 Mode 1 25 Mode 1 Mode 2 Mode 2 8 20 Mode 3 Mode 3 Mode 4 Error (Hz) Error (Hz) 6 15 Mode 4 Mode 5 Mode 5 4 10 Mode 6 Mode 6 2 Mode 7 5 Mode 7 Mode 8 Mode 8 0 0 Mode 9 Mode 9 0 0,5 1 0 0,5 1 Mode 10 Mode 10 Mesh Element size (mm) Mesh Element size (mm) Figure 2.7: Element size h-convergence test on FEM models of the piezoelectric flapping wings. If a too big mesh size is used in the optimization simulations, the plots would be unsmooth and unreliable. In the following simulations a mesh size of 0.4 mm was used. 2.4.2.2 Optimization of piezoelectric fans For this part, only the FEM model will be utilized, this way more complex geometries can be introduced. Figure 2.8: Basic piezo fan model with the definition of the length and width parameters. 32
  • 48. The following parameters of the system are allowed to vary: The length of the patch − 1. The location of patch while the length of the patch is kept fixed at 6.5mm; The height of the patch ℎ and elastic plate ℎ 2. 3. 4. The material properties of the elastic plate 5. The thickness of the bonding layer The relationships between the above parameters in a piezo fan and piezo flapping structure were modeled in order to achieve the maximum dynamic tip displacement, frequency and EMCF. By optimizing these quantities we want to obtain the maximum flow and efficiency generated by the flapping structure. Ultimately the parameter configuration needs to be chosen in such a way that a maximum thrust generation and energy efficiency is reached. Two parameter variations were considered simultaneously: one parameter variation was the length of the PZT patch while keeping the length of the passive elastic plate (stainless steel shim) constant. The PZT patch length was varied from 0 to 50 mm (the width 15 mm, the voltage and the thickness were kept constant), the stainless steel shim dimensions (50mm x 15mm x 60µm) were kept constants. The other parameter variation was changing the thickness of the piezoelectric patch, while keeping the thickness of the elastic plate constant at 60µm. Since there are two parameter variation, the optimization results could be plot in the convenient surf plots. From these plots most of the time a conclusion can be made for the optimal parameter configuration. By dividing the length of the patch by the length of the elastic plate (or the total length of the piezo fan), one obtains a dimensionless quantity. In the same manner the thickness of the piezoelectric patch was divided by the thickness of the elastic plate. 2− 1 ℎ ℎ ℎ = = 3 ℎ ℎ ℎ ℎ ℎ = = ℎ ℎ In this way one can use these results as design parameters for piezoelectric fans with other dimensions. The influence of the width of the piezo fan on the optimization results is limited. The simulations were done for both bimorph/unimorph and rectangular/triangular configurations. In Figure 2.9 the results of the optimization for a piezo fan with rectangular piezoelectric patches with bimorph and unimorph configuration is given. It can be seen that the dynamic tip deflection gets maximal to a value of approximately 20mm for the bimorph configuration when the piezoelectric patch is very thin (infinitely) and covering the whole passive elastic plate. For the unimorph patch configuration the tip deflection falls back to approximately 15mm. The surf plot is very similar to that of the bimorph case. The length of the piezoelectric patch must be as small as possible to reduce the weight and fabrication cost of the construction. The weight is very important for the implementation in MAVs as a propulsion system. This resulting optimal design is quite different from the design based on maximal EMCF. The resonant amplitude at resonance is inversely proportional to the modal damping. The modal damping being assumed proportional to the short-circuited natural frequency, an optimal design attempts to reduce the SC natural frequency to maximize resonant amplitude. This occurs when the patch entirely covers beam and if its thickness nearly vanishes. This leads to physically unacceptable optimal designs. The EMCF gets maximal when a length ratio between 0.6 and 1 is chosen with a thickness ratio between 0.67 and 2.5. Keeping in mind that the thickness ratio has to be as small as possible to obtain 33
  • 49. the largest tip deflections, the optimal length and thickness ratios for the bimorph case become respectively 0.6 and 0.67. For the unimorph patch configuration the EMCF gets maximal for a length and thickness ratio of 0.64 and 1.33, respectively. This optimal ratios are further away from the respectively. optimal ratios for the tip deflection than for the case of bimorph patch configuration A compromise configuration. has to be made and one needs to determine what quantity has the greatest priority in the application. Another important quantity is the frequency of the piezo fan. A greater oscillating frequency of the her 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 important ( ) as well. From the surf plots it can be seen that for the bimorph configuration well. gets maximized for a length ratio and thickness ratio of 0.36-1 and 0-0.83 respectively; for the unimorph configuration 0.36- 0.83 this becomes 0.36 1 and 0 1.17. The maximum is 0.36-1 0-1.17. for the bimorph case is about twice the size of the unimorph case. The first resonance frequency for the bimorph case gets maximized for a length and thickness ratio of 0.68 and 2.5, respectively. For the unimorph case this is 0.64 and 2.5. unimorph The computations for a single optimization given in Figure 2.9 took approximately 8 days on a PC with Intel i7 (2.5Ghz) processor, 12GB 1600Mhz DDR3 RAM and 2x 10k rpm HD in raid0. processor, Bimorph patch configuration imorph Unimorph patch configuration 0,02- -0,03 0,01-0,015 0,01 0,01- -0,02 0,03 0,005-0,01 0,005 0,015 0-0,01 0,01 0,02 0-0,005 0 0,01 0,01 0,005 0 0 0,17 0,17 0,04 0,04 0,2 0,24 Thick-- 1,50 1,50 0,36 0,44 Thick- Thick 0,52 0,64 ness 0,68 0,84 ness 0,84 ratio Length ratio ratio Length ratio 0,2-0,3 0,3 0,2-0,3 0,2 0,1-0,2 0,2 0,3 0,1-0,2 0,1 0,3 0,2 0-0,1 0 0,2 0-0,1 0,1 0,1 0,1 0 0 0,17 0,17 0,04 0,04 0,2 0,2 0,36 1,50 0,36 Thick- Thick 1,50 0,52 Thick-- 0,52 0,68 0,68 ness ness 0,84 0,84 Length ratio Length ratio ratio ratio 34
  • 50. 4-6 2-3 2-4 6 1-2 3 0-2 4 0-1 2 2 1 0 0 0,17 0,17 0,04 0,04 0,2 0,2 0,36 1,50 0,36 1,50 0,52 0,52 0,68 0,68 Length 0,84 Thickness ratio Thickness ratio Length 0,84 ratio ratio 80-100 100 60-80 100 60-80 80 80 40-60 60 40-60 80 60 60 20-40 40 40 20 0-20 20 0 0 1,50 0,04 1,50 0,04 0,16 0,2 0,28 0,36 0,4 0,17 Thicknes 0,52 0,52 0,17 0,64 0,68 0,76 0,84 Thicknes 0,88 Length ratio s ratio Length ratio s ratio Figure 2.9: Optimization results for the piezoelectric fans with rectangular PZT-5H patches. First row: dynamic tip 9: 5H deflection in meters; Second row: EMCF (%); Third row: fA in m/s; Fourth row: first resonance frequency in Hz. meters; ; ; Similar results are obtained for the configuration with triangular patches (see Figure 2 2.10). As expected the dynamic tip deflection is lower for the configuration with triangular patches (reduced with 38%). However the covered area (and thus the weight) is lower and the EMCF is slightly higher 38%). for the triangular patch configuration. The is slightly lower and gets maximal when the patch is infinitely thin and with length . The frequency gets maximal for a length ratio of 0.8 and thickness ratio of 2.5. Bimorph patch configuration Unimorph patch configuration 0,02- -0,03 0,01-0,015 0,01 0,01- -0,02 0,03 0,005-0,01 0,005 0,015 0-0,01 0,01 0,02 0-0,005 0 0,01 0,01 0,005 0 0 0,17 0,17 0,04 0,04 0,2 0,2 1,50 0,36 1,50 0,36 0,52 0,52 0,68 0,68 Thick- Thick Thick- Thick 0,84 0,84 Length ratio Length ratio ness ness ratio ratio 35
  • 51. 0,2-0,3 0,3 0,2-0,3 0,2 0,1-0,2 0,2 0,3 0,3 0,1-0,2 0,1 0-0,1 0,1 0,2 0-0,1 0 0,2 0,1 0,1 0 0 0,17 0,17 0,04 0,04 0,2 0,2 0,36 1,50 1,50 0,36 0,52 0,52 Thick- Thick 0,68 Thick- 0,68 0,84 0,84 ness Length ratio ness Length ratio ratio ratio 4-6 2-3 2 2-4 1-2 1 6 3 0-2 0-1 0 4 2 2 1 0 0 0,17 0,17 0,04 0,04 0,2 0,2 0,36 0,36 1,50 1,50 0,52 0,52 0,68 0,68 Length 0,84 0,84 Thickness ratio Length Thickness ratio ratio ratio 100 80-100 100 80 60-80 80 60-80 80 60 40-60 60 40-60 60 40 20-40 40 20-40 40 20 0-20 20 0 0-20 0 1,50 1,50 0,04 0,04 0,2 0,16 0,17 0,28 0,36 0,4 0,17 Thicknes 0,52 0,52 0,64 Thicknes 0,76 0,68 0,88 0,84 s ratio s ratio Length ratio Length ratio Figure 2.10: Optimization results for the piezoelectric fans with triangular PZT 5H patches. First row: dynamic tip 10: PZT-5H deflection in meters; Second row: EMCF (%); Third row: fA in m/s; Fourth row: first resonance frequency in Hz. resonance There is always a gap of a few millimeters ( ) needed between the clamping and the edge of the PZT patch, to prevent the PZT being broken by the clamping base during vibration. PZT is ceramic material and is fragile. It can be seen that for the location of the PZT patch (as represented by the le left of the PZT patch), further away from the clamping position leads to a smaller maximum tip displacement under a certain driving voltage (see Figure 2.11). So, if a larger tip displacement is ( ). required, the PZT patch should be placed as close to the clamping position as possible. The first patch resonant frequency of the fan also decrease with the PZT patch moving away from the clamping location, thus leading to a significant drop of . It can be observed that the EMCF is also decreased. 36
  • 52. 0,3 0,008 0,007 0,25 0,006 0,2 Tip deflection (m) 0,005 EMCF and fA 0,15 0,004 EMCF 0,1 0,003 fA 0,002 Tip deflection 0,05 0,001 0 0 0 0,005 0,01 0,015 0,02 0,025 0,03 -0,05 -0,001 Distance between piezo patch and clamping (m) 300 Resonance frequency (Hz) 250 First resonance 200 frequency 150 Second resonance 100 frequency 50 Third resonance 0 frequency 0 0,005 0,01 0,015 0,02 0,025 0,03 Distance between piezo patch and clamping (m) 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. In addition, the clamp-patch distance affects significantly, both the optimal thickness ratio and the length ratio for maximal EMCF. Analogue surf and contour plots are acquired and it can be seen that for the chosen material properties and patch length, the patch should be placed as close as possible to the clamp. The corresponding optimal patch thickness ratio tends to reduce to 0.35 for the bimorph configuration with rectangular patches. Next to the geometrical properties also the material properties of the elastic plate play an important role in the performance of the piezo fan. However not all materials can be made in very thin sheets, so = 15 , = = the thinnest commercially available sheets have been looked up and applied on the models. The µ 2.97). The results for an applied voltage of 120V can be seen in Figure 2.12. damping ratio and other geometrical properties have been kept constant ( µ 37
  • 53. 0,07 0,06 ABS Acrylic glass 0,05 Aluminium Tip deflection (m) Balsa wood 0,04 Brass 0,03 Copper Mylar 0,02 Nickel Plywood 0,01 Polyethylene terephthalate Polystyrene 0 Titanium 0 0,2 0,4 0,6 0,8 1 Length ratio (Lpatch/Ltotal) 0,25 ABS 0,2 Acrylic glass Aluminium 0,15 Balsa wood EMCF Brass Copper 0,1 Mylar Nickel 0,05 Plywood Polyethylene terephthalate Polystyrene 0 Titanium 0 0,2 0,4 0,6 0,8 1 Length ratio (Lpatch/Ltotal) 38
  • 54. 10 9 ABS 8 Acrylic glass 7 Aluminium 6 Balsa wood Brass fA 5 Copper 4 Mylar 3 Nickel 2 Plywood Polyethylene terephthalate 1 Polystyrene 0 Titanium 0 0,2 0,4 0,6 0,8 1 Length ratio (Lpatch/Ltotal) 180 160 ABS 140 Acrylic glass First resonance frequency (Hz) 120 Aluminium Balsa wood 100 Brass 80 Copper Mylar 60 Nickel 40 Plywood Polyethylene terephthalate 20 Polystyrene 0 Titanium 0 0,2 0,4 0,6 0,8 1 Length ratio (Lpatch/Ltotal) Figure 2.12: The influence of the elastic plate material on the Tip deflection, EMCF, fA and first resonance frequency of the piezo fan. A realistic thickness ratio is applied, leading to very characteristic curves for varying length ratios. The curves tend to increase first, and then reach a maximum for a certain length ratio and then fall back with increasing length ratio. Using a non-metal elastic plate gives better results, however for the 39
  • 55. implementation of 2 piezo fans in a flapping wing structure to obtain 2DOF we will later see that the use of metal elastic plates still can be advantageous (Figure 2.23). Piezoelectric patches add distributed elastic and piezoelectric stiffness as well as mass to the fan. For small patch lengths, the added stiffness effect dominates and increases the natural frequencies while at longer patch lengths the added mass effect reduces the natural frequencies. Therefore the natural frequencies and consequently EMCF are maximal at an optimal value of patch-to-beam length ratio. 0,016 Tip deflection (m) 0,014 0,012 0,01 0,008 0,006 0 50 100 150 200 Thickness bounding layer (µm) 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. In the reviewed models the effect of the bounding layer between the passive plate and piezo patch was ignored. This layer has to be as small as possible, since it can reduce the tip deflection significantly (Figure 2.13). The effect of the bounding layer will get bigger when the width of the piezo fan is increased. The influence of the width on the dynamic tip deflection and frequency is small, until the point is reached where the first mode shape changes from bending mode to torsion mode (Figure 2.14). 1 0,1 0 100 200 300 400 500 0,01 Basic piezo fan model 0,001 (width=0,0015m, 2% Amplitude (m) 0,0001 damping)) 0,00001 Basic piezo fan model 0,000001 (width=0,003m, 2% 0,0000001 damping)) 1E-08 1E-09 Frequency (Hz) Figure 2.14: The amplitude of the dynamic tip deflection in function of the frequency for different widths (under 120V). 2.4.2.3 Optimization of piezoelectric flapping wings A pair of spars made of carbon fiber reinforced plastic (CFRP) connected with a flexible polymer skin formed a wing and the wing was attached to the two piezo fans clamped in parallel to form the coupled piezo fans. These piezoelectric flapping wings are parametric modeled in ANSYS and the influence of various parameters on the dynamic tip deflection, EMCF, and the first resonance frequency is being investigated (Figure 2.15). 40
  • 56. 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. Basically these models consist of a piezo actuation part (two piezo fans) and a wing structure. The length of the piezo actuation part is held constant or the piezo fan is held constant at 50mm. The basic length of the piezoelectric patch is set to 20mm. The spars are glued on the piezo fans with an offset of 2 mm from the tip. The distance from the tip of the fan to the flexible wing material between the spars is 10mm. The elastic plate is made of stainless steel, the spars from CFRP and the wing from a polymer. Material CFRP Stainless steel Polymer Young's Modulus (GPa) 200 210 8 Poisson ratio 0.27 0.3 0.27 Density (kg/m3) 1750 8400 1534 Thickness 100 60 50 Width 2 5 30 Table 2.4: Material properties of the elastic plate, wing and spars used in the FEA of piezoelectric flapping wings. The dynamic properties can be retrieved by doing a finite element modal analysis and harmonic analysis (Figure 2.16). 1,E+00 1,E+00 Model 2a Model 2a 0 50 (2% -20 30 80 1,E-02 (other damping) 1,E-02 Amplitude (m) dimensions, Amplitude (m) Model 4a 2% damping) 1,E-04 1,E-04 (2% damping) Model 2a 1,E-06 1,E-06 (2% Model 6a damping) (2% 1,E-08 1,E-08 Frequency (Hz) damping) Frequency (Hz) Figure 2.16: The amplitude of the dynamic tip deflection in function of the frequency for the diverse models under 120V. 41
  • 57. The damping ratio is kept constant at a value of 2% for all the simulations. Later the damping of prototype structures will be determined experimentally. We will then be able to conclude that this value is a realistic estimation (see §4.3.1). The influence of the damping ratio on the dynamic properties of the structure is given in Figure 2.17. 0,1 0 50 100 150 0,01 0,001 Model 2a (0,5% damping) Amplitude (m) 0,0001 Model 2a (5% damping) Model 2a (10% damping) 0,00001 Model 2a (2% damping) 0,000001 0,0000001 Frequency (Hz) Figure 2.17: The amplitude of the dynamic tip deflection in function of the frequency for different damping ratios under 120V. According to the analytical solution for the basic piezo fan model, the tip deflection is proportional to the applied voltage on the electrodes of the piezoelectric patches. Similar results are obtained from the FEA simulations (Figure 2.18), but applied on the proposed piezoelectric flapping wing structure models (Figure 2.6). These graphs give the possibility to use a linear extrapolation against voltage to estimate the static tip displacement of the piezoelectric flapping structures at other voltages. The EMCF is independent of the applied voltage, because the resonance frequencies are not affected. The results for the tip deflection are very similar for the different models, the difference in performance can be better observed by looking at . Model 5a gives the greatest results. 0,05 0,7 Model 2a (unimorph) Model 2a (bimorph) 0,045 0,6 Model 2b (unimorph) 0,04 Model 2b (bimorph) 0,5 Model 3a (unimorph) Tip deflection (m) 0,035 Model 3a (bimorph) 0,03 0,4 Model 3b (unimorph) 0,025 Model 3b (bimorph) Model 4a (unimorph) fA 0,3 0,02 Model 4a (bimorph) 0,015 0,2 Model 4b (unimorph) Model 4b (bimorph) 0,01 Model 5a (unimorph) 0,1 0,005 Model 5a (bimorph) 0 6E-16 Model 5b (unimorph) Model 5b (bimorph) 0 100 200 0 50 100 150 200 Model 6a (unimorph) -0,1 Model 6a (bimorph) Voltage (V) Voltage (V) Model 6b (bimorph) Figure 2.18: The influence of the voltage on the tip deflection and fA. The influence of the patch length was also investigated to obtain the optimal length ratio (Figure 2.19). All the important optimization quantities increase with increasing patch length ratios. This implies that 42
  • 58. for all the models the patch length for these structures needs to be covering the whole passive plate to obtain the maximum tip deflection, EMCF, and first resonance frequency. Model 2a (bimorph) gives the highest tip deflection and first resonance frequency, and thus the highest . The highest EMCF is obtained by Model 5b, Model 5a and Model 2b. The values are very related. 0,07 Model 2a (unimorph) Model 2a (bimorph) 0,06 Model 2b (unimorph) Model 2b (bimorph) 0,05 Model 3a (unimorph) Tip deflection (m) Model 3a (bimorph) 0,04 Model 3b (unimorph) Model 3b (bimorph) 0,03 Model 4a (unimorph) Model 4a (bimorph) 0,02 Model 4b (unimorph) Model 4b (bimorph) 0,01 Model 5a (unimorph) Model 5a (bimorph) 0 Model 5b (unimorph) 0 0,2 0,4 0,6 0,8 1 Model 5b (bimorph) Model 6a (unimorph) Length ratio (Lpatch/Lbase) Model 6a (bimorph) Model 6b (bimorph) 0,16 Model 2a (unimorph) Model 2a (bimorph) 0,14 Model 2b (unimorph) Model 2b (bimorph) 0,12 Model 3a (unimorph) Model 3a (bimorph) 0,1 Model 3b (unimorph) EMCF 0,08 Model 3b (bimorph) Model 4a (unimorph) 0,06 Model 4a (bimorph) Model 4b (unimorph) 0,04 Model 4b (bimorph) Model 5a (unimorph) 0,02 Model 5a (bimorph) Model 5b (unimorph) 0 Model 5b (bimorph) 0 0,2 0,4 0,6 0,8 1 Model 6a (unimorph) Model 6a (bimorph) Length ratio (Lpatch/Lbase) Model 6b (bimorph) 43
  • 59. 2 Model 2a (unimorph) Model 2a (bimorph) 1,8 Model 2b (unimorph) 1,6 Model 2b (bimorph) Model 3a (unimorph) 1,4 Model 3a (bimorph) 1,2 Model 3b (unimorph) fA (m/s) 1 Model 3b (bimorph) Model 4a (unimorph) 0,8 Model 4a (bimorph) 0,6 Model 4b (unimorph) Model 4b (bimorph) 0,4 Model 5a (unimorph) 0,2 Model 5a (bimorph) Model 5b (unimorph) 0 Model 5b (bimorph) 0 0,2 0,4 0,6 0,8 1 Model 6a (unimorph) Model 6a (bimorph) Length ratio (Lpatch/Lbase) Model 6b (bimorph) 30 Model 2a (unimorph) Model 2a (bimorph) Model 2b (unimorph) First resonance frequency (Hz) 25 Model 2b (bimorph) Model 2b (bimorph) 20 Model 3a (bimorph) Model 3b (unimorph) 15 Model 3b (bimorph) Model 4a (unimorph) Model 4a (bimorph) 10 Model 4b (bimorph) Model 5a (unimorph) 5 Model 5a (bimorph) Model 5b (unimorph) Model 5b (bimorph) 0 Model 6a (unimorph) 0 0,2 0,4 0,6 0,8 1 Model 6a (bimorph) Model 6b (bimorph) Length ratio (Lpatch/Lbase) Model 4b (unimorph) Figure 2.19: The influence of the length ratio on the different optimization parameters for the different models. As in Figure 2.9 in §2.4.2.2 the optimization results can be obtained for the other models. For Model 2a and 6a in bimorph configuration this is given in Figure 2.20. The optimal length and thickness ratio can be extracted from the results. Similar to the previous cases for obtaining the maximum tip deflection the thickness ratio must be as small as possible while the length ratio equal to 1. As expected the tip deflection of Model 2a is significantly larger than for Model 6a due to the geometrical properties. The wing area of Model 6a is significantly greater than that of Model 2a. This leads also to a smaller first resonance frequency. Thus is significantly smaller for Model 6a. The EMCF for both models is very comparable. 44
