- 1. FACULTEIT INGENIEURSWETENSCHAPPEN Vakgroep Toegepaste
Mechanica The design of a piezoelectric fan system for the flapping
wing mircro-air-vehicle application Het ontwerp van een pizo fan
systeem voor de flapping wing micro-air- vehicle toepassingEindwerk
voorgelegd voor het behalen van de academische graad van Master in
de Ingenieurswetenschappen door:Mohammad Ahmadi
BidakhvidiAcademiejaar: 2008-2009 Promotor: Prof. Steve
Vanlanduit
2. Dankwoord Ik wens iedereen die mij in de loop van dit
academiejaar heeft geholpen om deze thesis tot een goed eind te
brengen, te bedanken.Hierbij bedank ik vooral mijn promotor, Prof.
Dr. Steve Vanlanduit, voor het scheppen van de mogelijkheid dit
onderzoek te verrichten. Ook zou ik hem willen bedanken voor zijn
vriendelijkheid, ondersteuning en advies. Mede door zijn
opmerkingen en goede raad heeft hij me steeds op het goede spoor
gezet.Prof. Dr. Ir. Dean Vucinic dank ik voor zijn toelichtingen en
opmerkingen over de CFD simulaties. Ook Jean-Paul Schepens dank ik
voor de technische ondersteuning die hij heeft gegeven bij de
talrijke experimenten.Tot slot wil ik nog mijn familie bedanken om
me gedurende heel mijn studie te steunen. i 3. Summary A micro
aerial vehicle (MAV) is a semiautonomous airborne vehicle which
measures less than 15 cm in any dimension. It can be used for video
reconnaissance and surveillance. As demonstrated by birds and
insects, flapping flight is advantageous for its superior
maneuverability and much more aerodynamically efficient at small
size than the conventional steady-state aerodynamics.Piezoelectric
actuators are easy to control, have high power density and can
produce high output force but typically the displacement is small.
By using appropriate amplification mechanisms the piezoelectric
actuators can generate enough thrust to be implemented in a MAV as
a propulsion system. They can also be used to drive the flapping
wings of MAVs.This research aims to develop a piezoelectric
flapping wing system with 2 piezoelectric fans for MAVs. Various
prototypes were made by attaching a flexible wing, formed by two
spars and a flexible membrane, to two piezoelectric fans to make
them coupled. The dynamic properties of the structures were
characterized by using laser doppler vibrometer
measurements.Theoretical models were used to analysis the
performance of the piezoelectric fans at both quasi-static and
dynamic operations, and the calculated results agreed well with the
finite element analysis (FEA) modeling results. Several FEA models
of the piezoelectric flapping wings were proposed and investigated.
Selected factors such as geometric ratios, material selection, etc
can affect the performances of the wing significantly. These
influences have been investigated and optimization results were
obtained using the FEA technique.Both numerical and experimental
flow analyses are carried out on a piezoelectric fan. A 3D fluid-
structure interaction computational fluid dynamics model was set up
with commercial codes (CFX and ANSYS) to predict the velocity
fields generated by the swinging movement of the piezoelectric fan.
The flow measurements were carried out by using hot wire
anemometry, particle image velocimetry and laser doppler
anemometry. Thrust measurements were conducted to determine the
feasibility of the use of piezoelectric flapping wing propulsion
systemss in MAV applications.Two sinusoidal voltages with phase
differences were then used to drive the coupled piezoelectric fans.
High speed camera photography was used to characterize the two
degrees of freedom motion of the wing. It has been observed that
the phase delay between the driving voltages applied to the coupled
piezoelectric fans play an important role in the control of the
flapping and twisting motions (rotation) of the wing.ii 4.
Samenvatting Een micro aerial vehicle (MAV) is een semiautonoom
vliegtuig dat minder dan 15 cm in om het even welke afmeting meet.
Het kan voor videoverkenning en toezicht worden gebruikt. Zoals
aangetoond door vogels en insecten, is de klappende vlucht
voordelig voor zijn superieure manoeuvreerbaarheid en veel meer
aerodynamisch efficint bij kleine grootte dan de conventionele
evenwichtstoestand aerodynamica.Pizo-elektrische actuatoren zijn
gemakkelijk te regelen, hebben een hoge energiedichtheid en kunnen
een hoge outputkracht veroorzaken, maar de verplaatsing is meestal
klein. Door aangewezen versterkingsmechanismen te gebruiken kunnen
pizo-elektrische actuatoren genoeg stuwkracht produceren die in een
MAV als aandrijvingsysteem moet worden uitgevoerd. Zij kunnen ook
worden gebruikt om de klappende vleugels van MAVs aan te
drijven.Met dit onderzoek wordt getracht met 2 pizo-elektrische
ventilatoren een pizo-elektrisch klappend vleugelsysteem te
ontwikkelen voor MAVs. Diverse prototypen werden gemaakt door een
flexibele vleugel, die door twee langsliggers en een flexibel
membraan wordt gevormd, aan twee pizo- elektrische ventilators vast
te maken om hen gekoppeld te maken. De dynamische eigenschappen van
de structuren werden geanalyseerd door midden van laser doppler
vibrometer metingen.De theoretische modellen werden gebruikt om de
prestaties van de pizo-elektrische ventilators bij zowel quasi
statische als dynamische werking te analyseren, en de berekende
resultaten kwamen goed overeen met de resultaten van de eindige
elementenanalyse (EEA). Verscheidene EEA modellen van de
pizo-elektrische klappende vleugels werden voorgesteld en
onderzocht. De geselecteerde factoren zoals geometrische
verhoudingen, de materiaalkeuze, enz. kunnen de prestaties van de
vleugel beduidend benvloeden. Deze invloeden zijn onderzocht en de
optimalisatieresultaten werden verkregen door de EEA techniek te
gebruiken.Zowel numerieke als experimentele stromingsanalyses
werden uitgevoerd op een pizo-elektrische ventilator. Een 3D
fluid-structure interaction computational fluid dynamics model is
opgesteld met commercile codes (CFX en ANSYS) om de
snelheidsvectorveld te voorspellen die door de harmonische beweging
van de pizo-elektrische ventilator worden gegenereerd. De
stromingsmetingen werden uitgevoerd door gebruik te maken van de
hot wire anemometry, particle image velocimetry en laser doppler
anemometry techniek te gebruiken. Metingen van de stuwkracht werden
uitgevoerd om de haalbaarheid van het gebruik van pizo-elektrische
klappende vleugelaandrijving systemen in MAV toepassingen te
bepalen.Twee sinusodale voltages met faseverschillen werden
gebruikt om de gekoppelde pizo-elektrische ventilators aan te
drijven. Een hogesnelheidscamera werd gebruikt om de beweging van
de vleugel met twee vrijheidsgraden te kenmerken. Er is vastgesteld
dat het faseverschil tussen de elektrische voedingspanningen, die
op de gekoppelde pizo-elektrisch ventilators worden toegepast, een
belangrijke rol spelen in de controle van het klappen en verdraaien
(rotatie) van de vleugel.iii 5. Resum Un micro aerial vehicle (MAV)
est un aronef semi-autonome qui mesure moins de 15 cm dans
n'importe quelle dimension. Il peut tre employ pour la
reconnaissance et la surveillance visuelles. Comme dmontr par des
oiseaux et des insectes, le battement du vol est avantageux pour sa
manuvrabilit suprieure et est beaucoup plus arodynamiquement
efficace de petite taille que l'arodynamique quilibre
conventionnelle.Les dclencheurs pizolectriques sont faciles
commander, ont la densit de puissance leve et peuvent produire une
force haute production mais le dplacement typiquement est petit. En
employant les mcanismes appropris d'amplification, les dclencheurs
pizolectriques peuvent produire d'assez pousss pour tre mis en
application dans un MAV comme systme de propulsion. Ils peuvent
galement tre employs pour conduire les ailes de battement de
MAVs.Ce recherche veut dvelopper un systme pizolectrique d'aile de
battement avec 2 ventilateurs pizolectriques pour MAVs. Des
prototypes divers ont t faits en attachant une aile flexible,
constitue par deux longerons et une membrane flexible, deux
ventilateurs pizolectriques pour les faire coupls. Les proprits
dynamiques des structures ont t caractrises en employant des
mesures de laser doppler vibrometer.Des modles thoriques ont t
employs pour analyser l'excution des ventilateurs pizolectriques
aux oprations quasi statiques et dynamiques, et les rsultats
calculs taient conformes bien finite element analysis (FEA)
modelant des rsultats. Plusieurs modles de FEA des ailes
pizolectriques de battement ont t proposs et tudis. Les facteurs
choisis comme des rapports gomtriques, le choix matriel, etc.
