ae_vos_20070811.

ae_vos_20070811.
Post-Buckled Precompressed elements: a new class
of control actuators for morphing wing UAVs
Roelof Vos
August 9, 2005
2
Summary
This report describes how Post-Buckled Precompressed (PBP) piezoelectric
bender actuators are employed in a deformable wing structure which changes
its camber and hence induces control. By applying axial compression to
these actuators, significantly higher deflections can be achieved. Classical
Laminate Plate Theory (CLPT) is shown to capture the behavior of the
unloaded elements. A deflection model, employing nonlinear structural relations is shown to predict the behavior of the PBP elements accurately.
Advantages of PBP actuators over conventional electro mechanical servo
actuators are a substantial reduction in slop, dead band, complexity, power
consumption and weight. Moreover, their corner frequency is an order of
magnitude higher.
Two concepts are investigated. The wing of the first concept is based
on the bending of a flat plate. The first 30% of the airfoil is shaped aerodynamically, whereas the remaining part is basically a flat plate with a total
thickness of 0.5 mm, which is able to deflect up and down. One wing consists of two individual controllable panels. The inboard panel is used for
direct lift control while the outboard panel induces roll control. This wing
is designed for a 300 gram MAV (span 0.5 m). Precompression of the piezoelectric actuators is induced by rubber bands at both sides of each panel.
A dynamic test article was fabricated, which showed an increase in end
rotation due to precompression of almost 100%, up to 14◦ peak to peak.
A static article of this thin wing was fabricated which did not employ the
piezoelectric actuators. This test article was used in the wind tunnel to
verify the lifting capability of the wing at an early stage. A lift curve slope
of Clδ = 3.80 [1/rad] was obtained and attached flow was present up to 15◦
angle of attack. Extensive wind tunnel tests showed good stall behavior of
the wing, and a maximum lift to drag ratio, (CL /CD )max = 5. Furthermore,
the lift as function of the shape deformation angle, δ, was measured to
be CLδ = 2.97 [1/rad], which translates to Clδ = 5.03 [1/rad] for a two
dimensional section of the wing.
The second wing design has a symmetric profile with finite thickness,
incorporating the PBP actuators on the aft 60% of the camber line. This
wing panel is used at the outboard parts of a wing for an existing UAV
(span 1.4 m) in order to induce roll control. Due to the elasticity of the skin
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4
the outer shape of the wing can be actively deformed. Moreover, the skin of
the wing induced the required amount of precompression to the piezoelectric
actuators.
For this thick wing, a proof-of-concept 100 mm wide wing section was
fabricated. The tension in the wing skin could be varied in order to adjust
precompression on the elements. Bench tests showed that with a wing chord
of 145 mm deflection levels were increased with more than a factor of 2, to
15 degrees peak to peak, at a maximum frequency of 34 Hz. A new wing,
employing the PBP morphing parts, was fabricated out of carbon fibre and
flight testing was carried out which showed excellent control characteristics.
The entire wing weight was reduced 9% with respect to the original wing,
which did employ any wing movables. Each morphing part of the wing
consisted of only 6 individual parts. Wind tunnel tests measured an average
CLδ = 1.42, which translates to Clδ = 4.16 for a two dimensional wing
section. The presence of wing loading caused a shift in deflection range at
higher absolute angles of attack.
To control the morphing wing panels (for both the thick wing and the
thin wing) an electronic circuit was designed. The PBP actuators demand a
maximum voltage difference of 100V. To increase the control voltage to this
value a linear amplifier was used. Battery voltage was increased with a direct
current converter. Due to problems with demodulating the control signal
and overheating of the amplifiers it was decided to separate the high voltage
circuit from the low voltage control circuit and to link them mechanically.
This led to a flight worthy electronic circuit which was suitable to control the
morphing wing panels employed on the UAV. Moreover, losses were reduced
and power consumption was limited to 0.1 W . However, the circuitry was
relatively heavy, big and the control frequency was low.
The research that is presented in this report has shown that a PBP actuated morphing wing is a feasible and flight worthy concept and possesses
significant benefits over conventional actuators. The PBP actuated morphing wing was demonstrated in benchtop experiments and its concept was
proven in wind tunnel tests and in flight.
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Nomenclature
Symbol
A
AR
B
b
C
c
Cd , CD
CDi
Cl , CL
Cm , CM
Cp
D
d
E
e
G
I
k
L
M
N
p
q
R
Re
S
SM
T
t
U
u
V
Z
Description
in-plane laminate stiffness matrix or aspect ratio
amplification ratio
coupled laminate stiffness matrix
span
capacitance
chord
2 and 3 dimensional drag coefficient, resp.
induced drag coefficient
2 and 3 dimensional lift coefficient, resp.
2 and 3 dimensional pitching moment coefficient,
resp.
pressure coefficient
bending laminate stiffness or drag force
piezoelectric constant or diameter
electric field or stiffness
Oswald factor
shear modulus of elasticity
2nd area moment of inertia
spring stiffness
lift force
applied moment vector
applied force vector
pressure
dynamic pressure
resistance
Reynolds number
wing surface area
static margin
trust force, temperature or thermal strain vector
thickness or time
internal energy
vertical displacement
velocity, potential energy, voltage or volume
Impedance
Units
N m−1 , −
N
m
F
m
-, -, -, N m, N
m V−1 , m
V m−1 , N m−2
N m−2
m4
N m−1
N
Nm
N
−2
Nm
N m−2
Ohm
m2
N, T, m, sec
N m m−1
m
−1
−1
m s , N mm , V, m3
Ohm
7
Nomenclature
Greek symbols
α
γ
δ
²
κ
Λ
λ
µ
Π
ρ
σ
τ
ωn
angle of attack or coefficient of thermal expansion
shear strain
PBP rotation angle
normal strain or downwash angle
curvature
virgin actuator strain or sweep angle
taper ratio
Poisson’s Ratio
potential energy
density
normal stress
shear stress
angular frequency
Subscripts
A
a
b
ac
cg
ex
f
h
l
s
t
w
aircraft
actuator or axial
bonding layer or buckling
aerodynamic center
center of gravity
external
fuselage
horizontal tail or hinge
laminate
substrate
thermal or tip
wing
Abbreviations
CLPT
CAP
CFD
GFRP
ISA
MAV
PBP
PZT
UAV
Classical Laminate Plate Theory
Conventionally Attached PZT
Computational Fluid Dynamics
Glass Fibre Reinforced Polymer
International Standard Atmosphere
Micro Aerial Vehicle
Post-Buckled Precompressed
Lead Titanate Zincronate
Unmanned Aerial Vehicle
deg, microstrain K−1
deg
-, deg
deg m−1
-, deg
N m m−1
kg m−3
N m−2
N m−2
s−1
8
Contents
Summary
3
Nomenclature
5
List of figures
15
1 Introduction
17
2 Background and Motivation
2.1 Piezoelectric Materials . . . . . . . . .
2.2 Morphing Wings . . . . . . . . . . . .
2.3 Micro Aerial Vehicle . . . . . . . . . .
2.4 Low Reynolds Number Aerodynamics
2.5 Synopsis . . . . . . . . . . . . . . . . .
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3 Post Buckled Precompressed Piezoelectric Actuator
3.1 Piezoelectric Actuator . . . . . . . . . . . . . . . . . .
3.2 Classical Laminated Plate Theory Model . . . . . . . .
3.3 Analysis of a PBP laminate . . . . . . . . . . . . . . .
3.4 Potential Energy Model . . . . . . . . . . . . . . . . .
3.5 PBP Experiments and Results . . . . . . . . . . . . .
3.6 Integration and Comparison . . . . . . . . . . . . . . .
3.7 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 Thin Wing Design, Fabrication and Testing
4.1 Application and Requirements of a Morphing Wing .
4.2 Airfoil Design . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Airfoil Geometry . . . . . . . . . . . . . . . .
4.2.2 Predicted Static Performance of Thin Airfoil
4.3 Wing Design . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Preliminary Design . . . . . . . . . . . . . . .
4.3.2 Lift Distribution . . . . . . . . . . . . . . . .
4.4 Fabrication . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Leading Edge Fabrication . . . . . . . . . . .
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CONTENTS
4.5
4.6
4.7
10
4.4.2 Actuator Fabrication and Integration .
Static Bench Testing . . . . . . . . . . . . . .
Wind Tunnel Tests . . . . . . . . . . . . . . .
4.6.1 Lift Curve Slope . . . . . . . . . . . .
4.6.2 Lift and Drag Measurements . . . . .
Synopsis . . . . . . . . . . . . . . . . . . . . .
5 Thick Wing Design, Fabrication and Testing
5.1 Aircraft Properties . . . . . . . . . . . . . . .
5.1.1 Aircraft Geometry . . . . . . . . . . .
5.1.2 Lift curve slope . . . . . . . . . . . . .
5.1.3 Static Margin . . . . . . . . . . . . . .
5.2 Airfoil Design . . . . . . . . . . . . . . . . . .
5.2.1 Airfoil Geometry . . . . . . . . . . . .
5.2.2 Predicted Static Performance of Thick
5.3 Wing Design . . . . . . . . . . . . . . . . . .
5.4 Proof of Concept . . . . . . . . . . . . . . . .
5.4.1 Skin of the Wing . . . . . . . . . . . .
5.4.2 Static Test Article . . . . . . . . . . .
5.4.3 Dynamic Test Article . . . . . . . . .
5.5 Wing Fabrication . . . . . . . . . . . . . . . .
5.5.1 Leading Edge Torque Box Fabrication
5.5.2 Static Wing Fabrication . . . . . . . .
5.5.3 Actuator Fabrication . . . . . . . . . .
5.5.4 Wing Assembly . . . . . . . . . . . . .
5.6 Flight Testing . . . . . . . . . . . . . . . . . .
5.7 Wind Tunnel Test . . . . . . . . . . . . . . .
5.8 Synopsis . . . . . . . . . . . . . . . . . . . . .
6 Electronics
6.1 Power Consumption . . . . . . . . . . .
6.2 Requirements on the Electronic Circuit
6.3 Design of the Electronic Circuit . . . . .
6.3.1 High Voltage Generation . . . . .
6.3.2 Amplifying circuit . . . . . . . .
6.3.3 Integration of Components . . .
6.4 Flight worthy electronics . . . . . . . . .
6.5 Synopsis . . . . . . . . . . . . . . . . . .
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Airfoil
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102
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7 Conclusions and Recommendations
103
7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . 104
11
CONTENTS
A Classical Laminate Plate Theory Model
A.1 Actuator Dimensions . . . . . . . . . . . . . . . . . .
A.2 Material Properties . . . . . . . . . . . . . . . . . . .
A.3 Stress-strain relations for plane stress in each lamina
A.4 Resultant Laminate Forces and Moments . . . . . .
A.5 Actuation . . . . . . . . . . . . . . . . . . . . . . . .
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107
. 107
. 108
. 109
. 110
. 112
B PBP Simulation
115
C Leading Edge Geometry
117
D Trailing Edge Analysis
119
CONTENTS
12
List of Figures
2.1
2.2
2.3
Piezoelectric materials and applications . . . . . . . . . . . .
Various types of wing morphing . . . . . . . . . . . . . . . . .
Schematic representation of the boundary layer. . . . . . . . .
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22
24
3.1
3.2
3.3
3.4
3.5
PZT actuator and atom structure. . . . . . . . . . . . . . . . 28
Geometry of an N-layered laminate.[12] . . . . . . . . . . . . 30
Rotation of the coordinate system. . . . . . . . . . . . . . . . 31
Pre-compression increases deflection [18]. . . . . . . . . . . . 34
Terms and conventions for analysis of the PBP actuator arrangement [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.6 Modeled relationship between δ0 and Fa at various field strengths. 37
3.7 Two applied forces on a cantilevered beam. . . . . . . . . . . 38
3.8 Vertical end force versus tip deflection . . . . . . . . . . . . . 39
3.9 Actuator and experimental setup [18]. . . . . . . . . . . . . . 40
3.10 Correlation of deflection angle, δ0 , applied voltage and axial
force, Fa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.11 Destructive experiment to determine maximum rotation. . . . 41
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
Geometry of the airfoil. . . . . . . . . . . . . . . . . .
Airfoil with highest Cl at α = 0. . . . . . . . . . . . .
Definition of the angle δ0 . . . . . . . . . . . . . . . . .
Maximum downward and upward deflection. . . . . . .
Lift coefficient as a function of δ and α. . . . . . . . .
Drag coefficient as a function of δ and α. . . . . . . . .
Moments coefficient as a function of δ and α. . . . . .
Compressive force by a rubber band. . . . . . . . . . .
3D image of the wing lay-out (all dimensions in mm).
Exploded view of the wing. . . . . . . . . . . . . . . .
The gaps cause additional vortices over the wing. . . .
The assumed spanwise lift distribution. . . . . . . . . .
Dimensions of the leading edge. . . . . . . . . . . . . .
Leading edge materials. . . . . . . . . . . . . . . . . .
Leading edge. . . . . . . . . . . . . . . . . . . . . . . .
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56
56
LIST OF FIGURES
4.16
4.17
4.18
4.19
4.20
4.21
4.22
Actuator lay-up on caul plate right before cure. . . . . . . . .
Total wing lay out. . . . . . . . . . . . . . . . . . . . . . . . .
Bench test experiment set-up and results . . . . . . . . . . . .
Schematic representation of forces on the wing. . . . . . . . .
Wind tunnel experiment set-up and results . . . . . . . . . .
Static deflections and wind tunnel setup. . . . . . . . . . . . .
Lift coefficient versus angle of attack at various deflection
angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.23 Lift over drag at various angles of attack. . . . . . . . . . . .
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
5.19
5.20
5.21
5.22
5.23
5.24
5.25
6.1
6.2
6.3
6.4
Radio Controlled UAV: Aerobird. . . . . . . . . . . . . . . . .
Change in airfoil camber due to PBP actuation. . . . . . . . .
Lift coefficient as a function of δ and α. . . . . . . . . . . . .
Drag coefficient as a function of δ and α. . . . . . . . . . . . .
Moment coefficient as a function of δ and α. . . . . . . . . . .
Topview of wing, including morphing outboard parts (all dimensions in mm). . . . . . . . . . . . . . . . . . . . . . . . . .
Thick wing design. . . . . . . . . . . . . . . . . . . . . . . . .
Model of the skin pre-compressing the adaptive laminate. . .
Experimental setup to determine the stiffness of the skin material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Specific force versus strain curves for various specimen. . . . .
Static test article . . . . . . . . . . . . . . . . . . . . . . . . .
Dynamic test article dimensions and features. . . . . . . . . .
Experimental test set-up. . . . . . . . . . . . . . . . . . . . .
First dynamic test article deflections. . . . . . . . . . . . . . .
Results quasi-static and dynamic experiments . . . . . . . . .
Torque box fabrication. . . . . . . . . . . . . . . . . . . . . .
Balsa wood wing. . . . . . . . . . . . . . . . . . . . . . . . . .
Bimorph actuator fabrication. . . . . . . . . . . . . . . . . . .
Adaptive wing assembly. . . . . . . . . . . . . . . . . . . . . .
Total assembled wing including required driver electronics. . .
PBP Wing mounted on UAV (topview). . . . . . . . . . . . .
PBP Wing mounted on UAV (sideview). . . . . . . . . . . . .
Flight testing. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wind tunnel test. . . . . . . . . . . . . . . . . . . . . . . . . .
End rotation of PBP versus lift coefficient. . . . . . . . . . . .
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81
83
83
84
85
85
86
86
88
89
90
The dc-dc converters increase the voltage. . . . . . . . . . . . 96
The electronic circuit should amplify the control signal. . . . 97
Electrical representation of a linear amplifier driving a piezoelectric actuator. . . . . . . . . . . . . . . . . . . . . . . . . . 98
Electrical representation of the autonomous circuit driving a
piezoelectric actuator. . . . . . . . . . . . . . . . . . . . . . . 100
15
LIST OF FIGURES
6.5
6.6
All panels of the wing are controlled by two amplifying circuits.101
Flight worthy electronics prior to integration into the aircraft. 102
A.1 Actuator dimensions. [18] . . . . . . . . . . . . . . . . . . . . 107
C.1 Definition of the leading edge geometry of the airfoil. . . . . . 117
D.1
D.2
D.3
D.4
Asymmetric loading of the adaptive laminate. . .
Pre-compression force on the trailing edge. . . . .
Model of trailing edge loading. . . . . . . . . . .
Fibre orientation in the trailing edge (top view).
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119
120
120
121
LIST OF FIGURES
16
Chapter 1
Introduction
For almost a century, piezoelectric materials have been used in various kinds
of applications, ranging from SONAR detectors to twist active helicopter
blades, to stereo tweeters. For more than fifteen years this unique branch
of adaptive materials has been successfully used to enhance aircraft flight
performance. Their relatively high mass specific energy and volume specific
energy make them suitable to aircraft application [4].
Traditional actuator schemes for piezoelectric actuators tend to trade stroke
at the expense of deflection. Recent studies at TU Delft and Auburn University have shown how the amount of deflection of piezoelectric actuators can
be increased without decreasing the applied force. This new class of flight
control actuators employ Post-Buckled Precompressed (PBP) piezoelectric
elements. These actuators have been recently patented by TU Delft [2]. In
the late 1990’s it was first demonstrated by Lesieutre that when using an
axial compression close to the buckling load of the actuator, the energy conversion efficiency becomes higher than the conversion efficiency of the raw
piezoelectric material itself [14].
In general, most aircraft classes have extremely tight weight, volume and
performance requirements. One of the most demanding types of aircraft are
the small Uninhabited Aerial Vehicles (UAVs), known as Micro Aerial Vehicles (MAVs). The missions envisioned for MAVs, such as urban surveillance,
require extreme agility. Although it has been shown that PBP actuators can
be used successfully to enable MAV control, they have not yet been used as
a structural part of either the wing or empennage to induce control [3]. It
has been shown that wing morphing (using conventional actuators) provides
excellent controllability and allows for rapid manoeuvring [16]. This report
presents how this new class of flight control actuators is integrated into a
flexible wing, allowing it to be deformed upon actuation.
Two individual wing morphing concepts are studied. Both concepts employ
PBP actuators to induce wing morphing and consequently control of the
17
18
aircraft. However, both concepts have slightly different applications. The
first concept contains a morphing wing designed for lift and roll control of
an MAV (span 0.5m). The second concept is the design of two outboard
morphing wing panels to provide roll control for a subscale UAV (span 1.4
m). An important similarity between both morphing wing concepts is the
requirements they impose on the electronic control circuit.
The body of this report is divided in five chapters. After this introduction,
background information on all the relevant topics is given in chapter 2. It
is explained why this new type of wing morphing is very suitable for small
aircraft with respect to larger aircraft. The basic principles of a Post-Buckled
Precompressed (PBP) piezoelectric actuator are made explicit in chapter 3.
Chapter 4 discusses the design, fabrication and testing of a thin morphing
wing for an MAV. In chapter 5 design, fabrication and (flight) testing of
a thick morphing wing for roll control on a small UAV is presented. The
fifth chapter is dedicated to the design and manufacturing of a flight worthy
electronic circuit for both morphing wing concepts. In the last chapter (7)
of this report conclusions and recommendations are presented.
Chapter 2
Background and Motivation
In this chapter an overview is given of the main areas of interest that concern the
investigation in so-called adaptive flight control. This includes an introduction in
adaptive materials in general, and piezoelectric materials in particular. Also the
concept of morphing wings is explained and why they should be used on Micro
Aerial Vehicles. The chapter closes with a section on low Reynolds number
aerodynamics.
2.1
Piezoelectric Materials
An adaptive material is defined to be a material which undergoes a change in
mechanical, thermal, optical, chemical, geometrical, electrical or magnetic
properties as a function of a given stimulus . The stimulus may be heat,
a magnetic field, or, in case of piezoelectric materials, an electric field. An
adaptive aerostructure is a structure which uses highly integrated, normally
load bearing, adaptive materials, to undergo a change in mechanical, thermal, optical, geometrical, electrical, or magnetical state as as function of a
given stimulus [6].
A piezoelectric material generates a mechanical motion or a change in stress
field when exposed to an electric field. Conversely they undergo a change
in electric charge state when exposed to a directional change in motion
or a stress field [6]. Consequently, they can be used as both actuator and
sensor. As a sensor, piezoelectric materials are not unfamiliar to the aircraft
industry. Using resonance techniques they are used in turbine engines to
detect small cracks in the turbine blades.
As an actuator the piezoelectric materials are less commonly used in (civil)
aircraft industry. Piezoelectric elements have been used on the F-18 to
alleviate buffet on the horizontal tail surface. The reason why active piezoelectric elements are not used as commonly as their passive counterparts
19
2.2. Morphing Wings
20
is twofold. Most importantly, they are relatively brittle and heavy. Using them as structural material is therefore not appealing. As part of an
actuator (so not integrated in the structure) is possible. Secondly, due to
the conservativeness of the aerospace world it is out of the question that
these materials are to be used as primary or secondary structure, in the
near future. Consequently, care must be taken when it comes to applying
these materials. It should always be kept in mind that using these materials
puts quite some constraints on the design. It must be verified that their use
actually contributes to a better performance, lower cost, or whatever it is
that needs to be optimized.
(a) Block cast Piezo ceramic [6]
(b) Piezoelectric sensor in
turbine blade [6]
(c) F-18 Active Aeroelastic
Wing [37]
Figure 2.1: Piezoelectric materials and applications
This does not mean that piezoelectric material does not have a relevant
application in modern aircraft industry. To see why this might be so, it is
important to understand the capabilities of this material. As established
above, active piezoelectric material shows strain when an electric field is
applied. In other words, when exposed to an electric field the material will
contract or expand, depending on the direction of the electric field. This has
to do with the fact that the material is poled in a specific direction (confer
section 3.1 for more detailed explanation). By altering the direction of the
electric field it is possible to make the material contract and expand very
rapidly. In fact, frequencies over 100 kHz are possible. This characteristic
makes it very suitable for applications which desire high speed actuation.
2.2
Morphing Wings
Wing morphing is actually quite an old concept in aircraft industry. The
Wright brothers used it in all their aircraft, including the Wright flyer II,
the first aircraft to actually fly. By pulling wires, which were attached to the
wing, the wing shape changed, inducing a change in pressure distribution
and consequently a change in lift. That way, they were able to steer the
aircraft [13].
21
2 Background and Motivation
That morphing is an efficient concept for control is proven by nature. Birds
all make use of their morphing wings to adapt to different flight conditions.
Those super morphing wings do not only change in twist and camber, but
also in aspect ratio, dihedral and sweep. Moreover, birds do not possess a
vertical tail surface for directional control. The morphing wings are used as
ailerons, flaps, slats and direct lift control. To be short, bird wings show
that there is still much to improve in aircraft wing design.
One step forward to resemble birds more closely is to apply morphing wings
to an aircraft. However, this is not easily done, since conventional aircraft
all have a rigid wing construction, which only have a limited amount of
flexibility. They must be able to carry the weight of the aircraft, when in
the air, and their own weight plus fuel, when on the ground. Morphing
wings would require a relatively flexible structure which is able to provide
sufficient strength to the wing on the one hand, but also give the wing
the ability to change its shape. This shape change can either be a change
in camber, thickness (t/c), aspect ratio, sweep, dihedral or twist. Each
of the afore mentioned characteristics typically induce weight, volume and
flexibility increments and penalties.
Wing morphing, to a certain extent, has been successful in the past and
the present. Increasing of chord and camber is the most regular way of
wing morphing in commercial and military aircraft. Flaps and slats are
deployed to adapt to the speed regimes occurring at, for instance, take off
and landing. This however, proves the wide interpretation that can be given
to the term ’morphing.’ The F-14 Tomcat is provided with a wing that can
change sweep angle (so-called ’swing wings’). It can therefore operate at
a larger speed range. Here the wing hinges around an hinge point and is
rotated backwards when the speed is increased. Above two examples both
contain a shape change of the wing. However, this shape change is in fact
due to the sliding or rotating of stiff elements of the wing. The structure
itself remains rigid. To be short, the current methods of wing morphing are
only an approximation of real structural morphing. Structural morphing is
possible when adaptive aerostructures are used.
Structural morphing for a 30 cm span aircraft was achieved by Lind at the
University of Florida. By pulling a string that runs between the wing tip
and the fuselage, the wing is deformed. Flight testing showed that wing
morphing can be an effective means to achieve roll control for this class of
small aircraft [16].
2.3
Micro Aerial Vehicle
During the past few years Unmanned Aerial Vehicles (UAVs) have become
a new major segment in aircraft industry. Especially defense projects are
2.3. Micro Aerial Vehicle
(a) Wright flyer [38]
22
(b) Eagle [39]
(c) Structural morphing of a 30
cm span aircraft [16]
Figure 2.2: Various types of wing morphing
more and more directed towards autonomous flight of robotic aircraft. Dangerous operations such as reconnaissance and post battle assessment can be
done by these aircraft without risking the life of pilots. On top of that are
the Uninhabited Combat Aerial Vehicles, or UCAVs, which can be used to
conduct pre-emptive and reactive suppression of enemy air defense missions
effectively and affordably.
The use of morphing wings on UAVs is a topic of interest for many research
institutions around the world (among others DARPA in their Morphing Aircraft Structure program [36]). Both structural morphing of wings by means
of adaptive materials, as shape change due to hinging or inflating parts of
the wing, is looked into. A lot of different configurations as well as material
applications are examined. On the one hand there is the investigation in
adaptive materials to induce camber changes or active twist in the wing.
On the other hand there is interest in extending, hinging, inflating and/or
folding wings. The applications of morphing wings are currently targeted to
UAVs because of their low wing loading and the ability to quickly perform
flight tests [27].
Since structural morphing is limited to camber variations, bending and active twist, this is where adaptive materials could be applied. An important
design parameter with respect to morphing wings on UAV aircraft is their
size. When piezoelectric materials are used to actuate the morphing wing,
there are some dimension related characteristics that should be considered
when designing a UAV. Currently, UAV dimensions range from several meters in span up to several centimeters [32]. In order to determine what size
would be appropriate for UAVs with morphing wings, actuated by piezoelectric materials, a comparison is made. Table 2.3 gives an overview of
size related characteristics for larger and smaller UAVs. The left column
considers a morphing wing on a micro aerial vehicle (span < 50 cm), the
right column on an ’ordinary’ UAV (span > 50 cm).
23
2 Background and Motivation
[!htp]
Atmospheric
instabilities
Agility
Manufacturing
MAV
Small moment of inertia: requires fast actuators to counteract this
Improves: small moment of inertia in combination with fast
actuators
Simple to make a proof-ofconcept model
Weight
Weight saving due to PZT elements will be considerable[3]
Volume
Relatively large volume reduction w.r.t. conventional actuators
Piezoelectric actuators will reduce power consumption substantially
Power
UAV
Larger moment of inertia, requires less fast actuators
Large moment of inertia prevents agility to improve
Limited size of available piezoelectric materials would yield
more complicated structure.
Weight saving due to PZT elements not ensured due to relatively complicated structure [9]
Relatively small volume reduction w.r.t. conventional actuators
Structural resistance could lead
to a higher power consumption.
Comparison MAV and UAV.
From Table 2.3 different conclusions on size could be drawn. An important
one is that the overall benefits from using PBP actuators are larger for the
MAV application than for the UAV. Weight and volume savings are more
substantial, power consumption will decrease and the aircraft becomes more
agile. From the MAV point of view, the capability of fast actuation is not so
much desired as it is an absolute necessity to operate. With ever decreasing
size of aircraft, they become more and more sensitive to wind gusts and
other atmospheric instabilities. Since their speeds are generally between
10 and 20 m/s, wind speed is not something trivial anymore [15]. To act
and react to these conditions the aircraft should possess fast and efficient
actuators. Since conventional actuators only go up to frequencies of 1 to 2
Hz, these will not be capable of coping with quick changes in flow field. The
piezoelectric actuators would be very suitable to perform changes in wing
geometry up to 25 Hz. Another reason to use the PZT laminates is the fact
that they can be integrated in membrane wings, which are commonly used
for MAV’s.
2.4
Low Reynolds Number Aerodynamics
To construct a wing for a UAV can be different from constructing a wing
for a MAV. The difference is caused by the difference in Reynolds number
between both. The Reynolds number is defined as in equation 2.1 [1].
2.4. Low Reynolds Number Aerodynamics
Re =
ρV c
µ
24
(2.1)
In equation (2.1) ρ is the density of the fluid. For air ρ = 1.225 [kg/m3 ],
at 0m International Standard Atmosphere (ISA). V is the velocity in [m/s]
and µ is the viscosity of the fluid in [N/s/m2 ]. At 0m ISA for air µ =
1.73·10−5 [N/s/m2 ]. Finally the c represents a characteristic length, usually
the chord of the wing in [m]. For the MAV configuration the speeds will
be around 15 m/s. Furthermore, a chord length of approximately 10 cm is
realistic. Substituting these numbers into equation 2.1 gives an approximate
Reynolds number of 1 · 105 .
Low Reynolds number aerodynamics is quite different from high Reynolds
number aerodynamics. Different flow problems arise. One of them is the
possible occurrence of laminar separation at the leading edge of the airfoil.
Due to high local curvature, the laminar boundary layer will not be able
to follow the contour of the airfoil. It will therefore separate. After separation, the boundary layer could become turbulent (more energized) and
could re-attach to the surface of the airfoil. This leaves the so-called laminar separation bubble, figure 2.3. The laminar separation bubble increases
drag considerably. In addition the lift is decreased. Re-attachment of the
flow only takes place if the boundary layer becomes turbulent, otherwise the
boundary layer will just separate, causing a large wake and consequently a
huge amount of drag. This form of leading edge separation destroys the lift
over the wing completely [1].
Figure 2.3: Schematic representation of the boundary layer.
Laminar separation bubbles are a common feature on airfoils at Reynolds
numbers between 50,000 and 200,000. They tend to be larger with decreasing
Reynolds number. They cause a substantial decrease in L/D. However, it
is possible to control the laminar separation bubble: artificial turbulators
can be placed on the leading edge to cause the boundary layer to trip. The
25
2 Background and Motivation
laminar flow in the boundary layer becomes turbulent and will have more
strength to follow the curvature of the airfoil. Determining the size and the
location of these turbulators is more an art than it is science [19].
Not only turbulators can energize the boundary layer, tip vortices also tend
to do this, causing the bubble to disappear near the tips of the wing. This
is a typical low Reynolds number phenomenon. The pressure at the top
side of the airfoil, near the tip of the wing, gets lower than at an average
spanwise location. This causes an increase in lift near the wing tips. This
is contradictory to what happens at high Reynolds numbers. In the latter
case the tip vortices tend to increase the pressure on the top side of the
wing. This deteriorates the lift at the tip. The increase in lift near the tip
is something the MAV can benefit from [22].
The low Reynolds number also puts constraints on the design of the airfoil. Thin airfoils (t < 0.05c) seem to give better results than thick airfoils.
Furthermore, the sharpness of the trailing edge, as well as turbulence does
not have much effect on the lift, drag and pitching moment. Also, slightly
cambered airfoil (around 4%) perform better than straight airfoils [19].
In combination with low Reynolds numbers, the aspect ratio (A) is an important feature with respect to aerodynamics of MAV’s. The aspect ratio
is defined to be the square of the span divided by the surface area of the
wing. The relatively low aspect ratios of MAV wings (down to A = 0.5)
cause again two special characteristics: a high stall-angle of attack, and a
non-linear CL /α curve. Both features are caused by the interference of the
tip vortices with the boundary layer on top of the wing [24][22].
2.5
Synopsis
This chapter has shown that piezoelectric materials are already being regularly applied in aerospace industry both as actuator and sensor. An introduction into ’morphing’ wings has shown that there is a difference between
structural deforming wings and wings that change shape due to sliding or
hinging components and that both types can induce aircraft control. Furthermore, it has been concluded that piezoelectric morphing wings could
best be applied to small UAVs rather than to larger ones. Finally, a qualitative aerodynamic analysis has shown that at low Reynolds number thin
airfoils with a sharp nose and a slight camber should perform best in terms
of lift and drag.
2.5. Synopsis
26
Chapter 3
Post Buckled Precompressed
Piezoelectric Actuator
In this chapter the theory behind the piezoelectric actuator and its capabilities
is explained in detail. First of all the principles of a piezoelectric actuator
are explained. In section 2 the actuator laminate is analyzed using Classical
Laminate Plate Theory. To be able to predict the deflection in the beam a
Newtonian model is presented in the third section. On top of that, a potential
energy model is laid out in section four, which can be used to calculate the
relation between tip force and deflection.
3.1
Piezoelectric Actuator
There are different kinds of piezoelectric materials. One commonly used is
the led-zincronate-titanium (PZT) ceramic. This material possesses a number of relevant characteristics which should be kept in mind when designing
a structure using PZT materials. High stress fields can de-pole the element
which means it becomes inactive. The same happens when the element
breaches its Curie temperature. Self heating can occur when the element
is exposed to high stress levels. Furthermore, the elements have low fracture toughness and a low ultimate tensile strength; they are very difficult to
handle in foil thicknesses because of their extreme brittleness. When cracks
occur in the surface of the PZT material, arcing can occur between the
electrically connected part and the disconnected part of the surface. And
finally, the performance of the elements tends to degrade with time after the
poling process (usually measured in decades). On the other hand, when the
element is poled again, it is as new.
The brittleness of the sheet material and the low ultimate tensile strength
make the manufacturing of PZT structures a laborious process. By attach27
3.1. Piezoelectric Actuator
28
ing the PZT sheets to a metal substrate, it is possible to make an active
bender element. If, during the curing of the adhesive, the laminate is heated,
the metal substrate will expand more than the PZT due to a mismatch in coefficient of thermal expansion. After the resin has set, the element will cool
again and compress the PZT which it is attached to. This means that the
PZT material is under constant compression. This internal precompression
makes the actuator much less brittle.
To avoid curvature in the laminate it is also possible to make a so-called
bimorph: a metal substrate with two PZT elements attached in the same
direction on either side (compare figure 3.1(a)). This class of actuators
are termed Conventionally Attached PZT (CAP) elements. The bimorph
structure is most simple to manufacture. If the sheets are attached correctly, applying an electric field results in straining the PZT sheets in equal
but opposite directions. This will induce an anti-symmetric motion of the
laminate about its axis of symmetry.
The maximum strain on a PZT-element is determined by two properties of
the material: the piezoelectric constants (dij ) and the electric field strength
at which the element de-poles. The piezoelectric constant dij is the ratio
between the induced strain in the j direction and the applied field in the
i direction, where i, j = 1, 2, 3. For example d31 is the ratio between the
electric field in direction 3 (the electrodes are perpendicular to axis 3) and
the induced strain or applied stress in direction 1. The three directions are
orthogonal and direction 3 is the poling direction of the element, confer figure
3.1(b). The field strength is given in V /mm and can go up to 600 [V /mm],
before the material is being de-poled.
(a) Example of a bimorph piezoelectric ac- (b) The Titanium atom is displaced during
tuator ??.
poling ??.
Figure 3.1: PZT actuator and atom structure.
Because the field levels exceed 200 [V /mm], the assumption that the thickness of the PZT sheet is independent of the electric field may not me used.
29
3 Post Buckled Precompressed Piezoelectric Actuator
The field strength is dependent on the thickness of the actuator (eqn. (5)
in [18]):
ta
(3.1)
Ej =
Vj
Equation 3.2 gives the relation between the free strain and the applied field
strength, E (eqn. (6) in [18]).
Λi = dij Ej
(3.2)
PZT 5A is a material with a strain/applied field ratio, d31 = d32 = −171 ·
10−9 [mm/V ], d33 = 374 · 10−9 [mm/V ]. This material is very suitable as
actuation element in a bimorph CAP laminate. From a practical point of
view the d31 coefficient is most interesting. It is relatively simple to apply
a sufficient electric field over a laminate in thickness direction, because it
is usually very thin. Expansion/contraction will occur perpendicular to the
field direction.
3.2
Classical Laminated Plate Theory Model
To model the behavior of a free piezoelectric laminate Classical Laminated
Plate Theory (CLPT) is used. The analysis presented in this section follows
the assumptions laid out in Jones [12]. Appendix A gives a more detailed
overview of the CLPT analysis applied to a piezoelectric actuator. In this
section the theory is laid out in a more general way. The resultant forces
and moments in the laminate are obtained by integrating the stress over the
thickness of the laminate (eq. (4.14) in [12]).
Z t/2
Z t/2
Nx =
σx dz
Mx =
σx zdz
(3.3)
−t/2
−t/2
For forces and moments in all directions and for N number of laminae,
equation (3.3) expands to the following expression (eqns. (4.15) and (4.16)
in [12]).