  • 60. Model 2a (bimorph) Model 6a (bimorph) 0,06- -0,08 0,02-0,03 0,04- -0,06 0,08 0,03 0,01-0,02 0,02- -0,04 0,06 0,02 0-0,01 0-0,02 0,02 0,04 0,02 0,01 0 0 0,04 0,04 0,16 0,16 0,25 0,25 0,28 0,28 0,4 0,4 1,5 1,5 0,52 0,52 0,64 0,64 Thick- 0,76 0,76 Length ratio Thicknes Length ratio 0,88 0,88 ness s ratio ratio 0,2-0,3 0,2 0,2-0,3 0,1-0,2 0,1 0,3 0,3 0,1-0,2 0-0,1 0,1 0,2 0-0,1 0,2 0,1 0,1 0 0 0,04 0,04 0,16 0,25 0,25 0,2 0,28 0,4 0,36 1,5 0,52 Thick- 0,52 0,64 Thicknes 0,68 0,76 ness Length ratio 0,84 s ratio 0,88 Length ratio ratio 6-8 1,5-2 1,5 8 1-1,5 1 2 4-6 6 0,5-1 0,5 1,5 2-4 4 0-0,5 0 1 0-2 2 0,5 0 0 0,04 0,04 0,16 0,16 0,25 0,25 0,28 0,28 0,4 0,4 1,5 1,5 0,52 0,52 0,64 0,64 Length 0,76 0,76 Thickness ratio Length Thickness ratio 0,88 0,88 ratio ratio 30 30 20 20- -30 20 20-30 10- -20 10-20 10 10 0-10 10 0-10 0 0 0,04 0,04 0,16 0,16 0,28 0,28 0,25 Thick Thick- 0,4 0,4 0,25 Thick Thick- 0,52 0,52 0,64 0,64 0,76 0,76 ness ness 0,88 0,88 Length ratio ratio Length ratio ratio Figure 2.20: Optimization results for M 20: Model 2a and Model 6a. First row: dynamic tip deflection in meters; Second row: odel EMCF (%); Third row: fA in m/s; Fourth row: first resonance frequency in Hz. 45
  • 61. Different piezoelectric fans with the same length, thickness and width were used to form coupled piezoelectric fans and were evaluated. The dynamic performance of a single piezoelectric fan attached and with a wingspan is also interesting to investigate, to understand the behaviors of a piezoelectric fan under different loading. The wingspan can be regarded as a loading to the piezoelectric fan conditions. the In Figure 2.21 the optimization results for the basic piezo fan model and Model 0a with the bimorph configuration is being compared. Therefore this is like comparing the piezo fan with and without a wing structure attached to it. The wing structure consists two spars and a thin sheet. It can be observed that there is a general shift to higher length ratios in the results, when a wing structure is added The results, added. tip deflection and EMCF have similar maximum values, however the frequency drops significantly have when a wing structure is added, leading to a lower for Model 0a. Basic piezo fan model (bimorph) Model 0a (bimorph) a Thickness ratio Thickness ratio 2,17 2,17 0,03-0,04 0,03 1,67 0,02-0,03 0,03 1,67 0,02-0,03 0,02 1,17 0,01-0,02 0,02 1,17 0,01-0,02 0,01 0,67 0-0,01 0,67 0,17 0,17 0-0,01 0,01 0,04 0,16 0,28 0,4 0,52 0,64 0,76 0,88 0,4 0,04 0,16 0,28 0,52 0,64 0,76 0,88 Length ratio Length ratio Thickness ratio Thickness ratio 2,17 0,2-0,25 0,2 2,17 0,2-0,25 0,25 1,67 1,67 0,15-0,2 0,15 0,15-0,2 0,2 1,17 1,17 0,1-0,15 0,1 0,1-0,15 0,15 0,67 0,67 0,05-0,1 0,05 0,05-0,1 0,1 0,17 0,17 0-0,05 0 0-0,05 0,05 0,4 0,04 0,16 0,28 0,52 0,64 0,76 0,88 0,04 0,16 0,28 0,52 0,64 0,76 0,88 0,4 Length ratio Length ratio Thickness ratio Thickness ratio 2,17 2,17 4-6 6 3-4 1,67 1,67 2-3 1,17 2-4 4 1,17 1-2 0,67 0-2 2 0,67 0,17 0,17 0-1 0,04 0,16 0,28 0,4 0,52 0,64 0,76 0,88 0,04 0,16 0,28 0,4 0,52 0,64 0,76 0,88 Length ratio Length ratio 46
  • 62. Thickness ratio Thickness ratio 2,17 2,17 30-40 30 80-100 100 1,67 60-80 80 1,67 20-30 20 1,17 40-60 60 1,17 10-20 10 20-40 40 0,67 0,67 0-10 0 0-20 0,17 0,17 0,4 0,04 0,16 0,28 0,52 0,64 0,76 0,88 0,04 0,16 0,28 0,4 0,52 0,64 0,76 0,88 Length ratio Length ratio Figure 2.21 Optimization results for the basic piezo fan model and model0a First row: dynamic tip deflection in meters; 21: ptimization model0a. Second row: EMCF (%); Third row: fA in m/s; Fourth row: first resonance frequency in Hz. It is expected that the material property of the wing also play an important role in the dynamic expected plays properties of the piezoelectric flapping structure. Therefore various simulations were conducted for Model 2a, Model 4a and Model 6a with different wing materials. The length ratio is here defined a as = , with the total length of the piezo fan being only a part of the total length of the flapping wing structure (piezo fan + wing structure). For (piezo structure) the majority of the length ratios the tip deflection and the EMCF are very analogous for the different materials, however starting from a length ratio of 0.7 the curves tend to spread out (Figure 2.22). Figure Model 2a with plywood wing gives the best results for all optimization quantities. In §4.2 prototypes will be presented with balsa wood wings. The stiffness of the wing plays a very important role for obtaining twisting motion of the wing ( g (§4.6). ). 0,008 Model 2a (ABS) Model 2a (Acrylicglass) 0,007 Model 2a (Plywood) Tip deflection (m) 0,006 Model 2a (Polyethyleneterephthalate) Model 2a (Polystyrene) 0,005 Model 4a (ABS) 0,004 Model 4a (Acrylicglass) 0,003 Model 4a (Plywood) Model 4a (Polyethyleneterephthalate) 0,002 Model 4a (Polystyrene) 0,001 Model 6a (ABS) Model 6a (Acrylicglass) 0 Model 6a (Plywood) 0 0,2 0,4 0,6 0,8 1 Model 6a (Polyethyleneterephthalate) Model 6a (Polystyrene) Length ratio (Lbase/Lpiezo) 47
  • 63. 0,14 Model 2a (ABS) Model 2a (Acrylicglass) 0,12 Model 2a (Plywood) Model 2a (Polyethyleneterephthalate) 0,1 Model 2a (Polystyrene) 0,08 Model 4a (ABS) EMCF Model 4a (Acrylicglass) 0,06 Model 4a (Plywood) 0,04 Model 4a (Polyethyleneterephthalate) Model 4a (Polystyrene) 0,02 Model 6a (ABS) Model 6a (Acrylicglass) 0 Model 6a (Plywood) 0 0,2 0,4 0,6 0,8 1 Model 6a (Polyethyleneterephthalate) Model 6a (Polystyrene) Length ratio (Lbase/Lpiezo) 3 Model 2a (ABS) Model 2a (Acrylicglass) 2,5 Model 2a (Plywood) Model 2a (Polyethyleneterephthalate) 2 Model 2a (Polystyrene) fA (m/s) Model 4a (ABS) 1,5 Model 4a (Acrylicglass) Model 4a (Plywood) 1 Model 4a (Polyethyleneterephthalate) Model 4a (Polystyrene) 0,5 Model 6a (ABS) Model 6a (Acrylicglass) 0 Model 6a (Plywood) 0 0,2 0,4 0,6 0,8 1 Model 6a (Polyethyleneterephthalate) Model 6a (Polystyrene) Length ratio (Lbase/Lpiezo) 45 Model 2a (ABS) First resonance frequency (Hz) Model 2a (Acrylicglass) 40 Model 2a (Plywood) 35 Model 2a (Polyethyleneterephthalate) 30 Model 2a (Polystyrene) 25 Model 4a (ABS) Model 4a (Acrylicglass) 20 Model 4a (Plywood) 15 Model 4a (Polyethyleneterephthalate) 10 Model 4a (Polystyrene) 5 Model 6a (ABS) Model 6a (Acrylicglass) 0 Model 6a (Plywood) 0 0,2 0,4 0,6 0,8 1 Model 6a (Polyethyleneterephthalate) Model 6a (Polystyrene) Length ratio (Lbase/Lpiezo) Figure 2.22: Optimization of the length ratio for the proposed models (bimorph configuration) with different wing materials. Also the material property of the passive plate plays an important role in the dynamic performance and properties of the flapping wing structure. A range of simulations were done with different material properties for the elastic plate and varying length ratios (Figure 2.23). For some plate materials the results jump suddenly to another value. This is due to the fact that the mode shape associated with the first resonance frequency changes (from first bending mode to torsion mode). This phenomenon did not occur with the basic piezo fan model, because of the geometrical properties. This effect also does not occur when metals are being used as passive plate. Overall stainless steel, titanium and copper give the best results. Using titanium a significant weight reduction could be established. 48
  • 64. Model 2a (ABS) 0,007 Model 2a (Acrylicglass) Model 2a (Aluminium) Model 2a (Brass) 0,006 Model 2a (Copper) Model 2a (Plywood) Model 2a (Polyethyleneterephthalate) Model 2a (Polystyrene) 0,005 Model 2a (Stainlesssteel) Model 2a (Titanium) Tip deflection (m) Model 4a (ABS) 0,004 Model 4a (Acrylicglass) Model 4a (Aluminium) Model 4a (Brass) Model 4a (Copper) 0,003 Model 4a (Plywood) Model 4a (Polyethyleneterephthalate) Model 4a (Polystyrene) 0,002 Model 4a (Stainlesssteel) Model 4a (Titanium) Model 6a (ABS) Model 6a (Acrylicglass) 0,001 Model 6a (Aluminium) Model 6a (Brass) Model 6a (Copper) 0 Model 6a (Plywood) Model 6a (Polyethyleneterephthalate) 0 0,2 0,4 0,6 0,8 1 Model 6a (Polystyrene) Model 6a (Stainlesssteel) Length ratio (Lpatch/Lbase) Model 6a (Titanium) Model 2a (ABS) 0,12 Model 2a (Acrylicglass) Model 2a (Aluminium) Model 2a (Brass) Model 2a (Copper) 0,1 Model 2a (Plywood) Model 2a (Polyethyleneterephthalate) Model 2a (Polystyrene) Model 2a (Stainlesssteel) 0,08 Model 2a (Titanium) Model 4a (ABS) Model 4a (Acrylicglass) Model 4a (Aluminium) EMCF 0,06 Model 4a (Brass) Model 4a (Copper) Model 4a (Plywood) Model 4a (Polyethyleneterephthalate) 0,04 Model 4a (Polystyrene) Model 4a (Stainlesssteel) Model 4a (Titanium) Model 6a (ABS) 0,02 Model 6a (Acrylicglass) Model 6a (Aluminium) Model 6a (Brass) Model 6a (Copper) 0 Model 6a (Plywood) 0 0,2 0,4 0,6 0,8 1 Model 6a (Polyethyleneterephthalate) Model 6a (Polystyrene) Length ratio (Lpatch/Lbase) Model 6a (Stainlesssteel) Model 6a (Titanium) 49
  • 65. Model 2a (ABS) 1,8 Model 2a (Acrylicglass) Model 2a (Aluminium) 1,6 Model 2a (Brass) Model 2a (Copper) Model 2a (Plywood) 1,4 Model 2a (Polyethyleneterephthalate) Model 2a (Polystyrene) Model 2a (Stainlesssteel) 1,2 Model 2a (Titanium) Model 4a (ABS) 1 Model 4a (Acrylicglass) fA (m/s) Model 4a (Aluminium) Model 4a (Brass) 0,8 Model 4a (Copper) Model 4a (Plywood) Model 4a (Polyethyleneterephthalate) 0,6 Model 4a (Polystyrene) Model 4a (Stainlesssteel) 0,4 Model 4a (Titanium) Model 6a (ABS) Model 6a (Acrylicglass) 0,2 Model 6a (Aluminium) Model 6a (Brass) Model 6a (Copper) 0 Model 6a (Plywood) 0 0,2 0,4 0,6 0,8 1 Model 6a (Polyethyleneterephthalate) Model 6a (Polystyrene) Length ratio (Lpatch/Lbase) Model 6a (Stainlesssteel) Model 6a (Titanium) 40 Model 2a (ABS) Model 2a (Acrylicglass) Model 2a (Aluminium) 35 Model 2a (Brass) Model 2a (Copper) Model 2a (Plywood) 30 Model 2a (Polyethyleneterephthalate) First resonance frequency (Hz) Model 2a (Stainlesssteel) Model 2a (Titanium) 25 Model 4a (ABS) Model 4a (Acrylicglass) Model 4a (Aluminium) 20 Model 4a (Brass) Model 4a (Copper) Model 4a (Plywood) 15 Model 4a (Polyethyleneterephthalate) Model 4a (Stainlesssteel) Model 4a (Titanium) 10 Model 6a (ABS) Model 6a (Acrylicglass) Model 6a (Aluminium) 5 Model 6a (Brass) Model 6a (Copper) Model 6a (Plywood) 0 Model 6a (Polyethyleneterephthalate) 0 0,2 0,4 0,6 0,8 1 Model 6a (Polystyrene) Model 6a (Stainlesssteel) Length ratio (Lpatch/Lbase) Model 6a (Titanium) Figure 2.23: Piezoelectric flapping wing models with diverse plate materials. 50
  • 66. The length of the wing construction in the different models has also been investigated. For all models the tip deflection seems to increase proportional to the wing length, however the first resonance frequency and the EMCF drop with an increasing wing length, until the mode shape changes. One should therefore apply a wing length that is short enough to keep the wing structure resonating at the first bending mode, with an acceptable EMCF. The maximum wing length to achieve this varies from 1.2 to 1.8 for the investigated models with the specific geometric and material properties. It is expected that these results will be very dependent on these system characteristics. 0,0045 Model 2a (unimorph) Model 2a (bimorph) 0,004 Model 2b (unimorph) 0,0035 Model 2b (bimorph) Tip deflection (m) Model 3a (unimorph) 0,003 Model 3a (bimorph) 0,0025 Model 3b (unimorph) Model 3b (bimorph) 0,002 Model 4a (unimorph) 0,0015 Model 4b (unimorph) 0,001 Model 4b (bimorph) Model 5a (unimorph) 0,0005 Model 5a (bimorph) 0 Model 5b (unimorph) Model 5b (bimorph) 0 0,5 1 1,5 2 2,5 3 Model 6a (unimorph) Model 6a (bimorph) Length ratio (Lwing/Lbase) Model 6b (bimorph) Model 2a (unimorph) 0,16 Model 2a (bimorph) 0,14 Model 2b (unimorph) Model 2b (bimorph) 0,12 Model 3a (unimorph) 0,1 Model 3a (bimorph) EMCF Model 3b (unimorph) 0,08 Model 3b (bimorph) 0,06 Model 4a (unimorph) Model 4b (unimorph) 0,04 Model 4b (bimorph) 0,02 Model 5a (unimorph) Model 5a (bimorph) 0 Model 5b (unimorph) 0 0,5 1 1,5 2 2,5 3 Model 5b (bimorph) Model 6a (unimorph) Length ratio (Lwing/Lbase) Model 6a (bimorph) Model 6b (bimorph) 51
  • 67. Model 2a (unimorph) 0,9 Model 2a (bimorph) 0,8 Model 2b (unimorph) 0,7 Model 2b (bimorph) Model 3a (unimorph) 0,6 Model 3a (bimorph) fA (m/s) 0,5 Model 3b (unimorph) Model 3b (bimorph) 0,4 Model 4a (unimorph) 0,3 Model 4b (unimorph) 0,2 Model 4b (bimorph) Model 5a (unimorph) 0,1 Model 5a (bimorph) 0 Model 5b (unimorph) Model 5b (bimorph) 0 0,5 1 1,5 2 2,5 3 Model 6a (unimorph) Length ratio (Lwing/Lbase) Model 6a (bimorph) Model 6b (bimorph) Model 2a (unimorph) 45 Model 2a (bimorph) First resonance frequency (Hz) 40 Model 2b (unimorph) 35 Model 2b (bimorph) Model 2b (bimorph) 30 Model 3a (bimorph) 25 Model 3b (unimorph) 20 Model 3b (bimorph) Model 4a (unimorph) 15 Model 4b (unimorph) 10 Model 4b (bimorph) Model 5a (unimorph) 5 Model 5a (bimorph) 0 Model 5b (unimorph) 0 0,5 1 1,5 2 2,5 3 Model 5b (bimorph) Model 6a (unimorph) Length ratio (Lwing/Lbase) Model 6a (bimorph) Model 6b (bimorph) Figure 2.24: The influence of the wing length on the performance of the piezoelectric flapping wing structure. From the previous results it follows that the maximum length of the wing structure is limited to a certain ratio between the wing length and the length of the piezo fan. However the width of the wing also plays an important role in the production of flow generated by the flapping wing. The width of the ⁄ wing has no influence on the dynamic tip deflection and causes the EMCF to decrease slowly, until at a certain width ratio ( ) where suddenly the tip deflection and the EMCF drop significantly while the first resonance frequency increases. This is again due to the change in the mode shape of the structure. Overall the width of the structure increases, but at the same time the frequency decreases, causing a decline of the curve. 52
  • 68. 0,0035 Model 4a (unimorph) Tip delfection (m) 0,003 0,0025 Model 4a (bimorph) 0,002 0,0015 Model 4b (unimorph) 0,001 Model 4b (bimorph) 0,0005 0 Model 6a (unimorph) 1,00 2,00 3,00 4,00 Model 6a (bimorph) Width ratio (wwing/wbase) Model 6b (bimorph) 0,08 Model 4a (unimorph) 0,06 Model 4a (bimorph) EMCF 0,04 Model 4b (unimorph) 0,02 Model 4b (bimorph) 0 Model 6a (unimorph) 1,00 2,00 3,00 4,00 Model 6a (bimorph) Width ratio (wwing/wbase) Model 6b (bimorph) 0,4 Model 4a (unimorph) 0,3 Model 4a (bimorph) fA (m/s) 0,2 Model 4b (unimorph) 0,1 Model 4b (bimorph) 0 Model 6a (unimorph) 1,00 2,00 3,00 4,00 Model 6a (bimorph) Width ratio (wwing/wbase) Model 6b (bimorph) 14 Model 4a (unimorph) First resonance 12 frequency (Hz) 10 Model 4a (bimorph) 8 6 Model 4b (unimorph) 4 2 Model 4b (bimorph) 0 Model 6a (unimorph) 1,00 2,00 3,00 4,00 Model 6a (bimorph) Width ratio (wwing/wbase) Model 6b (bimorph) Figure 2.25: Optimization results for models 4 and 6 with different width ratios. Until now the simulation were done with a constant distance between the two piezo fans. Thus the influence of this width ratio (≜ 2 ∙ ⁄ ratio between the total width of the piezo patch and the width of the base was kept constant. The ) has been investigated by changing the piezo patch width and keeping all other parameters unchanged. The tip deflection increases until at a certain point (width ratio of 0.5) it virtually stays constant. Because of the extra stiffness added by the piezo patches the first resonance frequency increases with increasing width ratios. This leads to an overall increase of the when a higher width ratio is used. For the models with the piezo patches connected with a flexible elastic plate (model 3 and 5) there is a drop of the EMCF with increasing width ratios. 53
  • 69. The EMCF stays quasi constant for the models with 2 independent piezo fans (Models 2, 4 and 6). Note that the increase of the first resonance frequency graph is steeper for these models, because the stiffness effect is dominating the mass effect more strongly. 0,04 Model 2a (unimorph) Model 2a (bimorph) 0,035 Model 2b (unimorph) Model 2b (bimorph) 0,03 Model 3a (unimorph) Tip deflection (m) Model 3a (bimorph) 0,025 Model 3b (unimorph) 0,02 Model 3b (bimorph) Model 4a (unimorph) 0,015 Model 4a (bimorph) Model 4b (unimorph) 0,01 Model 4b (bimorph) Model 5a (unimorph) 0,005 Model 5a (bimorph) 0 Model 5b (unimorph) Model 5b (bimorph) 0,00 0,20 0,40 0,60 0,80 1,00 Model 6a (unimorph) Model 6a (bimorph) Width ratio (2wpiezofan/wbase) Model 6b (bimorph) 0,12 Model 2a (unimorph) Model 2a (bimorph) Model 2b (unimorph) 0,1 Model 2b (bimorph) Model 3a (unimorph) 0,08 Model 3a (bimorph) Model 3b (unimorph) EMCF 0,06 Model 3b (bimorph) Model 4a (unimorph) Model 4a (bimorph) 0,04 Model 4b (unimorph) Model 4b (bimorph) 0,02 Model 5a (unimorph) Model 5a (bimorph) 0 Model 5b (unimorph) Model 5b (bimorph) 0,00 0,20 0,40 0,60 0,80 1,00 Model 6a (unimorph) Model 6a (bimorph) Width ratio (2wpiezofan/wbase) Model 6b (bimorph) 54
  • 70. 0,7 Model 2a (unimorph) Model 2a (bimorph) 0,6 Model 2b (unimorph) Model 2b (bimorph) 0,5 Model 3a (unimorph) Model 3a (bimorph) Model 3b (unimorph) fA (m/s) 0,4 Model 3b (bimorph) 0,3 Model 4a (unimorph) Model 4a (bimorph) 0,2 Model 4b (unimorph) Model 4b (bimorph) 0,1 Model 5a (unimorph) Model 5a (bimorph) 0 Model 5b (unimorph) Model 5b (bimorph) 0,00 0,20 0,40 0,60 0,80 1,00 Model 6a (unimorph) Model 6a (bimorph) Width ratio (2wpiezofan/wbase) Model 6b (bimorph) 20 Model 2a (unimorph) Model 2a (bimorph) 18 First resonance frequency (Hz) Model 2b (unimorph) 16 Model 2b (bimorph) Model 2b (bimorph) 14 Model 3a (bimorph) 12 Model 3b (unimorph) 10 Model 3b (bimorph) Model 4a (unimorph) 8 Model 4a (bimorph) 6 Model 4b (unimorph) Model 4b (bimorph) 4 Model 5a (unimorph) 2 Model 5a (bimorph) Model 5b (unimorph) 0 Model 5b (bimorph) 0,00 0,20 0,40 0,60 0,80 1,00 Model 6a (unimorph) Model 6a (bimorph) Width ratio (2wpiezofan/wbase) Model 6b (bimorph) Figure 2.26 The influence of the width of the piezo fans on the performance of the flapping wing structure. 26: ⁄ℎ At last the thickness of the spar was investigated while changing the length of the spar (wing length). The influence of the thickness ratio ((ℎ ) is small while longer spars let the tip deflection ⁄ increase. The first resonance frequency and EMCF is being decreased. This trend is followed until the first mode shape changes at a length ratio ( ) o 1.9. of Tip deflection (m) EMCF 0,006- 0,006 0,15 0,008 0,008 0,006 0,004- 0,004 0,1 0,1-0,15 0,1 0,004 0,006 0,05 0,05 0,05-0,1 0,002 0,002- 0,002 Thick- 0 0,004 Thick- Thick 0 0-0,05 0 ness 0,25 0,25 ness 0- ratio 1,5 0,3 ratio 1,5 0,3 1,1 0,002 1,1 (hspar/hb 1,9 (hspar/hb 1,9 2,7 2,7 eam) eam) Length ratio (Lspar/Lbase) Length ratio (Lspar/Lbase) 55
  • 71. fA (m/s) 1st resonance frequency (Hz) 2 1,5 60 1 1,5-2 1,5 40 40-60 40 0,5 1-1,5 1,5 20 20-40 20 0 0,5-1 0,5 Thick- Thick 0-20 0 0,3 0-0,5 0,5 ness 0,25 0 0,25 1,1 ratio 1,5 1 1,9 0,3 (hspar/hb 1,75 1,1 2,7 Length ratio (Lspar/Lbase) 2,5 1,9 eam) 2,7 Length ratio (L /L ) Thickness ratio (hspar/hbeam) spar base Figure 2.27 The influence of the thickness ratio and length ratio of the spar on the optimization quantities. 27: 2.4.2.4 Conclusion Various simulations have been conducted on the basic piezo fan model and other alternative flapping wing models to investigate the influence of diverse parameters. The optimization of the geometrical and physical properties of these models was pursued. In many cases the optimization quantities (e.g. tip deflection, EMCF, , first resonance frequency) reached a maximum within some ranges of the geometrical ratios. These results could be used when designing an in house made prototype of the in-house piezoelectric flapping wing structure. 56