peuvent affecter les excutions de l'aile de manire significative.
Ces influences ont t tudies et des rsultats d'optimisation ont t
obtenus utilisant la technique de FEA.Des analyses de flux
numriques comme exprimentales sont effectues sur un ventilateur
pizolectrique. Un modle informatique de dynamique des fluides
d'interaction de la fluide-structure 3D a t install avec des codes
commerciaux (CFX et ANSYS) pour prvoir les champs de vitesse
produits par le mouvement d'oscillation du ventilateur
pizolectrique. Les mesures d'coulement ont t effectues en employant
hot wire anemometry, particle immage velocimetry et laser doppler
anemometry. Des mesures de pousse ont t conduites pour dterminer la
praticabilit de l'utilisation des systmes pizolectriques de
propulsion d'aile de battement dans des applications de MAV.Deux
tensions sinusodales avec des diffrences de phase ont t alors
employes pour conduire les ventilateurs pizolectriques coupls. La
photographie de camra grande vitesse a t employe pour caractriser
les deux degrs de mouvement de libert de l'aile. On l'a observ que
le retard de phase entre les tensions motrices s'est appliqu
pizolectrique coupl de ventilateurs a un rle important dans la
commande du battement et des mouvements de vrillage (rotation) de
l'aile. iv 6. Contents List of
Figures..........................................................................................................................................
vii List of Tables
............................................................................................................................................
xi List of symbols
........................................................................................................................................
xii Abbreviations
........................................................................................................................................
xiv 1Introduction and overview
..............................................................................................................
1 1.1Motivation of research
............................................................................................................
1 1.2Micro Aerial Vehicles
...............................................................................................................
11.2.1 Fixed wing MAV
...............................................................................................................
31.2.2 Rotary wing MAV
.............................................................................................................
41.2.3 Flapping wing MAV
..........................................................................................................
5 1.3Piezoelectric actuators
............................................................................................................
61.3.1 Piezoelectricity
................................................................................................................
61.3.2 Piezo
fans.......................................................................................................................
11 1.4Piezoelectric actuated flapping wings
...................................................................................
13 1.5Research objectives
...............................................................................................................
15 2Optimization of piezoelectric actuated wing structures
............................................................... 17
2.1Introduction
...........................................................................................................................
17 2.2Theoretical analysis of piezoelectric fans
..............................................................................
172.2.1 Introduction
...................................................................................................................
172.2.2 Analysis for bimorph actuators at quasi-static operation
............................................. 182.2.3 Analysis for
unimorph actuators at quasi-static operation
........................................... 192.2.4 Analysis of the
dynamic peak amplitude
.......................................................................
222.2.5 Electromechanical coupling factor (EMCF)
...................................................................
24 2.3FEM analysis of piezoelectric fans
.........................................................................................
252.3.1 Piezoelectric FEM Equations
.........................................................................................
252.3.2 Finite element software: ANSYS
....................................................................................
27 2.4Parametric optimization
........................................................................................................
282.4.1 Validation of the finite element models
.......................................................................
282.4.2 Optimization Results and
Discussion.............................................................................
30 3CFD Simulations
.............................................................................................................................
57 3.1Introduction
...........................................................................................................................
57 3.2Fluid Structure Interaction: coupling of CFD and FE analysis
................................................ 573.2.1 Defining
the problem
....................................................................................................
59 v 7.
3.2.2Modeling........................................................................................................................
603.3Numerical model
...................................................................................................................
623.3.1Analysis settings
............................................................................................................
633.3.2Sensitivity analysis and convergence test
.....................................................................
653.3.3Results and discussion
...................................................................................................
683.3.4Conclusion
.....................................................................................................................
73 4Experiments
...................................................................................................................................
744.1Introduction
...........................................................................................................................
744.2Prototype design
...................................................................................................................
744.3Laser Doppler Vibrometer measurements
............................................................................
754.3.1Measurements
..............................................................................................................
774.4Propulsion and energy consumption measurements
...........................................................
854.5Flow experiments
..................................................................................................................
874.5.1Introduction
...................................................................................................................
874.5.2Hot wire Anemometry
measurements..........................................................................
874.5.3Laser Doppler Anemometry measurements
.................................................................
934.5.4Particle Image Velocimetry measurements
..................................................................
994.6High speed camera visualization
.........................................................................................
108
5Conclusions..................................................................................................................................
1125.1Conclusions and final
remarks.............................................................................................
1125.2Recommendations for future work
.....................................................................................
113 A. Time-history solution of the velocity of the flow (CFD)
.............................................................. 116
B. Time-history solution of the velocity of the flow (LDA)
.............................................................. 134
C. Contents of the DVD
....................................................................................................................
137
Bibliography.........................................................................................................................................
138vi 8. List of Figures Figure 1.1: Basic concept of a MAV flight
for surveillance applications.
................................................ 2 Figure 1.2:
Black widow MAV
..................................................................................................................
4 Figure 1.3: Micro Flying Robot
................................................................................................................
4 Figure 1.4: Microbat MAV
.......................................................................................................................
6 Figure 1.5: Piezoelectric material deformation depending upon the
electric field and the polarization direction of the piezo
material.
...............................................................................................................