Z t/2
N Z zk
σx
σx
Nx
X
 σy  dz
 σy  dz =
 Ny  =
(3.4)
z
−t/2
k−1
k=1
τxy k
τxy
Nxy






Z t/2
N Z zk
σx
σx
Mx
X
 σy  zdz
 σy  zdz =
 My  =
(3.5)
z
−t/2
k−1
k=1
τxy k
τxy
Mxy
The thicknesses in equations 3.4 and 3.5 refer to figure 3.2.
The PZT sheets that are used in the laminate is so-called orthotropic material. An orthotropic material possesses symmetry in stiffness properties with
3.2. Classical Laminated Plate Theory Model
30
Figure 3.2: Geometry of an N-layered laminate.[12]
respect to two or more directions perpendicular to another. For orthotropic
materials the relation between stress and strain can be written as follows
(eqn. (2.59) in [12]):

 


σ1
Q11 Q12
0
²1
 σ2  =  Q12 Q22
0   ²1 
τ12
0
0 Q66
γ12
(3.6)
Where the matrix components, Qij , are a function of the stiffness of the
material (Ei ) and Poisson’s ratio (µij ) (eqn. (2.61) in [12]):
E1
1 − µ12 µ21
µ12 E2
=
1 − µ12 µ21
E2
=
1 − µ12 µ21
= G12
Q11 =
(3.7)
Q12
(3.8)
Q22
Q66
(3.9)
(3.10)
The above stiffness components can be transformed to a rotated coordinate
system, xyz. This coordinate system is rotated over an angle θ, using the
3-axis as rotational axis (cf. figure 3.3). The rotated stiffness components
then become (eqn. (2.79) and (2.80) in [12])

 


Q11 Q12 Q16
σx
²x
 σy  =  Q12 Q22 Q26   ²y 
τxy
γxy
Q16 Q26 Q66
(3.11)
31
3 Post Buckled Precompressed Piezoelectric Actuator
Figure 3.3: Rotation of the coordinate system.
Where:
Q11 = Q11 cos4 θ + 2(Q12 + 2Q66 )sin2 θcos2 θ + Q22 sin4 θ
2
2
4
(3.12)
4
Q12 = (Q11 + Q22 − 4Q66 )sin θcos θ + Q12 (sin θ + cos θ)
4
2
2
(3.13)
4
Q22 = Q11 sin θ + 2(Q12 + 2Q66 )sin θcos θ + Q22 cos θ
3
(3.14)
3
Q16 = (Q11 − Q12 − 2Q66 )sinθcos θ + (Q12 − Q22 + 2Q66 )sin θcosθ
(3.15)
Q26 = (Q11 − Q12 − 2Q66 )sin3 θcosθ + (Q12 − Q22 + 2Q66 )sinθcos3 θ
(3.16)
Q66 = (Q11 + Q22 − 2Q12 − 2Q66 )sin2 θcos2 θ + Q66 (sin4 θ + cos4 θ) (3.17)
The stress strain relations in an arbitrary coordinate system (equation 3.11)
are useful for determining the stiffness of the entire laminate. Following
Jones the strain in the laminate can be split up into vectors: an in plane
and an out of plane strain vector. The first of these vectors gives the middle
surface strain of the laminate and the second vector gives the middle surface
curvature (eqn. (4.13) in [12]).