  • 72. 3 CFD Simulations 3.1 Introduction Computational fluid dynamics (CFD) is a new "third approach" in the philosophical study and development of the whole discipline of fluid dynamics. Throughout most of the twentieth century the study and practice of fluid dynamics involved the use of pure theory on the one hand and pure experiment on the other hand. This was the "two way approach". However, the advantage of the high- speed digital computer combined with the development of accurate numerical algorithms for solving physical problems on these computers has revolutionized the way we study and practice fluid dynamics today. It has introduced a fundamentally important new third approach in fluid dynamics: the approach of CFD. So it has become an equal partner with pure theory and pure experiment in the analysis and solution of fluid dynamic problems. While CFD gives us a new approach, it will never replace either of the other two approaches. There will always be a need for theory and experiment. The advancement of a study will depend on the balance of all three approaches, with CFD helping to interpret and understand the results of theory and experiment, and vice versa. CFD is playing a strong and important role as a design and research tool in this work. Figure 3.1: The three dimensions of fluid dynamics. ANSYS CFX provides the ability to solve, or take part in the solution of cases that involve the coupling of solution fields in fluid and solid domains. This coupling is commonly referred to as Fluid Structure Interaction (FSI) [38]. 3.2 Fluid Structure Interaction: coupling of CFD and FE analysis Fluid structure interaction (FSI) occurs when fluid flow generates forces on a solid structure, causing it to deform and potentially perturb the initial fluid flow. It represents the class of multiphysics where we consider the effects of fluid flow on compliant structures and their subsequent interactions. This type of interaction causes the deformation of an aircraft wing during flight, for example, or the vibration of a civil engineering structure due to airflow. The primary fields interacting across the fluid and the structure domain are pressures and displacements, respectively. For problems where thermal effects are significant, temperature is an additional field in both the domains. In addition to these primary fields, there are secondary fields such as piezoelectric effects in the structural domain or cavitation effects in the fluid domain that indirectly contribute to the fluid-structure interaction. Two or more physical systems frequently interact with each other, where the independent solution of one system is impossible without a simultaneous solution of the others. Such systems are known as coupled, where coupling may be weak or strong, depending on the degree of interaction. FSI 57
  • 73. simulations require the solution of multiple coupled fields. Optimal coupling methods for the execution of FSI simulations are determined by the complexity of the fluid and solid models and their physical coupling. A one-way coupling method and information stream is sufficient when the physical coupling between the fluid and solid models is steady and predominantly from one model to the other (weakly coupled systems). For example, an accurate structural analysis that is based upon static loads from a completed fluid flow simulation is possible if the resulting structural deformations do not feed back to the fluid flow field. Strongly coupled systems necessitate a full exchange of information between two or several interacting physical systems (two-way coupling method). For example, this method is required to simulate the potentially transient deformation of an aircraft wing as air flows around it. Tight integration of advanced fluid and structural analysis tools is required for the efficient and accurate solution of large and physically complex FSI problems. In this work this integration is obtained by using the ANSYS MFX Multi-field solver. In this chapter a computational analysis of the fluid-structure interaction of an oscillating plate in a container filled with fluid is presented. The fluid can have a significant influence on the deformation of wing during operation, depending on the frequency of the flapping, the amplitude and the fluid viscosity. A computational two-way coupling analysis method has been performed. The multiphase fluid field for the flapping wing was first analyzed with the structural code ANSYS, wich is based on the finite element method (FEM). The flapping wing deformed and interacted with the fluid, which was then analyzed with the CFD code CFX. This was done for every time step. Solution data from the solid model structural field is required for the fluid model field. Two approaches, direct and iterative, exist for addressing this coupling. The direct approach involves assembling a single monolithic equation set, with coefficients to actively couple the individual field equations. In spite of solving all field equations together, the direct approach still requires iteration to resolve the nonlinearities that exist in all FSI simulations. The iterative or load transfer approach involves assembling and solving an equation set for each field, with coupling data transferred at field interfaces. At least one coupling iteration, consisting of data transfer and equation set solution, is needed for each field to achieve a coupled response. MFX simulations are executed as a series of multi-field load or time steps. Within each multi-field step, multiple stagger iterations are performed until each of the fluid and solid field equations and the coupling data transferred at the field interfaces have converged. This implicit coupling ensures that fluid and solid solution fields are consistent with each other at the end of each multi-field step, which leads to improved numerical solution stability. Overall efficiency improves since time step sizes are limited only by the physical processes being modeled. Conversely, an explicit coupling does not ensure solution field consistency, and time step sizes can be severely limited by the numerical solution process. Additional controls over the ordering of field solver execution and the under-relaxation of coupling data transferred at field interfaces allow fine-tuning of the solution process. The MFX Multi- field solver uses fully automatic methods for treating dissimilar meshes at solid fluid interfaces, with both profile-preserving and conservative interpolation options available at each interface. Two-way FSI applications require data transfer between the fluid and solid models throughout the simulation. This is achieved by the MFX solver through a native communication infrastructure that is based on a client/server protocol using standard internet sockets. This infrastructure provides an independence from third-party coupling software. 58
  • 74. 3.2.1 Defining the problem The solid physics simulation is first defined in ANSYS Workbench. This is done by creating a new geometry file that includes both the fluid and solid domains with their material properties defined. The ANSYS Multi-field simulation is a transient mechanical analysis, with a certain timestep and time duration, therefore a Flexible Dynamic (time history) simulation is chosen. Loads are applied to an FEA analysis as the equivalent of boundary conditions in ANSYS CFX. A fixed support on the piezoelectric wing and a time dependent pressure function is defined on one side of the piezo fan as given in Figure 3.2: = ∗ ∙2 ∙ This allows us to set the amplitude and the frequency or period of the wing. From the chosen function and time step settings tabular data will be created with the pressure for each time step. The plate has a length, width and thickness of 65mm, 16mm and 0.8 mm, respectively. Figure 3.2: A time dependent pressure function can be applied on the plate to let it move similar to the first bending mode. Data is exchanged across this interface during the execution of the simulation. The faces of the geometry (region in the solid) that form the interface between the solid in ANSYS and the fluid in CFX are selected and defined as the Fluid-Solid Interface. The simulation settings are written in an ANSYS Batch file (APDL), and will be read by CFX so that the software knows which Fluid Solid Interfaces are available. In ANSYS CFX Preprocessor the fluid physics and the ANSYS Multifield settings are defined. The CFD mesh is generated in Workbench and can be imported. Once the timesteps and time duration are specified for the ANSYS Multi-field run (coupling run), ANSYS CFX automatically picks up these settings and it is not possible to set the timestep and time duration independently. In order to allow the ANSYS Solver to communicate mesh displacements to the CFX Solver, mesh motion must be activated in CFX. Initially a 2D representation of the flow field is being modeled (using a 3D mesh with one element thickness in the Z direction), so symmetry boundaries must be created on the low and high Z 2D regions of the mesh. The interface between ANSYS and CFX is defined as an external boundary in CFX that has its mesh displacement being defined by the ANSYS Multi-field coupling process. 59
  • 75. A custom fluid is created with user-specified properties: Parameter Property Thermodynamic State Gas Molar Mass (kg/kmol) 28.96 Density (kg/m3) 1.185 Specific Heat Capacity J/(kg K) 1004.4 Dynamic Viscosity (kg/(m s) 1.831e-5 Table 3.1: Material properties of air at 20°C used in the numerical flow simulations. Since a transient simulation is being modeled, initial values are required for all variables. Within each timestep, a series of “coupling” or “stagger” iterations are performed to ensure that CFX, ANSYS and the data exchanged between the two solvers are all consistent. Within each stagger iteration, ANSYS and CFX both run once each, but which one runs first is a user-specifiable setting. In general, it is slightly more efficient to choose the solver that drives the simulation to run first. In this case, the simulation is being driven by the pressure function applied in ANSYS, thus ANSYS is set to solve before CFX within each stagger iteration. A second order backward Euler transient scheme is used for defining the discretization algorithm for the transient term. This is an implicit time-stepping scheme that is second order accurate. This scheme is generally recommended for most transient runs. At each timestep in a transient simulation, the ANSYS CFX-Solver performs several coefficient iterations or loops, either to a specified maximum number or to the predefined residual tolerance. The maximum number of iterations per timestep may not always be reached if the residual target level is achieved first. Using a large number of coefficient iterations for transient runs is not recommended. Improved accuracy is much more efficiently achieved by reducing the timestep size. If the solution is not converging within each timestep, which is required to maintain a conservative discretization, it may signify that the timestep is too large for good transient accuracy. Decreasing the timestep size typically makes convergence within a timestep easier to achieve, and also improves the transient accuracy of the solution. For this particular application, it is essential to make a compromise between timestep size, number of coefficient iterations and residual tolerance to get the most cost effective solution, i.e. to obtain a balance between accuracy and solution time. The minimum number of iterations that the solver will complete per timestep is set to 2, the maximum is set to 3. The residual is set to 10-4, therefore the solver will terminate the run when the equation residuals calculated are below this value. 3.2.2 Modeling The set of equations solved by CFX are the unsteady Navier-Stokes equations in their conservation form. The governing equations that describe the fluid flow are conservation of mass, conservation of momentum and conservation of energy [38]. These partial differential equations were derived in the early nineteenth century and have no known general analytical solution but can be discretized and solved numerically. The equations are given below. The solution method used by CFX is the finite volume technique. In this technique, the region of interest is divided into small sub-regions, called control volumes. The equations are discretized and solved iteratively for each control volume. As a result, an approximation of the value of each variable at specific points throughout the domain can be obtained. In this way, one derives a full picture of the behavior of the flow. The governing equations are generally solved using Cartesian spatial coordinates and velocity components. 60
  • 76. The general form of conservation of mass or continuity equation is: + =0 Where is the fluid density, the velocity component. This equation is valid for both compressible and incompressible flows. Conservation of momentum in -th direction in an inertial reference frame is: + =− + + + Where is the static pressure, is the viscous stress tensor and and respectively the gravitational acceleration and the external body force in the direction. The energy equation can be written as: ℎ + ℎ = Here h is the static enthalpy, T the temperature and k the thermal conductivity. Deforming meshes are simulated through the use of dynamic meshes. These dynamic meshes may be used to model flow where the shape of the domain changes with time due to motion of the domain boundaries. The integral form of the transport equation for a general scalar Φ on an arbitrary control volume on a moving mesh is: Φ + Φ − = Γ∇Φ + Φ Where and are the flow velocity vector and grid velocity of the moving meshes, respectively. The first term on the left of the equation is the time derivative term. The second term is the convective term. The first term on the right side is the diffusive term; the second term is the source term. Γ is the diffusion coefficient and Φ is the source term of Φ. The term is used to represent the boundary of the control volume and is the area movement. Non-slip boundary conditions are applied at all solid surfaces. Consequently, mesh deformation is an important component for solving problems with moving boundaries or changing domain geometries. This capability is available in fluid and solid domains. Motion can be specified on selected regions via an external solver coupling. The motion might be imposed, or might be an implicit part of a coupled fluid-structure simulation. Large mesh deformations are often required in fluid-structure interaction analysis. The control volumes are subjected to translation or deformation and the Navier-Stokes equations have to be extended by a term, which describes the relative motion of the surface element with respect to the fixed coordinate system. Additionally the Geometric Conservation Law (CGL) has to be fulfilled [39]. Such deformations are handled without remeshing. Moving mesh works essentially by moving the mesh as prescribed on certain boundaries, and then solving a diffusion equation to calculate the new mesh positions throughout the domain. If the mesh is moved large distances with too large timesteps, this can lead to mesh folding or negative mesh volumes. If the mesh movement is very large over time it can lead to poor quality meshes even if folding is avoided. Therefore a combination of a fine CFD mesh with small time steps are essential for reliable results. Both can be calculated by doing a sensitivity and 61
  • 77. convergence test. In general, the desired total mesh deformation should be split up so that regions where motion is specified move through less than approximately 5 adjacent elements per step [38]. During each timestep, the mesh displacement equations are solved to the specified convergence level and the resulting displacements are applied to update the mesh coordinates. This occurs before proceeding to solve the general transport equations. Mesh folding occurs and is detected when the displacements are used to update the mesh coordinates. Folded meshes can occur if the displacement equations are incompletely solved. In this case, the unconverged displacement solution field does not change smoothly enough to ensure that adjacent mesh nodes move by similar amounts. 0,01 Displacement (m) 0,008 Total Mesh 0,006 0,004 0,002 0 0 200 400 600 800 Time step Figure 3.3: Total Mesh Displacement of the tip of the wing in CFX Solver. The variable Total Mesh Displacement becomes available when executing simulations with mesh deformation. This variable represents the displacement relative to the initial mesh. By using this variable the amplitude and the motion of the wing at the tip can be checked, while the simulation is running (see Figure 3.3). 3.3 Numerical model The case study for this work is defined as a thin plate, clamped on one side, vibrating at resonance in an enclosure filled with an incompressible and viscous fluid. Although the shim used for vibration is actuated by a piezoelectric patch, the geometry has been simplified for numerical consideration to a plate structure. It is expected that the influence of the thickness of the piezoelectric patches is very small. To perform the CFD analysis of the posed physical problem, the enclosure containing the fluid is discretised with finite volumes. The piezoelectric wing is placed in the container and via a pressure function deformed with a certain deflection amplitude and resonance frequency at the first bending mode. The model was analyzed with the finite volumes CFD code CFX. The vibrating amplitude and the flapping frequency of the piezoelectric wing are varied. A beam model of the piezoelectric fan displacement was used by Acikalin et al. [40]. They described the first mode shape (resonance frequency of 60Hz) of a commercial piezoelectric fan with an equation dependant on the dimensions/material properties. Using the equation for beam location during vibration, a user-defined function was called in Fluent to provide the boundary along the beam. The equation was set up for 60Hz, but for numerical stability they used a resonance frequency of 62.5Hz. It was assumed that the error made in using the equation derived for 60Hz was small for small change in vibration. Variation of heater plate proximity and fan deflection amplitude were considered through a design of experiment approach and compared to available experimental data. Four regions of circulation could be seen in the velocity plots presented when the heater plate was close to the fan tip. The smaller two circulation regions near the object are characterized by stronger velocities due to their proximity to the driving force, i.e. moving wall boundary. Symmetric circulation zones above and below the vibrating beam are observed it moves up and down forcing fluid to displace more rapidly as 62