6 Figure 1.6: Force-Deflection characteristics of piezoelectric
actuators. ................................................. 8
Figure 1.7: Structure of bimorph piezo patches for (a) Bimorph in
series connection (b) Bimorph in parallel connection.
.................................................................................................................................
9 Figure 1.8: Comparison between (a) bimorph bending actuator and
(b) shear actuator. ..................... 9 Figure 1.9: A
commercial bimorph piezo fan from [15].
.......................................................................
11 Figure 1.10: Set-up and basic principle of an operating piezo
fan. ....................................................... 11
Figure 1.11: Four-bar mechanism from [25]
.........................................................................................
13 Figure 1.12: Schematic of the coupled piezoelectric fans for MAV
applications. ................................. 14 Figure 1.13:
Piezoelectric flapping wing propulsion system.
................................................................ 14
Figure 1.14: Methodology used in this research to obtain the
parameters for an optimal piezoelectric flapping wing prototype
design.............................................................................................................
16 Figure 2.1: The effect of the thickness ratio on , fr, 0 and Fbl
............................................................. 21
Figure 2.2: Flow chart to determine the deflection at resonance.
....................................................... 23 Figure
2.3: Definition of short circuited and open circuited configuration
for piezoelectric bending
actuators................................................................................................................................................
24 Figure 2.4: First three normalized bending modes for ANSYS and
the analytical calculation. ............. 29 Figure 2.5: Amplitude
of the tip of the piezo fan in function of the frequency.
Comparison between the analytical and FEM (ANSYS)
results.................................................................................................
30 Figure 2.6: Definition of the various models used in the FEA
simulations............................................ 31 Figure
2.7: Element size h-convergence test on FEM models of the
piezoelectric flapping wings. ..... 32 Figure 2.8: Basic piezo fan
model with the definition of the length and width parameters.
............... 32 Figure 2.9: Optimization results for the
piezoelectric fans with rectangular PZT-5H patches. First row:
dynamic tip deflection in meters; Second row: EMCF (%); Third row:
fA in m/s; Fourth row: first resonance frequency in Hz.
...................................................................................................................
35 Figure 2.10: Optimization results for the piezoelectric fans
with triangular PZT-5H patches. First row: dynamic tip deflection
in meters; Second row: EMCF (%); Third row: fA in m/s; Fourth row:
first resonance frequency in Hz.
...................................................................................................................
36 Figure 2.11: The influence of the distance between the piezo
patch and the clamping of the piezo fan on the Tip deflection,
EMCF, fA and first resonance frequency of the piezo fan.
................................ 37 Figure 2.12: The influence of
the elastic plate material on the Tip deflection, EMCF, fA and
first resonance frequency of the piezo
fan...................................................................................................
39 Figure 2.13: The influence of the thickness of the boundary
layer, between the piezoelectric patch and passive plate, on the
tip deflection of the piezo fan.
.....................................................................
40 Figure 2.14: The amplitude of the dynamic tip deflection in
function of the frequency for different widths (under 120V).
.............................................................................................................................
40 Figure 2.15: A meshed FEA model of a piezoelectric flapping wing
structure (Model 6a, bimorph). The mesh size is 0.5mm obtained
after a convergence analysis.
................................................................ 41
vii 9. Figure 2.16: The amplitude of the dynamic tip deflection in
function of the frequency for the diverse models under 120V.
..............................................................................................................................
41 Figure 2.17: The amplitude of the dynamic tip deflection in
function of the frequency for different damping ratios under 120V.
..................................................................................................................
42 Figure 2.18: The influence of the voltage on the tip deflection
and fA. ............................................... 42 Figure
2.19: The influence of the length ratio on the different
optimization parameters for the different models.
...................................................................................................................................
44 Figure 2.20: Optimization results for Model 2a and Model 6a.
First row: dynamic tip deflection in meters; Second row: EMCF (%);
Third row: fA in m/s; Fourth row: first resonance frequency in Hz.
. 45 Figure 2.21: Optimization results for the basic piezo fan
model and model0a. First row: dynamic tip deflection in meters;
Second row: EMCF (%); Third row: fA in m/s; Fourth row: first
resonance frequency in Hz.
.....................................................................................................................................
47 Figure 2.22: Optimization of the length ratio for the proposed
models (bimorph configuration) with different wing materials.
.......................................................................................................................
48 Figure 2.23: Piezoelectric flapping wing models with diverse
plate materials. .................................... 50 Figure
2.24: The influence of the wing length on the performance of the
piezoelectric flapping wing structure.
...............................................................................................................................................
52 Figure 2.25: Optimization results for models 4 and 6 with
different width ratios. .............................. 53 Figure
2.26: The influence of the width of the piezo fans on the
performance of the flapping wing structure.
...............................................................................................................................................
55 Figure 2.27: The influence of the thickness ratio and length
ratio of the spar on the optimization
quantities...............................................................................................................................................
56 Figure 3.1: The three dimensions of fluid dynamics.
............................................................................
57 Figure 3.2: A time dependent pressure function can be applied on
the plate to let it move similar to the first bending mode.
.........................................................................................................................
59 Figure 3.3: Total Mesh Displacement of the tip of the wing in
CFX Solver. .......................................... 62 Figure
3.4: Velocity vectors colored by velocity magnitude (m/s)
(Time=1.5280s) [40]. ..................... 63 Figure 3.5:The
solution of the FSI model with the piezo fan oscillating at 60Hz
with a tip deflection of 2cm. It can be observed that the solution
converges to a certain result when smaller time steps are applied
for the simulation.
....................................................................................................................
66 Figure 3.6: Time step size convergence plots for the velocity.
(A) Relative error for velocity v. (B) Relative error for velocity
vu. (C) Relative error for velocity
vv.............................................................. 67
Figure 3.7: Comparison between the velocity vector field obtained
by using a fine (left) and coarse (right) mesh.
..........................................................................................................................................
67 Figure 3.8: Geometrical properties of the enclosed space and the
definition of the several locations in the fluid domain. The
Cartesian coordinate system is placed in the base of the piezo fan.
................ 68 Figure 3.9: The velocity in function of the
time in point 3.
...................................................................
68 Figure 3.10: The velocity in function of the time for the
different positions........................................ 69
Figure 3.11: Velocity vector plot of the induced flow by the
harmonic movement of the piezo fan at time=0.1s.
..............................................................................................................................................
70 Figure 3.12: Velocity vector field induced by a piezo fan [40]
.............................................................. 70
Figure 3.13: Streamline for the flow pattern that has developed
after time=0.1s. .............................. 71 Figure 3.14: The
influence of the tip deflection on the velocity of the flow in
point 3. ....................... 71 Figure 3.15: The influence of
the frequency on the induced velocity by a piezo fan with a tip
deflection of 2
cm..................................................................................................................................
72viii 10. Figure 3.16: Influence of the frequency (averaged).
.............................................................................
72 Figure 3.17: 3D velocity vector field plot for a piezoelectric
fan at time=0.12s. .................................. 73 Figure
4.1: First prototype of a piezoelectric flapping wing built for
this work, using two coupled piezo fans.