  0 
²x
κx
²x
 ²y  =  ²0y  + z  κy 
0
κxy
γxy
γxy

(3.18)
The results of equations 3.11 and 3.18 can now be substituted in equations
3.4 and 3.5. This gives the following expression for the force and moment
3.2. Classical Laminated Plate Theory Model
32
vectors (eqns. (4.17) and (4.18) in [12]):



 (
 0 


)
Z zk
Z zk
²x
Q11 Q12 Q16
N
Nx
κx
X
 Q12 Q22 Q26 
 Ny  =
 ²0y  dz +
 κy  zdz
0
z
z
k−1
k−1
k=1
Nxy
κxy
γxy
Q16 Q26 Q66 k
(3.19)




 (

 0 
)
Z zk
Z zk
²x
Q11 Q12 Q16
N
Mx
κx
X
 Q12 Q22 Q26 
 My  =
 κy  z 2 dz
 ²0y  zdz +
0
z
z
k−1
k−1
k=1
Mxy
κxy
γxy
Q16 Q26 Q66 k
(3.20)
Keeping in mind that ²0i and κi are middle surface strains and therefore
independent of z, these equations can be written as follows (eqns. (4.19) to
(4.21) in [12]):

 
 0  


²x
Nx
A11 A12 A16
B11 B12 B16
κx
 Ny  =  A12 A22 A26   ²0y  +  B12 B22 B26   κy 
0
Nxy
A16 A26 A66
γxy
B16 B26 B66
κxy
(3.21)