  • 78. the beam passes a given plate position (see Figure 3.4). Due to proximity to the object, fluid velocities are relatively larger in magnitude than the other cases with larger beam to plate distances. Figure 3.4: Velocity vectors colored by velocity magnitude (m/s) (Time=1.5280s) [40]. Unlike the case in which an object is at a small distance from the beam, two standing circulation regions of fluid near the object do not seem to form for large distances and small fan amplitudes. Two other larger circulation zones similar to those in case 1 are observed far away from the fan. So, a different flow regime developed as this distance increased. The difference in these flow regimes contributed to differing heat transfer coefficients. It had also been shown that heat transfer rates decreased with increasing fan distance and decreasing fan amplitude as expected. Experimental and simulation studies were made for a piezo fan for two different heights above the heat source in Abdullah et al. [41]. The piezoelectric fan operation was modeled using the dynamic meshes coupled with a harmonic motion in FLUENT 6.2.3 software and predicted the induced flow and heat convection involved. The heat transfer coefficient predicted by CFD simulation had shown good agreement with the experimental data. 3.3.1 Analysis settings Complexity of the moving wall boundary condition required a mesh domain for the incompressible fluid that would accommodate shifting of the boundary. This aspect proved to be somewhat problematic, especially with large amplitudes in which node boundaries were significantly deformed. Fluid cells surrounding the beam are prescribed to adjust their size to during beam movement. This mesh motion is the primary enabler of a large beam deflection boundary condition. However, during beam and fluid cell movement, the fluid cells can be greatly distorted from the equilateral triangle condition resulting in faulty fluid cells for individual control volume analysis. The physical parameters varied in these cases were the operating frequency, deflection amplitude and geometric parameters. An attempt was made to include different cases of a flapping wings in a container in this study. The following assumptions were made for the numerical models: incompressible flow, 2-dimensional, no buoyancy, no radiation. The second order upwind discretization scheme is used both for momentum and energy for better accuracies. Different turbulence models were compared [38]: • The Laminar Model Laminar flow is governed by the unsteady Navier Stokes equations. This model does not 63
  • 79. apply a turbulence model to the simulation and is only appropriate if the flow is laminar. This typically applies at low Reynolds number. Energy transfer in the fluid is accomplished by molecular interaction (diffusion). In the case of high speed flows, the work of the viscous stresses can also contribute to the energy transfer. If a simulation is set up using laminar flow, but the real flow is turbulent, convergence is difficult and the simulation will not reach the correct solution. • The k-ε Model One of the most prominent turbulence models, the k-ε (k-epsilon) model, is considered the industry standard model. It has proven to be stable and numerically robust and has a well established regime of predictive capability. For general purpose simulations, the k-ε model offers a good compromise in terms of accuracy and robustness. While standard two-equation models, such as the k-ε model, provide good predictions for many flows of engineering interest, there are applications for which these models may not be suitable. Among these are: • Flows with boundary layer separation. • Flows with sudden changes in the mean strain rate. • Flows in rotating fluids. • Flows over curved surfaces. A Reynolds Stress model may be more appropriate for flows with sudden changes in strain rate or rotating flows, while the Shear Stress Transport (SST) model may be more appropriate for separated flows. • The Standard k-ω and Shear Stress Transport (SST) k-ω Models One of the main problems in turbulence modeling is the accurate prediction of flow separation from a smooth surface. Standard two-equation turbulence models often fail to predict the onset and the amount of flow separation under adverse pressure gradient conditions. In general, turbulence models based on the ε-equation predict the onset of separation too late and under- predict the amount of separation later on. This is problematic, as this behavior gives an overly optimistic performance characteristic for an airfoil. The prediction is therefore not on the conservative side from an engineering stand-point. The models developed to solve this problem have shown a significantly more accurate prediction of separation in a number of test cases and in industrial applications. The standard and the Shear-Stress Transport (SST) k-ω models have both transport equations for k and ω. The major differences between the SST model and the standard model are: • The gradual change from the standard k-ω in the inner region of the boundary to a high-Reynolds-number version of the k-ε model in the outer part of the boundary layer. • The modified turbulent viscosity formulation to account for the transport effects of the principal shear stress. This feature gives the SST k- ω model an advantage in terms of performance over both the standard k-ω model and the standard k-ε model. Other modifications include the addition of a cross-diffusion term in the ω equation and a blending function to ensure that the model equations behave appropriately in both the near-wall and far-field zones. The SST turbulent model is used for the conducted flapping wing FSI simulations in this work. • The Reynolds Stress Model Two-equation turbulence models (k-ε and k-ω based models) offer good predictions of the characteristics and physics of most flows of industrial relevance. In flows where the turbulent transport or non-equilibrium effects are important, the eddy-viscosity assumption is no longer 64
  • 80. valid and results of eddy-viscosity models might be inaccurate. Reynolds Stress models naturally include the effects of streamline curvature, sudden changes in the strain rate, secondary flows or buoyancy compared to turbulence models using the eddy-viscosity approximation. In the following cases the Reynolds Stress Model may be considered: • Free shear flows with strong anisotropy, like a strong swirl component. This includes flows in rotating fluids. • Flows with sudden changes in the mean strain rate. • Flows where the strain fields are complex, and reproduce the anisotropic nature of turbulence itself. • Flows with strong streamline curvature. • Secondary flow. • Buoyant flow. Reynolds Stress models have shown superior predictive performance compared to eddy- viscosity models in these cases. This is the major justification for Reynolds Stress models, which are based on transport equations for the individual components of the Reynolds stress tensor and the dissipation rate. These models are characterized by a higher degree of universality. The penalty for this flexibility is a high degree of complexity in the resulting mathematical system. The increased number of transport equations leads to reduced numerical robustness, requires increased computational effort and often prevents their usage in complex flows. Theoretically, Reynolds Stress models are more suited to complex flows, however, practice shows that they are often not superior to two-equation models. It is based on the ω-equation and automatic wall treatment. Compared to the k-ε model, the Reynolds Stresses model has six additional transport equations that are solved for each time step or outer coefficient loop in the flow solver. The source terms in the Reynolds Stress equations are also more complex than those of the k-ε model. As a result of these factors, outer loop convergence may be slower for the Reynolds Stress model than for the k-ε model. In principle, the same time step can be used for all turbulence model variants, but the time step should be reduced for the Reynolds Stress Model due to the increased complexity of its equations and due to numerical approximations made at general grid interfaces and rotational periodic boundary conditions. • The ω-Reynolds Stress (SMC-ω) and BSL Reynolds Stress Models Some of the main deficiencies of the Reynolds Stress models for the simulation of boundary layers are an inheritance from the underlying ε-equation. Particularly the accurate prediction of flow separation is problematic when the ε-equation is used. Furthermore, low-Reynolds number formulations for the ε-equation are usually complex and difficult to integrate, a deficiency which is exaggerated in combination with a Reynolds Stress model formulation. In order to avoid these issues, a Reynolds Stress model has been implemented which uses the ω- equation instead of the ε-equation as the scale-determining equation. 3.3.2 Sensitivity analysis and convergence test As the beam deflects to upper and lower directions, vortices of local fluid rotation form and shed away from the beam tip away of the wing. These fluid vortices alternate with direction of beam deflection similar to the vortex shedding situation developed in flow around a cylinder. As the wing continues to vibrate, pressure gradients formed by the displaced beam, push the vortices away toward the nearest object causing fluid present near the object to be forced upward and downward into the larger circulation present in the fluid domain. If no object is present the vortices slowly fade away and a somewhat constant flow is generated. The frequency of the vortex shedding is twice the frequency of the piezoelectric wing, i.e. two vortices are shed during one full cycle of fan vibration. The piezoelectric wing moves from the undeflected position to the upper extreme position and then returns back to the undeflected position, shedding one vortex. When it travels further down to the lower extreme position and again passes through the undeflected position, it will shed the second vortex. 65
  • 81. In order to capture the flow field correctly a very small time step is required. For all the cases that are presented in this work the time step was chosen in such a way that in one cycle of fan vibration more than 100 iterations in time were required. A time step convergence test (sensitivity test) was conducted. For the sensitivity analysis the step time was varied and the influence on the velocity field was determined. In Figure 3.5 a part of the solution for the velocity in a point with coordinates (0.0005;0.08;0) with different time steps is presented. It can be seen that the velocity solution converges using a time step smaller than 0.0001 s, which means 167 steps per period. Therefore in the following simulations the time step was chosen in such a way that there were always more than 167 steps per oscillation of the fan. V with step time 5e-5 (334 step 2,3 times per period) V with step time 1e-4 (167 step times per period) V with step time 2e-4 (83 step times per period) V with step time 4e-4 (42 step times per period) 1,8 V with step time 8e-4 (20 step times per period) Vu with step time 5e-5 (334 step Velocity (m/s) times per period) Vv with step time 5e-5 (334 step times per period) 1,3 Vu with step time 1e-4 (167 step times per period) Vv with step time 1e-4 (167 step times per period) Vu with step time 2e-4 (83 step times per period) 0,8 Vv with step time 2e-4 (83 step times per period) Vu with step time 4e-4 (42 step times per period) Vv with step time 4e-4 (42 step times per period) V with step time 8e-4 (20 step 0,3 times per period) 0,01 0,011 0,012 0,013 0,014 0,015 V with step time 8e-4 (20 step times per period) Time (s) 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. The results of the sensitivity analysis are given in Figure 3.6. There can be concluded that a time step size of 0.0001s is adequate to obtain reliable results. All the further simulations were done with this time step size. 66
  • 82. 0,09 0,2 0,1 0,08 Velocity Error (m/s) 0,08 Time 0,07 0,15 0,06 0,0104s 0,05 0,06 0,1 Time 0,04 0,04 0,0112s 0,03 0,02 0,05 Time 0,02 0,01 0,012s 0 0 0 Time 0 2 4 0 2 4 0 2 4 0,0128s Time step size (x10-4s) Time step size (x10-4s) Time step size (x10-4s) (A) (B) (C) 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. Next to the correct selection of the time step, the mesh size is also an important aspect for obtaining accurate results. In Figure 3.7 a comparison has been made between the velocity vector field obtained by using a fine (left) and coarse (right) mesh. In the model with coarse mesh only the tip has a fine mesh, causing the solution to be inaccurate for locations far away from the tip. The maximum velocity reaches about 12m/s and 14m/s in the model with coarse and fine mesh, respectively. This maximum is not reached at the tip of the fan, but at approximately 30mm from the tip. The model with the fine mesh is used for the further simulations. Figure 3.7: Comparison between the velocity vector field obtained by using a fine (left) and coarse (right) mesh. 67
  • 83. 3.3.3 Results and discussion The size of the computational domain is similar to those used in the experiments (30cm x 30cm). The clamp of the piezoelectric fan is neglected and simply replaced by a wall. Different point locations of interest are defined in the 3D fluid domain with 40779 elements and 197651 nodes (Figure 3.8). A single simulation with 4000 time steps (and time step size 0.0001s) and a fine mesh took approximately 7 days on a PC with Intel i7 2.5Ghz processors, 12GB 1600Mhz DDR3 RAM and 2x 10k rpm HD in raid0. 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. The plate is being oscillated at a frequency of 60Hz. The velocity in point 3 is plotted in function of time in Figure 3.9. The oscillating frequency of is half the oscillating frequency of , because per period of the flapping wing 2 vortices are being shed. Therefore is oscillating with a frequency of 120Hz. 12 10 8 Velocity (m/s) 6 4 Velocity V 2 Velocity Vu 0 Velocity Vv -2 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 -4 -6 Time (s) Figure 3.9: The velocity in function of the time in point 3. 68
  • 84. The velocity in function of time in the different positions defined in the domain in Figure 3.8 is given Figure 3.10. The oscillation of the velocity can be clearly observed. To get a better view on the velocity in all the points in the domain, velocity vector fields can be plotted. 12 10 Position 1 Position 2 8 Position 3 Velocity (m/s) Position 4 6 Position 5 Position 6 4 Position 7 Position 8 2 Position 9 Position 10 0 0,3 0,305 0,31 0,315 0,32 0,325 0,33 0,335 Time (s) Figure 3.10: The velocity in function of the time for the different positions. Figure A.1 (in Appendix A) shows the transient flow generated by the vibrating fan (velocity vectors for different time intervals). This shows that the flow induced by the piezoelectric fan is a highly unsteady phenomenon. At the time intervals the position of the fan changes with time when the fan is swinging either in a downward (right in the figure) or upward (left in the figure) direction. The velocity vector induced is very much dependent on the fan swinging. It can be seen that a vortex is being shed when the flapping wing has already made a downward sweep and is heading back up towards the upper extreme position. When the fan gets at its right extreme position for the second time the boundary layers around the fan are seen to start forming a distinct pattern. Another vortex is shed, this time on the upper side of the flapping wing, while the first one is displaced downwards. When the flapping wing is passing through the neutral position it is apparent that the first vortex is moving counter-clockwise while the second vortex is moving clockwise. The boundary layer formed over the wing is clearly visible and the obtained flow patterns suggest that the vortices are shed slightly before the tip of the wing. When the wing keeps oscillating more vortices can be observed. These vortices continue to emerge and move away from the point of origin, while being combined with other vortices. The vortex in the top left in Figure 3.11 is a conglomerate of 4 vortices. The vortices are pushed away from the tip by air flow at higher time intervals. The boundary layers over the fan appear to be fully developed and unchanging after some time. As time passes, vortices of opposite circulation continue to be shed from each side of the fan, traveling in opposite directions. All the vortices that are created at the left side of the piezo fan move counter- clockwise and conversely all the vortices at the right side of the piezo fan move clockwise. Suction occurs both near the clamp on the left and near the tip on the right. The maximum fluid rejection velocity is observed not at the tip but slightly before the tip of the wing. 69
  • 85. Figure 3.11: Velocity vector plot of the induced flow by the harmonic movement of the piezo fan at time=0.1s. The velocity vector field generated by a piezo fan in [40] is qualitatively and quantitavely very similar to the velocity field in Figure 3.11. Figure 3.12: Velocity vector field induced by a piezo fan [40] Streamlines for the flow pattern that has developed after time=0.1s is given in Figure 3.13. The vortices are clearly visible. Because the flapping wing first makes a downward sweep (in the picture to the right), it causes an asymmetrical flow in the beginning. This effect vanishes when enough time is passed. 70
  • 86. Figure 3.13 Streamline for the flow pattern that has developed after time=0.1s 13: time=0.1s. The influence of the tip deflection is investigated by changing the pressure function. By applying the investigated right pressure the tip deflection can be controlled. Four different simulations with different tip deflections were conducted keeping all the other parameters constant. The velocity in Point 3 is given oint in Figure 3.14(a). Like expected an increase in the tip deflection of the flapping wing causes a . proportional increase in the velocity of the flow. In Figure 3.14(b) the averaged velocity is given. (b) averaged (a) 20 (b) 12 y = 27,902x + 2,0308 18 11,5 R² = 0,9988 16 11 Averaged Velocity (m/s) 14 10,5 Velocity (m/s) 12 10 2cm 10 9,5 8 2,5cm 9 6 3cm 8,5 4 3,5cm 8 2 7,5 0 7 0 0,05 0,1 0,2 0,25 0,3 0,35 Time (s) Tip Deflection (m) Figure 3.14: The influence of the tip deflection on the velocity of the flow in point 3. 14: 71
  • 87. It is also expected that the frequency plays an important role in the velocity of the flow field. The flow result for the velocity in Point 3 is given in Figure 3.15. The simulations were conducted for 6 flapping Figur . oscillations of the wing, therefore the curves for the higher frequencies are shorter. 30 25 20 40Hz Velocity (m/s) 50Hz 15 55Hz 60Hz 10 80Hz 100Hz 5 150Hz 0 -0,01 0,01 0,01 0,03 0,05 0,07 0,09 0,11 0,13 0,15 Time (s) Figure 3.15 The influence of the frequency on the induced velocity by a piezo fan with a tip deflection of 2 cm. 15: nfluence A proportional relationship between the frequency and the velocity of the flow can be observed when the velocity in the point is being time-averaged for the different cases with other flapping frequencies averaged (Figure 3 3.16). 20 y = 0,1206x + 0,1961 Averaged Velocity (m/s) R² = 0,9987 15 10 5 0 0 50 100 150 Frequency (Hz) Figure 3.16: Influence of the frequency ( 16: (averaged). A 3D CFD model is used, giving the possibility to present 3D velocity vector fields (Figure 3. used, .17). Due to the large calculation times and time constraints, no simulations of the 2DOF piezoelectric flapping wing structures with flapping and twisting motion have been run, yet However by changing several , yet. geometrical properties these simulations can be easily scheduled in the future. 72
  • 88. Figure 3.17: 3D velocity vector field plot for a piezoelectric fan at time=0.12s. 3.3.4 Conclusion The piezoelectric wing operation that was modeled by an oscillating wall boundary condition with large deflection in an enclosure has been studied using two way fluid structure interaction. Numerical models with variable flapping frequency and the deflection amplitude of the wing have been set up an simulated. In the next stage the 3D motion of the flapping wing (twisting and flapping) will be simulated as part of further research (and will be presented at the International Symposium On Coupled Methods In Numerical Dynamics, Croatia, 2009. See Appendix 0). 73