.......................................................................................................................................................
74 Figure 4.2: The prototype with balsa wood operating at
resonance. ................................................... 75
Figure 4.3: Basic set-up principle for the LDV measurement applied
on the piezoelectric flapping wing.
...............................................................................................................................................................
75 Figure 4.4: First two mode shapes of a commercial piezoelectric
fan with the frequencies for the open and short circuited
configuration.................................................................................................
77 Figure 4.5: Results from the SLDV measurements. The tip
deflection is obtained by driving the piezoelectric flapping wing
at resonance and 130 VAC.
.......................................................................
83 Figure 4.6: Determination of the quality factor for Prototype 2.
......................................................... 84 Figure
4.7: Calculated damping ratios for the different piezoelectric
flapping wing prototypes (by using the LDV measurement data).
.......................................................................................................
84 Figure 4.8: The electrical current and tip deflection in
function of the applied voltage for the prototype in Figure 4.2.
.........................................................................................................................
85 Figure 4.9: The experiment set-up for the thrust measurements.
....................................................... 85 Figure
4.10: Definition of the geometrical variables for the thrust
measurements. ............................ 86 Figure 4.11: Results
of the thrust measurements.
................................................................................
86 Figure 4.12: Hot wire probe measurements in a piezo fan flow.
.......................................................... 88
Figure 4.13: Scheme of the CTA principle.
............................................................................................
89 Figure 4.14: Set-up for the calibration of the hot wire
anemometer. .................................................. 90
Figure 4.15: Calibration of the hot wire anemometer.
.........................................................................
91 Figure 4.16: Hot wire probe placed in the flow of the piezo fan.
......................................................... 92 Figure
4.17: Location of different measurement position for the hot wire
experiment. ..................... 92 Figure 4.18: Results of the
hot wire measurements.
............................................................................
93 Figure 4.19: The LDA principles
[44]......................................................................................................
94 Figure 4.20: Location of the measurement grid for the LDA
experiment. ............................................ 96 Figure
4.21: Results of the flow velocity in the different positions
obtained using LDA measurements.
...............................................................................................................................................................
96 Figure 4.22: Results of the flow velocity using CFD simulation
(the point is located in position 1 defined for the LDA
measurements).
....................................................................................................
97 Figure 4.23: LDA velocity vector field of the flow generated by
the piezo fan. The piezo fan is placed horizontally pointing in the
positive y-direction.
..................................................................................
98 Figure 4.24: The time-averaged velocity vector field of the
generated flow by the piezo fans over 1 oscillation (LDA
measurement). The piezo fan is placed vertically pointing in the
positive y-direction.
...............................................................................................................................................................
98 Figure 4.25: The basic set-up principle of particle image
velocimetry. .............................................. 100
Figure 4.26: Basic working principle of PIV.
........................................................................................
100 Figure 4.27: The employed experiment set-up for the PIV
measurements. ....................................... 103 Figure
4.28: Residues on the glass enclosure due to the generated smoke
during measurements. The bending of the piezo fan can clearly be
observed.
.............................................................................
103 Figure 4.29: The piezoelectric flapping wing prototype in the
Plexiglas enclosure. ........................... 104 Figure 4.30: A
recorded image pair with a separation time of 50s (PIV
measurement). The velocity vector field near the tip is obtained.
...................................................................................................
105 ix 11. Figure 4.31: Post processing of the results of the PIV
measurements. .............................................. 106
Figure 4.32: The velocity vector field of the flow generated by the
harmonic motion of a piezo fan with a tip deflection of 3cm,
simulated with CFX (see 3.3.3).
........................................................... 107
Figure 4.33: Velocity vector field of the flow induced by a
flapping piezo fan obtained with PIV
measurements.....................................................................................................................................
107 Figure 4.34: Two piezoelectric fans with a phase delay of 180
degrees. ............................................ 108 Figure
4.35: Set-up for the high-speed camera
recordings.................................................................
109 Figure 4.36: The tip deflection and used electric current in
function of the phase delay for the prototype in Figure 4.2.
.......................................................................................................................
109 Figure 4.37: High-speed camera recordings of a prototype moving
at the first bending mode. ....... 110 Figure 5.1: Stacking of
multiple piezoelectric flapping wings.
............................................................ 113x
12. List of Tables Table 1.1: Comparison of PZT and PVDF material
properties.
................................................................ 7
Table 1.2: Compressed Matrix Notation.
..............................................................................................
10 Table 2.1: Material properties of PZT-5H for analytic and FEA
calculations. ........................................ 28 Table
2.2: Validation of the FEM model: comparison between the analytical
and ANSYS solution..... 29 Table 2.3: Validation of the FEM model.
...............................................................................................
30 Table 2.4: Material properties of the elastic plate, wing and
spars used in the FEA of piezoelectric flapping
wings........................................................................................................................................
41 Table 3.1: Material properties of air at 20C used in the
numerical flow simulations. ........................ 60 Table 4.1:
Definition of the geometrical variables of the models for the
thrust measurements. The first resonance frequency is also
specified.
..........................................................................................
86 Table 4.2: Parameters for the hot wire anemometer.
..........................................................................