 
 0  


²x
Mx
B11 B12 B16
D11 D12 D16
κx
 My  =  B12 B22 B26   ²0y  +  D12 D22 D26   κy 
0
Mxy
B16 B26 B66
γxy
D16 D26 D66
κxy
(3.22)
Where:
Aij
Bij
Dij
n
n
n
=
=
=
N
X
(Qij )k (zk − zk−1 )δkn
k=1
N
X
1
2
1
3
2
(Qij )k (zk2 − zk−1
)δkn
k=1
N
X
3
(Qij )k (zk3 − zk−1
)δkn
(3.23)
(3.24)
(3.25)
k=1
In the above equations δkn is the Kronecker delta where δ = 1 for k = n and
δ = 0 for k 6= n. The subscript n denotes the A,B or D-matrix for the n’th
layer in the laminate. The laminate consists of five layers: two actuator
layers (k, n = 1 and k, n = 5), two bonding layers (k, n = 2 and k, n = 4)
and the substrate (k, n = 3). Adding the ABD-matrices for the individual
layers in the laminate (with subscript l) results in the A,B and D-matrices
for the entire laminate:
33
3 Post Buckled Precompressed Piezoelectric Actuator
Aij l =
Bij l =
Dij l =
N
X
n=1
N
X
n=1
N
X
Aij
n
(3.26)
Bij
n
(3.27)
Dij
n
(3.28)
n=1
With these components equation (3.2) can also be written as:
µ
N
M
¶
·
¸ µ
A B
B D
=
l
¶
²0
κ
(3.29)
l
The forces and moments in equation (3.29) can be sub-categorized in actuator in-plane forces and moments (a), external forces and moments (ex) and
forces and moments due to a mismatch in coefficients of thermal expansion
(t) [3].
µ
N
M
¶
µ
+
a
N
M
¶
µ
+
ex
N
M
¶
·
=
t
A B
B D
¸ µ
l
²0
κ
¶
(3.30)
l
If external forces and moments are ignored, that is the element is free to
move, and when thermally induced stresses are not considered, this equation
can be reduced to (eqn. 4 in [18]):
·
A B
B D
¸ µ
a
Λ
0
¶
·
=
a
A B
B D
¸ µ
l
²0
κ
¶
(3.31)
l
Λ represents the virgin strain of the actuator, which is a function of the
electric field, Ej , and the actuator thickness, dij , as in equation (3.2). The
ABD-matrix for the two actuators in the laminate carry the subscript (a).
This matrix is a superposition of the two matrices which result from inserting n = 1 and n = 5 in equations (3.23) to (3.25). Equation (3.31)
shows how the in-plane strain and the curvature of the laminate (l) are
related to the free strain of the actuator (a). The curvature is of main importance. The laminate that is used in this actuator consists of five layers:
[CAP(+Λ)/bond/isotropic sustrate/bond/CAP(−Λ)]. Since this laminate
is symmetric in both material as geometrical properties, with respect to the
midplane of the laminate, all the coupling stiffnesses, (Bij )l , become zero
(sec. 4.3.2 in [12]). Applying this to equation (3.31) the curvature of the
3.3. Analysis of a PBP laminate
34
laminate can be directly coupled to the free strain in the actuators (eqn. (4)
in [3]):
Ba
κ=
Λ
(3.32)
Dl
For the bending actuator κ11 represents the curvature w.r.t. the 1-axis, and
can be deduced from equation (3.32). It is assumed that the two piezoelectric
actuators are attached at either side of an isotropic substrate with a bond of
finite thickness and low stiffness. Lateral contraction is therefore neglected.
When all the appropriate substitutions are made, θs = θa = 0, the expression
for κ11 is as in equation (3.33), in which subscript s denotes substrate and
b stands for bonding layer (eqn. (4) in [18]).
κ11 = ³
3.3
´
3
Es ts
12
E (t t + 2t t + t2a )
nha s a 2 ib a
o Λ1
b)
2 + 2 t3
+ Ea ta (ts +2t
+
(t
+
2t
)t
s
b
a
a
2
3
(3.33)
Analysis of a PBP laminate
The previous section showed how in a free pzt-laminate the free strain of
each of the actuator elements is related to the curvature of the laminate. In
this section it is shown that the curvatures in the laminate can be increased
considerably when a compressive force is applied at the center line of the
laminate. This type of actuator is termed a Post Buckled Precompressed
(PBP) piezoelectric actuator and the principle that it is based upon is shown
in figure 3.4.
(a) Without compressive
force.
(b) With
force.
compressive
Figure 3.4: Pre-compression increases deflection [18].
The following analysis of the PBP actuator is laid out in [3]. All the formulae
appearing in the following analysis are taken from this source. Referring to
figure 3.5, it is assumed that the element is only loaded in pure bending and
35
3 Post Buckled Precompressed Piezoelectric Actuator
that the rotations are moderate. The normal strain in the laminate at any
distance y from the neutral axis through its thickness is expresses as:
²=
ydδ
σ
=
ds
E
(3.34)
Figure 3.5: Terms and conventions for analysis of the PBP actuator arrangement [3].
For a beam element in pure bending the following holds:
σ=
My
I
(3.35)
Combining equations (3.34) and (3.35) and inserting CLPT conventions and
terminology the following can be obtained:
My
ydδ
=
ds
Db
(3.36)
The moment that is externally applied comes from the compressive force,
Fa :
M = −Fa y
(3.37)
Substituting equation (3.37) in equation (3.36) yields:
dδ
Fa y
=−
ds
Db
(3.38)
When (3.38) is differentiated with respect to s the following expression is
obtained:
d2 δ
Fa
=−
sin δ
(3.39)
2
ds
Db
Multiplying this result with an integrating factor 2 dδ
ds yields:
2
dδ d2 δ
Fa
dδ
= −2
sin δ
ds ds2
Db
ds
(3.40)
3.3. Analysis of a PBP laminate
36
Integrating equation (3.40):
³ dδ ´2
ds
=2
Fa
cos δ + a
Db
(3.41)
To determine the integration constant, a, it is considered that the applied
moment via the piezoelectric elements generates an imperfection across the
beam. It is then given that at x = 0:
dδ
=κ
ds
δ = δ0
Substituting these expressions yields for the integration constant, a = κ2 ,
and accordingly equation (3.41) results in:
³ dδ ´2
ds
=2
Fa
(cos δ − cos δ0 ) + κ2
Db
(3.42)
Using the appropriate trigonometric substitutions and considering the negative root because dδ is always negative:
s
r
³ ´
³ ´ κ2 Db
dδ
Fa
2 δ0
2 δ
= −2
sin
(3.43)
− sin
+
ds
Db
2
2
4Fa
To solve this equation a change of variables is imposed:
sin
³δ ´
2
= c sin ξ
(3.44)
Where ξ is a variable with the value π/2 when x = 0 and the value of 0 at
x = L2 . Consequently at x = 0:
c = sin
³δ ´
0
(3.45)
2
Solving for δ and differentiating with respect to ξ yields:
¡ δ0 ¢
´
³
³δ ´
2
sin
0
2 cos ξ
sin ξ
dδ = q
δ = 2 sin−1 sin
ξ
¡ ¢
2
1 − sin2 δ0 sin2 ξ
(3.46)
2
When equations (3.43) to (3.46) are combined, this leads to the following
result:
r
r
Z L
2
Fa
L Fa
=
ds =
Db 0
2 Db
¡ ¢
Z π
2
sin δ20 cos ξ
=
³q
´³q
¡ ¢
¡ ¢ 2 ´ dξ (3.47)
2
2 δ0
0
sin2 δ20 cos2 ξ + κ4FDb
1
−
sin
2 sin ξ
a
37
3 Post Buckled Precompressed Piezoelectric Actuator
Equation (3.47) does not have a closed form solution but it can be solved
numerically. For a given precompressive force, Fa , both the left and the
right side are solved. Depending on the difference between each of these
solutions δ0 is increased or decreased. After this alteration the routine is
run again and eventually it will converge to the point where both sides of
the equation are equal. This is the value of δ0 that belongs to the specific
value of Fa . Appendix B presents the Matlab code that was written to
solve this equation. Figure 3.6 shows a result of applying this model at
various potential levels to a piezoelectric element with dimensions as shown
in figure A.1. In this model it is assumed that the total length (92 mm) of
the laminate is covered by the PZT actuators.
Figure 3.6:
strengths.
3.4
Modeled relationship between δ0 and Fa at various field
Potential Energy Model
In section 3.3 the deformation of a piezoelectric laminate is shown as a
function of the applied voltage. However, when the laminate is not able
to attain this strain state due to counteracting forces or moments, part of
the strain is exchanged for applied stress. This section will show how the
amount of applied stress and the amount of strain are related. To obtain this
relation a so-called semi-analytical approach is followed, after De Breuker
(section 4.1.4 in [8]).
3.4. Potential Energy Model
38
To describe the state of the laminate the theory of minimum potential energy
is used. Potential energy consists of internal (U ) and external (V ) energy
(eqn. (4.37) in [8]):
Π=U +V
(3.48)
Where the internal energy is defined as (eqn. 4.37 in [8]):
Z
U=
σd²
(3.49)
V
For a symmetric beam that is bent and actuated by the piezoelectric elements, the internal energy expands to:
U=
1
2
Z
Z
L
(N ² + M κ)dx −
0
0
L
Ba Λκdx
(3.50)
External energy is generated by any force or moment that causes a deflection
or rotation in the beam. Using the PBP actuators in a morphing wing means
that they will be be used in a cantilevered configuration. In the previous
sections the results for the simply supported adaptive beam are directly
applicable to the cantilevered beam. Since this section is also concerned
with resistive forces and moments (as opposed to the previous sections), the
cantilevered beam will behave differently from the simply supported one.
A cantilevered beam is considered, being precompressed by a force Fa , on
which a tip force Ft is exerted (compare figure 3.7). For this case the external
energy is expressed as:
p
(3.51)
V = −Fa ( (L + u)2 + w2 − L) + Ft w
Figure 3.7: Two applied forces on a cantilevered beam.
For a structure to be in equilibrium the virtual work done on the structure
equals zero. Assuming that Fa and Ft are both conservative forces, the
virtual work is equivalent to the first variation of the potential energy. Using
Hamilton’s principle, the first variation of the potential energy is zero:
δΠ = 0
(3.52)
39
3 Post Buckled Precompressed Piezoelectric Actuator
This equation is solved numerically. One of the results is a relation between
the tip force, Ft , the precompressive force, Fa , and the free strain of the
actuator, Λ. Since Λ is linearly related to the voltage, via equation 3.2,
it is now possible to show a relationship between tip force, Ft and the tip
deflection, w, at a constant voltage level. This relationship is shown in
figure 3.8. These figures give an overview of different force deflection-curves
at a constant voltage of 100V (3.8-a), and constant precompression force of
0.9Fb (3.8-b). Where Fb is the buckling load of the actuator.. This data is
obtained for a 15 mm wide specimen with a length of 72.4 mm (standard
length of PZT sheet) [8].
(a) At constant voltage of 100V .
(b) At constant axial force of 0.9 Fb .
Figure 3.8: Vertical end force versus tip deflection
3.5
PBP Experiments and Results
To show that the modeling of the PBP actuator is accurate, the results
shown in figure 3.6 are to be compared to practical measurements. These
measurements were taken in the spring of 2004 and are laid out in [18].
Figure 3.9 shows the wired test actuator and the experimental setup that
was used to determine the precompression force, Fa , and the end rotation,
δ0 . To measure Fa , the actuator is placed on a scale. By screwing down the
top bar, this force can be adjusted. A laser beam is used to magnify the end
rotation so it can be read accurately from a screen (not displayed). A more
detailed description of the test setup can be found in [18]. The dimensions
of the test specimen are displayed in figure A.1 in appendix A.
Using this experimental setup a series of experiments were carried out. Eight
different specimen were tested successfully. The result of one of the specimen
is displayed in figure 3.10. The experimental values in this graph are close to
3.5. PBP Experiments and Results
40
Figure 3.9: Actuator and experimental setup [18].
the predicted values, indicating that the model as laid out in section 3.3 gives
a good representation of the behavior of a PBP piezoelectric actuator. The
discrepancies between the two plots are the result of the assumption that the
total laminate length is covered by the PZT laminate, while in reality the
end parts are glass fibre reinforced epoxy. Figure 3.10 shows that curvatures
can be magnified up to a factor of four, when precompression is applied.
Furthermore, this figure shows that the buckling load for this specimen is
found between 550 and 600 [gmf ], which translates to 55 − 60 [gmf /mm].
Figure 3.10: Correlation of deflection angle, δ0 , applied voltage and axial
force, Fa .
41
3 Post Buckled Precompressed Piezoelectric Actuator
Maximum Curvature To find out at what curvatures the specimen fails
a destructive experiment is carried out. This is done by means of a three
point pressure test. A simply supported specimen (length= 40mm, width=
12 mm, thickness= 0.66 mm, cure temperature= 176◦ C) is being deflected
by a point force, acting at the center. Figure 3.11 shows the destructive test
set-up. The load is introduced using a small vise. A laser beam is reflected
by a mirror at the end of the specimen and projects onto the reflection board
from which the rotation can be determined. To determine when a crack is
formed the resistivity over the tensioned face is measured by a voltmeter.
When the resistivity suddenly increases orders of magnitude, this would
indicate the element has cracked.
(a) Resistivity
ment
measure-
(b) Specimen in vise
(c) Total set-up with laser
Figure 3.11: Destructive experiment to determine maximum rotation.
The test shows an increase in tip deflection of 5.4 degrees. Since this is a
simply supported set-up and only small rotations are present, the curvature
of the specimen is prescribed by:
κ=
2δ0
l
(3.53)
In equation 3.53 l denotes the length of the specimen and δ0 denotes the deflection angle as presented in Figure 3.5. The length of the specimen amounts
to 40 mm. Substituting both values into equation 3.53 gives a maximum
curvature of 0.27◦ /mm. A 72 mm long simply supported specimen should
be able to handle a peak to peak deflection of 19.4◦ . Translating this to a
clamped specimen, would give a maximum peak to peak tip deflection of
38.8◦ .
Note that the values are only applicable for actuator laminates that are cured
at 176◦ C. Lower temperatures will induce less internal precompression of the
3.6. Integration and Comparison
42
PZT sheets. Accordingly, the sheets will break at lower curvatures. To make
a robust actuator it is therefore important to cure at high temperatures.
3.6
Integration and Comparison
Significant benefits are obtained by switching from conventional electromechanical servo actuators to PBP actuators. They do not employ any linkages, gears, or heavy motors, and are therefore significantly lighter. Since
PBP actuators operate under a high voltage but very low current, power
consumption is decreased dramatically [7]. This in turn leads to a reduction
in battery capacity and consequently battery weight. Section 6.1 will go
more into detail about power consumption. Contradictory to conventional
servo actuators, the PBP actuator is solid state so part count, slop and
deadband are one to two orders of magnitude lower [3]. Moreover, using a
PBP actuated morphing wing can increase the actuation frequency by an
order of magnitude, with excellent control authority. Table 3.1 shows how
PBP actuators compare to conventional electromechanical servo actuators
used for identical actuation purposes [3].
Max Power
Max Current
Mass
Slop
Corner Frequency
Part Count
Conventional Servoactuator
24W
5A
108 g
1.6◦
3 Hz
56
PBP Actuator
100mW
1.4mA
14 g
0.02◦
34 Hz
6
Table 3.1: Comparison of electromechanical servo actuator and PBP actuator [3].
An additional benefit of these type of solid state actuators is the fact that the
actuator can form an integral part of the structure. Therefore, the weight of
the actuator is not added to the structure, but could already be perceived
as part of the structural weight. By introducing axial loads in a simple way,
complexity can be greatly reduced. The following two chapters will go more
into detail about how a PBP piezoelectric actuator can be integrated into
the wing of an aircraft and be used to induce control.
3.7
Synopsis
This chapter has presented the basic principles and model of a post-buckled
precompressed (PBP) piezoelectric bending actuator. It has been shown
43
3 Post Buckled Precompressed Piezoelectric Actuator
how piezoelectric strain is related to the atomic structure of the material and
and how a piezoelectric bender actuator is actuated by applying a voltage
difference over the individual piezoelectric elements. Classical Laminated
Plate Theory has shown to capture the behavior of the unloaded bender
element really well. A Newtonian model has been developed to predict
curvatures of the axially loaded PBP actuator. Experiments showed good
a correlation with this model. It has been shown that by applying an axial
load, curvatures can be magnified up to a factor of 4. A comparison between
a conventional servo actuator and a PBP actuator has shown that power
consumption, slop, dead band, part count and weight are all substantially
lower for the PBP actuator, while at the same time the corner frequency is
increased.
3.7. Synopsis
44
Chapter 4
Thin Wing Design,
Fabrication and Testing
This chapter will show how a PBP actuator can be integrated in a wing and be
used to change its camber and consequently control the aircraft. This chapter
starts off with a set of requirements that are imposed on the design. The
following section will give an overview of all the important features of the wing,
their function and their dimensions. Also, the performance characteristics of the
airfoil are investigated. The design and fabrication of the wing are presented
in the two subsequent sections and this chapter ends with a section on bench
testing and a section on wind tunnel testing.
4.1
Application and Requirements of a Morphing
Wing
Integrating PBP piezoelectric actuators into a morphing wing configuration
has not been done before. The main goal of this project is to show that a
morphing wing can induce control successfully. Applications of a (partly)
morphing wing can be various. Deforming wing parts could for instance
replace conventional ailerons or flaps for small UAVs. They could be applied
on the entire wing or just on a small part. In this chapter it is assumed that
a morphing wing is to be used on a subscale UAV (span 0.5 m) in order to
supply it with roll control and direct lift control.
The requirements on the aircraft mission are partly drawn from an annual
MAV contest that is held in the US. The flight time of the aircraft must
be 10 minutes, its ceiling 100 meters and an average cruise velocity of 15
m/s. Furthermore it must be capable of carrying a small camera that takes
pictures at regular intervals. The stall speed may not be lower than 7 m/s,
45
4.2. Airfoil Design
46
which makes it possible to throw the aircraft manually to have it take off.
The maximum take off weight amounts to 300 grams [31].
From these requirements it can be easily established that each of the wings
must supply the aircraft with approximately 1.5 N of gross lift (without
being actively deformed). Since no specific aircraft is yet available to fly
with this new wing, no performance calculations of the aircraft as a whole
can be made.
4.2
4.2.1
Airfoil Design
Airfoil Geometry
A small deformable wing with a static gross lifting capability of 1.5 N is to
be designed. To supply the wing with both lift control and roll control, it is
decided to divide each of the wings into two individual panels. The inboard
panels for lift control and the outboard panels for roll control.
From section 2.4 it is known that a low Reynolds number airfoil should be
thin (t < 0.05c) and that a slight cambering is beneficial (approximately
4%) The leading edge of the airfoil should be relatively sharp, while the
trailing edge geometry does not significantly alter the pressure distribution
[19]. Furthermore curvature on the leading edge may not be too large with
respect to the size of the laminar separation bubble (confer section 1.4).
What makes this airfoil design different from all the other airfoil designs,
is of course the bimorph piezoelectric element (cf. figure 3.9 (a)). From
the trade study between the UAV and the MAV (section 2.3) it has become
clear that using the bimorph to directly influence the flow is most efficient.
No losses in structural deformation of a thick airfoil occur. The camber line
virtually forms both top and bottom face of the profile. This is the most
important constraint on the geometry. Furthermore, a bimorph, uncharged,
is basically a flat plate.
Combining the desired shape with the constraints of the bimorph laminate,
yields an airfoil with an aerodynamically shaped leading edge and a flat
plate behind it. How the aerodynamic shape should look like is part of an
aerodynamic study, which is shown later in this section. For now, figure 4.1
gives a generic view of how the airfoil should look like, when the bimorph is
uncharged. This airfoil will be referenced to as the ”thin airfoil”.
As shown in figure 4.1 the geometry of the airfoil is fixed for 70 to 80% of the
airfoil. This part is just a flat plate (in uncharged condition). The remaining
part (the leading) edge has to be shaped in such a way that it gives an as high
as possible lift coefficient. To calculate this analytically is a laborious job,
so the panel code Xfoil is used. This panel code is based on potential flow
47
4 Thin Wing Design, Fabrication and Testing
Figure 4.1: Geometry of the airfoil.
in combination with the Kutta condition but is able to incorporate viscous
effects. The boundary layers and wake are described with a two-equation
lagged dissipation integral boundary layer formulation and an envelope en
transition criterion [25]. Xfoil is able to calculate pressure distributions
over airfoil sections, determine the aerodynamic coefficients, and give an
impression of boundary layer development over the airfoil. It incorporates
viscous effects in the boundary layer and also includes the Reynolds number.
On the other hand, it is not able to incorporate three dimensional effects like
tip vortices. It must also be noted that with Reynolds numbers lower than
200.000 Xfoil results must be interpreted as an indication rather than an
exact simulation of reality. Wind tunnel tests are the only way to determine
the true characteristics of a low Reynolds number wing.
In appendix C it is explained how the basic geometry of the leading edge is
defined by using simple geometric shapes. By using the appropriate boundary conditions (Reynolds number of 105 and a velocity of 15 m/s) it is then
possible to alter the shape within certain physical constraints. This way numerous of different shapes are examined in Xfoil. The airfoil is examined in
uncharged condition, meaning that 70-80% is flat. The leading edge geometry that produces the highest lift coefficient at zero angle of attack, is the
one that is most suited. Excessive values of the drag coefficient would also
indicate that a particular shape is not desirable, however the lift coefficient
in cruise condition is dominant.
A number of iterations of the shape finally indicated that the shape presented
in figure 4.2 performed best. A lift coefficient of Cl = 0.25 is the largest
value, obtained at zero angle of attack. From figure 4.2 it can be seen that
this airfoil has a rather small nose radius and is therefore fairly sharp. On
the other hand, the top side of the leading edge has little curvature because
it is quite stretched. The flow will therefore remain attached more easily,
also at larger angles of attack.
4.2. Airfoil Design
48
Figure 4.2: Airfoil with highest Cl at α = 0.
4.2.2
Predicted Static Performance of Thin Airfoil
This section will investigate the predicted performance of the airfoil in terms
of lift, drag and moment coefficient. Besides the normal effect of the angle
of attack on these coefficients, the influence of morphing is of particular
interest. To express the amount of morphing the trailing edge deflection is
measured. Figure 4.3 shows how the deflection of the tip is converted to
the angle δ. Angle δ corresponds with δ0 in figure 3.5. The graphs in this
section are produced using Xfoil.
Figure 4.3: Definition of the angle δ0 .
Opposite to the simple supported beam in section 3.3, in this case the bimorph is clamped to one side at the leading edge. As shown in section 3.3,
δ0 is determined by the amount of curvature κ. By changing the curvature
the airfoil shape is changed subsequently. In this section delta0 is shortened
with δ, for convenience. It is assumed that the maximum value of δ amounts
to +/-6◦ . The two most pronounced shape deformations are displayed in
figure 4.4.
An angle of attack sweep, ranging from 0 to 15 degrees is carried out. Using
Matlab, the curvature is altered for each airfoil configuration. Between
maximum and minimum curvature seven different deflections are examined
with a step size of 2 degrees. The results of all the calculations are presented
in figures 4.5 to 4.7.
Figure 4.5 shows the change in lift coefficient with the deflection of the
49
4 Thin Wing Design, Fabrication and Testing
Figure 4.4: Maximum downward and upward deflection.
airfoil. As can be seen from the graph, Cl varies linearly with the deflection.
At zero angle of attack Clδ = 7.95 [1/rad]. For higher angles of attack Clδ
only declines a little to 6.58 [1/rad] at α = 8◦ . Finally, note that some of
the points are not lined up with the rest. This indicates that Xfoil is not
capable to produce a closed solution at that specific condition.
(a) Predicted Cl vs. δ
(b) Predicted Cl vs. α
Figure 4.5: Lift coefficient as a function of δ and α.
Contradictory to the lift coefficient, the drag coefficient does not show a
linear behavior with the tip deflection (compare figure 4.6-a). Moreover, the
curves also differ in shape at different angles of attack. It can generally be
seen that at small angels of attack (α < 4◦ ) Cdδ is negative, but decreasing
with deflection. For larger angles of attack (α > 4◦ ) Cdδ appears to be
positive, and increasing with deflection. Consequently, two totally different
ways of behavior for small and large angles of attack. Note that for α = 4◦
the drag coefficient is approximately constant for every deflection angle. As
can be seen from figure 4.6-b the drag rise due to separation starts at lower
angles of attack for higher deflection angles.
The moment coefficient (point of application = quarter chord point [25]) as
a function of angle of attack, α, and deflection angle, δ is presented in Figure
4.7. The moment coefficient shows a negative linear trend with increasing
4.2. Airfoil Design
(a) Predicted Cd vs. δ
50
(b) Predicted Cd vs. α
Figure 4.6: Drag coefficient as a function of δ and α.
deflection angle. From Figure 4.7-a it can be seen that for higher angles of
attack Cmδ becomes a little less (Cmδ = −0.57 [1/rad]) At zero angle attack
Cmδ amounts to −0.82 [1/rad]. Figure 4.7-b shows that the moment line at
various deflection angles is relatively flat up to 8 degrees angle of attack.
(a) Predicted Cm vs. δ
(b) Predicted Cm vs. α
Figure 4.7: Moments coefficient as a function of δ and α.
In conclusion, this paragraph will repeat the most important features of
this airfoil. Clmax = 0.9 at zero angle of attack; at maximum deflection
Clmax = 1.45; Clδ = 7.95 [1/rad] for zero angle of attack; Cdδ is negative up
to α < 4◦ , at higher angle of attack it is positive. At α = 4◦ Cdδ is constant
for all delta. Cmα is fairly constant at various values of δ up to 8◦ angle of
attack.
51
4.3
4.3.1
4 Thin Wing Design, Fabrication and Testing
Wing Design
Preliminary Design
Now that the airfoil shapes are more or less determined, the layout of the
total wing is considered. The wing should be able to perform both roll
control and direct lift control. Also, each wing should supply 1.5 N of lift
during cruise (V = 15 m/s) in order to carry the aircraft. Most desirable
would be a wing with a continuously variable spanwise camber distribution.
This is practically what a bird does with its wings when it is soaring. The
inboard part of its wing remains at a constant camber, while at the tips the
shape is altered constantly to manoeuvre. Between the tip and root of the
wing a continuous change in camber can be observed.
The question is whether this is also possible for this MAV wing. A bird
uses its feathers to slide over each other when the wing morphs. To mimic
this in a mechanical wing would be very difficult. One must consider that
the airfoil shape is already established (compare previous section). It would
be possible to divide the wing in spanwise direction into a finite number of
panels. Each panel would than independently of the rest change its camber.
A semi-continuous spanwise camber change could then be achieved.
The PBP actuator requires a tool to pre-compress the PZT laminate. In
the thin wing concept this force is easily applied by using a rubber band
at either side of each wing panel (compare figure 4.8). This compression
mechanism is described in [2]. This implies that the wing will not have a
continuous cross section but will have gaps to store the rubber bands. The
gaps will increase induced drag and profile drag. It is therefore decided that
the wing is divided into two spanwise panels, which each can be operated
separately. The inboard panel will be used for direct lift control, while the
outboard panel induces roll control.
Figure 4.8: Compressive force by a rubber band.
Figure 4.9 shows an impression of the wing and all the components it consists
of. The total wing area amounts to 2.64 · 10−2 m2 .
4.3. Wing Design
Figure 4.9: 3D image of the wing lay-out (all dimensions in mm).
Figure 4.10: Exploded view of the wing.
52
53
4 Thin Wing Design, Fabrication and Testing
Figure 4.10 also suggests the materials to be used in the wing. The leading
edge is fabricated out of foam, covered with thin Aluminum sheet. Spanwise rigidity is increased due to the application of unidirectional carbon
fibres that are laminated in between the foam core and the Aluminum. The
trailing edges of the wing are also fabricated out of this material. The relatively large width of the trailing edges (10 mm) is necessary to ensure that
the load introduction on the piezoelectric elements (PZT, in figure 4.10) is
constant over the width of the specimen (compare appendix D for trailing
edge structural analysis).
The substrate material, that is used between the bimorph PZT, is extended