  • 89. 4 Experiments 4.1 Introduction 4.2 Prototype design Different prototypes were made by looking to the FEA models. The quantitative analysis of the effect of the phase-delay on the flapping and twisting motions, especially at frequencies away from their respective resonate frequencies, would be hampered greatly by the difficulty in obtaining identical piezo fans with the identical first resonate frequency and vibration amplitude. This situation can be improved, at least partially, by manufacturing the piezo fans on industry scale. Therefore by purchasing piezo fans from an industrial supplier, two nearly identical piezo fans could be obtained. Commercial piezo fans from Mide, Inc [42] were used to build the flapping wing structures. This piezo has a length and width of 73.9mm and 15.9mm, respectively. It has a zone where the width is decreased to 11.9mm. Cyanoacrylate (superglue) is used to bond the wing to the piezo fans. In the initial prototypes the wing consisted of stiff spars that held a flexible plastic sheet (Figure 4.1). Figure 4.1: First prototype of a piezoelectric flapping wing built for this work, using two coupled piezo fans. For the initial prototypes two ABS spars with dimensions (70 x 2 x 2) mm were fabricated using 3D printing (Young modulus= 2473 MPa; Poisson ratio= 0.3; Density= 1025 kg/m3). Later CFRP spars were used, because better results could be obtained due to their higher stiffness. The wing was made of a sheet of Polyethelyne with a thickness of 50µm (see Table 2.4 for material properties). However these prototypes led to a significant drop of the peak to peak amplitude (1.2 cm) and resonance frequency (16 Hz) when operating at 120 VAC. The original piezoelectric fans have a peak to peak amplitude of 3 cm and resonance frequency of 60Hz. The lack of performance is mainly due to the small patch length for the new structure (see §2.4.2.3). Other prototypes using no spars gave much better results. A flexible plate was glued between the two piezo fans. Balsa wood caused a small decrease of the amplitude and the resonance frequency, while the generated flow was significantly increased. 74
  • 90. Figure 4.2: The prototype with balsa wood operating at resonance. The peak to peak amplitude was 2.2 cm and resonance frequency 55.4 Hz. More results will be discussed in the following section. The clamping of the piezoelectric flapping structures is very stiff; this is achieved by using a block of metal with significantly greater weight than the resonating structure (see Figure 4.2) to prevent the clamping mass to resonate and interfere with the measurement results. 4.3 Laser Doppler Vibrometer measurements The Scanning Laser Doppler Vibrometer (SLDV) can automatically collect complete vibration data of thousands individual points in a region defined by the user. This system eliminates the placement of numerous accelerometers. Moreover we obtain a contactless measuring without any physical influence on the dynamic behavior of the structure. Figure 4.3: Basic set-up principle for the LDV measurement applied on the piezoelectric flapping wing. 75
  • 91. The resonance frequencies of the piezoelectric wing can easily be determined with the LDV. The flapping wing was clamped on a structure with enough mass and stiffness. The speed is measured in the user defined point on the object, which is being excited. To acquire this information, a laser beam must be aimed at each measurement point. The reflection of this jet is again acquired by the LDV. By using the Doppler effect the speed can be determined. In general, if the intensity of the reflected beam is too low, reflecting tape should be added to the test object. The laser beam will then be better reflected and less noise will be measured. The distance between the different measurement points must be as low as possible to avoid the possibility that only measurements in the node point are done. Less problems with aliasing will also occur. We can excite the object in three ways: • Using a stinger: It is possible to use a stinger and a shaker to let the flapping wing vibrate. The vibration of the shaker is controlled by a function generator. However this method is not useful for little structures like a piezoelectric flapping wing. • Using a loudspeaker: a small loudspeaker can be used for the acoustic actuation of the flapping wing. This speaker is connected with a power supply that is connected with the computer where the signal and amplitude is verified. • Using the piezoelectric actuators: it is possible to excite the structure by actuating the piezoelectric patches. The signals from the computer are amplified and connected with the piezoelectric electrodes. For the measurements the software of the LDV was used. The used frequency range was 0 to 2 kHz. The resolution must be high enough to be able to clearly distinguish all modes in the FRF. The used signal was a periodic chirp. When setting up the LDV measurement the following parameters are important: • Excitation signal • Sensitivity • Measurement range • Frequency range When chosing a frequency bandwidth the following problems can occur: • Noise: this can be reduced by averaging over different measurements • Aliasing: this can be prevented by sampling with two times the maximal frequency that occurs • Leakage: this effect can be reduced by using the right window. There will be leakage if the signal is not periodic. There will be an overlap of the measured signals which caused leakage. To reduce this effect a window can be introduced, by example the Hanning or the Exponential window. In this work a rectangular window was used. The laser is equipped with a video camera that is used to define the grid of measurement points. The points of this grid are scanned one by one by the laser and the bad measurements are automatically redone until good measurements are obtained. By means of the measurements of the speed and the force the H1 estimated FRF (=V/F) is determined by the software. On this FRF peak-picking is applied to obtain the bending mode frequencies. Peak-picking a method based on the modal model in the frequency domain to determine the eigenfrequencies of a structure. The method is based on the fact that the response of a structure on an excitation reaches a maximum in 76
  • 92. the area of the eigenfrequencies. The frequencies where the maximums are located are considered as the eigenfrequencies. It is possible to visualize the vibration mode associated with the eigenfrequencies. One must however realize that these are not always modeshapes, because this method cannot distinguish coincident modes, or modes that are very close to each other. So it is possible that it will just visualize the vibration of the structure with that frequency, which is an operational deflection shape. 4.3.1 Measurements The POLYTEC Scanning Head PSV-400 has been used to do the SLDV measurements on the prototypes. The main objective of these measurements is to determine how the modal properties of the piezoelectric flapping wing structure changes when certain parameters are altered. Another objective is to find an interesting prototype that can be further optimized with FEA and CFD analysis, so later on this can lead to an improved prototype. The main optimization criteria are the dynamic tip deflection, the area, the frequency and the EMCF. First the commercial piezo fan with no alternations is examined. So the results of the used prototypes can be compared to that reference results. The SLDV has an inbuilt function generator. For the experiments it was set to generate frequencies from 0 to 0.2 kHz, because this is the main area of interest. The first resonance frequency of the prototypes will be smaller than the initial resonance frequency of 60 Hz of the unaltered piezo fan. The signals from the SLDV function generator (periodic chirp) is given to the external piezo fan amplifier, which is then connected to the electrodes of the piezo patches, or the acoustic speaker if the EMCF needs to be calculated. The EMCF can only be calculated if the SC and the OC resonance frequencies are known. Therefore the electrodes need to be open circuited or short circuited, so they cannot be connected to the function generator. The main disadvantage of the speaker is that results are significantly polluted with noise for frequencies under 20 Hz. Both techniques were used to perform the measurements. The = − ⁄ was calculated by measuring the short and open circuited resonance frequency of the piezoelectric flapping prototypes (§2.2.5). Figure 4.4: First two mode shapes of a commercial piezoelectric fan with the frequencies for the open and short circuited configuration. The results of the LDV measurements on a commercial piezoelectric fan (Piezo Systems, Inc.) are given in Figure 4.4. This commercial fan is different to the other one that has mainly been used in this work [42]. It uses a low DC input voltage (12V). The piezoelectric material used in these fans is PSI- 77
  • 93. 5H4E and the flexible plate material is stainless steel. The fan is mounted on a system that transforms the DC input to an AC output needed for oscillation. The fan has a plate length, patch length and width of respectively 50.4mm, 29.2mm and 12.7mm. The total thickness is 2.03mm and the thicknesses of one piezo patch and the PCB being respectively 0.19mm and 1.6mm, makes the thickness of the steel blade 0.05mm. Further technical specification of this fan is given by the supplier [15]. The EMCF was calculated by using an acoustic actuation of the piezo fan construction and applying the SC and OC configuration on the piezo fans like in Figure 2.3. The EMCF of respectively the first and the second mode was 0.0567 and 0.0568. In this study we are mainly interested in the EMCF of the first mode, therefore for the results presented in Figure 4.5 only the first EMCF is given. Geometrical properties Remarks Results The basic piezo fan, Resonance frequency: that plays a crucial role 59.4 Hz for the flapping of the piezoelectric flapping Tip deflection at first wing. resonance frequency: 32 mm Mode shape: Quality factor: 31.56 Damping ratio: 0.0158 EMCF: 0,219 1 The piezo fans have Resonance frequency: balsa wood glued at 19.4 Hz and 19.8 Hz the tip. The two fans are clamped to the Tip deflection at first same structure, but are resonance frequency: not connected and can 20 mm move independently. Quality factor: Mode shape: 37.5 Damping ratio: 0.0133 EMCF: 0,0675 2 78
  • 94. The two piezo fans are Resonance frequency: glued to a balsa 24.8 Hz wooden wing. The connection is like the Tip deflection at first previous case resonance frequency: established at the tip 16 mm of the two piezo fans. Quality factor: Mode shape: 19.9 Damping ratio: 0.0251 EMCF: 0,0404 3 Similar to the previous Resonance frequency: case, but the balsa 44.6 Hz wood wing has other geometrical Tip deflection at first properties. resonance frequency: 13 mm Mode shape: Quality factor: 33.9 Damping ratio: 0.0147 EMCF: 0,0369 4 Similar to the previous Resonance frequency: case, but the balsa 50.45 Hz wood wing has other geometrical Tip deflection at first properties. resonance frequency: 16 mm Mode shape: Quality factor: 34.79 Damping ratio: 0.0144 EMCF: 0,128 5 79
  • 95. Similar to the previous Resonance frequency: case, but the balsa 21.23 Hz wood wing has other geometrical Tip deflection at first properties. The base of resonance frequency: the wing (glued to the 18 mm piezo fans and tip wing) has a width like Quality factor: in prototype2 of 22.1 29mm. Damping ratio: Mode shape: 0.0226 EMCF: 0,087 6 Similar to the previous Resonance frequency: case, but the balsa 30.25 Hz wood wing has other geometrical Tip deflection at first properties. The base of resonance frequency: the wing (glued to the 19 mm piezo fans and tip wing) has a width like Quality factor: in prototype2 of 34.38 29mm. Damping ratio: Mode shape: 0.0145 EMCF: 0,091 7 80
  • 96. Two spars from CFRB Resonance frequency: were glued to the balsa 27.8 Hz wood wing and the two piezo fans. Tip deflection at first resonance frequency: Mode shape: 18 mm Quality factor: 23.17 Damping ratio: 0.0216 EMCF: 0,103 8 Similar to the previous Resonance frequency: case, but the balsa 33.08 Hz wood wing has other geometrical Tip deflection at first properties. resonance frequency: 20 mm Mode shape: Quality factor: 37.17 Damping ratio: 0.0135 EMCF: 0,108 9 81
  • 97. Similar to the previous Resonance frequency: case, but the balsa 36.26 Hz wood wing has other geometrical Tip deflection at first properties. resonance frequency: 25 mm Mode shape: Quality factor: 33.57 Damping ratio: 0.0149 EMCF: 0,130 10 The wing material is Resonance frequency: now made of 50.05 Hz cardboard of 400 μm. Tip deflection at first Mode shape: resonance frequency: 18 mm Quality factor: 24.18 Damping ratio: 0.0207 EMCF: 0,0509 11 82
  • 98. The wing material is Resonance frequency: now made of paper of 50.2 Hz 200 μm. Tip deflection at first Mode shape: resonance frequency: 11 mm Quality factor: 17.68 Damping ratio: 0.0283 EMCF: 0,0879 12 A rib from CFRB was Resonance frequency: placed between the 48.65 Hz two piezo fans. The material of the wing is Tip deflection at first balsa wood. resonance frequency: 22 mm Mode shape: Quality factor: 29.49 Damping ratio: 0.0204 EMCF: 0,106 13 The wing material is Resonance frequency: Polyvinylchloride with 47.85 Hz a thickness of 200 μm. Tip deflection at first Mode shape: resonance frequency: 18 mm Quality factor: 20.63 Damping ratio: 0.0242 EMCF: 0,0550 14 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
  • 99. The quality factor Q can be a measurement of the sharpness or frequency selectivity of a resonant equal to = ∆ , where is the resonance frequency and ∆ = − vibratory system having a single degree of freedom. In the mechanical system Q is approximately single is the bandwidth between the half-power points. In Figure 4.6 the situation at resonance is considered for prototype 2 in Figure -power 4.5. Figure 4.6 Determination of the quality factor for Prototype 2. 6: quality rototype The relation between the quality factor and damping ratio is = 1⁄ 2 ∙ , where is the damping ratio of the mechanical system. The damping ratios for the investigated prototypes are summarized in tio Figure 4 . The minimum and maximum damping ratios are respectively 1.33% and 2.83%. The mean 4.7. of the damping ratios is approximately 1.9%. This confirms that the chosen damping ratio of 2% in the FEM models is well estimated. The air damping is expected to be proportional to the cross section area cross of the wing. The dependence of the damping ratio on the area of the wing is clearly observable in the the results. 3 2,5 Damping ratio (%) 2 1,5 1 0,5 0 Figure 4.7 Calculated damping ratios for the different piezoelectric flapping wing prototypes (by using the LDV 7: amping measurement data) data). 84
  • 100. 4.4 Propulsion and energy consumption measurements The generated thrust is proportional to the tip deflection, which is proportional to the applied voltage. A greater tip deflection of the fan requires more electrical current (Figure 4.8). 8 25 7 Electric current (mA) 20 Tip deflection (mm) 6 5 15 4 Current (mA) 3 10 Tip Deflection (mm) 2 5 1 0 0 45 65 85 105 125 Voltage (V) Figure 4.8: The electrical current and tip deflection in function of the applied voltage for the prototype in Figure 4.2. An idea of the actual thrust provided by the piezoelectric flapping wing propulsion unit is required. Hence, a setup for thrust measurement was designed. The propulsion unit was powered using a function generator and an amplifier. It was clamped on an aluminum plate mounted vertically on a wooden base, as shown in Figure 4.9. Figure 4.9: The experiment set-up for the thrust measurements. The whole setup was then put on a digital weighing scale and the difference in the weight before and after turning on the piezoelectric actuators of the flapping wing structure was recorded. A wooden support was constructed to let the prototype face upwards, in opposite direction of the gravitational force. The extra mass sensed by the weighing scale could easily be converted in a force (thrust). There was no possibility that the readings would be affected by ground effects. The experiments were conducted on 5 prototypes. The geometrical properties of the wing of these flapping wings is 85
  • 101. described in Figure 4.10. Two parameters Var1 and Var2 are defined in Table 4.1 for the different models. The first resonance frequency of the models is also specified. As expected the frequency increases when (wing) mass is removed from the resonating structure. Figure 4.10: Definition of the geometrical variables for the thrust measurements. Parameter Model 1 Model 2 Model 3 Model 4 Model 5 Var 1 (mm) 85 45 15 0 0 Var 2 (mm) 57 57 57 57 39 fres (Hz) 34.5 41 47 49 53.5 Table 4.1: Definition of the geometrical variables of the models for the thrust measurements. The first resonance frequency is also specified. The results of the thrust measurements are given in Figure 4.11. The electrical current needed by the prototypes increases when a higher voltage is applied. This higher voltage causes a greater tip deflection, and thus more thrust. The thrust increases linearly with the applied voltage. The thrust generated by the different models depends on the area of the wing, the frequency and the tip deflection. Note that the size of the piezo patches was not optimized for the design of these prototypes, and thus in reality a greater thrust could be generated. For the construction of these prototypes in-house made piezo fans have to be fabricated. This could be part of future work. 0,004 6 0,0035 5 Electrical current (mA) 0,003 0,0025 4 Thrust (N) Model1 0,002 3 Model2 0,0015 Model3 2 0,001 Model4 0,0005 1 Model5 0 0 70 90 110 130 70 90 110 130 Voltage (V) Voltage (V) Figure 4.11: Results of the thrust measurements. 86
  • 102. 4.5 Flow experiments 4.5.1 Introduction One of the oldest know techniques to quantatively measure fluid flow is the pitot tube, named after H. de Pitot. He was the first person to measure velocity with an upstream pointed tube. Pitot tubes are widely used. The major disadvantages of this measurement method are: • The tube has to be placed into the fluid flow, and thus disturbs the flow. • Only one point can be simultaneously measured and thus it can take a lot of time to measure the complete profile of a flow. In the late 1950’s hot-wire anemometers were introduced. As the name implies, hot-wire anemometers use a very thin wire, which is placed into the flow and through convective cooling by the flow of a wire which is heated by an electric current, the flow velocity can be measured. Most hot-wires have a diameter of 5 µm and a length of approximately 1 mm and are made of tungsten and can take thousands of velocity measurements per second, allowing to study the details of fluctuations in turbulent flow. The major drawbacks of this method are the calibration before each experiment, disturbance of the flow by the probe and the possibility to measure one point at a time. The temporal resolution is a lot higher than the pitot tube and the disturbance is smaller. In the mid 1960’s, the Laser Doppler Anemometer (LDA) was developed. This is an optical technique to measure flow velocity in a desired point without disturbing the flow. The operating principle of LDV is based on sending a highly coherent monochromatic light beam (laser beam) toward the target, collecting the light reflected by small particles in the target area, determining the change in frequency of the reflected radiation due to the Doppler effect, and relating this frequency shift to the flow velocity of the fluid in the target area. The major advantage of this method over hot-wire anemometry is that is non-intrusive. Nowadays systems which can measure the three components of velocity at once become more and more available. However this method also has its disadvantages: • The desired target area has to be reachable by the laser beams • The major cost of a system • Difficult to measure close to a surface • Only 1 point can be measured at once To overcome this last drawback, other measuring methods where developed, called particle-imaging techniques (most important was Particle Image Velocimetry or PIV). An overview of these methods are described in literature [43]. The flow generated by the piezoelectric fans and certainly piezoelectric actuated flapping wing MAVs can be very complex and needs to be well understood for the fans to be incorporated into MAVs. To better understand the physics of the flow patterns generated by these fans, several flow experiments were performed using these 3 different techniques: hot wire anemometer, LDA and PIV. 4.5.2 Hot wire Anemometry measurements 4.5.2.1 Introduction The hot wire anemometer can be applied in both gases and liquids. The sensor is placed in the flow (liquid or gas) and gets cooled (Figure 4.12). The amount of heat that is lost by convection is function of certain parameters of the flow, as in particular the temperature, the pressure and the speed of the fluid. If only the speed is varied or if the influence of the other parameters is compensated by adjusting 87