90xi 13. List of symbols ROMANArea of cross section A Amplitude of
the tip of the piezoelectric actuated structure,Width of beam and
piezoelectric actuator Components of the mechanical stiffness
Components of the electric flux density vector Components of the
piezoelectric coupling (electrical field/stress) Components of the
electric field vector Youngs modulus Components of the
piezoelectric coupling (electrical field/strain) Frequency
Components of body force vector Resonance frequency Blocking force
Shear modulus h Enthalpy Moment of inertia = 1 Current Imaginary
unit Length, Electromechanical coupling factor, Electromechanical
coupling factors of piezo material Electromechanical coupling
factors of a bimorph/unimorph Dynamic/effective electromechanical
coupling factor Power, Pressure Surface change Quality factor
Components of the strain Components of the mechanical compliance
tensor , Components of the stress Thickness/height of layer k, ,
Internal energy density Displacements relative to , , respectively
Voltage across electrodes, , Volume Cartesian coordinates
GREEKDielectric permittivity in vacuum (= 8.85 10 ) Components of
the dielectric permittivity tensor Variation (of length), distance
Tip deflection Electromechanical coupling factor Dynamic viscosity
Poisson ratio Normalized frequencyxii 14. FrequencyNatural
frequency of mode kDensityDamping ratio MATRICES Dielectric
permittivity matrixStructural damping matrixMechanical stiffness
matrixPiezoelectric coupling matrix (electrical
field/strain)Dielectric permittivity matrixPiezoelectric coupling
matrix (electrical field/stress)Piezoelectric coupling matrix
(electric flux density/strain)Piezoelectric coupling matrix
(electric flux density/stress) Structural stiffness
matrixEquivalent stiffness matrix ( Piezoelectric stiffness
matrixDielectric stiffness matrixStructural mass matrixMechanical
compliance matrix VECTORS Electric flux density vectorElectric
field vectorForce vector Vector with nodal structural
forcesEquivalent structural forcesVector with nodal chargesPart
ofwith prescribed charge boundary distributionPart ofwith
prescribed voltage boundary distributionPart of with prescribed
charge boundary distributionVector with nodal chargesPart of with
described voltage boundary distributionPolarization vectorStrain
vectorStress vectorVector with nodal structural displacements
MISCELLANEOUS MatrixMatrix transposedVector ,Vector transposedFirst
and second time derivatives of, First and second spatial
derivatives ofSpatial derivation operatorxiii 15. Spatial
derivative of with respect to , Short notation for the spatial
derivative of with respect to Complex conjugate of Abbreviations
ABS Acrylonitril butadieen styreen ACAlternating current CCD
Charge-coupled device CFD Computational fluid dynamics CFRPCarbon
fiber reinforced polymer CTA Constant Temperature Anemometry DARPA
Defence Advanced Research Projects Agency DCDirect current DOF
Degree of freedom EAPElectroactive Polymers EMCFElektromechanical
coupling factor FEFinite element FEA Finite element analysis FEM
Finite element method FFT Fast fourrier transformation FRF
Frequency response function FSI Fluid-structure interaction LDA
Laser doppler anemometry LDV Laser doppler vibrometry MFI Micro
flying insect MAV Micro aerial vehicle OCOpen circuited PIV
Particle image velocimetry PTV Particle tracking velocimetry
PVDFPolyvinylidene Fluoride PZT Lead zirconate titanate RMS Root
mean square SCShort circuited SLDVScanning laser doppler vibrometer
UAV Unmanned aerial vehicle VAC Volts Alternating Currentxiv 16. 1
Introduction and overview1.1 Motivation of research The recent
advances of small CCD cameras, infrared sensors, etc have led to
significant interest in small flying vehicles called Micro Air
Vehicles (MAVs), which can perform as highly portable platforms for
the tiny cameras and sensors. These aerial vehicles were originally
proposed as extremely portable observation platforms for military
applications. The potential of these flapping wing MAVs has
resulted in extensive work in recent years.It is demonstrated by
flying birds and insects that flapping flight and thus flapping
wing MAV is advantageous for its greater maneuverability and
lifting capability at low flight speeds in indoor environments.
Insects can commence complex maneuvers like taking off backwards,
flying sideways and landing upside down. Small flapping wing MAVs
would not only move like insects, but with typical dimensions in
only the millimeter range can also function almost unnoticed. The
aerodynamic mechanisms allowing the high lift forces and
maneuverability of insect flight are complex. Dickenson et al. [1]
addressed and modeled three separate aerodynamic lift mechanisms in
fruit flies. These mechanisms have been named delayed stall,
rotational lift and wake capture. Delayed stall is a leading edge
vortex on the wing due to a high angle of attack that would
eventually cause the wing to stall. However, before stall occurs, a
large increase in lift force is observed. Since the wing soon
reverses direction, the leading edge vortex does not separate
(stall). Rotational lift occurs when the wing is simultaneously
translating and rotating. Finally, wake capture occurs when the
wing reverses direction; since it has rotated, when the wing now
meets the vortex that was attached to the wing during the previous
stroke, a significant inertial lift spike is observed.The main goal
of this present work is to obtain a propulsion system that can let
a MAV achieve autonomous flight; specifically designing the power
plant in such a way that the maximum thrust to power ratio is
obtained.This work commenced with two difficult constraints. The
first constraint was that the flying construction must be a MAV. By
definition, MAV must have a total wingspan less than 15 cm. In our
case this was an essential constraint for the flapping wing. The
second constraint was that the flying object must fly by flapping
wings or using flapping wings to maintain flight. The aerodynamics
of flapping-wing flight, especially MAV size, is still not a
fully-explored subject. There have been studies of insect flights,
but unlike fixed-wing aerodynamics there have not yet been any
available design rules for flapping-wing aerodynamics for MAV
size.1.2 Micro Aerial Vehicles DARPA (Defense Advanced Research
Projects Agency), the research and development organization for the
Ministry of Defense of America, introduced the concept of an insect
like miniature vehicle. The purpose of such flying objects was
originally for military applications [2]. They defined a MAV to be
sized less than 15 cm in length or width or height, weight less
than 50 grams and capable of staying in flight for 20 to 60 minutes
for a distance of 10 km. These size and weight restrictions put
MAVs in a size class which is at least an order of magnitude
smaller than other Unmanned Air Vehicles (UAVs). The MAVs could be
applied to enter environments which are too risky for direct human
intervention, for instance, searching for disaster survivors or
detecting explosive devices planted in buildings. Other
applications are communications, traffic monitoring, inspections,
etc (Figure 1.1). This would necessitate a highly maneuverable
capability of evading obstacles to access the targets. So next to
the military applications a large number of commercial applications
exist for this technology. The low1 17. detectability, low noise
production, the ability to broadcast real-time data from an area of
observation and the ability to maneuver within confined spaces,
makes MAVs perfect for those applications. In recent years the size
and weight constraints set in the definition of MAVs by DARPA have
become quite flexible, with MAVs ranging from less than 10 grams to
more than 300 grams. Figure 1.1: Basic concept of a MAV flight for
surveillance applications.The Reynolds number is a ratio of the
inertial to viscous aerodynamic forces used to characterize flight
regimes, and is defined as: == =where is the fluid density, is the
characteristic length (in this case the chord), is the fluid(10 or
below) as compared to the conventional aircrafts (over 10 for
fast-flying commercial aircraft) viscosity, and is the dynamic
viscosity of the fluid. MAVs fly at a extremely low Reynolds
numberdue to their small dimensions and low speed.A number of
aerodynamic challenges exist for designing a MAV to obtain enough
lift and low drag. The MAV must have only small amounts of
material. This could also give the possibility to manufacture them
very economically, meaning a swarm of MAVs could be used to deal
with the problem at hand without requiring an optimal performance
of each vehicle.Latest studies in the understanding of aerodynamics
of flapping wing flight have led to new methods to realize the
flapping wing flight. A vital challenge in creation of
bird/insect-mimicking flapping machine is to select an actuator
which could produce sufficient wing deflections.Existing MAVs can
be classified into three broad categories based on the aerodynamic
mechanisms used to generate lift: fixed wing, rotary wing and
flapping wing. In the development of MAVs, an analogy can be drawn
with the development of their larger, manned counterparts during
the last century. Fixed wing technology was always a step ahead of
rotary wing technology because of the added complexities involved
in rotary wing flight. Likewise, among the existing MAVs,
fixed-wing MAVs perform better than both rotary and flapping wing
MAVs. Flapping wings, with their unsteady wing beating, establish
an extra level of complexity above and beyond rotary wings, and
hence their2 18. development seems to be the slowest. Fixed-wing
MAVs have a better endurance than rotary and flapping wing MAVs.