throughout the wing panel. The substrate is a thin Aluminum sheet (3/4
hard Al 1018) of 0.078 mm thickness. It is a relatively tough material and
because it is so thin it is very flexible. The additional weight to the total
wing will be less than 2 grams, so it is light as well. Moreover, because it
is supported by the frame, it will be able to carry the aerodynamic loads.
Another advantage is that the wing panel can be produced more easily. The
substrate/foil is used as base material. The PZT specimen and the trailing
edge can simply be attached to it.
The complex interaction between aerodynamic loads on the panel and the
panel deflection, make it hard to analytically determine the combination of
end force and deflection. For practical reasons it is therefore chosen to give
each piezoelectric actuator a width of 15 mm. From figure 3.10 it can be
deduced that the buckling load for such an element will be between 800 and
900 gmf . Available rubber bands have shown a compressive force between
250 and 300 gmf , at the required elongation. This would mean three rubber
bands should be able to give the required precompression. Increasing the
width of the elements means more rubber bands are needed, which can be
obstructive and require stronger (heavier) attachment points.
4.3.2
Lift Distribution
The simplest way to design a wing from the airfoil is just adding a third
dimension, yielding a straight wing with no in change section geometry in
spanwise direction. To a certain extend this can be applied to the wing
sketched in section 4.3. The wing is straight: no taper, sweep or twist is
present. There are, however, discontinuities in the form of gaps between the
wing panels (compare figure 4.9).
The effect of the gaps on lift is uncertain. The gaps will allow air from below
the wing (high pressure) to flow to the top side of the wing (low pressure).
As a result a vortex is produced over the each of the wing panels. Figure 4.11
gives a schematic impression of this phenomenon. The vortices between the
panels are similar to the tip vortices which are present at all regular aircraft.
4.3. Wing Design
54
In general tip vortices cause a loss in lift, because the negative pressure on
top of the wing is reduced by the flow over the wing tip [23]. However,
investigation in low Reynolds number, low aspect ratio wings shows that tip
vortices are not always detrimental for lift [22].
Figure 4.11: The gaps cause additional vortices over the wing.
A way to mimic the influence of the gaps between the panels is by assuming
that one panel can be treated as a separate wing. In this assumption, the
continuity of the leading edge of the wing is neglected. The aspect ratio of
a wing is defined as follows:
b2
(4.1)
A=
S
The dimensions of one panel are b = 11 [cm], S = b · c = 12 · 10 = 120 [cm2 ].
Substituting these values in equation 4.1 yields an aspect ratio of 0.92. Investigation has shown that for a wing with aspect ratio 1 and a Reynolds
number around 8.5 · 104 , the tip vortex causes an increase in lift near the
tip of the wing. Theory behind this increase is the disappearance of the
separation bubble, due to energizing of the local boundary layer. The effect
increases with increasing angle of attack [22].
Beside an increase in lift near the tip of the wing, the vortices also induce
an off-elliptical lift distribution, meaning more induced drag. The exact
influence of the gaps is very hard to predict at this stage. Wind tunnel
experiments are required to establish this. To calculate some effective forces
over the individual panels it is assumed that the lift distribution in spanwise
direction is constant and equals that of the airfoil Cl (compare figure 4.12).
From the assumption of constant lift distribution, the required lift coefficient can be determined. The lift coefficient can be altered by changing the
angle of attack or by changing the deflection of the adaptive laminates (or a
combination). Accordingly, for every lift coefficient there will be a range of
combinations of angle of attack and deflection angles. At cruise condition,
the required lift is 1.5 N and the lift coefficient is given by equation (4.2)[21]:
CL =
2L
ρV 2 S
(4.2)
55
4 Thin Wing Design, Fabrication and Testing
Figure 4.12: The assumed spanwise lift distribution.
Substituting for the total wing area, S = b · c = 0.22 [m] · 0.12 [m] =
0.0242 [m], the required lift coefficient, CL = 0.41. This means that an
angle of attack of 1.5◦ is needed at zero deflection. At zero angle of attack,
a deflection of 1.4 degrees is required to attain the same lift coefficient (cf.
figure 4.5).
4.4
4.4.1
Fabrication
Leading Edge Fabrication
Manufacturing the thin wing will be divided into two distinct parts: the
leading edge part and the adaptive part. This section will describe how
the leading edge is made. The cross sectional width of the leading edge is
31 mm, the height amounts to 8 mm (compare figure 4.13). The length of
the leading edge amounts to 230 mm.
Figure 4.13: Dimensions of the leading edge.
The leading edge is fabricated as a sandwich structure. It consists of a foam
core with an aluminum sheet attached to it. To give the laminate additional
stiffness, unidirectional fibres are positioned at the thickest point between
the aluminum sheet and the core. This requires that the shape of the core
matches exactly the shape of the leading edge itself. This can be done fairly
simple by cutting the foam with a hot wire. To attach the aluminum sheet
4.4. Fabrication
56
in the concave part of the leading edge vacuum must be applied during the
curing of the adhesive (compare figure 4.14).
Figure 4.14: Leading edge materials.
To save weight it would be best to keep the aluminum skin as thin as possible.
The thinnest plates at hand are the same sheets that are used as substrate
materials for the PZT’s. These measure a mere 0.0762 mm in thickness.
To prevent the leading edge from buckling additional unidirectional carbon
fibres are applied between the foam and the aluminum around the thickest
location of the wing.
Figure 4.15: Leading edge.
4.4.2
Actuator Fabrication and Integration
The actuator and the aft part of the wing are one integrated part. A wet layup is used to attach the PZT elements onto the Aluminum substrate (figure
4.17). The trailing edge on the wing consists of a glass fibre reinforced
plastic layer on the top and bottom of the aluminum substrate. The trailing
edge covers the last 10 mm of the wing and is made a little wider than the
individual wing panels such that the rubber bands can be easily attached
later.
57
4 Thin Wing Design, Fabrication and Testing
Figure 4.16: Actuator lay-up on caul plate right before cure.
The wires to connect the PZT sheets are integrated in a narrow band of glass
fibre reinforced plastic that touches the foam of the leading edge. Integration
of the leading edge and the deformable panels is done by affixing them to
the aluminum sheet of the leading edge. A slight deviation from the original
design was made by omitting the steel rods to hook the rubber bands to.
Instead, the glass fibre at the trailing edge and near the leading edge is used
to put the rubber bands around. Figure ?? shows the entire wing, including
the rubber bands.
Figure 4.17: Total wing lay out.
4.5. Static Bench Testing
4.5
58
Static Bench Testing
Static bench tests were carried out to determine the deflections of the wing.
Since the wing consists of two individual panels, each of the panels was tested
individually. End rotations of the trailing edge are measured by reflecting a
laser off the trailing edge onto a reflection board. The experiments are done
statically, meaning voltage is constant during each measurement. Figure
4.18-a shows the experimental test setup.
To achieve the right amount of precompression, a combination of two different kinds of rubber bands is used. Each panel is being compressed by two
sets of rubber bands, each set consisting of three rubber bands: two orthodontal high quality elastics and one ordinary ’desktop’ rubber band. Each
set provides 800 gmf of precompression, which is just below the expected
buckling load (825 − 900 gmf ).
Two sets of experiments were undertaken. The first one involving only
the unloaded actuators, the second one with axial loading of the elements,
introduced by rubber bands. Figure 4.18-b shows the end rotation of the
elements as a function of the applied voltage. It can be seen that by applying
the precompression end rotations are almost doubled, up to 13.6 degrees
peak to peak at 95V.
(a) Bench test setup
(b) Static end rotations as a function of voltage.
Figure 4.18: Bench test experiment set-up and results
4.6
4.6.1
Wind Tunnel Tests
Lift Curve Slope
In order to verify the lifting capability of the wing in an early stage, a static
test article was fabricated. This wing has the exact same geometry as the
59
4 Thin Wing Design, Fabrication and Testing
designed wing, but does not employ the piezoelectric actuators. Instead,
copper substrates of the same dimensions as the actuators are used as surrogates. The wing was hinged and put into an open wind tunnel. During the
wind tunnel tests the angle of attack and the velocity were measured. The
wing could rotate freely about the hinge point. Figure 4.19 gives a schematic
overview of the wing in the airflow and all the forces and moments that are
acting on it.
Figure 4.19: Schematic representation of forces on the wing.
The wing’s center of gravity (cg) had been determined beforehand. The
aerodynamic center (ac) was assumed to be positioned at a quarter chord
and also the hinge point (h) was measured in accurately. Since the wing is
hinged freely, the air velocity will determine the angle of attack of the wing.
The moment balance around the hinge point can be written as:
0 = Mac − L(xac − xh ) − D(xac − xh )(α − α0 ) + W (xcg − xh )
(4.3)
= CM qSc − CL qS(xac − xh ) − CD qS(xac − xh )(α − α0 ) + mg(xcg − xh )
In equation (4.3) it is assumed that only small angles of attack are attained,
implying cos α = 1 and sin α = α. Remembering that CL = CLα (α − α0 ),
and assuming that CD << CLα , equation (4.3) can be rearranged to:
1
Cm Sc + CLα S(xac − xh )α0 CLα S(xac − xh )
=−
+
α
q
mg(xcg − xh )
mg(xcg − xh )
(4.4)
This equation can be written as:
1
= η + ξα
q
(4.5)
4.6. Wind Tunnel Tests
60
With:
ξ=
CLα S(xac − xh )
mg(xcg − xh )
(4.6)
Rewriting this for CLα results in:
CLα =
mg(xcg − xh )
ξ
S(xac − xh )
(4.7)
All the terms on the right hand side of equation (4.7) are either known, or
measured during the wind tunnel tests. As is shown in figure 4.20-b, the
inverse of the dynamic pressure, q is linearly related to the angle of attack,
α. The slope of this line is ξ = 0.1345. Furthermore, xac = 0.0340 [m],
xcg = 0.0692 [m], xh = 0.0130 [m], m = 0.032 [kg], S = 2.97 · 10−2 [m2 ] and
g = 9.81 [m/s2 ]. Substituting this in equation 4.7 yields:
CLα = 3.80 [1/rad]
These wind tunnel tests have shown qualitatively that the wing geometry
does induce lift and that attached flow is present up until an angle of 15◦ .
Figure 4.20-a shows a picture of the hinged wing during the wind tunnel
tests.
(a) Static wing in wind tunnel
(b) Relation between the dynamic pressure, q,
and the angle of attack, α
Figure 4.20: Wind tunnel experiment set-up and results
4.6.2
Lift and Drag Measurements
The static test article gives a good indication of the behavior of the wing
under a range of angle of attacks. To determine the actual lift and drag forces
a set of nine static wing articles are fabricated. Each of the test articles has
an identical leading edge. However, they each exhibit a differently curved
61
4 Thin Wing Design, Fabrication and Testing
rear part. The curvatures each represent a tip rotation of respectively 0,
2.5, 5, 7.5 and 10 degrees (both positive and negative), which translates to a
deflection of +/-5◦ in steps of 1.25◦ (cf. figure 4.21-a). The static tests have
shown that the maximum peak to peak end rotation of the wing is close to
14◦ peak to peak (cf. figure 4.18), which translates to a maximum deflection
of +/-3.5◦ . Actuator improvement might lead to even higher curvatures, so
it is therefore chosen to measure beyond this maximum value of deflection.
Deflection is defined in figure 4.3 and is denoted here with δ, for convenience.
Tests are performed in the Low Speed Wind Tunnel of the TU Delft. The
wing is positioned on a rotating part of the top wall of the wind tunnel.
Due to the presence of this wall, a half model of the total wing is simulated.
Since the test article is only one of the two wings for the MAV, this setup
is a good estimation of what happens during actual flight. A picture of the
test setup is shown in figure 4.21-b.
(a) Set of deflections for aft part of wing
(b) Experimental setup of wing in wind tunnel
Figure 4.21: Static deflections and wind tunnel setup.
Measurements were taken at an average airspeed of 15 m/s, at a Reynolds
number of 1.24 · 105 . At each deflection a angle of attack sweep was carried
out in steps of 1◦ between α = −5◦ and α = 25◦ . At each measurement
point CL , CD and CM were recorded. Figure 4.22 shows the lift coefficient
plotted against the angle of attack at different deflection angles.
From figure 4.22 the overall lift performance is shown to be linear up until
an angle of attack of 18◦ , with a lift curve slope of CLα = 3.52 [1/rad]. For
straight wings at low Reynolds numbers and low aspect ratio a reduced version of the Polhamus equation can be used to calculate the relation between
the wing lift curve slope and the lift curve slope of the profile [20]:
Clα =
2+
√
A2 + 4
CLα
A
4.6. Wind Tunnel Tests
62
Figure 4.22: Lift coefficient versus angle of attack at various deflection angles.
460
Substituting for A = cb = 127
= 3.62, and CLα = 3.52 [1/rad] yields Clα =
5.97 [1/rad]. Note that since this wing should be mirrored in the wind
tunnel wall, the span is increased with a factor of 2. The corrected Clα is
close to the theoretical value of 2π.
The increase in lift as function of the deflection angle, δ, is non-linear. However, it is possible to make an estimate of CLδ . Since the CL − α curves are
parallel up to 18◦ , the average CLδ is constant over this range and amounts
to CLδ = 2.97 [1/rad]. Using the same transformation factor as for the
lift curve slope, the section Clδ = 5.03 [1/rad]. Although this value is well
below the predicted values from section 4.2.2 (Clδ between 7.95 [1/rad] and
6.58 [1/rad]), it is still significant.
The stall behavior of this wing is fairly good. No steep decreases in lift, even
for high deflection angles. The stall angle of attack is around 18◦ . At higher
angles of attack, the lift gradually decreases to a constant value (between
CL = 0.9 and CL = 1.1, depending on the deflection).
During cruise the wing should be able to supply 1.5 Newton of lifting force.
This translates to a lift coefficient of CL = 0.37 at the specified conditions.
Assuming no deformation of the wing, this required lift coefficient is obtained
at an angle of attack of α = 6◦ . The relation between lift and drag at a
particular angle of attack is displayed in figure 4.23. At an angle of attack of
63
4 Thin Wing Design, Fabrication and Testing
6 degrees the lift to drag ratio amounts to 4. The maximum lift to drag is a
little over 5 for the positive deflections, and shifts between α = 5◦ (at δ = 5◦ )
and α = 10◦ (at δ = 0◦ ). For the negative deflections the maximum lift to
drag ratio declines to 4.4 at an angle of attack of α = 10◦ at a deflection of
δ = −5◦ .
Figure 4.23: Lift over drag at various angles of attack.
In section 3.4 the relation between applied tip force and deflection was discussed. As can be seen from figure 3.8 the deflections of the actuator will be
lower when tip force is present. This also holds for the deforming part of the
wing, which is loaded by a pressure distribution. This pressure distribution
will help the wing deflect upward (negative), but will counteract downward
(positive) deflections. These wind tunnel tests have not been carried out
with the actual dynamically morphing wing, so no data is yet available on
the influence of wing loading on the deflection angles. Further wind tunnel
tests with a dynamic test article are required to gain insight in this effect.
4.7
Synopsis
A thin airfoil morphing wing, applicable to a 300 gram MAV has been
designed, built and tested. The wing consists of an aerodynamically shaped
leading edge which spans 30% of the wing chord and is used to transfer all
the aerodynamic loads to the fuselage. The leading edge is made out of a
foam core with thin aluminum sheet attached to it. Unidirectional fibres give
it additional stiffness. The aft 70% of the airfoil is basically a flat plate with
4.7. Synopsis
64
the piezoelectric elements attached on top and bottom. The wing consists of
two individual panels which can be controlled separately in order to create
lift control and roll control. In order to magnify rotations, each panel is
being precompressed by rubber bands. By applying this precompression,
end rotations were magnified with almost a factor of 2, up to 14◦ peak
to peak. A static test article was fabricated in order to verify the lifting
capabilities of the wing. A lift curve slope of CLα = 3.64 was found and
attached flow was present up to 15◦ angle of attack. Extensive wind tunnel
tests of a static half model of the (total) wing showed a lift curve slope
of CLα = 3.52. Furthermore, the influence of the deflection was measured
which resulted in a CLδ = 2.97, which translates to Clδ = 5.03 for a two
dimensional section of this wing. The wing showed good stall behavior. The
maximum lift to drag ratio, (CL /CD )max = 5 and occurs between α = 5 and
α = 10, depending on the amount of deflection. Further wind tunnel tests
are required to determine the effect of wing loading on the deflections.
Chapter 5
Thick Wing Design,
Fabrication and Testing
A second application of the PBP actuator is found in a thick wing. An existing
radio controlled UAV is used to proof the concept of the thick morphing wing.
By replacing the normal wing with the modified one, it will be shown that the
change in camber of the airfoil will be sufficient to control the rolling motion
of the aircraft. This chapter will first give an overview of some aircraft characteristics that are important for the modified wing. Then a preliminary design is
presented in terms of general geometry of the wing and airfoil characteristics.
Based on the preliminary design each of the wing details are determined and
laid out. Then wing fabrication and experiments are presented and the chapter
ends with sections on flight and wind tunnel testing.
5.1
5.1.1
Aircraft Properties
Aircraft Geometry
The thick wing concept is implemented in an existing radio controlled UAV.
The model type is called the aerobird (compare figure 5.1). Before the
thick wing is implemented the characteristics of the unmodified aircraft are
examined. This section will show how important properties as aerodynamic
center and center of gravity are determined. First of all some basic geometric
features of the aircraft are laid out below. Since the wing possesses a curved
form of taper, the method of equivalent wing as laid out in section 8.2.3 of
[20] is used to determine the equivalent taper ratio and the sweep angle.
65
5.1. Aircraft Properties
66
Figure 5.1: Radio Controlled UAV: Aerobird.
cw = 145
[mm]
ch = 80 [mm]
5
2
Sw = 2.08 · 10
bw = 1392
Λc/2w = 6
Sh = 1.56 · 104
[mm ]
[mm]
bh = 372
[deg]
[mm]
Λc/2 = 18 [deg]
Λc/4w = 10
[deg]
lh = 776
df = 75
[mm]
hh = −165
λw = 0.31
5.1.2
[mm2 ]
[mm]
[mm]
λh = 0.2
Lift curve slope
To determine the lift curve slope of the wing Polhamus equation is used
(eqn. (8.22) [20]):
CLαw =
Where,
q
2+
2πA
A2 β 2
k2
(1 +
tan2 Λc/2
)
β2
(5.1)
+4
67
5 Thick Wing Design, Fabrication and Testing
A=
β=
k=
Clαat
M
b2
S
p
(5.2)
1 − M2
Clαat
(5.3)
M
(5.4)
2π
β
Clαat M
0
=√
1 − M2
(5.5)
These equations are equations (8.23) to (8.25) and (8.1)respectively from
[20]. For a low speed MAV with a straight wing the following assumptions
are made:
β=
p
1 − M2 ∼
=1
Clαat
M0
∼
= 2π
With these assumptions equation 5.1 reduces to
CLαw =
2πAw
p
2 + 1.01A2w + 4
(5.6)
For the horizontal tail a similar analysis can be exercised. Because of the
sweep angle of the horizontal tail the equations for the lift curve slope of the
horizontal tail reduce to
CLαh =
2πAh
q
2 + 1.11 · A2h + 4
(5.7)
When the aircraft geometry parameters are substituted in equations (5.2)
and (5.6) than the following is obtained.
5.1.3
Aw = 9.32
CLαw = 5.06 [1/rad]
Ah = 4.45
CLαh = 3.94 [1/rad]
Static Margin
The aerodynamic center of the aircraft is calculated as follows (eqn. (8.82)
[20]):
5.1. Aircraft Properties
xacA =
68
xacwf CLαwf + {ηh CLαh (1 −
d² Sh
dα ) Sw xach
− ηc CLαc (1 +
d²c Sc
dα )( S xacc )}
CLα
(5.8)
Where (eqn. (8.83) [20]):
xacwf = xacw + ∆xacf
(5.9)
CLαw and CLαh are determined by equation 5.1. When no canard is present
and ∆xacf is assumed 0, equation 5.8 is reduced to
xacA =
xacw CLαw + {ηh CLαh (1 −
d² Sh
dα ) Sw xach }
CLα
(5.10)
Where (eqn. (8.34) [20]):
ηh =
qh
q
(5.11)
With q being the dynamic pressure in the free stream and q h being the
dynamic pressure at the horizontal tail. CLα is the lift curve slope of the
total aircraft. Equations (5.12) to (5.18) correspond to equations (8.42) to
(8.48) in [20]:
CLα = CLαw f + CLαh ηh (1 −
d² Sh
)
dα Sw
(5.12)
In this equation the following terms appear:
CLαw f = Kwf CLαw
(5.13)
where Kwf is the fuselage interference factor given by:
Kwf = 1 + 0.025
d²
dα
³ d ´2
df
f
− 0.35
b
b
(5.14)
is the downwash gradient at the horizontal tail:
q
d²
(CLαw )at M
= 4.44[KA Kλ Kh cosΛc/4 ]1.19
dα
(CLαw )at M =0
(5.15)
69
5 Thick Wing Design, Fabrication and Testing
where:
1
1
−
Aw
1 + A1.7
w
10 − 3λw
Kλ =
7
1 − hbwh
Kh = ³ ´ 1
KA =
2lh
bw
(5.16)
(5.17)
(5.18)
3
Substituting the aircraft geometry parameters in these equations yields the
following results:
Kwf = 1.000
d²
= 0.35
dα
KA = 0.085
CLαw f = 5.06 [1/rad]
Kλ = 1.296
CLα = 5.25 [1/rad]
Kh = 1.079
ηh = 1
It is assumed that the aerodynamic center of the wing and the horizontal
tail is at the quarter chord of their mean chord. The center of gravity is
determined experimentally by balancing the aircraft on the wing. When the
numerical values are substituted in equation 5.10 the mean aerodynamic
center and center of gravity are obtained. The positions of the aerodynamic
center and the center of gravity in terms of fuselage station w.r.t. the nose
of the aircraft are displayed below the mean values.
xacA = 0.44
xcg = 0.33
xacA = 235 [mm]
xcg = 219 [mm]
The static margin is good measure for the longitudinal stability of the aircraft. Using the above stated numerical values gives the following:
SM = xacA − xcg = 0.11
5.2
5.2.1
Airfoil Design
Airfoil Geometry
Following a top down approach different airfoil geometries are examined
on their aerodynamic properties. This is done using Xfoil, at a Reynolds
number of 105 and a velocity of 15 m/s. Existing low Reynolds number
airfoil geometries are investigated and compared (NACA, Selig, Eppler). In
order to be able to apply an axially precompressed piezoelectric element
on the camber line a symmetric airfoil is required. Lift, drag and moment
are examined at angles of attack ranging from 0 to 15 degrees. For this
5.2. Airfoil Design
70
morphing wing an NACA 0012 airfoil geometry is proposed (Clmax = 0.95,
(l/d)max = 33 and occurs at α = 5◦ ).
In the previous chapter it was determined that the piezoelectric element
would form both inner and outer skin of the thin airfoil. This way no
deforming of the rest of the structure would be necessary and there would
be no additional resistance. For a profile with finite thickness it is impossible
to not deform the structure. There will be structural morphing of at least the
skin of wing. The first 40% of the chord is rigid and provides the wing with
stiffness to transfer the aerodynamic loads to the remaining wing structure.
A bender element is placed over the aft 60% of the chord line of the airfoil.
When the element bends, it will deflect the trailing edge, and consequently
deform the entire airfoil section. This does require some sort of flexible skin
that is able to expand and contract when the airfoil deflects.
Whereas the thin airfoil makes use of a rubber band to pre-compress the
piezoelectric laminate, the thick airfoil might use another instrument. Instead of using a flexible skin that produces an as little as possible resistance
to bending, the skin could also be used as a precompression tool for the
laminate. This way the skin contributes to the deformation of the wing
without being actively controlled. A consequence of this is that a flexible
skin (e.g. natural rubber) over a substantial part of the wing is required.
This also means that it is not possible to follow the entire contour of the
NACA 0012 because the second half of the profile will be straight due to the
precompression characteristic of the skin.
To check what would be the influence of such a design, the NACA 0012
geometry is adjusted and run in the panel code Xfoil. The characteristics
are very much alike. The lift coefficient is only slightly less than for the
NACA 0012. The stall characteristics are not as good as for the original
profile. Stall will originate at a lower angle of attack (12◦ )than for the
original airfoil (15◦ ).
The shape of the NACA 0012 is prescribed by a set of coordinates. The
thickest point lies at 30% of the chord. The maximum thickness is 12% of
the chord. After the first 30% of the chord the contour of the NACA 0012
cannot be followed anymore. From there on, the skin could follow a straight
line to the trailing edge of the airfoil. However, this would lead to a kink in
the skin. This kink is un-beneficial since it is likely to cause separation and
consequently a lot of additional drag and a loss in lift. When the trailing
edge is moved downwards, the kink on the upper side would increase even
more.
Instead of the kink, the straight after part of the airfoil should be tangent
to the front part. Then there will be no discontinuities in the skin geometry
and the flow is less likely to separate. Moreover, when the trailing edge is
deflected the tangency condition should remain. The structure of the airfoil
71
5 Thick Wing Design, Fabrication and Testing
must be rigid up to the point where the straight part of the skin meets the
curved part. When the trailing edge is deflected up or downward this point
will move also. This is shown in figure 5.2.
Figure 5.2: Change in airfoil camber due to PBP actuation.
5.2.2
Predicted Static Performance of Thick Airfoil
This section will give an overview of the main characteristics of the thick
airfoil. As shown in section 3.3 the actuator deflection, δ0 is determined by
the amount of curvature κ. By changing the curvature the actuator will
deflect and the outer shape of the airfoil will deform. In this section delta0
is shortened with δ, for convenience. The deflection angle, δ in these plots
is limited to +/-6◦ .
Figure 5.3 shows the predicted lift coefficient plotted against the deflection
angle and angle of attack. Both graphs show a more or less linear relationship between the lift coefficient and the deflection angle (δ) and angle of
attack (α), respectively. At zero angle of attack Clδ = 5.54 [1/rad], while at
α = 8◦ this has declined to Clδ = 3.05 [1/rad].
(a) Predicted Cl vs. δ
(b) Predicted Cl vs. α
Figure 5.3: Lift coefficient as a function of δ and α.
Figure 5.4 shows the drag coefficient of a function of delta and alpha, re-
5.2. Airfoil Design
72
spectively. From 5.4-a it can be seen that at angles of attack higher than
six degrees the deflection of the airfoil can induce early separation and consequently a drag rise.
(a) Predicted Cd vs. δ
(b) Predicted Cd vs. α
Figure 5.4: Drag coefficient as a function of δ and α.
The moment lines are drawn in figure 5.5. As can be seen from 5.5-b the
moment line has a positive slope for each deflection angle. This indicates
that the airfoil is unstable and a horizontal tail is required to balance the
aircraft. From 5.5-a it can be observed that Cmδ is negative, although
decreasing in an absolute sense with increasing angle of attack.
(a) Predicted Cm vs. δ
(b) Predicted Cm vs. α
Figure 5.5: Moment coefficient as a function of δ and α.
73
5.3
5 Thick Wing Design, Fabrication and Testing
Wing Design
The wing of the Aerobird has a mean aerodynamic chord of 145 mm and
a span of 140 cm. The replacing wing has these exact same dimensions, in
order to stay close to the original design. However, the new wing does not
exhibit any taper or sweep, contradictory to the original wing. This is done
to make incorporation of the morphing part easier. According to equation
5.2 the aspect ratio of this wing amounts to 9.65.
The morphing parts of the wing are supposed to induce a motion about the
longitudinal axis of the aircraft (rolling). They will act the same way as
ailerons would do on conventionally controlled aircraft. To produce a large
moment around the longitudinal axis, the morphing parts are placed at the
outboard side of the wing (confer figure 5.6).
Figure 5.6: Topview of wing, including morphing outboard parts (all dimensions in mm).
The influence on the stability by using this new wing geometry is determined
using the model as laid out in section 5.1. Inserting the new values for
wing geometry and keeping in mind that the rest of the aircraft remains
unchanged, the following values are obtained:
Clαw = 5.11 [1/rad]
d²
= 0.25
dα
KA = 0.082
CLαw f = 5.11 [1/rad]
Kλ = 1.000
CLα = 5.34 [1/rad]
Kh = 1.079
ηh = 1
The neutral position is shifted backward with 2.5% with respect to the
original wing and is now positioned at 46.5% of the wing chord. Assuming
5.3. Wing Design
74
the position of the center of gravity remains unchanged, this yields a static
margin of 13% for this new wing. Consequently, the application of this
straight wing indicates that the aircraft would be slightly more stable than
with the original wing.