  • 103. the settings of the electronic circuit, than is the instantaneous heat loss of the measurement probe a measure for the instantaneous speed of the flow. There are two different circuits for the measurement of the thermal losses, namely the Constant-Temperature Anemometer and the Constant-Current Anemometer. A thin platinum wire is placed in the middle of the discharge meter. This wire is heated to a constant temperature. As soon as an air flow is established the wire is cooled and there is more electric energy needed in order to keep the wire at a same temperature, whereby the resistor is constant (Constant-Temperature Anemometer). An increase in the air flow will lead to an increase in the cooling of the wire. This extra electrical energy is depending on the air flow speed. These discharge meters can easily be linearized with a computer. It is also possible to keep the electrical current constant (Constant-Current Anemometer). Here the electrical resistance of the wire changes proportional with the variation of this discharge. The constant-temperature anemometer (CTA) is used for the hot wire measurements conducted for this work. Figure 4.12: Hot wire probe measurements in a piezo fan flow. The main disadvantage of this method is that it is not possible to derive the flow direction. This disadvantage can be avoided by using two platinum employed wires side by side. It is then possible to see which one is cooled first. Another disadvantage of this method is the temperature sensitivity. These systems are also very vulnerable and not very user-friendly. Moreover a calibration procedure needs to be prepared for every new measurement setup. The hot wire anemometer measures an average velocity of a flow. This speed is formed by the true mean speed (in the first order approximation) and the standard deviation. The mean velocity is composed for 50% by the velocity component normal to wire and probe axis, 49.5% by the velocity component normal to wire and parallel to the probe axis, 0.5% by the velocity component parallel to the wire axis. cold wire resistance (1.6 < < 1.8). The overheat ratio should be put as high as possible to achieve The wire temperature is determined by the so-called overheat ratio, which is the ratio between hot and the largest frequency response, highest sensitivity and highest independency from the flow temperature. For the experiments was set to 1.6 because at higher values the wire risks to break too fast. 88
  • 104. 4.5.2.2 Measurement principle of CTA In this section the measurement equipment for the hot wire anemometer is discussed. The signal that is measured by the hot wire anemometer is amplified via a electronic circuit, processed and forwarded to the computer where all the necessary parameters can be calculated. The CTA technique utilizes a very small measurement device with a short response time and a high sensitivity. It is an intrusive method, but the measurement device (probe) is very small. Therefore in most cases the interference in the flow can be neglected. Information regarding the microstructure of the flow is obtained, which forms a center for the interpretation of the flow phenomena. The main advantage of the CTA is that it allows us to measure high frequency turbulences, without the use of complex circuits. With the DISA 55M fluctuations up to 1.2Mhz can be measured. The CTA has a Wheatstone bridge and a servo amplifier. The active part of the Wheatstone bridge consists a sensor and an electric resistance (Figure 4.13). The passive part has the other electric resistance and a circuit that compensates the influence of the cable. Figure 4.13: Scheme of the CTA principle. There is no potential difference between the points on the horizontal diagonal of the bridge if the bridge is in balance. Every variation of the flow on the location where the sensor is placed will adjust the temperature of the sensor. The change of the electrical resistance of the hot wire will hinder the balance of the bridge and as a result a potential difference between the points of the horizontal diagonal of the bridge will emerge. This potential difference is amplified by the Servo amplifier. The output of the amplifier is connected to the bridge unit (55M10 CTA Standard Bridge van DISA) that fixes the temperature back to the original temperature by adjusting the potential over the Wheatstone bridge. A higher amplification and a higher reaction speed of the amplifier gives a high sensitivity of the DISA bridge. This means that the bridge will react to small potential differences to control the temperature. 4.5.2.3 Different types of probes A great advantage of the hot wire anemometers is that the sensor can be made very small. As a result this measurement device can have some important requirements for scientific research, namely: • The effect of the sensor on the flow is small. • A high spatial resolution can be achieved. • The short response time allows us to do research on high frequency fluctuations. • The high sensitivity allows us to detect and measure very low flow speeds. 89
  • 105. A distinction is made between four type of probes with different structural characteristics, namely: • The hot wire probe • The gold-plated probe • The hot-film probe • The fiber-film probe For our measurements a hot wire probe is applied. This is the most used probe type because it has a low price and better response time compared to the other probes. In many cases it is also possible to fix this probe type. The hot wire makes use of a very thin wire of platinum plated tungsten that is attached to a sensor. Depending on the application the wire can have different diameters and lengths. The most common diameter is 5µm and the length is typically approximately 1mm. 4.5.2.4 Measurement The velocity is calculated via LabVIEW software from National Instruments. This software gives the possibility to automate the post processing and the measurements. In LabVIEW certain parameters had to be specified for the measurements. Parameter Value Temperature (K) 292 Pressure (Pa) 98100 Density (kg/m3) 1,17 Humidity (%) 55% Table 4.2: Parameters for the hot wire anemometer. Before the actual measurements could be performed the following steps had to be completed: • Calibration of the hot wire • Placement of the hot wire in the flow The purpose of the calibration of the hot wire is to find the relationship between the measured voltage over the hot wire and the speed of the flow over the sensor. Figure 4.14: Set-up for the calibration of the hot wire anemometer. 90
  • 106. The Bernouilli equation can be used in combination with the conservation of mass law to determine the velocity. The first point is located in the pressure chamber and the second point is above the jet opening (Figure 4.14). The pressure difference between the static pressure in the chamber below the jet and the atmospheric pressure is being measured in height alcohol. 1 1 + = + 2 2 1 1 + = 2 2 = Hence: 2. ∆ = 1− with = 25 the diameter of the jet and = 100 the diameter of the pressure chamber. The pressure and the speed are being changed by controlling the motor of the pump. Next the velocity is being plot in function of the voltage. 40 y = -21,86x4 + 175,1x3 - 475,2x2 + 551,2x - 236,2 35 30 Velocity (m/s) 25 20 15 10 5 0 1,2 1,4 1,6 1,8 2 2,2 2,4 Voltage (V) Figure 4.15: Calibration of the hot wire anemometer. By calculating a polynomial fit of the 4th order the coefficients are obtained that are being used later to calculate the velocity out of the measured voltage. The sample rate is the number of samples per second taken from a continuous signal to make a discrete signal. The duration of such interval, which is the inverse of the sampling frequency, is depending on the application, but is limited by the Nyquist-Shannon sampling theorem. For the conducted measurements the mean and RMS velocity converged when used more than 64000 samples (sample rate was 50 kHz). 91
  • 107. After setting all the parameters and completing the calibration of the probe, the probe is placed in the flow of the piezo fan (see Figure 4.16). Time series of velocity were recorded by a data acquisition card in a PC. Figure 4.16: Hot wire probe placed in the flow of the piezo fan. The measurements were conducted on different locations near the piezo fan. This fan had a tip deflection of 32mm and was operating at a frequency of 59Hz. First a measurement was conducted 20 mm from the tip of the fan on the symmetry line of the fan as presented on Figure 4.17 (1), and then this distance was set to 2 mm while the distance from the symmetry line was changed in steps of 2.5mm (2, 3, 4 and 5). Figure 4.17: Location of different measurement position for the hot wire experiment. The other measurements were carried out on the positions located at the side of the piezo fan. The measurement first started at 15mm (6), then 5mm (7) and finally 2mm (8) from the side of the piezo fan. Then this distance was set to constant and the probe was moved in the direction of the clamping with steps of 12mm (9, 10, 11, 12). The results are summarized in Figure 4.18. The turbulence Intensity (TI) or turbulence level is also calculated. TI relates the standard deviation of the turbulent velocity fluctuations to that of the mean flow velocity: ∑ ∑ − = = = 92
  • 108. 4,00 45,0 Velocity (m/s) and RMS (m/s) 3,50 40,0 Turbulence intensity (%) 3,00 35,0 30,0 2,50 25,0 2,00 20,0 1,50 15,0 1,00 10,0 0,50 5,0 0,00 0,0 1 2 3 4 5 6 7 8 9 10 11 12 Mean (m/s) 1,84 3,14 3,62 2,78 1,11 0,44 0,54 0,77 1,29 0,88 0,41 0,53 RMS (m/s) 1,19 1,25 1,33 1,71 1,21 0,85 0,84 1,10 1,29 1,11 0,89 0,66 Turbulence Intensity (%) 31,6 24,5 23,7 36,3 39,8 36,0 34,0 40,5 39,9 39,4 37,9 26,6 Figure 4.18: Results of the hot wire measurements. Similar to what was expected the velocity increases when the probe was placed close to the tip of the fan. There is a significant drop in the velocity of the flow when the probe is placed outside the flapping region of the fan tip (measurement 4 to 5). Keeping in mind that with the hot wire technique the average velocity of a flow is measured, we can observe that the results match the CFD simulations quite well. Another uncertainty is the exact position of the hot wire probe relative to the tip of the piezoelectric fan. There were also simplifications of the geometrical dimensions of the fan in the CFD model, and the operating frequency was slightly lower than 60Hz. There is however a minor over- prediction by the numerical flow simulation. This over-prediction will also appear when comparing the results of the other flow experiments with the CFD results. 4.5.3 Laser Doppler Anemometry measurements 4.5.3.1 Introduction and principles The basic idea underlying LDA is to measure the velocity of tiny particles transported by the flow [44]. If these particles are small enough, their velocity is assumed to be that of the stream and LDA provides a measure of the local instantaneous velocity, the mean velocity as well as the turbulent quantities. Laser anemometers offer unique advantages in comparison with other fluid flow instrumentation: • Non-contact optical measurement. LDA probes the flow with focused laser beams and can determine the velocity without disturbing the flow in the measuring volume. The only necessary conditions are a transparent medium with a suitable concentration of tracer particles (or seeding) and optical access to the flow through windows, or via a submerged optical probe. • No calibration – absolute measurement technique. The laser anemometer has a unique intrinsic response to fluid velocity. The measurement is based on the stability and linearity of optical electromagnetic waves, which can be considered unaffected by other physical parameters such as temperature and pressure. • Well-defined directional response. The quantity measured by LDA is the projection of the velocity vector on the measuring direction defined by the optical system. • High spatial and temporal resolution. The optics of the laser anemometer is able to define a very small measuring volume and thus provides good spatial resolution and allows local 93
  • 109. measurement of velocity. The small measuring volume in combination with fast signal processing electronics also permits high bandwidth, time-resolved measurements of fluctuating velocities, providing excellent temporal resolution. Figure 4.19: The LDA principles [44]. The basic configuration of an LDA consists of (see Figure 4.19): • A continuous wave laser • Transmitting optics, including a beam splitter and a focusing lens • Receiving optics, comprising a focusing lens, an interference filter and a photodetector • A signal conditioner and a signal processor. A Bragg cell is often used as the beam splitter. It is a glass crystal with a vibrating piezo crystal attached. The vibration generates acoustical waves acting like an optical grid. The output of the Bragg cell is two beams of equal intensity with frequencies and . These are focused into optical fibers bringing them to a probe. In the probe, the parallel exit beams from the fibers are focused by a lens to intersect in the probe volume. The probe volume is typically a few millimeters long. The light intensity is modulated due to interference between the laser beams. This produces parallel planes of high light intensity, so called fringes. The fringe distance is defined by the wavelength of the laser light and the angle between the beams θ: = 2 ∙ sin ⁄2 Each particle passage scatters light proportional to the local light intensity. Flow velocity information comes from light scattered by tiny "seeding" particles carried in the fluid as they move through the probe volume. The scattered light contains a Doppler shift, the Doppler frequency , which is proportional to the velocity component perpendicular to the bisector of the two laser beams, which corresponds to the x-axis shown in the probe volume. 94
  • 110. The scattered light is collected by a receiver lens and focused on a photo-detector. An interference filter mounted before the photo-detector passes only the required wavelength to the photo-detector. This removes noise from ambient light and from other wavelengths. The photo-detector converts the fluctuating light intensity to an electrical signal, the Doppler burst, which is sinusoidal with a Gaussian envelope due to the intensity profile of the laser beams. The Doppler bursts are filtered and amplified in the signal processor, which determines for each particle, often by frequency analysis using the FFT algorithm. frequency provides information about the time = 1/ . The expression for velocity thus becomes The fringe spacing, provides information about the distance travelled by the particle. The Doppler = ∙ . The frequency shift obtained by the Bragg cell makes the fringe pattern move at a constant velocity. Particles which are not moving will generate a signal of the shift frequency . The velocities and will generate signal frequencies and , respectively. LDA systems without frequency shift cannot distinguish between positive and negative flow direction or measure 0 velocity. LDA systems with frequency shift can distinguish the flow direction and measure 0 velocity. Liquids often contain sufficient natural seeding, whereas gases must be seeded in most cases. Ideally, the particles should be small enough to follow the flow, yet large enough to scatter sufficient light to obtain a good signal-to-noise ratio at the photo-detector output. 4.5.3.2 Technical specifications of LDA The power source is an Argon Ion Laser with rated output of 4 watts, of which 1.7 watt is emitted in the green colour at the wavelength 514.5 nm and 1.3 watt is emitted in the blue color at the wavelength 488.0 nm. The beam exits from the laser tube with a diameter of 1.34 mm and the beam divergence is 0.03 degree. The LDA system used for the purpose is based on fiber optics, with back scatter method and has processors based on spectrum analysis of the back scattered signals. This gives the processor ability to process signals with very poor quality (very low signal to noise ratio). The measurements are operated using data acquisition software (from DANTEC). A series of measurement parameters are fixed using this software. We precise some of these parameters: all the measurements were taken with the 300 mm focal length front lens. The two beams were then focused at a point within the enclosure of the piezo fan, the measuring point. The angle θ between the two converging beams was 7.4°. The crossing of the beams generates an ellipsoidal probe volume, of radius approximately 0.12 mm and length 0.9 mm. The back-scattered light from particles present in the water was collected by the focusing lens and produced an image of the measuring volume on the EMI model 9813 photo multiplier. The Bragg cell introduces frequency shift driven by a 40 MHz signal from one of the processors. A PC workstation drives the controllers and the processors. The LDA processors are connected through the IEEE 488 bus and the traverse controller is connected to RS232 port. A digital storage oscilloscope is used as the on-line monitor of the Doppler signals. In all the studies conducted in this project high intensity laser beams are used. The measurement point can be chosen and moved using a three-axis traverse system. The entire laser-Doppler system could be traversed relative to the pump impeller via a large horizontal traversing table. The precision of this system in the x and y directions (horizontal plane) are 0.0125 mm and in the z direction (vertical) is 0.00625 mm. The green and blue 95
  • 111. beams belong to 2 planes whose intersection is called the LDA probe axis. This axis was always chosen horizontal and vertical, so that the axial and tangential velocities are measured. For each particles, velocity, transit time across the probe volume, and arrival time, are recorded. 4.5.3.3 Measurements and results The LDA measures the velocity in a point; therefore an area (or volume) must be scanned to get a velocity vector field. As a result the measurement is significantly slower than measurements with the hot wire anemometry and particle image velocimetry. For the measurement of the flow generated by the piezo fan a grid with an area of 120mm x 120mm and 18x18=324 points was defined at the end of the tip. The measurement of one point took about 11 seconds, so the total measurement over all the grid points took approximately 1 hour. Figure 4.20: Location of the measurement grid for the LDA experiment. The post processing of the results was completed with MATLAB. The piezo fan was operating at resonance (60Hz) with a tip amplitude of 32mm. The results for the velocity in the 3 different positions can be seen in Figure 4.21. (1.5,0), (85,0) and (1.5,40) are approximately the coordinates of the measurement positions 1, 2 and 3 in millimeters, respectively. 3 2,5 2 1,5 Vx in position 1 Velocity (m/s) 1 Vy in position 1 0,5 V in position 1 0 Vx in position 2 0 10 20 30 40 50 -0,5 Vx in position 3 -1 -1,5 -2 Time (ms) Figure 4.21: Results of the flow velocity in the different positions obtained using LDA measurements. 96
  • 112. These results for Position 1 can be compared with the results obtained with the CFD simulations in §3.3.3. The difference in the velocity can be due the uncertainty of the measuring position. Like observed from the hot wire measurements, also here it can be seen that the numerical simulation somewhat over-predicts the velocity of the flow. 4 3 2 Velocity (m/s) 1 Velocity V Velocity Vu 0 0,3 0,31 0,32 0,33 0,34 0,35 Velocity Vv -1 -2 -3 Time (s) Figure 4.22: Results of the flow velocity using CFD simulation (the point is located in position 1 defined for the LDA measurements). The periodicity of the velocity can be clearly seen when the measurement point is close enough to the tip of the piezo fan. Next to the possibility to show the velocity in one point in function of the time, it is also possible to show the complete velocity vector field (Figure 4.23). The contour plots of the x and y components of the velocity and the RMS velocity can be found in Appendix B (Figure B.1, Figure B.2, Figure B.3). A symmetrical result of the velocity vector field is obtained when the velocity vectors are time- averaged over 1 oscillation of the piezo fan (Figure 4.24). 97