However, their major shortcoming is the lack of hover capability,
which allows an MAV to maneuver in much smaller spaces.Propulsion
mechanisms continue to stay an important limitation of MAV
advancement. Most recent MAVs are electric-powered.
Electric-powered systems efficiently convert stored energy into
usable energy, but present battery equipment has a low energy
density. Gasoline has a very high energy density, but combustion
engines are very inefficient at the MAVs scale and produce a lot of
noise. Existing propulsion systems obtainable for MAVs are not
appropriate for long endurance, allowing less than 10 minutes of
flight in many cases. MAV endurance is limited primarily by the
efficiency of the system and by the propulsive efficiency.
Rotary-wing MAVs are especially limited in endurance as they
consume great amounts of power in order to hover. Thus, scientists
have begun to investigate rotary and flapping-wing MAV designs,
which can securely operate at low speeds and offer the possibility
to hover. However, unlike fixed wings, these MAVs function in a
more complex aerodynamic environment.1.2.1 Fixed wing MAV Most of
the MAV have fixed wings. While the small size of MAVs is
attractive, there are associated technology barriers. The most
obvious are the complexity linked with miniaturization and our
imperfect understanding of the complex low Reynolds number
aerodynamic regime where MAVs operate. The MAV in [3] could cruise
at the speed of 65 km/h at Reynolds number of about 130,000.
However, as the size of a MAV reduces, the Reynolds number of the
flow surrounding the MAV also decreases. The challenge to the fixed
wing aircraft is that at low Reynolds number, the aircraft lacks of
maneuverability and needs a large turning radius to navigate and
avoid obstacles in a confined space. The fixed-wing designs are
based on the conventional scaled-down aerodynamics and flight
control approaches, and despite the ongoing research, these MAVs
are not suitable for operations in constrained areas because of
their relatively high stall speeds.A well-known fixed wing MAVs is
Black Widow MAV [4], which has a wing span of 15.2 cm and can
achieve a fly speed to 48.2 km/h, the maximum fly range to 1.8 km
and the maximum fly altitude to 769 feet, and the endurance to 30
minutes. Multidisciplinary design optimization was employed to
determine the battery, motor, gearbox, power requirements,
propeller diameter, wingtip chord, and cruise velocity combination
that would result in the best configuration. Black Widow can be
used for missions such as target tracking and video monitoring. It
could deliver live images in real-time via a custom-made color
camera and transmitter. The model is based on a disc shape
structure as shown in Figure 1.2, and many other MAVs have been
developed with slight modification using a similar shape and
concept.3 19. Figure 1.2: Black widow MAV1.2.2 Rotary wing MAV The
main purpose to develop MAVs is for the military surveillance
applications; hence an agile MAV would be more beneficial. If the
MAV could hover or fly slowly, then it could explore and relay
clear images back to the control centre. Helicopter MAVs are
interesting because of their capability to hover. They have been
constructed and studied by many researchers. Except carefully
designed, rotors can suffer from performance degradation at low
Reynolds numbers since their airfoils operate in a more challenging
environment.The Micro Flying Robot (see Figure 1.3) is a rotary
wing MAV developed by Seiko Epson in Japan, which weighs 8.9 gram.
It has the ability to transmit images to a control centre via
Bluetooth technology. Figure 1.3: Micro Flying RobotThe Pixel and
Proxyflyer Micron are other types of rotary wing MAVs which can
hover and fly in every direction. The Pixel weighs 6.9 grams and is
a fully functional helicopter controlled by an infrared signal. The
Proxyflyer has the identical weight as the Pixel, but designed
under a different technical concept.The MAVs mentioned above are
scaled down versions of helicopters. The smallest scaled down
version of the helicopter is the Small Flying Helicopter, developed
by Microtechnology in Germany with the dimensions of 24 x 8 x 0.4
mm and weight of 0.4 grams. It is powered by a 5mm long motor with
a diameter of 2.4 mm. This tiny flying helicopter could take off at
40,000 revolutions per minute, but did not include remote control
capability.4 20. 1.2.3 Flapping wing MAV For the miniature scale of
MAVs, flapping wing vehicles could be the favored approach because
of their presence in nature, and their capability to harness low
Reynolds number unsteady vortex lift effects. That is why
scientists are trying to mimic the wing motions of birds and
insects to build flapping wing MAVs. There is a great amount of
biological inspiration offered by nature. With the introduction of
a continuously accelerating and decelerating wing, the aerodynamics
of such vehicles is highly unsteady. Because they operate at low
Reynolds numbers where high viscous effects dominate, they need
high flapping frequencies and consume large amounts of power. Their
tiny size also restricts their payload capacity. Additionally, the
highly evolved motions involved with insect flight renders
mechanical replication difficult and costly in terms of weight.
Flapping is much more aerodynamically efficient than the
conventional steady-state aerodynamics at small size [5].The wing
kinematics of insects and birds are both based on flapping wings,
but there is a fundamental difference between both [6]. Birds
primarily utilize wing flapping for propulsion, while lift is
generated by a combination of forward speed and wing flapping,
causing the lack of hover capability. Most birds flap their wings
in a vertical plane with small changes in the pitch of the wings
during a flapping cycle. Since birds are much larger than insects,
incorporating muscles, feathers and other moving parts into the
wings is easier. Birds can control the shape and even the span of
their wings to adapt to different flight modes. However, without
large changes in pitch, this kind of flapping cannot produce
sufficient vertical force to support the weight in the absence of
any forward velocity. As a result most birds cannot hover. There
are a great number of insects that can hover. These insects flap
their wings in a nearly horizontal plane, accompanied by large
changes in wing pitch angle to produce lift even in the absence of
any forward velocity. Birds like the hummingbird, which are capable
of hovering, have wing motions very similar to hover capable
insects. Thus, insect-based bio-inspired flight may present a
hover-capable and highly maneuverable solution for MAVs. There
exist large differences between the flight kinematics of diverse
insect species.It has been reported that the mass of the system has
a large influence on the performance of the flapping wing MAV.