For the static part of the wing a conventional balsa wood structure is proposed. This section will not go into the details of that part of the wing
because it does not exhibit any extraordinary features. Section 5.5 will
show the structure of this part of the wing in more detail.
The morphing part of the wing is designed as simple as possible. Including
the skin the entire morphing wing part consists of only 6 components: a
carbon fibre shell that acts as torque box, three PZT bending actuators, a
trailing edge and the skin. The span of each of the morphing parts amounts
to 230 mm, which implies that 33% of the will be morphing. Figure 5.7
shows the design of the wing.
Figure 5.7: Thick wing design.
The most interesting part of this design is the carbon fibre shell. It follows
the contour of the NACA 0012 airfoil exactly, up till 30% of the chord. After
this point it converges until the top and bottom face almost touch. Between
these faces the bender actuators can be clamped. The rubber skin goes over
the carbon fibre shell and over the trailing edge thereby forming the outer
shape of the airfoil.
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5 Thick Wing Design, Fabrication and Testing
The width of the piezoelectric actuators is chosen based on the available
stock of PZT 5A with a width of 0.192 mm. The two outboard actuators
both have a width of 10 mm while the center element has a width of 35 mm.
Accordingly, the total width amounts to 55 mm, which is a slightly smaller
than the total width of the two panels for the thin wing concept (60 mm).
The total buckling load for these elements lies between 3.0 and 3.3 kgf
(compare figure 3.10). The skin precompression should remain well below
this value. This means that the force applied by the skin may not me higher
than 13 gmf /mm.
5.4
5.4.1
Proof of Concept
Skin of the Wing
As has been said before, the skin of the wing is one of the key features of
this design. It fulfills two purposes. First of all it will transfer the pressure distribution to the structure of the wing. Additionally it will also
pre-compress the adaptive laminate inside the wing. When the laminate is
properly compressed, the deflections will be larger than for the free piezoelectric laminates.
A model of the precompression of the laminate is not too difficult to construct. When the laminate is loaded by the skin, this can be modeled as a
cantilevered beam loaded by two springs (compare figure 5.8-a). Since the
skin is likely to made out of rubber, the stiffness, k, of the spring is not constant, but will be linear with the amount of stretching. The forces induced
by the springs can be decomposed into a force in plane with the laminate
and one orthogonal to this plane. This is shown in figure 5.8-b.
(a) Model of elastic skin.
(b) Decomposed forces.
Figure 5.8: Model of the skin pre-compressing the adaptive laminate.
The two forces shown in figure 5.8-b are dependent on the spring stiffness,
k, and the amount of deflection, u. The force F1 will always work in the opposite direction of the deflection. The other force, F2 , will help the laminate
deflect, analogous to the rubber band in the thin airfoil concept. The ratio
5.4. Proof of Concept
76
between those two forces will determine whether the tip of the airfoil will deflect. Naturally these two forces are coupled, because they are produced by
the same skin. The amount of initial stretch in the rubber skin determines
the amount of deflection of the laminate. The total deflection will probably
be less than for the thin airfoil wing, because there is an additional resistive
force, F1 .
Skin experiment The stiffness of the skin relates the elongation of the
skin to the force it applies. Since the skin of the wing is going to be made out
of natural rubber (latex), its modulus of elasticity (E) will not be constant.
Therefore an experiment was carried out to determine the relation between
the stress and strain in the material. The test setup is shown in figure 5.9.
The skin is wrapped over two identical circular bars. One bar is connected
to a scale and one bar is connected a vertical displacement measurement
device. By turning the wheel on this device the elongation of the skin can
be measured up to an accuracy of 0.001”. The force on the scale is measured
in grams.
Figure 5.9: Experimental setup to determine the stiffness of the skin material.
A series of experiments were carried out in the following way: After proper
installment of the skin in the test setup the skin was elongated over an interval of 12.7 mm (0.5 inch). At each interval the specimen was given time
to adjust and settle down to a constant force level. When this level was
77
5 Thick Wing Design, Fabrication and Testing
reached the force measurement was taken. This was done until a maximum
elongation level was reached. Then the specimen was relaxed again at regular intervals and the force was obtained. When the specimen was again in
a total relaxed state the run was completed. After one run of experiments
a second run was carried out to see what the effect of hysteresis is on the
stiffness properties of the material. The same sequence was carried out for
this run as for the first one.
During the first set of experiments it was observed that the rubber showed
non-uniform lateral contraction and local wrinkling during relaxing of the
specimen. Since this is not a desired characteristic it was determined to
grease the top and bottom bar so the specimen would be able to slide over
them. The next set of experiments therefore showed a uniform lateral contraction (which was also measured) and no wrinkling. Three sets of experiments were carried out. The first one using a specimen from a high
quality helium balloon. The second set used specimen from the same type
of balloon, only after it had been inflated (polymers are oriented). The third
specimen came from a Trojan condom.
Results The results of the experiment are laid out as specific force vs.
strain curves in figure 5.10. Curves are shown for three individual specimen
with different properties. The first specimen was made out of a high quality
helium balloon. The original length of this specimen was 32 mm (4). A
second specimen consisted of the same balloon material only after inflation.
This pretreatment gave the specimen a new initial length of 50 mm. A
third specimen made out of a (Trojan very sensitive) condom had an initial
length of 51 mm. The experiments were carried out several times to verify
the measurements. The specimen showed very consistent behavior.
Figure 5.10: Specific force versus strain curves for various specimen.
5.4. Proof of Concept
78
From figure 5.10 it can be seen that the three specimen had different characteristics. The normal (that is: not pre-tensioned) balloon specimen exerts a
much higher force per mm than the pre-inflated specimen. The third specimen (condom) shows distinctly lower specific forces than the other two.
Hysteresis is defined as the ratio between the surface area under the upper
part of the curve and the surface area between the two curves. It turns
out that the hysteresis for both balloon specimen is virtually identical and
amounts to 34%. However, for the condom, hysteresis only amounts to 26%.
As stated in section 5.3 a maximum axial force of 13 gmf /mm can be
applied by the skin. Both the balloon specimen gave a force that was too
high. It was therefore determined that the condom had the most desirable
characteristics. It provided a relatively low compressive force per millimeter.
Furthermore, the condom showed the lowest amount of hysteresis, comes in
uniform quality, is easy to purchase and is relatively cheap.
5.4.2
Static Test Article
Before the actual wing is manufactured, a static test article is build. This
test article has the same shape and chord length as the real wing. However,
it is made out of normal plywood, and instead of using the expensive PZT
actuators, they are replaced by aluminum bars with a similar EI product.
The major goal of this test article is to show that precompression of the skin
does decrease the effective stiffness of the surrogate actuators. To be able
to stretch the skin and wrap it around the frame in a controlled manner, a
stretching mechanism built. This ensures that the force-elongation curve as
shown in figure 5.10 can be followed.
(a) Stretch contraption.
(b) Deforming wing.
Figure 5.11: Static test article
The static test article gives a good insight in the result of axial compression
on the surrogate actuators. By making a mechanical contraption, the chord
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5 Thick Wing Design, Fabrication and Testing
length is made variable. By varying the cord, the tension in the skin can
be adjusted. This is used to tweak the precompressive force such that by
applying only 0.1 N of a vertical tip force the tip deflection is over 10 mm.
5.4.3
Dynamic Test Article
For the dynamic test article the basic set-up is analogous to the static test
article. However, the aluminum strips are replaced by piezoelectric actuators. The product of moment of inertia and stiffness is the same for both
the aluminum strip and the actuator. It is assumed that the total laminate
stiffness of the actuator can be approximated by the stiffness of the PZT
elements.
Ea Ia = EAL IAL
(5.19)
A second deviation from the static test article is the fabrication of the trailing
edge. To comply with the Kutta condition a fairly sharp trailing edge is
fabricated. It consists of a sandwich of 0.058 mm steel, and a balsa wood
core. Between the steel outside and the balsa wood a thin layer of 45 degree
oriented glass fibre fabric is integrated to provide torsional stiffness and also
insulate the aluminum core of the bending actuator. Figure 5.12 shows the
dynamic test article and its dimensions.
Figure 5.12: Dynamic test article dimensions and features.
The influence of the pre-compression of the skin is investigated. To do this,
the end rotation of the specimen is measured for both the skin-on and the
5.4. Proof of Concept
80
skin-off case. A series of quasi static tests is carried out. The test setup
for this experiment is displayed in figure 5.13 and is analogous to the set-up
that is described in section 3.5. A small laser mirror is mounted on the
trailing edge. A laser beam is used to magnify the end rotations so it can be
accurately read from the screen. The laser dot can be seen in figure 5.13-b,
near the right edge of the picture.
(a) Power supply and signal generator
(b) Deflection measurements
Figure 5.13: Experimental test set-up.
The first test article failed because of an erroneous signal that was created
by the power supply upon connecting it to the actuators. Both actuator
specimen broke. A second test article was manufactured to continue testing.
To prevent this test article from breaking, a bump stop was put in the center
of the airfoil section. This acts as a physical boundary for the skin. When
the specimen tends to over-rotate the skin will hit the bump stop and will
prevent the actuators from breaking. Figure 5.14 shows how the shape of
the wing changes when the PBP elements are actuated.
Two series of experiments were carried out. A quasi static test was done
to determine the relation between the applied voltage and the end rotation
of the specimen. A dynamic test was executed to investigate the change in
natural frequency due to the application of the precomprressing skin. Both
experiments were carried out on the skin-on and skin-off test article under
the same conditions. Figure 5.15 shows the peak-to-peak end rotations of
the trailing edge.
From figure 5.15-a it can be noticed that due to the application of the skin
the end rotations are more than doubled, to 15◦ peak to peak. In 5.15-b
it is shown that the natural frequency has shifted from 31 Hz to 26 Hz.
This indicates that the geometric stiffness of the total assembly, being the
actuators plus the skin, has decreased.
ω2
kskin on
= 2skin on
kskin of f
ωskin of f
(5.20)
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5 Thick Wing Design, Fabrication and Testing
(a) Deflection to the left
(b) Neutral position
(c) Deflection
right
to
the
Figure 5.14: First dynamic test article deflections.
(a) Quasi static end rotations as a func- (b) End rotations as a function of actuation frequency.
tion of voltage.
Figure 5.15: Results quasi-static and dynamic experiments
5.5. Wing Fabrication
82
Substituting the above mentioned values for the natural frequency shows
that the remaining geometric stiffness in the actuator assembly is only 70%
of the original geometric stiffness.
From the dynamic tests it can be concluded that the pre-compression in
the skin could still be further increased in order to decrease the effective
stiffness even further. A result of that would be higher curvatures. However,
if curvatures become too high the skin will eventually break the specimen.
The deflections in the test article specimen go up to 40% of the maximum
deflection (compare section 3.5). Furthermore it can be seen that the corner
frequency of the actuator is shifted from 38 Hz to 34 Hz. Above the corner
frequency, the deflections decrease substantially and can therefore not be
used anymore.
5.5
5.5.1
Wing Fabrication
Leading Edge Torque Box Fabrication
The carbon fibre torque box is manufactured in three subsequent steps.
First a positive mold is made out of oak wood. To ensure symmetry, a thin
metal sheet is positioned in the middle of the mould. From the wooden
mold a silicone female mold is produced. This is done for each of the two
halves separately. After the two molds are finished a carbon fibre pre-preg
is positioned in between the two. A vacuum bag is used to apply pressure.
Then the entire assembly is cured in the oven. After removal of the molds the
carbon fibre torque box remains. Figure 5.16 shows the subsequent steps.
5.5.2
Static Wing Fabrication
The morphing wing part of the wing is intended to replace the ailerons. This
means that the static part of the wing (b = 940 mm, c = 145 mm) can be
manufactured using a conventional structure and conventional materials. It
is chosen to make the wing structure out of balsa wood and use covering
material for the skin. A relatively simple structure is designed and built.
The structure consists of a main spar, a leading edge (which also acts as front
spar), a trailing edge (which also acts as rear spar), and ribs in between. All
the individual parts are bonded together using cyanoacrylate resin.
To give additional stiffness to the structure, the first 30% of the airfoil is
covered with balsa wood sheeting (both top and bottom face). An extension
of the wing is created to attach the carbon fibre shell, which has to be
positioned at the tips of the balsa structure. Holes in the ribs allow for
electrical wiring to the piezoelectric laminates. The wing has a dihedral of 2
degrees on each side. A picture of the wooden structure is displayed in figure
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5 Thick Wing Design, Fabrication and Testing
(a) Positive mold
(c) Vacuum bagging
(b) Mold sandwich
(d) Finished product
Figure 5.16: Torque box fabrication.
5.17-a. The final static part of the wing weighs 120 grams and is shown in
figure 5.17-b.
(a) Wing structure.
(b) Finished static part.
Figure 5.17: Balsa wood wing.
5.5.3
Actuator Fabrication
Contradictory to the actuator fabrication of the thin wing, for these actuators a dry lay-up is carried out. The bimorph element consists of five layers:
[CAP(+Λ)/Hysol bond/aluminum sustrate/Hysol bond/CAP(−Λ)]. The
5.5. Wing Fabrication
84
thickness of each of the PZT sheets is 0.191 mm (7.5 mil). The thickness of
the aluminum and the pre-preg Hysol bonding layer amount to 0.0765 mm
(3 mil) and 0.101 mm (4 mil), respectively. Total thickness of a well manufactured laminate adds up to 0.66 mm (26 mil). Figure 5.18-a shows the
different layers, along with the fibre glass end tabs.
The dry lay-up is a very neat way of manufacturing the bimorph laminates.
The finished actuators are placed on a caul plate (figure 5.18-b). A second
caul plate is positioned on top of the first one. The whole assembly is being
clamped as it goes into the oven at a temperature of 176 degrees Celsius,
for two hours. As the laminates come out, they are trimmed to the required
size and wired up subsequently. Each element is tested individually before
it is applied in the wing structure.
(a) Different layers of the laminate.
(b) PZT lay-ups on caul plate before curing.
Figure 5.18: Bimorph actuator fabrication.
5.5.4
Wing Assembly
With all the main components present, the entire wing is assembled. The
PZT parts are glued into the carbon shell. However, in addition to the
original design, an extra fibre glass slot for the PZT laminates to slide in is
fabricated. This prevents electrical charging of the carbon fibre shell during
operation. Furthermore, a simple sandwich trailing edge is made analogous
to the one used in the test article. The assembled adaptive part is shown in
figure 5.19-a and weighs 45 grams.
The adaptive part of the wing is first assembled, then the skin is put around
it and a test is run to determine the maximum deflection. Also a frequency
sweep is carried out to find the shift in eigen frequency due to the precompression of the skin (compare figure 5.19-b). This is all analogous to the
tests done on the test article.
85
5 Thick Wing Design, Fabrication and Testing
(a) Assembled adaptive part.
(b) Deflection testing.
Figure 5.19: Adaptive wing assembly.
The skinned adaptive parts are then bonded to the balsa wood wing. Physical bump stops are connected to the balsa wing and to the tip of the carbon
shell. These bump stops prevent the adaptive part from over rotating. The
outer bump stop acts as a winglet at the same time and also protects the
morphing part during a heavy landing. Figure 5.20 gives an overview of the
wing in assembled state. In figure 5.21 the total aircraft is shown with the
morphing wing on top and all the electronics included in the body of the
aircraft. The total weight of the wing amounts to 210 grams, which is a
reduction of 9% with respect to the original wing (230 grams).
Figure 5.20: Total assembled wing including required driver electronics.
5.6
Flight Testing
To close the design loop a flight test is carried out to see how the wing
behaves in flight. This is one way to prove that the concept of a morphing
wing is realistic and feasible. Since a symmetric airfoil is used, the wing has
to fly at relatively high angles of attack and high speeds to gain enough lift.
5.6. Flight Testing
Figure 5.21: PBP Wing mounted on UAV (topview).
Figure 5.22: PBP Wing mounted on UAV (sideview).
86
87
5 Thick Wing Design, Fabrication and Testing
However, this does also mean that a high speed take off is required to get it
flying. The aircraft is being hand launched.
The actual flight test was carried out on April 29th , 2005 in Auburn, Alabama. To let the pilot, mr. Christoph Burger, get used to the aircraft
characteristics, the aircraft was first flown with the conventional wing on.
The original wing does not have any ailerons and is just made out of foam.
Pictures and video were taken throughout the maiden flight of the first piezoelectric actuated morphing wing. Figure 5.23 gives an overview of the flight
testing.
The first flight with the adaptive wing, showed that the center of gravity
was shifted too much to the front. By adding weight to the tail this problem
was solved quickly and easily (fig. 5.23-c). The second run the aircraft did
not pick up enough speed to take off (it is hand launched) and so it crashed
causing the propeller to chop off a small part of the trailing edge of the wing.
Flashing tape was used to repair this (fig. 5.23-d).
The third try was successful. The plain took off (fig. 5.23-e) and the pilot
manoeuvred it using the conventional controls first. After a few laps of
normal control, the morphing parts of the wings were used to induce a
rolling motion. According to the pilot the aircraft responded ”real nice”
to the roll control input. A five minute flight was carried out in which the
aircraft rolled really well in both directions (fig. 5.23-f).
Because the wing is symmetric, the plane had to fly fast and under a hight
angle of attack. The symmetric profile was chosen because the morphing
part of the wing needs symmetry in order to operate well. However, from
the test flight it became clear that the static part of the wing should have
been made cambered in order to produce enough lift at zero or small angles
of attack.
The most important conclusion is however, that it is possible to integrate
a piezoelectrically actuated morphing wing on a UAV and make it flight
worthy. Moreover, it is shown that this way of wing morphing is a feasible,
efficient and light weight alternative for roll control.
5.7
Wind Tunnel Test
To determine the actual behavior of the morphing wing panel under aerodynamic loading, a wind tunnel test article is fabricated. This test article
is basically the morphing wing with additional end plates. The end plates
prevent tip vortices to distort the flow over the top of the wing. This way
a two dimensional wing section is being simulated. Figure 5.24-a shows the
wind tunnel test article.
5.7. Wind Tunnel Test
88
(a) Ready for take off
(c) Increasing tail load
(b) First take off
(d) Repairing trailing edge
(e) Successful take off
Figure 5.23: Flight testing.
(f) In flight
89
5 Thick Wing Design, Fabrication and Testing
The wing panel is tested in the Dobbinga Vertical Wind Tunnel of the TU
Delft. The panel is positioned with the leading edge pointing downwards
into the airflow. A balance, attached to the wing measures the lift force.
Deflections of the trailing edge are measured by a scale which is attached
to the inside of one of the endplates. Voltages are kept track of by a voltmeter. Angle of attack can be easily altered by rotating the wing over its
longitudinal axis. Air velocity is measured by a pitot tube at the exit of
the tunnel. Figure 5.24-b depicts the wing in the test configuration over the
wind tunnel.
(a) Wind tunnel test article.
(b) Test setup in wind tunnel.
Figure 5.24: Wind tunnel test.
Wind tunnel measurements were taken on July 12th 2005. The wing was
being positioned at seven different angles of attack, from −15◦ to +15◦ in
steps of 5◦ . At each position a voltage sweep was carried out from +100V
to -100V in steps of 10V. At each voltage point both lift force and deflection
were recorded. From these parameters a relation is plotted between the lift
coefficient and the deflection, δ, of the piezoelectric actuator in figure 5.25.
The angle δ is the same as was used in section 5.2.2 and is equal to δ0 as
defined in figure 4.3.
From figure 5.25 shows seven more or less parallel lines that represent the
change in lift coefficient due to the shape change of the wing. The attempt
to do a two dimensional wind tunnel test on this wing panel partly failed
because slots remained present between the end plates and the wing in order
to let the wing deform with obstruction of the plates. Therefore the slope
of the lines should be corrected. The expression for the two dimensional lift
curve slope as a function of δ, Clδ can now be expressed using the reduced
Polhamus equation [20]:
Clδ =
2+
√
A2 + 4
CLδ
A
5.8. Synopsis
90
Figure 5.25: End rotation of PBP versus lift coefficient.
Substituting for A = cb = 230
145 = 1.59, and CLδ = 1.42 [1/rad] (average value
over all angles of attack) yields: Clδ = 4.06 [1/rad]. At α = 5 the predicted
Clδ = 3.94 [1/rad] (cf. section 5.2.2) whereas the wind tunnel experiment
shows that Clδ = 4.16 [1/rad]. The discrepancy between prediction and
experiment is larger at α = 0 where it was predicted that Clδ = 5.54 [1/rad]
while in reality it only amounts to Clδ = 3.93 [1/rad].
From figure 5.25 the influence of the pressure distribution on the deflection
of the wing can be seen clearly. At an angle of attack of 15◦ , the wing is
not capable of deflecting in downward anymore. Maximum voltage must
be applied to keep the wing at its center position. However, in the other
direction the negative pressure helps the wing deflect and much higher deflections than during static bench tests are obtained (δ = −5.5◦ in wind
tunnel versus δ = −3.75◦ at bench test). This effect is caused by the fact
that the piezoelectric actuators trade deflection for blocked force and must
be taken into consideration when a control system is designed. Increasing
the maximum blocked force can be done easily by applying actuators with
a larger width. However, this does increase weight so a good trade off is
required.
5.8
Synopsis
A thick airfoil morphing wing has been designed, built and flight tested to
replace ailerons on a subscale UAV (span = 1.4 m). The outboard parts can
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5 Thick Wing Design, Fabrication and Testing
change their camber in order to induce roll control. In total one third of
the wing can be actively deformed. A NACA 0012 airfoil has been chosen
for the morphing wing section. The leading 40% of the airfoil chord consists
of a carbon fibre torque box which transfers the aerodynamic loads from
the deforming wing parts to the remaining (static) wing structure. The
piezoelectric bending actuators are incorporated on the camber line, over
the aft 60% of the airfoil. By using an elastomeric skin, the actuators are
being precompressed in order to increase trailing edge deflections. Bench
tests on a test article showed that end rotations of the PBP actuator could
be increased by a factor of two with respect to the unloaded actuators, up
to end rotations of 15◦ peak to peak. A total new wing was designed and
built, incorporating the morphing outboard parts. Flight tests proved that
the PBP morphing wing induced excellent roll control. Wind tunnel tests
showed an average CLδ = 1.42, which translates to Clδ = 4.16 for a two
dimensional section of this wing. The presence of wing loading shows a shift
in deflection range. At an angle of attack of 15◦ , the wing is not capable
of deflecting downward anymore due to the blocking force of the negative
pressure.
(5.21)
5.8. Synopsis
92
Chapter 6
Electronics
Both morphing wing concepts that were discussed in the two previous chapters
require the same electronic circuit to power their actuators. The piezoelectric
actuators that are used in both these concepts require voltages that range between +/ − 100 V . Normal electromechanical servo actuators work at 4.8 V .
Accordingly, a relatively high voltage should be generated out of normal batteries in a very small aircraft. This chapter will present the design, fabrication and
testing of the electric circuit.
6.1
Power Consumption
One of the most substantial advantages of using PZT actuators over conventional servo actuators, is their extremely low power consumption. The
reason for this lies in the nature of the actuators. They act as capacitors,
meaning very little current is actually flowing trough the actuator. The
potential between the two electrodes on the PZT sheets is what makes the
actuator expand or contract.
The PZT material between the two electrodes of the actuator is a dielectric.
The capacitance can be calculated as follows (p. 111 [10]):
C=
²0 ²r Sa
ta
(6.1)
The relative dielectric constant for PZT 5A, ²r = 1800 [−]. Furthermore,
²0 = 8.8542 · 10−12 [F/m], and ta = 0.1902 [mm]. For the thin wing, the
total actuator area amounts to 8, 69 · 10−3 m2 . Substituting these values
yields for a total wing actuator capacitance of 728 nF . The thick wing has
a slighter smaller total actuator area (7.96 · 10−3 [m2 ]), however, to make a
conservative comparison, the actuator area of the thin wing is chosen.
93
6.2. Requirements on the Electronic Circuit
94
It is assumed that the maximum potential over the individual sheets is 100V .
From the dynamic test, described in section 5.4.3, a corner frequency of
38 Hz is obtained for the non precompressed actuators. Since a conservative
estimation of the power consumption is made, this frequency is used.
The consumed power is a product of the applied current (I) and the voltage
(V ). Using Ohm’s law and substituting impedance (Z) for resistance (R)
the following can be obtained [7]:
P =VI =
V2
V2
=
R
Z
(6.2)
The impedance is a function of the actuator frequency and capacitance (eqn.
(10.37) in [10]):
1
(6.3)
Z=
jωC
Substituting the following values
ω = 2π · 38 = 238 [rad/s]
C = 728 [nF ]
V = 100 [V ]
yields a total maximum power consumption of
P = 1.73 [W ].
To compare this actuator with an average servomechanical actuator of a
similar scale and designed for analogous use it is necessary to evaluate their
power consumption at equal frequencies. The corner frequency for such a
servo actuator is maximally 2 Hz. It draws a constant 5 V and current of
500 mA. This results in a power consumption of 2.5W . A PZT actuator,
actuated at 2 Hz will only consume 0.091 W The proposed piezoceramic
actuated morphing wing would therefore represent a power consumption
that is only 3.6% that of currently used electromechanical servo actuators.
For the thick wing, the power consumption will be even slightly lower than
that [7].
6.2
Requirements on the Electronic Circuit
The requirements on the electronic circuit are all very different in nature.
Originating from performance, electronics or material perspective, some of
the requirements may be conflicting. Consequently, compromises are to be
made in order to stay within reasonable boundaries.
The first en most obvious requirement is the fact that the circuit should be
able to generate a 100 V potential over the adaptive laminate. Whether this
95
6 Electronics
is done using only a stack of batteries or using a sophisticated is unimportant at the moment. To generate maximum deflection of the piezoelectric
laminate this voltage is required.
The second requirement comes from the performance perspective. The electronic circuit, including the batteries, should be as light weight as possible.