  • 113. 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. 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
  • 114. 4.5.4 Particle Image Velocimetry measurements 4.5.4.1 Introduction Particle image velocimetry (PIV) method is a contact free optical capture and analysis of velocity arrays or pattern of intensity in fluids but also in granular material. In contrast to the particle tracking velocimetry (PTV) method where the track of a single particle is recorded, the idea of the PIV method is to record a group of particles which are affected by the flow. By analyzing the alteration of the position of a particle group within a defined interval one gets a value of the velocity. In combination with conventional techniques to visualize a current one gets a complete description about quality and quantity of a flow. With other instruments one can measure the velocity at particular points or cross sections. The PIV method provides an instant, two dimensional vector field on the order of a few mm2 to a few km2. The high resolution of those vector fields also enable the direct visualization and interpretation of flow conditions regarding the characteristics and dimension. It can also be used as a numerical model validation. By using several cameras to create a stereoscopic system one can get quasi sterical results. In the present experiment the PIV method was employed to measure the velocity vector field at the tip of the piezoelectric fan and wing. 4.5.4.2 Elements of PIV To visualize the flow in a fluid for the unit of measurement it is necessary to add tracers to the medium. This is called the seeding of the flow. Which tracer material and size is needed depends on the characteristics of the fluid (density, viscosity), dimension of the investigation area and the required resolution. There are different concepts about the value of the particle density, but with 10-15 particles per interrogation area one obtains a reliable estimator of the particle image displacement. Up to a certain point we can say the more particles are in an interrogation area the lower is the mean deviation. Each particle should be pictured on at least four pixels, otherwise one gets the so called peak-locking- effect. That means one can determine the particle image displacement only in a range of integer displacement quantities. The possibility to determine a particle image displacement up to a tenth part pixel is no longer given. On the other hand the particle size should be much smaller than the smallest flow structure. Some particle material have the leaning to accumulate to conglomerates which affects the interpretation negatively. The second step is the illumination of a sheet in the flow field. A high power laser New Wave MinilaseII Nd-Yag laser (532 nm wavelength, 100mJ/pulse) is used with an optical arrangement to convert the laser output light to a light sheet (normally using a cylindrical lens). The laser acts as a photographic flash for the digital camera, and the particles in the fluid scatter the light. It is this scattered light that is detected by the camera. Black curtains were used to homogenize the light and to prevent reflections of light in the background from reaching the camera. 99
  • 115. Figure 4.25: The basic set-up principle of particle image velocimetry. The third step is the recording of 2 successive images. The essential part for the recording unit is a charge-coupled device (CCD) camera with controllable spatial and temporal resolution. Such cameras can record 5-15 double frames per second with a resolution up to 1000x1000 pixels. The synchronization of the camera with the illumination unit is transferred by a trigger system which can be contained as timing board on the PC or as external trigger. In the current experiment a CCD camera PCO Sensicam QE 5Hz was used. As opposed to the PTV method where one investigates single particles, the average displacement of a particle group will be evaluated. In this step the images are being decomposed into interrogation windows to obtain one velocity vector per interrogation window. Basically there are two different techniques to calculate the particle displacement, auto-correlation and cross-correlation. The former one uses the data of one double exposed single frame. More common is to use cross-correlation where the displacement of one particle is calculated by two images separated by a time distance. The velocity is obtained by the determination of the position of the correlation peak. The advantage of that method is that one cannot only calculate the direction but also the algebraic sign. In modern cameras the exposure speed is not a limiting factor anymore, so almost exclusive cross-correlation technique is used [43]. Figure 4.26: Basic working principle of PIV. 100
  • 116. Assume that there are images I1 and I2, separated by a time distance ∆t. Both images are split into smaller regions, interrogation areas. Each sub-window of one image will be compared with the corresponding one in the second image. The aim now is to see if a displacement of the pattern in the sub-window of the first image can be identified. To do this one can evaluate the squared Euclidean distance between the two sub-windows which is defined as: , = , − − , − Where denotes sub-window number k in the first image and the corresponding sub-window in the second image. R stands for the intensity of correlation. The above equation tells us that the sum of the squared difference between all couples of sub-windows will be calculated. The equation can be expanded: , = , −2 , ∙ − , − + − , − The first term , is constant since it does not depend on m and n. The last term − , − seems to depend on m and n, but it is just dependent on the second image. Only the middle term deals with both of the images and as a matter of fact this term is usually referred to as a cross- correlation and defined as: , = , ∙ − , − The above equation is the basic form of many algorithms in PIVview (PIVTEC GmbH), used for the post processing of the measurements. The cross-correlation algorithm is in fact a feature matching algorithm. After a run of the algorithm one can get a vector field of possible particle displacements. In considering the known time interval the software outputs a velocity vector for each interrogation area. The vector fields were generated using cross correlation fast Fourier transform (FFT) with a multi-grid procedure combined with a sub-pixel based image shifting or image deformation with a third order interpolation scheme [45]. With this a pyramid approach is used by starting off with larger interrogation windows on a coarse grid and refining the windows and grid with each pass. This is especially useful in PIV recordings with both a high image density and a high dynamic range in the displacements. The program runs through several iterations starting from a bigger sub-window (64x64 pixel) and taking the results into another iteration with a scaled down sub-windows (32x32 and 16x16 pixel) finishing at a subwindow size of 16x16 pixel. The received values from the larger sub-window serve as a reference for the smaller one and increase the chance of a successful correlation. Starting directly with an 16x16 sub-window can cause failures in the output because the probability to define sub-windows without particles is much higher. A least squares 3-point Gauss fit algorithm was used to recover the sub-pixel displacement of the correlation. Overlapping the interrogation regions can furthermore improve the results. An overlap of 50% was used. Another important component of the image processing is the calibration of the recorded image. That means you need to define the size of the area one pixel represents in the image. To improve the results or to eliminate false vectors there are diverse filters. The quality of one vector is compared by 101
  • 117. comparing it with the surrounding vectors and if necessary using the nearest value of correlation peak. By overlapping interrogation areas a higher vector density can be achieved. But note that the higher information density is only visual not spatial. 4.5.4.3 Error estimation and detection of outliners An important aspect of measurements is the estimation of the errors, which always arise within an : experiment [46]. The absolute measurement error, , can be decomposed into a group of systematic errors, , and a group of residual errors, = + Systematic errors are those which arise due to the inadequacy of the statistical method of cross- correlation in the evaluation of a PIV record, such as the use of an inappropriate sub-pixel peak estimator. These errors follow a consistent trend which makes them predictable. The second type of errors remain in a form of measurement uncertainty, even when all systematic errors are removed. In literature detailed information can be found for optimizing a number of experimental parameters [47]. It is shown that the optimal particle image diameter is slightly higher than 2 pixel. After automatic evaluation of the PIV recordings, frequently a number of false vectors are found back on processed images. These vectors deviate unphysically in magnitude and direction from nearby ”valid” vectors and often appear at the edges of the data field. (near the surface of the model, at edges of drop-out areas, at the edges of the illuminated area). Mostly, they appear as a single incorrect vector. This can be caused by insufficient particle images in the interrogation area, noise or artifacts not due to the correlation of matched image pairs. Most of the false vectors can easily be detected by visual inspection, but when dealing with thousands of recorded images it is not obvious to deal with these outliers in an interactive way. These erroneous results have to be removed, certainly when wanting to calculate for instance, vorticity. Because of the great amount of data involved, this can only be done by an automatic algorithm. This algorithm must ensure with a high level of confidence that no questionable data is stored in the data set. The most easy algorithm used to solve this issue is the dynamic mean value operator. This algorithms checks each velocity vector individually by comparing its magnitude with the average value over its nearest (mostly 8) neighbors. The velocity vector will be rejected if the absolute difference between its magnitude and the average over its neighbors is above a certain threshold. Problems will arise at the edges of the data field when there are less than eight neighbors available for comparison. 4.5.4.4 Measurements Approximately 1000 image pairs at were recorded for the experiments. In Figure 4.27 the basic set up for the PIV measurements is shown. A glass enclosure is used to keep the tracer particles in the area of the piezoelectric flapping wing. The laser was placed in such a way that the laser sheet coincided as much as possible with the area of interest, i.e. the area at the end of the flapping wing. By moving the camera different zones could be measured. 102
  • 118. Figure 4.27: The employed experiment set-up for the PIV measurements. Three important components of the flow visualization experiments are the tracer particles, the illumination used, and the recording procedure. In the flow visualization experiments, different kinds of tracer particles were used as alternatives to seed the flow. These particles include smoke particles obtained by burning an incense stick, and fog from a fog generator which heats up N2 gas. Since the smoke produced from a burning incense stick is limited in quantity, it is easy to control the amount of smoke introduced into the experimental domain by this method. However, incense sticks are unreliable, and can extinguish during the experiment. Furthermore, since only a limited amount of smoke can be produced with a single stick, they have to be replaced during the experiments. The conglomerate effects are greater than the alternative methods to create tracer particles. Water droplets from a humidifier offer a reliable source of tracers, but condensation in the domain and high scattering are the associated disadvantages. The water droplets are not well-suited when images need to be obtained at some depth into the domain. The fog generator was found to be the best alternative. The only disadvantage is that the fog leaves an oily residue in the domain (glass boundary), which needs to be cleaned after each experiment. 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
  • 119. The tracer particles are illuminated by a sheet of light. The light sheet can be generated by using a slide projector and a specially built slide that is opaque all over except for a thin slit in the middle, or by using a laser beam in conjunction with a cylindrical lens. The latter method results in better quality of visualization since the laser sheet can be positioned where desired, and it is also possible to control the intensity of the laser beam. The light sheet is used to illuminate the area of interest in the flow field. Recording can be performed in two ways, either with still images or by capturing on video. The still images are of better quality. Hence, movies are preferred over still photographs. The camera used in these experiments was a PCO Sensicam camera. Since this is a digital camera, the recorded video clips can be transferred to a computer via an IEEE 1394 connection for editing and for producing still images. The piezoelectric fans and piezoelectric flapping wings were placed in a glass enclosure (box) of dimensions 60x30x30cm, built from Plexiglas for the purpose of visualization. The main purpose of the box was to keep the tracer particles concentrated. The resonance frequency of this prototypes was always under 100 Hz at which the tip deflection was at maximum under a 130 Volt AC input. The box was filled with fine particles from the fog generator. The prototypes were placed such that the influence of the Plexiglas boundary on the flow was limited. A sheet of light is introduced from the left side, parallel to the left and right surfaces of the box, and the camera is placed at the right side, for normal viewing through the laser sheet surface. Figure 4.29 shows a photograph of a constructed piezoelectric flapping wing prototype clamped in the domain. Figure 4.29: The piezoelectric flapping wing prototype in the Plexiglas enclosure. 4.5.4.5 Evaluation of the PIV results The following equation can define the locally measured velocity [46]: ∆ = + ∆ ∆ with ∆ distance traveled by the particle images within the pulse seperation time ∆ . are the residual errors of the measured image displacements. These errors are not affected by the alteration of 104
  • 120. the pulse separation time. Therefore, the second term on the right hand side of the above equation will increase if the pulse separation time decreases: lim =∞ ∆ → ∆ On the other hand, the particle image displacement decreases if the pulse separation time decreases. This leads to: ∆ lim = ∆ → ∆ As been shown, the accuracy of PIV measurements can be increased by increasing pulse separation time between the exposures. But for high values of the separation time the measurement noise increases. For very large pulse separation time, the particle displacement will exceed the extent of the interrogation volume. So the separation time ∆ must be chosen correctly to obtain stochastic independent images. Therefore a rough prediction of the velocity must be made. Since the tip deflection is approximately 3.5cm and the operating frequency approximately 60Hz, we get a tip velocity of 2.1m/s. Now having estimated the velocity, the separation time must be determined in such a way that the displacement is approximately 2 pixels. From the calibration procedure followed that approximately 1000 pixels becomes: ∆ = ≈ 50 represented 0.05m., so in the same way 2 pixels represents 0.1mm. Hence the chosen separation time . ∙ . ⁄ During the separation time of 50 the particles in the area of the tip get displaced by the motion of . In Figure 4.30 one image pair of the PIV measurements is given. the piezo fan. The particles far away from the tip of the piezo fan did not have sufficient time to translate large distances, and therefore mainly in the area near the tip of the piezo fan a velocity vector field is obtained. Figure 4.30: A recorded image pair with a separation time of 50µs (PIV measurement). The velocity vector field near the tip is obtained. 105
  • 121. The post processing of the results is very important when conducting PIV measurements. The correct grid size together with correlation filtering and sub pixel interpolation must be applied to obtain a better representation of the real flow. In Figure 4.31 the post processing is done iteratively. As mentioned above a multi-grid procedure has been used. The end grid size can be configured by the user and it was found that a grid size of 8 pixels with 50% overlap gave the best results. Figure 4.31: Post processing of the results of the PIV measurements. The final result of the velocity vector field obtained by the PIV measurements is given in Figure 4.33. The bending of the piezo fan is clearly visible. The fan is moving upwards and sheds a vortex below the tip, while the earlier shed vortex is pushed away the upward motion. The obtained velocity vector field is very similar to the velocity vector field obtained by the FSI CFD simulations. The velocity vector field of the flow generated by the harmonic motion of the piezo fan with a tip deflection of 3cm 106
  • 122. obtained in §§3.3.3 is presented in Figure 4.32. We can see that the numerical model somewhat over 32. model predicts the velocity o the flow relative to the measurements, but the simulated results are generally in of measurements, good agreement with the experimental findings. Figure 4.32: The velocity vector f ld of the flow generated by the harmonic motion of a piezo fan with a tip deflection of 32: field 3cm, simulated with CFX (see § §3.3.3). Figure 4.33 Velocity vector field of the flow induced by a flapping piezo fan obtained with PIV measurements. 33: induced 107
  • 123. 4.6 High speed camera visualization The two coupled piezo fans can give the wing a rotational movement when the piezoelectric patches are driven by a sinusoidal voltage with a phase difference (Figure 4.34). This twisting can be important when these wings are being used to generate lift for a MAV. Consequently the influence of the phase difference is being investigated. To obtain the phase difference two function generators needed to be coupled and synchronized with each other (Hewlett Packard 33120A and Agilent 33220A Function). Another high amplifier was needed besides the Piezo Linear Amplifier (EPA-104 from Piezo Systems, Inc.). This amplifier was built using an in-house made amplifier circuit (based on LME49830). An oscilloscope was used to check the phase difference between the sinusoidal voltages. Multimeters were used to measure the electric current. Figure 4.34: Two piezoelectric fans with a phase delay of 180 degrees. The prototype was constructed by using the commercially available piezo fans (see §4.2). These fans were placed in parallel with a centre distance of 40mm. The CFRP spars had a length of 60mm and were glued with superglue on the piezo fans with an offset of 15mm. The wing was made of a sheet of Polyethelyne with a thickness of 50µm. The wing had a width of 60mm and a length of 40mm, pasted on the spar starting at the end (top right image in Figure 4.37). The piezo fans were clamped perpendicularly in parallel and both the flapping and twisting motions are in the horizontal direction. This enabled a high speed camera (AOS X-PRI 769) fixed parallel with the piezo fans to record both the flapping and the twisting motions of the wing. A frame rate of 1000 frames/s was used with a shutter time of 1000µs. The camera was controlled by a computer system to record 1 s of data corresponding to 1000 images. The 1000 images covered several full cycles of the vibration (the frequency was 21Hz). A 50W light source was used to illuminate the flapping wing structure. The frame resolution was 800x600 pixels. The set-up can be seen on Figure 4.35. The maximum voltage which could be applied using the in-house made amplifier was 50V. Therefore the output of the other amplifier had to be also set to 50V. This gave a significant decrease of the tip deflection, since normally a voltage of 130V has to be applied on the piezo fans. The tip deflection is proportional to the applied voltage on the piezoelectric patches (§2.4.2.3). 108