Singh et al. [7] reported that when the mass of the flapping wing
MAV increases, the maximum frequency of the mechanism needs to
increase as well due to high inertial power requirements. Also,
wing tests showed a decrease in thrust at high frequencies.Nature
had millions of years to optimize its designs through the process
of natural selection, therefore the understanding of the
fundamental physics of flapping flight is important. However,
simply copying biological morphology, kinematics, or behaviors
could not certainly lead to an optimum system [8], because even if
such an optimum system could be achieved, it may not be practical
(and economical) due to external constraints such as availability
of suitable material or nonexistent manufacturing techniques. In
flapping flight, a mechanism that can imitate insect wing
kinematics is also a major obstacle which requires newer materials
such as Electroactive Polymers (EAP) for artificial muscles [9].The
latest flapping wing MAV projects have adopted a battery powered
electrical motor as an actuator. The rotary motion of the
electrical motor is converted to a linear or flapping motion by a
mechanism. The most common mechanism that can convert rotary motion
to linear or flapping motion is a slider crank type four-bar
mechanism.The flapping wing MAVs fly like birds in which the wings
function as static lifting surfaces similar to conventional
airplanes. Flapping is used along with changing angle of attack to
generate forward thrust. One of the first flapping MAVs with static
lifting surface wings was the Caltech Microbat [10].5 21. The
Microbat was a 23-cm span, electric-powered ornithopter, developed
in response to DARPAs original MAV initative. Microbat was built
primarily of carbon fiber and Mylar, weighted 12.5 g and an
endurance of 22 min, and was remotely piloted. This MAV flew like a
bird however several claims of wake capture have been made for the
vehicle. A super capacitor powered electric motor was applied. Some
enhanced models have been derived from the original Microbat
(Figure 1.4). Figure 1.4: Microbat MAVResearchers at Delft
University of Technology in the Netherlands have created a micro
UAV they have termed DelFly [11]. The DelFly had two sets of
flapping wings, which allowed it to fly both fast forward flight
missions and very slow (almost hovering) missions. It had a 35 cm
wingspan, flapping frequency of 6 Hz, and weight of 17 g. The
DelFly carried a video camera payload, allowing it to identify
targets. It had an endurance of 12 minutes at a cruise velocity of
1.8 m/s. The latest version, Delfly II is a 30cm device and can fly
horizontally at 15m/s but can also hover. Delflys performance is
certainly noteworthy but with such a large vehicle, constrained
indoor flight is still difficult due to maneuverability issues. At
Harvard Micro Flying Insect (MFI) technology is used to realize
takeoff of a tethered 60 mg flapping vehicle [12]. Woods vehicle
flapped at approximately 110Hz. Though the vehicle was tethered and
uncontrolled, it is the first vehicle of its size to produce thrust
greater than its weight.1.3 Piezoelectric actuators 1.3.1
Piezoelectricity Piezoelectricity is a coupling between a materials
mechanical and electrical behaviors. When a piezoelectric material
is squeezed, an electric charge collects on its surface (direct
effect). Conversely, when a piezoelectric material is subjected to
an electric field, it exhibits a mechanical deformation (inverse
effect). A basic illustration of converse piezoelectricity is shown
in Figure 1.5. Applying an electric voltage to the electrodes of
piezoelectric material will induce a mechanical deformation
according to the magnitude and sign of applied voltage [13]. Figure
1.5: Piezoelectric material deformation depending upon the electric
field and the polarization direction of the piezo material.6 22.
The piezoelectric effects can be seen as transfers between
electrical and mechanical energy. Such transfers can only occur if
the material is composed of charged particles and can be polarized.
For a material to exhibit an anisotropic property such as
piezoelectricity, its crystal structure must have no centre of
symmetry. Most of the piezoelectric materials are crystalline
solids. They can be single crystals, either formed naturally or by
synthetic processes, or polycrystalline materials like
ferroelectric ceramics. Certain polymers can also be made
piezoelectric by stretching under an electrical field.Piezoelectric
ceramics are formed by conventional ceramic processing techniques,
such as dry pressing, casting or extrusion. The ceramic material is
then sintered, machined into the desired dimensions and pasted on
electrodes. Polarization of the ceramic element is the final step
in processing which involves heating the ceramic above the Curie
temperature and subsequently cooling the material in the presence
of a strong DC electric field. This poling process aligns the
molecular dipoles of the ceramic in the direction of the applied
field and thus induces its piezoelectric properties. Piezoelectric
ceramics are hard, chemically inert and completely insensitive to
humidity or other atmospheric influences. Their mechanical
properties resemble those of the better known ceramic insulators
and they are manufactured by much the same processes. Furthermore
they are extremely stiff. They are capable of exerting or
sustaining great stresses. One of the principal advantages of Lead
Zirconate Titanate (PZT) ceramics is that their properties can be
optimized to suit specific applications by appropriate adjustment
of the zirconate-titanate ratio. They can be tailored to suit
specific applications.Piezoelectricity can also be obtained by
orientating the molecular dipoles of polar polymers such as
Polyvinylidene Fluoride (PVDF) in the same direction. The PVDF can
be made piezoelectric because fluorine is much more electronegative
than carbon. The fluorine atoms will attract electrons from the
carbon atoms to which they are attached. A sequence of processes,
including elongation, annealing, evaporation of electrodes and
poling, has to be performed to make the material piezoelectric.
PVDF differs in many ways from the conventional crystalline and
polycrystalline materials. In particular, PVDF is characterized by
such properties as flexibility, ruggedness, softness, lightweight,
relatively low acoustic impedance and low mechanical quality
factor. The material is also available in thin films and in large
sheets and is inexpensive to produce.Material propertiesPZTPVDF d33
(10-12 m/V)300-25 d31 (10-12 m/V)-150 15Relative permittivity d32
(10-12 m/V)-150 31800 12 (0 = 8.854 x 10-12 F/m) Young modulus50 5
Maximum operating14090 1 10 500 10 temperature (C) Maximum electric
field (V/m) Density (kg/m3)7600 1800 Table 1.1: Comparison of PZT
and PVDF material properties.A comparison of some of the physical
properties of PVDF with those of PZT is given in Table 1.1 It is
clear from the table that the piezoelectric strain constant d31,
which relates the induced in-plane strain due to the electric field
in the thickness direction, of PVDF is considerably smaller than
the constant of PZT. Also the maximum operating temperature of PVDF
is much lower than that of PZT which makes it less useful working
in high temperature environment. The advantage of PVDF over PZT is
that the7 23. maximum electric field strength that can be applied
to the polymer without danger of depolarization is much
greater.Actuator is a device which transforms energy into
controllable motion. The primary performance characteristics of any
linear actuator are displacement, force, frequency, size, weight
and electrical input power. Piezoelectric materials are known for
their excellent operating bandwidth and can generate large forces
in a compact size, but traditionally they have very small
displacements. They cannot be used directly as actuators in their
raw form. So amplification is required.Piezoelectric actuators are
usually specified in terms of their free deflection and blocked
force. Free deflection (Xf) refers to displacement obtained at the
maximum recommended voltage level when the actuator is completely
free to move. Blocked force (Fb) refers to the force exerted at the
maximum recommended voltage level when the actuator is totally
blocked and not allowed to move. Deflection is at a maximum when
the force is zero, and force is at a maximum when the deflection is
zero. All other values of simultaneous displacement and force are
determined by a line drawn between these two points on a force
