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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. ⁄
⁄
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

16. ∅⁄ 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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.
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31. 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

32. Figure 1.14: Methodology used in this research to obtain the parameters for an optimal piezoelectric flapping wing
prototype design.
16

33. 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

34. 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