  • 124. Figure 4.35: Set-up for the high-speed camera recordings. When the phase delay is 0° the tip deflection is maximal (Figure 4.36). When the phase delay is increased the tip deflection decreases until the piezo fans are driven in anti parallel (180°). It decreases to about a quarter of the value when the phase-delay is 0°. In a symmetrical manner the tip deflection increases back to its original value at 0° when 360° is reached. The total electrical current slightly increases and peaks at 180°. 12 1,8 Tip deflection 1,6 (mm) 10 Electrical current (mA) 1,4 Tip deflection (mm) 8 1,2 Electric current 1 first piezo fan 6 0,8 (mA) 4 0,6 Electric current 0,4 second piezo fan 2 0,2 (mA) 0 0 Total electri 0 100 200 300 400 current (mA) Phase delay (°) Figure 4.36: The tip deflection and used electric current in function of the phase delay for the prototype in Figure 4.2. In Figure 4.37 an oscillating voltage of 21Hz and 50V is applied on the piezo patches causing it to resonate at the first bending mode. Because of the low voltage that could be applied the phase difference caused only a small rotation of the wing that was not clearly observable by the high speed 109
  • 125. recordings. The recordings for different time steps in one period with a phase difference of 0° are given. It can be noticed that the tip deflection is relatively small. Figure 4.37: High-speed camera recordings of a prototype moving at the first bending mode. The flapping motion of the wing and the coupled piezo fans around the flapping resonance reduces with the increasing phase delay. The twisting motion around the twisting resonance increases with the increasing phase delay. Because of the small deflections due to the reduced input voltages, the effect could not be clearly presented in this document via still images. However the included recordings of the high speed camera are included in the DVD and will present the effect in a better way. It is extremely hard to fabricate two identical piezo fans in terms of their resonance frequencies and vibration amplitude. Since piezo fans are fabricated individually, a small variation in width, lengths of the PZT patch and passive plate, the relative position of the PZT patch on this passive plate, exact clamping positions, piezoelectric coefficient of the PZT, condition of the bonding layer, etc, can lead to the deviation of the resonance frequency and amplitude. These piezo fans act as the actuator of the flapping wing and therefore are important for the dynamic behavior. If the piezo fans are identical and there is no phase difference between the two input signals, this would lead to a perfect parallel motion around the first resonance frequency. However when the piezo fans are slightly different, this would mean that at any frequency, even with a phase delay of 0°, the two piezo fans would not vibrate in parallel; one would vibrate with a larger amplitude than the other. Therefore this would give a visible twisting motion for the wing. It is obvious that the flapping and the twisting motions of the wing also depend very much on the stiffness of the wing skin materials, as well as the piezo fan actuators. If the wing material is infinitely soft, the whole structure will act like two independent piezo fans. This would mean that the two piezo fans would vibrate in parallel when the phase delay is 0° and in anti phase when the phase delay is 180°. The twisting motion would be maximal at a phase difference of 180° and diminish to 0 when a phase delay of 0° would be applied. Thus by controlling the phase delay, it is possible to change the flapping and twisting amplitude of the wing. But a variation of the of the two first resonant frequencies 110
  • 126. of the two piezo fans causes the flapping and twisting of the structure be dependent on the operating frequency and the difference between the two resonance frequencies. On the other hand if the wing material has an infinitely stiffness, then the complete structure could be treated as a single body with a unique resonant frequency. The structure would be designed in such a way that the first and second mode is respectively bending (flapping) and torsion (twisting). In this case the influence of the resonance frequency of the individual piezo fans has little effect on the performance of the wing. Naturally the actual wing has a finite stiffness. The performance of the structure can be affected by the difference in resonant frequencies of the individual piezo fans. Therefore every attempt must be made to build the piezo fans as identical as possible. Different skin materials were used in this study. The prototypes with balsa wood wings gave a decent tip deflection and first resonance frequency, while the prototypes with a very thin polymer skin gave lower tip deflections and first resonance frequencies. Therefore it is interesting to use the balsa wood wings for propulsion systems, with limited control of the twisting, while the polymer skins for flapping wings that would imitate insect wings. 111
  • 127. 5 Conclusions 5.1 Conclusions and final remarks With an aim to develop an easy to control method for the actuation and controlling of two DOF motions, namely the flapping and twisting motions for a wing of MAV, investigations were carried out for piezoelectric actuated flapping wing systems in this project. This involved the analysis, modeling, optimization and characterization of piezoelectric cantilever at first, then for the piezo fan, and finally for the coupled piezo fans structures. The bending amplitude of a piezo fan at its bending resonant frequency is boosted significantly. Therefore these systems are driven at resonance. Analytical and finite element modeling were used to study the performance of the piezoelectric unimorph cantilever and piezo fan structures. Good agreements were obtained between the analytical and FEM results. For a piezoelectric fan made by a stainless steel passive layer with a fixed length and a PZT patch layer which was shorter than the stainless steel, a longer PZT patch lead to larger tip displacement. Depending on the geometrical configuration of the piezo fan, the first resonant frequency of the piezoelectric fan gets maximal at a certain length ratio (0.6-0.7) and thickness ratio (>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 configuration gets maximized for a certain length ratio (0.36-1) and thickness ratio (0-1.17). The maximum for 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 127µm thick PZT-5H and a stainless steel plate with a thickness greater than 110µm. This thickness combination can produce the largest value with f<100Hz for both the flapping resonate frequencies. For a piezoelectric fan, made by gluing (superglue) a passive layer and a PZT patch with a fixed length shorter than the elastic layer, a smaller tip displacement and lower resonant frequencies are resulted, when the location of the PZT patch is further away from the clamping position. However the patch cannot be clamped because it can brake because it is a ceramic material. Therefore, if a larger tip displacement is required, the PZT patch should be placed as close to the clamping position as possible. The clamping stiffness plays also a very important role in the performance of the piezoelectric fans. Another effect that could reduce the performance of the piezoelectric fans is the thickness of the bounding layer between the piezoelectric patches and the elastic passive plate. This adhesive layer has to be as thin as possible. Next to the geometrical properties the material properties of the elastic plate play a very important role for the piezoelectric fans. The application of these propulsion systems must be taken into account, so the weight of the materials plays also an important role. The material for the spars needs to be as stiff as possible (CFRP). Different materials for the wing and plate material were investigated. It must be avoided that the first mode shape changes to the twisting mode shape and reducing the performance of the piezoelectric flapping wing structure drastically. Therefore the material selection must be done with care. 112
  • 128. Other geometrical properties of the flapping wing, like the width and length ratios play an import role, however also here one needs to pay attention to the possible change of the first bending mode shape into the torsion mode shape. Another point is that the air damping is proportional to the cross section area of the wing. Various prototypes were built using spars made of ABS and CFRP, and wings made of plastics, balsa wood and paper. Balsa wooden wings gave the best results for the first bending mode (flapping). The dynamic properties of the prototypes were investigated by conducting measurements with the Laser Doppler Vibrometer. Thrust measurements were conducted to investigate the feasibility of the piezoelectric flapping wings as a propulsion system in MAVs. The maximum thrust generated by the investigated prototypes was 0.004 N (at 130VAC). The piezoelectric wing operation that was modeled by an oscillating wall boundary condition with large deflection in an enclosure has been studied using two way fluid structure interaction. Numerical models with variation of geometrical parameters, the flapping frequency and the deflection amplitude of the wing have been set up and compared to experimental data conducted with Particle Image Velocimetry, Laser Doppler Anemometry and Hot Wire Anemometry. The phase delay between the driving voltages supplied to the two coupled piezo fans was found capable to change the flapping and twisting motions of the wing attached to the two piezo fans. The harmonic bending motion of the wing structure and the coupled piezo fans around the first resonance frequency (first bending mode) reduces notably with an increasing phase delay. The torsion bending around the second resonance frequency (torsion mode) increases with the increasing phase delay for the designed prototype. The phase delay between the driving voltages supplied to the two coupled piezo fans can also change the flapping and twisting motions at frequencies far away from their respective resonate frequencies. The exact format of the effect depends on the stiffness of the wing itself. 5.2 Recommendations for future work There is a fundamental need for improving our understanding of the fluid physics of biology-inspired mechanisms that simultaneously provide lift and thrust, enable hover, and provide high flight control authority, while minimizing power consumption. First principles-based computational modeling and analysis capabilities are essential in support of the investigation of issues related to fluid-structure interactions, laminar-turbulent transition, unsteady free stream, and time dependent aerodynamics. We need to conduct further exploration of flexible, lightweight, multifunctional materials and structures for large displacement and suitability for actuators and sensors. Figure 5.1: Stacking of multiple piezoelectric flapping wings. 113
  • 129. Bio-inspired mechanisms need to be developed for flapping wings. Most importantly, the motion produced by these mechanisms should be experimented based on what we can learn from biological systems performing flapping in gusty conditions. The fluid flow associated with these mechanisms need to be detailed using a rigorous set of experiments. These measurements will include simultaneous flow field and structural deformation measurements in order to better understand the relationships between them. The following potential future works are suggested: • A clamping construction has to be designed to reduce the effect of non-perfect clamping of the piezo fan. This is one of the great challenges in the future research about this subject. • The integration of the propulsion system in the fixed wing MAV is expected not to be easy. If multiple piezoelectric flapping wing systems need to be used, they can be stacked like proposed Figure 5.1. However there is a great possibility that these mechanisms will interfere with the normal operation of the fixed wing, and therefore the design of the fixed wing has to be adapted. • A power supply system needs to be designed that can be implemented in the MAV. The control system to dynamically find the resonance frequency and changing the phase difference for rotation is also a great challenge. Low Reynolds number aerodynamics and the small size of MAVs make control more complex. Birds and insects use a variety of subtle movements, not only to generate lift, but also to control their flight course. • In this study, only one type of PZT ceramic (PZT-5H) was used to produce piezoelectric fans, for its relatively high transverse piezoelectric coefficient and electromechanical coupling factor values, high Curie temperature and affordable cost. However, single crystals like Lead magnesium niobate-lead titanate (PMN-PT) have much higher , and values, but they are not ordinary piezoelectric materials, they are relaxors (have extraordinarily high electrostrictive constants), and their Curie temperatures are lower. To increase the tip displacement, blocking force, and the electrical mechanical energy transformation of the piezoelectric fans, it is worthwhile to use these single crystals to investigate their relationships between their , values and their dynamic behaviours. • Electroactive Polymers or EAPs are polymers whose shape is modified when a voltage is applied to them. They can like piezoelectric materials be used as actuators and sensors. As actuators, they are characterized by being able to undergo a large amount of deformation while sustaining large forces. Due to the similarities with biological tissues in terms of achievable stress and force, they are often called artificial muscles. These materials have the potential for application in the field of flapping wing MAVs, where large linear movement is often needed. The possibility to use EAPs as an alternative to piezoelectric materials can be investigated [9]. • Wind tunnel test for the aerodynamic of the coupled piezoelectric fan systems can be conducted. Aerodynamic force produced by the coupled piezoelectric fan systems has not be measured yet in this study. It is suggested to obtain the aerodynamic force of the coupled piezoelectric fan to investigate the relationships between the aerodynamic force produced and the whole piezoelectric flapping wing systems. • The use of ultra thin PVDF actuator sheets as flexible wing skin can be interesting for extra control over the movement of the actual wing. This material can also be used as sensors in the system. 114
  • 130. 115
  • 131. Appendix A. Time-history solution of the velocity of the flow (CFD) In this work, the physical process of the flapping wing is being modeled as transient. In a transient simulation, the behavior of a physical system as a function of time is investigated. The CFX solution shows real transient effects as the flow field evolves with time. 1. t= 0 s 2. t= 1,52e-03 s 3. t= 3,03e-03 s 4. t= 4,55e-03 s 5. t= 6,06e-03 s 6. t= 7,58e-03 s 116
  • 132. 7. t= 9,09e-03 s 8. t= 1,06e-02 s 9. t= 1,21e-02 s 10. t= 1,36e-02 s 11. t= 1,52e-02 s 12. t= 1,67e-02 s 117
  • 133. 13. t= 1,82e-02 s 14. t= 1,97e-02 s 15. t= 2,12e-02 s 16. t= 2,27e-02 s 17. t= 2,42e-02 s 18. t= 2,58e-02 s 118
  • 134. 19. t= 2,73e-02 s 20. t= 2,88e-02 s 21. t= 3,03e-02 s 22. t= 3,18e-02 s Figure A.1: Velocity vector field plots for different time steps. 1. t= 0 s 2. t= 1,52e-03 s 119
  • 135. 3. t= 3,03e-03 s 4. t= 4,55e-03 s 5. t= 6,06e-03 s 6. t= 7,58e-03 s 7. t= 9,09e-03 s 8. t= 1,06e-02 s 120
  • 136. 9. t= 1,21e-02 s 10. t= 1,36e-02 s 11. t= 1,52e-02 s 12. t= 1,67e-02 s 13. t= 1,82e-02 s 14. t= 1,97e-02 s 121
  • 137. 15. t= 2,12e-02 s 16. t= 2,27e-02 s 17. t= 2,42e-02 s 18. t= 2,58e-02 s 19. t= 2,73e-02 s 20. t= 2,88e-02 s 122
  • 138. 21. t= 3,03e-02 s 22. t= 3,18e-02 s Figure A.2: Velocity contour plots for different time steps. 1. t= 0 s 2. t= 1,52e-03 s 3. t= 3,03e-03 s 4. t= 4,55e-03 s 123
  • 139. 5. t= 6,06e-03 s 6. t= 7,58e-03 s 7. t= 9,09e-03 s 8. t= 1,06e-02 s 9. t= 1,21e-02 s 10. t= 1,36e-02 s 124
  • 140. 11. t= 1,52e-02 s 12. t= 1,67e-02 s 13. t= 1,82e-02 s 14. t= 1,97e-02 s 15. t= 2,12e-02 s 16. t= 2,27e-02 s 125
  • 141. 17. t= 2,42e-02 s 18. t= 2,58e-02 s 19. t= 2,73e-02 s 20. t= 2,88e-02 s 21. t= 3,03e-02 s 22. t= 3,18e-02 s Figure A.3: Velocity vu contour plots for different time steps. 126
  • 142. 1. t= 0 s 2. t= 1,52e-03 s 3. t= 3,03e-03 s 4. t= 4,55e-03 s 5. t= 6,06e-03 s 6. t= 7,58e-03 s 127
  • 143. 7. t= 9,09e-03 s 8. t= 1,06e-02 s 9. t= 1,21e-02 s 10. t= 1,36e-02 s 11. t= 1,52e-02 s 12. t= 1,67e-02 s 128
  • 144. 13. t= 1,82e-02 s 14. t= 1,97e-02 s 15. t= 2,12e-02 s 16. t= 2,27e-02 s 17. t= 2,42e-02 s 18. t= 2,58e-02 s 129
  • 145. 19. t= 2,73e-02 s 20. t= 2,88e-02 s 21. t= 3,03e-02 s 22. t= 3,18e-02 s Figure A.4: Velocity vv contour plots for different time steps. 1. t= 0 s 2. t= 1,52e-03 s 130
  • 146. 3. t= 3,03e-03 s 4. t= 4,55e-03 s 5. t= 6,06e-03 s 6. t= 7,58e-03 s 7. t= 9,09e-03 s 8. t= 1,06e-02 s 131
  • 147. 9. t= 1,21e-02 s 10. t= 1,36e-02 s 11. t= 1,52e-02 s 12. t= 1,67e-02 s 13. t= 1,82e-02 s 14. t= 1,97e-02 s 132
  • 148. 15. t= 2,12e-02 s 16. t= 2,27e-02 s 17. t= 2,42e-02 s 18. t= 2,58e-02 s 19. t= 2,73e-02 s 20. t= 2,88e-02 s 133
  • 149. 21. t= 3,03e-02 s 22. t= 3,18e-02 s Figure A.5: Streamline plots for different time steps. B. Time-history solution of the velocity of the flow (LDA) The velocity contour plots obtained with the LDA measurements are presented below. The presented pictures capture the velocity induced by one oscillation of the piezoelectric fan. 1. t= 8,77e-04 s 2. t= 1,75e-03 s 3. t= 2,63e-03 s 4. t= 3,51e-03 s 5. t= 4,39e-03 s 6. t= 5,26e-03 s 7. t= 6,14e-03 s 8. t= 7,02e-03 s 9. t= 7,89e-03 s 10. t= 8,77e-03 s 11. t= 9,65e-03 s 12. t= 1,05e-02 s 13. t= 1,14e-02 s 14. t= 1,23e-02 s 15. t= 1,32e-02 s 134
  • 150. 16. t= 1,40e-02 s 17. t= 1,49e-02 s 18. t= 1,58e-02 s 19. t= 1,67e-02 s Figure B.1: Velocity field in the vertical direction. 1. t= 8,77e-04 s 2. t= 1,75e-03 s 3. t= 2,63e-03 s 4. t= 3,51e-03 s 5. t= 4,39e-03 s 6. t= 5,26e-03 s 7. t= 6,14e-03 s 8. t= 7,02e-03 s 9. t= 7,89e-03 s 10. t= 8,77e-03 s 11. t= 9,65e-03 s 12. t= 1,05e-02 s 13. t= 1,14e-02 s 14. t= 1,23e-02 s 15. t= 1,32e-02 s 16. t= 1,40e-02 s 17. t= 1,49e-02 s 18. t= 1,58e-02 s 19. t= 1,67e-02 s Figure B.2: Velocity field in the horizontal direction. 135
  • 151. 1. t= 8,77e-04 s 2. t= 1,75e-03 s 3. t= 2,63e-03 s 4. t= 3,51e-03 s 5. t= 4,39e-03 s 6. t= 5,26e-03 s 7. t= 6,14e-03 s 8. t= 7,02e-03 s 9. t= 7,89e-03 s 10. t= 8,77e-03 s 11. t= 9,65e-03 s 12. t= 1,05e-02 s 13. t= 1,14e-02 s 14. t= 1,23e-02 s 15. t= 1,32e-02 s 16. t= 1,40e-02 s 17. t= 1,49e-02 s 18. t= 1,58e-02 s 19. t= 1,67e-02 s Figure B.3: RMS velocity field 136
  • 152. C. Contents of the DVD With this thesis a DVD is included. The following contents can be found on the DVD: • Thesis in PDF format • Presentation given at UAS-LW Conference (Symposium on Light Weight Unmanned Aerial Vehicle Systems and Subsystems) • Paper submitted for International Symposium On Coupled Methods In Numerical Dynamics (Croatia) • MATLAB, ANSYS, CFX and MATHEMATICA codes and programs • Results data from the experiments/measurements and simulations • Technical information, pictures, unprocessed data, etc of the experiments. • The most important papers directly related to this subject. 137
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