versus deflection line, as shown in Figure 1.6. For the
piezoelectric actuators, the focus of research has been on an
attempt to amplify the deflection of the material.Figure 1.6:
Force-Deflection characteristics of piezoelectric actuators.There
are different types of piezoelectric actuators: Stack actuator: a
large number of piezo layers can be stacked to linearly increase
their overall deflection while maintaining a low voltage
requirement. The displacement and force of a stack actuator are
directly proportional to the actuator length and cross-sectional
area, respectively. Unimorph actuator: a composite beam is formed
by attaching a plate with one active layer and one inactive layer,
or substrate. Bimorph actuator: two thin ceramic plates bonded
together and driven with opposite electric field. One plate expands
while the other contracts. The net result is a lateral deflection
of the plates. Piezoelectric bimorph is a bending element that
generates horizontal displacement at the drive of electric field
using the converse piezoelectric effect. There are two different
electrical connections which are usually used in bimorph
fabrication: one is series connection in which two piezoelectric
plates have opposite polarization directions and the actuator is
driven by applying electrical field between the top and bottom
electrodes (see Figure 1.7(a)); the other is parallel connection in
which two piezoelectric plates are of the same polarization
directions and the actuator is driven by applying electrical field
between surface electrodes and the bonding layer (see Figure
1.7(b)). In the latter case, two ceramic plates are electrically
connected in parallel and driven voltage is applied across half the
actuator thickness, thus enabling half driving voltage to achieve
the same electrical field as in the series case.8 24. Figure 1.7:
Structure of bimorph piezo patches for (a) Bimorph in series
connection (b) Bimorph in parallel connection. Usually a metallic
sheet or middle shim is sandwiched between the two piezoelectric
platesto increase the reliability and mechanical strength. Unlike
the PZT stack, bimorphs areoperated in the d31 mode.Shear actuator:
This mechanism is another way to deflect the beam and create the
wellknown fan moves. Piezo fans with this actuation mechanism are
more difficult to make. Theproposed configuration is such that,
this time, the d15 coupling coefficient dictates the design.In this
situation the electric field is applied perpendicularly to the
poling direction, inducing atransverse shear strain. The sandwich
plate exists of the following components: the top andbottom layers
are for example aluminum and the core is a shear actuated
piezolayer. In casethe patch of this material does not cover the
whole length of the sandwich plate, the coreshould be filled with a
rigid foam material. The core should be softer than the faces and
thickenough to produce shear stresses.Figure 1.8: Comparison
between (a) bimorph bending actuator and (b) shear actuator.The use
of stack actuators as bending actuators and shear actuators have
been investigated in [14] and led to smaller tip deflections,
therefore in this work only bending actuators were used.1.3.1.1
Constitutive equations An important characteristic of piezoelectric
materials compared to other smart materials is its linear behavior
within a certain range. The constitutive relations are based on the
assumption that the total strain in the actuator is the sum of the
mechanical strain induced by the stress and the controllable
actuation strain caused by the electric voltages. The strain (S) -
stress (T) - electric field (E) - electric 9 25. displacement (D)
relationships of the piezoelectric materials can be approximated to
have linear behavior:= += +where s and represent short-circuited
elastic compliance and free electric permittivity of the materials,
respectively. Note that IEEE compressed matrix notations (IEEE
1978) are used to denote the tensor variable. This consists of
replacing subscripts ij and kl by p and q according to Table 1.2Ij
or kl p or q 11 1 22 2 33 3 23 of 32 4 31 of 13 5 12 of 21 6 Table
1.2: Compressed Matrix Notation.The symbol dip represents the
electro-mechanical coupling and is called the piezoelectric strain
constant. Here, the first subscript i refers to the direction of
applied electric field and the second subscript p refers to the
direction of resulting strain. The piezoelectric strain constants
of PZT and PVDF are (IEEE 1978):0 00 00=0 000 000 0where= ,=. It is
clear that PZT and PVDF have three normal strains (S1, S2 and S3)
when an electric field is applied in the thickness direction
(subscript 3). When an electric field is applied in the inplane
direction (subscript 1), the materials can also have shear strain
(S5). In linear piezoelectricity, the constitutive relationships
are often expressed with matrix notations as= += +where {S}, {T},
{E} and {D} represent strain, stress, electric field and electric
displacement vector, respectively.=, = ,=, = Here, the matrices
(SE), (d) and (T) represent the short-circuited (i.e., {E}={0})
elastic compliance, piezoelectric strain constants and free (i.e.,
{T}={0}) dielectric permittivity, respectively. 10 26. The
alternate forms using alternative choices of independent variables
for the above representation are: ==+ = + = + = = + Where the
matrices (cE) and (S) represent the short-circuited stiffness and
clamped (i.e., {S} = {0}) dielectric permittivity, respectively.
The matrices (sD), (cD), (T) and (S) represent the open-circuited
(i.e., ({D} = {0}) elastic compliance and stiffness, free and
clamped dielectric impermittivity, respectively. The matrix (e) is
the piezoelectric stress constants. The element dij of (d)
represents the coupling between the electric field in the direction
i (if a poling occurred, its direction is taken as direction 3) and
the strain in the j direction. Sj = dijEi.1.3.2 Piezo fans The two
most remarkable characteristics of the piezoelectric fans are their
low noise levels and their low power consumption. These qualities
make the piezoelectric fans well-suited for applications in the
thermal management of portable electronic devices (see Figure 1.9).
Figure 1.9: A commercial bimorph piezo fan from [15].A
piezoelectric fan is fabricated by bonding a piezoelectric patch or
several patches to a shim stock. If only one patch is used, this
patch may be in any orientation on any side of the fan. However, in
a two- patch configuration (one on each side) the patches are
oriented such that when one expands the other contracts. A
two-patch configuration is shown in Figure 1.10. An alternating
voltage is applied to the piezoelectric patch in the fan. This
causes the patch to alternately expand and contract. As this
happens, the blade bonded to the piezoelectric patch flaps back and
forth like a Japanese fan, but much faster to create a fluid flow.
Figure 1.10: Set-up and basic principle of an operating piezo
fan.11 27. The applied alternating voltage is at the frequency of
the first resonance mode of the piezoelectric fan, driving the fan
in resonance and letting it move in the first bending mode.
Therefore the power consumption is minimized for the maximum tip
deflection. Since the piezoelectric fans are driven at resonance,
they are designed such that their fist mode of resonance will be
outside the audible range (2.5). The thickness ratio must be as
small as possible to obtain the maximal tip deflection. The EMCF
gets maximal for a certain length ratio (between 0.6 and 1) and a
thickness ratio (between 0.67 and 2.5). This optimal ratios are not
coincident with the optimal ratios for the tip deflection and
frequency. A compromise has to be made and one needs to determine
what quantity has the greatest priority in the application. A
greater oscillating frequency of the piezo fan creates a greater
flow when the same dimensions are used. Multiplication of the
resonance frequency with the dynamic tip amplitude gives an
important quantity ( ) that needs to be maximized as well. From the
obtained surf plots it can be seen that depending on the
configurationgets maximized for a certain length ratio (0.36-1) and
thickness ratio (0-1.17). The maximumfor the bimorph design is
about twice the size of the unimorph configuration.It was found for
the application of flapping wing actuation application, the best
thicknesses of the materials available were the 127m thick PZT-5H
and a stainless steel plate with a thickness greater than 110m.
This thickness combination can produce the largest value with f