The circuit creates an electric field over the piezo ceramics.
The third requirement is closely related to the second one and addressed the
volume of the electronics. Naturally, a circuit which inhabits all the space
in the fuselage is unwanted. One of the reasons to use the piezoelectric
actuators in the first places, is the fact that servo motors in the fuselage
become obsolete. It would be undesirable to substitute their space with the
electronics for the piezoelectric actuators. So, another requirement on the
electronics is to make it as small as possible.
From a production point of view there are some additional requirements that
are not as strong as the first four, but should be kept in mind during the
design process. The size of the components should be such that it can still be
handled properly, so not on a miniature scale. The number of components
should be minimized. Furthermore, it would be desirable if the batteries
that power the electronic circuit could also power the receiver and maybe
additional electronics that are to be placed on board the aircraft. The
batteries should be either rechargeable or they should last very long. And
finally, the components should be commercially available and affordable.
Summing all the above mentioned requirements produces the following circuit. An electronic circuit is required that produces a 100 V electric field,
at as low as possible weight and volume. Batteries should also be able to
power the additional electronics. Furthermore, everything should be at at
a proper size and the batteries should be either rechargeable or very long
lasting.
6.3
6.3.1
Design of the Electronic Circuit
High Voltage Generation
There are basically two ways to produce an electric field of 100 V . One is
creating a battery of 100V . Although this may seem odd, it can be a good
option when the batteries are small and light. The second option is to use
a small stack of batteries, say for example 4 of 1.2 Volts. The resulting 4.8
volts should be amplified with a factor of 20 to get to the required voltage.
This second option has been used successfully in the past [5]. To decide
which of those two options is best for this application a comparison must be
made.
6.3. Design of the Electronic Circuit
96
Option one considers two battery packs of 3.6 Volts, each stack containing
three 1.2 V rechargeable batteries. Two dc-dc converters are used to amplify
the voltage to + 65 V . This means that the potential between the two
amounts to 130 V . The weight of each of the batteries is estimated to be
3 grams. The weight of the lightest dc-dc converters amount to 3.5 grams.
Consequently, the total weight of this circuit amounts to 6 · 3 + 2 · 3.5 =
25 grams. The volume of the batteries is small. They are about 1.5 cm
high and 1 cm in diameter. The converters are about 2 cm3 . Total volume
then adds up to approximately 15 cm3 . The circuit is drawn in figure 6.1.
The additional capacitors of 10 pF will be very light weight and are not
considered here.
Figure 6.1: The dc-dc converters increase the voltage.
The second option incorporates eight 12 V batteries, together generating a
potential of 96 Volts. The 12 Volt batteries are really a stack of eight 1.5
Volt non-rechargeable batteries. So, what is basically done is stacking 64
1.5 Volt batteries together. The weight of each of the batteries amounts to 7
grams. Since no dc-dc converter is required the total weight of this battery
stack is 8 · 7 = 56 grams. The batteries are about 3 cm high and 1 cm in
diameter. So the entire pack inhabits a space of roughly 30 cm3 .
A third option is added as an interpolation of the first two options. This
option incorporates two non-rechargeable 12 Volt batteries and two dc/dc
converters. The converters will crank up the voltage to + 48V . Two batteries
are required because with one battery it is impossible to impose a ground
voltage. In this case the connection between the two batteries functions as
ground. The voltage output of the batteries will consequently be + 12V .
Total weight of this circuit amounts to 21 grams and the total volume will
be about 12 cm3 . The circuit is analogous to the one drawn in figure 6.1
only the input voltage is increased from two times 3.6 V to two times 12 V .
From a weight and volume point of view the last option would be the best,
closely followed by the first one (only 4 grams weight difference). Option
number two is definitely worse because of the high weight and volume. From
97
6 Electronics
a practical point of view option one has additional benefits over option three.
First of all the batteries are rechargeable, so in the long run this will be
both less expensive and easier to maintain. Second of all, the 7.2 voltage
difference between the batteries could be used to power the receiver of the
aircraft. A voltage difference of 12 or 24 volts is not standard to power
the receiver and would therefore require additional electronic components
or even a separate battery pack. Accordingly option number one is the best
compromise between all the requirements.
6.3.2
Amplifying circuit
Apart from generating a high voltage difference, an additional circuit is
required to amplify the control signal to this high voltage (compare figure
6.2). The input voltage is generated by the receiver and will be in the order
of 5 volts. Amplifying this signal requires so-called operational amplifiers or
op-amps.
Figure 6.2: The electronic circuit should amplify the control signal.
The way an op-amp amplifies the input signal is clarified using a relevant
example. Figure 6.3 gives a schematic representation of the circuit. The
triangle in the middle is the symbol of the op-amp. An input signal (Vin ) of
1.5 Volts is assumed (think of an ordinary Nickel-Cadmium battery). The
negative pole of the battery is connected to ground. The supply voltage (Vs )
amounts to + 30 V . The amount of amplification is determined by the ratio
between two resistors in the circuit (compare equation 6.4). However, when
the product of the input voltage and the amplification factor exceed the
supply voltage, the op-amp saturates and output voltage equals the supply
voltage.
AR =
Rf
Ri
(6.4)
The resistance Ri = 2 [kΩ] and Rf = 40 [kΩ]. The amplification ratio then
amounts to 20 and the output voltage will be -30 Volts, because the op-amp
inverts the voltage. This voltage is equal to the supply voltage and the
op-amp is saturated.
The output of the op-amp drawn in figure 6.3 is always negative and the
maximum range is from zero to the supply voltage. This circuit is capable of
6.3. Design of the Electronic Circuit
98
Figure 6.3: Electrical representation of a linear amplifier driving a piezoelectric actuator.
creating an electric field in only one direction over the capacitor (Ceq ). The
direction of this field has been drawn with an arrow pointing upwards. This
means that piezoelectric ceramic will be only capable of either contraction
or expansion, not both at the same time. In order to be able to create an
electric field in the opposite direction, another op-amp should be added to
the circuit (figure 6.3.2-a).
(a) Electrical representation of two linear amplifiers driving a piezoelectric actuator.
(b) Amplifying circuit including two
linear amplifiers.
The circuit drawn in figure 6.3.2-a will be able to let the ceramic expand
and contract. Moreover, the potential over the ceramic is now doubled. So
the maximum voltage difference amounts to 2Vs . Assuming a supply voltage
of 65 V this would mean a maximum of 130 V . This would be even higher
than the de-poling voltage (114 V ). Therefore the amplification factor, still
determined by equation 6.4, must be determined properly to avoid de-poling.
The amplifying circuit is based on the use of (dual) op-amps. Op-amps
are as diverse in specifications as in price. There are op-amps that would
99
6 Electronics
be suitable for this circuit (e.g. APEX PA-41). However, they are very
expensive (in the order of 100 Euros), due to their extreme specifications.
A disadvantage of this op-amp is the fact that it has a high trickle current.
This implies that the circuit will consume a substantial amount of power,
even when nothing is being activated. On the other hand there are the TL
741 type of op-amps: cheap, easy to purchase and a low trickle current. The
supply voltage of these op-amps is 22 V maximally.
Although the maximum supply voltage is far less than the required supply
voltage of 65 V , the TL 741 type op-amps could still be used. This is because
the actuators act as a dielectric in a capacitor. This implies that only very
little current is drawn, resulting in values two orders of magnitude lower than
the maximum allowed current draw. The heat dissipation in the op-amps
is the result of internal resistances, R. Since the heat, Q, is proportional
to the square of the current (Q = RI 2 ) only little heat is produced [10]. A
high voltage in combination with a low current might therefore not be as
disastrous for the op-amp as would be expected before hand. There will
probably be more heat generated than with a low voltage but this could be
dissipated using a dedicated heat sink. Since this scenario has been applied
successfully in the past it has proven to be a feasible option [5].
In order to make the amplifying circuit more compact, the two op-amps can
be integrated into one IC (integrated circuit). These so-called dual op-amps
work exactly the same as two separate op-amps. The circuit as drawn in
figure 6.4 remains applicable. However, since two circuits are integrated
now, the heat production might be doubled as well. Consequently it must
be examined if the dual op-amps can withstand the high voltages. This can
only be done by experiment, since there is no data available on overcharged
dual op-amps. The op-amp circuit that was made to control the movements
of the PBP actuators, only measured 10 × 10 × 20 mm and weighed less
than 5 grams (including all wiring and connectors). Figure 6.3.2-b shows a
picture of this circuit.
6.3.3
Integration of Components
Integration of the supply voltage circuit and the amplifying circuit does not
make the circuit complete. The receiver has been mentioned before, but will
now be treated more extensively. The receiver is feeded by a constant voltage
from the battery pack. It receives a control signal at a certain frequency
(for model airplane the 72 MHz band is often used). It translates the radio
signal to a pulsewidth modulated signal. This signal cannot be used as an
input for the op-amp because of the pulsewidth nature of it. Accordingly,
an additional component should be added to transform this signal into a
control signal. Those components are so-called servo controllers and are
also integrated in conventional servo motors.
6.4. Flight worthy electronics
100
By adding this component to the two circuits, the entire circuit for controlling one piezoceramic is complete. The schematic representation is shown
in figure 6.4. Naturally this circuit can be expanded easily to driving a
piezoelectric laminate. Then, the circuit drives two piezoelectric elements
in parallel. This way, the same circuit can be used to drive one panel of the
wing. To control the different panels on the wing independently, only the
amplifying circuit and the servo controller have to be copied. The supply
circuit can supply a number of op-amps and therefore does not need to be
copied.
Figure 6.4: Electrical representation of the autonomous circuit driving a
piezoelectric actuator.
The wing should be capable of performing roll and direct lift control. The
inboard panels of two wings can perform direct lift control, while the output
panels can perform roll control. As a first approach, the inboard panels of
both wings can operate simultaneously. That implicates that one amplifying
circuit is required to operate two panels. This can also be applied to the
outboard panels. The only difference is that their deflection must be antisymmetric. Consequently, only two circuits are required to control all of the
control surfaces. In figure 6.5 these two circuits are shortened with ”Direct
Lift Control” and ”Roll Control.”
6.4
Flight worthy electronics
When building the electronic circuit as displayed in figure 6.4, a number
of problems arose. The first problem had to do the with the pulsewidth
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6 Electronics
Figure 6.5: All panels of the wing are controlled by two amplifying circuits.
modulated control signal. In order to get a voltage regulated input signal
for the op-amp, this control signal should be demodulated. This became
a problem since commercial available demodulators come in miniaturized
form, which means they cannot be soldered by hand. Using demodulators
from existing electronics did not seem to work.
Overheating of the op-amps was another problem. The high voltage caused
the op-amp to overheat and eventually the internal circuitry was critically
damaged . Using a different type of op-amp, in combination with a heat
sink may be the solution to this problem. However, a more fundamental
obstacle is the fact that the op-amp dissipates all the energy whether the
pzt is activated or not. This takes away the advantage of the low power
consumption of pzt actuators with respect to conventional electromechanical
servo actuators.
Due to these two problems a new circuit was designed, built and tested.
In the new setup the high voltage circuit is electrically separated from the
low voltage circuit. The amount of output voltage is controlled by a 100kΩ
dual potentiometer which is mechanically linked to a electromechanical servo
actuator. The servo actuator is controlled by a 4.8V control signal from the
receiver.
The advantages of this circuit are that high voltages can be obtained with
almost no losses in the circuit, it is relatively simple to manufacture and
it does not require additional elements such as a heat sink. On the other
hand, it is relatively big, heavy and the maximum operating frequency is
determined by the servo actuator. It is therefore definitely not a well suited
circuit for a PBP actuated morphing wing. However, it still is able to control
the pzt actuators during flight and with that proof the concept of a piezo-
6.5. Synopsis
102
electrically actuated morphing wing. Figure 6.6 shows all the electronics
that were used to power the PBP actuators in the thick wing.
Figure 6.6: Flight worthy electronics prior to integration into the aircraft.
6.5
Synopsis
It has been shown that piezoelectric actuators, integrated in a morphing
wing, consume an order of magnitude less power than electro mechanical
servo actuators for aileron control. While power consumption is low, a high
potential voltage (maximum of +/- 100V) has to be generated to control the
PZT sheets. A linear amplifier was used to increase the control signal (4.8V)
to this high voltage. A direct current converter was intentioned to increase
the supply voltage from the batteries up to a potential difference of 130V.
When integrating the electronics two major problems arose: demodulation
of the control signal, and overheating of the linear amplifiers. To be able to
make flight worthy electronics it was decided to use a 96V battery pack in
combination with a dual potentiometer linked to a electro mechanical control
servo. This way no interference between the high and low voltage circuit
occurred and losses were minimized. On the other hand, the circuit was
relatively heavy, big and the maximum operating frequency was determined
by the maximum operating frequency of the control servo.
Chapter 7
Conclusions and
Recommendations
7.1
Conclusions
It can be concluded that the application of Post-Buckeled Precompressed
(PBP) piezoelectric elements in a morphing wing configuration is feasible
and has significant benefits over conventional control surface actuation.
PBP actuators It is shown that axial precompression of conventional bimorph piezoelectric bender elements (PBP actuators) can increase end rotations up to a factor of four. Classical Laminate Plate Theory (CLPT) models
are shown to capture the behavior of bimorph piezoelectric laminates. A deflection model, employing nonlinear structural relations, is shown to predict
the behavior of axially compressed elements very well. A series of actuator specimen were tested to demonstrate the utility of the PBP elements.
Advantages of PBP actuators over conventional electro mechanical servo actuators are a substantial reduction in slop, dead band, complexity, power
consumption and weight. Moreover, their corner frequency is an order of
magnitude higher.
Thin wing A thin airfoil, employing PBP actuators on the aft 70% of
the camber line, was designed, built and tested. Precompression of the
piezoelectric elements was ensured by using rubber bands at each side of the
individual wing panels. A static test article was fabricated in order to verify
the lifting capability of the wing. Wind tunnel tests showed a lift curve
slope of 3.80 and attached flow up to 15◦ angle of attack. It has been shown
that by using precompression end rotations were increased with almost a
factor of 2, up to 14◦ peak to peak. Wind tunnel tests showed that for
103
7.2. Recommendations
104
this wing CLα = 3.52 [1/rad] and CLδ = 2.97[1/rad], which translates to
Clα = 5.97[1/rad] and Clδ = 5.03[1/rad] for a two dimensional section of
this wing. The maximum lift to drag ratio, (CL /CD )max = 5. Furthermore,
the profile shows a gradual stall behavior, which starts at α = 18◦ .
Thick wing Two morphing wing panels of finite thickness, employing
PBP actuators on the aft 60% of the camber line, were designed, built and
flight tested on a subscale UAV (span 1.4 m). A NACA 0012 airfoil was
used to provide the first 30% of the airfoil shape. Bench tests have shown
that by using the skin of the wing to precompress the piezoelectric actuators, trailing edge deflections are increased with a factor of two, up to 15◦
peak to peak, with respect to the unloaded actuators. A maximum actuation frequency of 34 Hz can be attained. The morphing wing panels were
positioned at the outboard side of the wing in order to induce roll control.
A carbon fibre torque box was fabricated to transfer aerodynamic loads to
the inboard (static) parts of the wing. Flight tests showed excellent control
characteristics. Wind tunnel tests measured an average CLδ = 1.42[1/rad],
which translates to Clδ = 4.16[1/rad] for a two dimensional wing section.
The presence of wing loading shows a shift in deflection range. At an angle
of attack of 15◦ , the wing is not capable of deflecting downward anymore
due to the blocking force of the negative pressure.
Electronics Power consumption of PBP actuators employed in a morphing wing panel has shown to be only 3.6% of the power consumption of their
electro mechanical counterparts. An electronic circuit was designed employing linear amplifiers to increase the control signal to the required +/-100V.
Due to problems with demodulating the control signal and overheating of
the amplifiers it was decided to separate the high voltage circuit from the
low voltage control circuit and to link them mechanically. This led to a
flight worthy electronic circuit which was suitable to control the morphing
wing panels employed on the UAV. However, they were relatively heavy, big
and the control frequency was low.
7.2
Recommendations
PBP actuators Further research is needed to increase deflections of the
PBP actuator and to make it suitable to integrate in an MAV/UAV control
system. Current deflections are limited by the ultimate tensile strain of the
PZT sheets. Whenever this limit is surpassed cracks occur at the face of
the sheet which makes part of the actuator inactive. Applying thin facing
sheets of a very high stiffness at both sides of the actuator will limit the
maximum tensile strain of both PZT elements. In this case the maximum
105
7 Conclusions and Recommendations
curvatures will not be limited by the PZT sheet that is loaded in tension,
but by the compressed PZT sheet. Since PZT has an ultimate compression
stress which is an order of magnitude higher than the ultimate tensile stress,
much higher curvatures could be attained.
Increasing deflections could also be done by deliberately making cracks in
the pzt sheets. The individual panels on the actuator that occur as a result
of the cracks should be interconnected by electric wiring. Accordingly, there
will be no inactive parts on the PZT sheets. Because of the cracks, the
actuator will be able to rotate more and deflections can be increased.
Apart from increasing deflections, integration of the actuators into a control
system is one of the most important steps towards a ”plug and play” control
actuator. At this moment, PBP actuators have open loop control, meaning
that a control input correlates with voltage output to the actuators. No
information on the amount of deflection is fed back into the control system.
Since deflections of the actuator will be different for different airspeeds or
air loads, it is important to control the voltage over the actuator in order to
ensure that the input deflection is maintained.
Thin wing The main drawback of the thin wing concept is the lack of a
physical stop to protect the actuators from over rotating and thus breaking. Hard gusts or heavy landings can cause a the actuators to break very
easily. Applying a facing sheet over the elements might be a solution to
this problem. This would imply no additional elements would stick out of
the wing like for example a trailing edge bump stop would do. Additional
wind tunnel tests are necessary to determine the behavior of the wing under
aerodynamic loading. Sizing of the piezoelectric elements can be done in
conjunction with the obtained results from these tests. This way a more
optimal balance can be found between the total mass of the actuators and
their performance.
Further wind tunnel tests are required to determine the effect of wing loading
on the deflections of the wing. A dynamic test article is required, which can
be actuated in the wind tunnel. This wind tunnel experiment should record
all the aerodynamic data plus the wing deflection and the voltage input.
This requires a measurement method to determine the deflections without
obstructing the flow and thereby influencing the aerodynamic measurements.
Small strain gauges at the top and bottom of the PZT elements might be a
solution for this. The boundary layer is already turbulent over the aft part
of the wing, so no danger of separation is present. Moreover, the application
of the strain gauges might be the first step towards a closed loop control
system for the actuator.
7.2. Recommendations
106
Thick wing It has been proven that using a morphing wing based on a
symmetric NACA 0012 airfoil is suitable for roll control on a subscale UAV.
A disadvantage of this airfoil is the fact that a relatively high (take off) speed
and angle of attack is required to get the aircraft in the air. Furthermore,
leading edge separation occurs at a relatively low angle of attack (12◦ ) which
makes the wing relatively sensitive to stall. In general, Selig airfoils are
very suitable for low Reynolds number wings. However, they are cambered,
which make them unsuitable to axially precompress the PBP actuators in
the wing. It is suggested to investigate the performance of an airfoil which
has the same thickness distribution as such a well performing low Reynolds
number airfoil. Camber can be induced actively by deforming the airfoil
with the PBP actuators. In practice this will imply an airfoil with a sharper
leading edge and a thickest point which is shifted more backwards.
Thorough research is required in the skin material. The hysteresis in the
total PBP actuator is being determined mainly by the hysteresis in the skin.
During this research, natural rubber was chosen as skin material because it
easy to purchase and relatively cheap. However, the natural rubber possesses
a large amount of hysteresis (26%), which is undesirable for the actuator.
Electronics Converting direct current of low voltage to 100V is inefficient.
Losses will occur in the conversion process which are really hard to overcome.
On the other hand, converting from 100V down to a low voltage (e.g. 4.8V)
can be done very efficient. To save electrical energy from dissipation it could
be efficient for the entire aircraft to have one 100V battery pack to control
both the PBP actuators and the additional electronics (e.g. receiver and
motor). Instead of using eight 12V (non-rechargeable) batteries it might
be more convenient to use small rechargeable Lithium Polymer Cells (3.7V)
and put them in series.
In addition, instead of using linear amplifiers (opams) the use of switching
amplifiers to control the PBP actuators should be investigated. It has been
shown that switching amplifiers can operate very efficiently up to high actuation frequencies [17]. Switching amplifiers control the piezoelectric actuators
by sending a pulse signal at very high frequency and constant voltage. By altering the actuation frequency the mean voltage will be changed which can
control the actuator. Since switching amplifiers should be custom made,
complexity, wight and volume might be substantially higher than for linear amplifiers. A fair comparison between linear amplifiers and switching
amplifiers in the low frequency range should be made.
Appendix A
Classical Laminate Plate
Theory Model
This appendix will give a more in-depth analysis of the application of the
CLPT on the piezoelectric laminate. As stated in chapter 3 the analysis
follows the analysis in Jones [12]. The reader must note that not all the
subscripts that are used in this section are in conjunction with the subscripts
used in chapter two.
A.1
Actuator Dimensions
Figure A.1: Actuator dimensions. [18]
Piezoelectric actuator thickness (m):
ta = 0.191 · 10−3
107
A.2. Material Properties
108
Substrate thickness (m):
ts = 0.0762 · 10−3
Bonding layer thickness (m):
tb = 0.0635 · 10−5
Length of PBP actuator (m):
L0 = 0.0724
Total length of the laminate (m):
L0tot = 0.0889
width of PBP laminate (m):
b= 0.010
A.2
Material Properties
Material properties PZT-5A
Stiffness (Pa):
Ea1 = 61 · 109
Ea2 = 61 · 109
Ga12 = 23 · 109
µa12 = 0.3 µa21 = 0.3
Thermal expansion coefficients:
αa1 = 9 · 10−6
αa2 = 9 · 10−6
Piezoelectric charge coefficients (m/V):
d31lina = −115 · 10−12
d31linb = −3.27 · 10−16
d32lina = −115 · 10−12
d32linb = −3.27e · 10−16
Orientation of the actuator:
θa1 = 0 θa2 = 0
Material properties of the Al 6064 substrate
Stiffness (Pa):
Es1 = 100 · 109
Es2 = 100 · 109
Gs12 = 38.5 · 109
109
A Classical Laminate Plate Theory Model
µs12 = 0.3 µs21 = 0.3
Thermal expansion coefficients:
αs1 = 22.05 · 10−6
αs2 = 22.05 · 10−6
Orientation of the substrate:
θs = 0
Material properties of the bonding layer
Stiffness (Pa):
Eb = 1 · 109
A.3
Gb = 0.4 · 109
µb = 0.3
Stress-strain relations for plane stress in each
lamina
PZT stiffness matrix components
Ea1
1 − µa12 µa21
Ea2
=
1 − µa12 µa21
µa12 Ea2
1 − µa12 µa21
Qxxa =
Qxya =
Qyya
Qzza = Ga12
Rotated stiffness matrix components of upper piezoelectric element
Q11a1 = Qxxa cos4 (θa1 ) + 2(Qxya + 2Qzza )sin2 (θa1 )cos2 (θa1 ) + Qyya sin4 (θa1 )
Q12a1 = (Qxxa + Qyya − 4Qzza )sin2 (θa1 )cos2 (θa1 ) + Qxya (sin4 (θa1 ) + cos4 (θa1 ))
Q22a1 = Qxxa sin4 (θa1 ) + 2(Qxya + 2Qzza )sin2 (θa1 )cos2 (θa1 ) + Qyya cos4 (θa1 )
Q16a1 = (Qxxa − Qxya − 2Qzza )sin(θa1 )cos3 (θa1 ) + (Qxya − Qyya + 2Qzza )sin3 (θa1 )cos(θa1 )
Q26a1 = (Qxxa − Qxya − 2Qzza )sin3 (θa1 )cos(θa1 ) + (Qxya − Qyya + 2Qzza )sin(θa1 )cos3 (θa1 )
Q66a1 = (Qxxa + Qyya − 2Qxya − 2Qzza )sin2 (θa1 )cos2 (θa1 ) + Qzza (sin4 (θa1 ) + cos4 (θa1 ))
Rotated stiffness matrix components of lower piezoelectric element
Q11a2 = Qxxa cos4 (θa2 ) + 2(Qxya + 2Qzza )sin2 (θa2 )cos2 (θa2 ) + Qyya sin4 (θa2 )
Q12a2 = (Qxxa + Qyya − 4Qzza )sin2 (θa2 )cos2 (θa2 ) + Qxya (sin4 (θa2 ) + cos4 (θa2 ))
Q22a2 = Qxxa sin4 (θa2 ) + 2(Qxya + 2Qzza )sin2 (θa2 )cos2 (θa2 ) + Qyya cos4 (θa2 )
Q16a2 = (Qxxa − Qxya − 2Qzza )sin(θa2 )cos3 (θa2 ) + (Qxya − Qyya + 2Qzza )sin3 (θa2 )cos(θa2 );
Q26a2 = (Qxxa − Qxya − 2Qzza )sin3 (θa2 )cos(θa2 ) + (Qxya − Qyya + 2Qzza )sin(θa2 )cos3 (θa2 );
Q66a2 = (Qxxa + Qyya − 2Qxya − 2Qzza )sin2 (θa2 )cos2 (θa2 ) + Qzza (sin4 (θa2 ) + cos4 (θa2 ))
A.4. Resultant Laminate Forces and Moments
110
Substrate stiffness matrix components:
Es1
1 − µs12 µs21
Es2
=
1 − µs12 µs21
µs12 Es2
1 − µs12 µs21
Qxxs =
Qxys =
Qyys
Qzzs = Gs12 ;
Rotated stiffness matrix components:
Q11s = Qxxs cos4 (θs ) + 2(Qxys + 2Qzzs )sin2 (θs )cos2 (θs ) + Qyys sin4 (θs )
Q12s = (Qxxs + Qyys − 4Qzzs )sin2 (θs )cos2 (θs ) + Qxys (sin4 (θs ) + cos4 (θs ))
Q22s = Qxxs sin4 (θs ) + 2(Qxys + 2Qzzs )sin2 (θs )cos2 (θs ) + Qyys cos4 (θs )
Q16s = (Qxxs − Qxys − 2Qzzs )sin(θs )cos3 (θs ) + (Qxys − Qyys + 2Qzzs )sin3 (θs )cos(θs )
Q26s = (Qxxs − Qxys − 2Qzzs )sin3 (θs )cos(θs ) + (Qxys − Qyys + 2Qzzs )sin(θs )cos3 (θs )
Q66s = (Qxxs + Qyys − 2Qxys − 2Qzzs )sin2 (θs )cos2 (θs ) + Qzzs (sin4 (θs ) + cos4 (θs ))
Bonding layer stiffness matrix components (always considered isotropic therefore no rotation required):
Eb
1 − µ2b
µb Eb
=
1 − µ2b
Q11b =
Q12b
Q22b = Q11b
Q66b = Gb
Q16b = 0
Q26b = 0
A.4
Resultant Laminate Forces and Moments
Thickness definitions:
t1 = ts /2 + tb + ta
t2 = ts /2 + tb
t3 = ts /2
t4 = −ts /2
t5 = −ts /2 − tb
t6 = −ts /2 − tb − ta
Determination of the ABD matrix elements:
Aija1 = (Qij )a1 (t1 − t2 )
1
Bija1 = (Qij )a1 (t21 − t22 )
2
1
Dija1 = (Qij )a1 (t31 − t32 )
3
111
A Classical Laminate Plate Theory Model
Aija2 = (Qij )a2 (t5 − t6 )
1
Bija2 = (Qij )a2 (t25 − t26 )
2
1
Dija2 = (Qij )a2 (t35 − t36 )
3
Aijb1 = (Qij )b1 (t2 − t3 )
1
Bijb1 = (Qij )b1 (t22 − t23 )
2
1
Dijb1 = (Qij )b1 (t32 − t33 )
3
Aijb2 = (Qij )b2 (t4 − t5 )
1
Bijb2 = (Qij )b2 (t24 − t25 )
2
1
Dijb2 = (Qij )b2 (t34 − t35 )
3
Aijs = (Qij )s (t3 − t4 )
1
Bijs = (Qij )s (t23 − t24 )
2
1
Dijs = (Qij )s (t33 − t34 )
3
Actuator ABD-matrices:


A11a1 A12a1 A16a1
Aa1 =  A12a1 A22a1 A26a1 
A16a1 A26a1 A66a1

B11a1 B12a1
Ba1 =  B12a1 B22a1
B16a1 B26a1

D11a1 D12a1
Da1 =  D12a1 D22a1
D16a1 D26a1

B16a1
B26a1 
B66a1

D16a1
D26a1 
D66a1

A11a2 A12a2 A16a2
=  A12a2 A22a2 A26a2 
A16a2 A26a2 A66a2

Aa2

B11a2 B12a2
Ba2 =  B12a2 B22a2
B16a2 B26a2

D11a2 D12a2
Da2 =  D12a2 D22a2
D16a2 D26a2

B16a2
B26a2 
B66a2

D16a2
D26a2 
D66a2
A.5. Actuation
112
Bonding layer ABD-matrices:


A11b1 A12b1 A16b1
Ab1 =  A12b1 A22b1 A26b1 
A16b1 A26b1 A66b1

B11b1 B12b1
Bb1 =  B12b1 B22b1
B16b1 B26b1

D11b1 D12b1
Db1 =  D12b1 D22b1
D16b1 D26b1

B16b1
B26b1 
B66b1

D16b1
D26b1 
D66b1
Substrate ABD-matrices:


A11s A12s A16s
As =  A12s A22s A26s 
A16s A26s A66s

Ab2

A11b2 A12b2 A16b2
=  A12b2 A22b2 A26b2 
A16b2 A26b2 A66b2

B11b2 B12b2
Bb2 =  B12b2 B22b2
B16b2 B26b2

D11b2 D12b2
Db2 =  D12b2 D22b2
D16b2 D26b2

B16b2
B26b2 
B66b2

D16b2
D26b1 
D66b2


B11s B12s B16s
Bs =  B12s B22s B26s 
B16s B26s B66s


D11s D12s D16s
Ds =  D12s D22s D26s 
D16s D26s D66s
Laminate ABD-matrices:
Al = Aa1 + Aa2 + Ab1 + Ab2 + As
Bl = Ba1 + Ba2 + Bb1 + Bb2 + Bs
Dl = Da1 + Da2 + Db1 + Db2 + Ds
Complete ABD-matrices:
·
¸
Aa1 Ba1
ABDa1 =
Ba1 Da1
·
Ab1
ABDb1 =
Bb1
·
As
ABDs =
Bs
A.5
Actuation
Thermal loading
Bb1
Db1
¸
·
ABDa2 =
·
Aa2 Ba2
Ba2 Da2
Ab2 Bb2
ABDb2 =
Bb2 Db2
¸
·
¸
Bs
Al Bl
ABDl =
Ds
Bl Dl
¸
¸
113
A Classical Laminate Plate Theory Model
In-plane Thermal strains
T1a1 = αa1 dT sin2 (θa1 ) + αa2 dT cos2 (θa1 )
T2a1 = αa2 dT sin2 (θa1 ) + αa1 dT cos2 (θa1 )
1
T3a1 = (αa1 dT − αa2 dT )sin(2θa1 )
2
T1a2 = αa1 dT sin2 (θa2 ) + αa2 dT cos2 (θa2 )
T2a2 = αa2 dT sin2 (θa2 ) + αa1 dT cos2 (θa2 )
1
T3a2 = (αa1 dT − αa2 dT )sin(2θa2 )
2
T1s = αs1 dT sin2 (θs ) + αs2 dT cos2 (θs )
Ta1
T2s = αs2 dT sin2 (θs ) + αs1 dT cos2 (θs )
1
T3s = (αs1 dT − αs2 dT )sin(2θs )
2





T1a1
T1a2
 T2a1 
 T2a2 






 T3a1 
 T3a2 





=
Ta2 = 
Ts = 



 0 
 0 

 0 
 0 

0
0
T1s
T2s
T3s
0
0
0








Strains:
²T 11a1 = T1a1
²T 11a2 = T1a2
²T s11 = T1s
Electrical loading
Polarization of the piezoelectric elements:
p1 = 1
p2 = −1
Determination of the electrical strain coefficients:
d31 = d31lina + d31linb (V3 /ta )
d32 = d32lina + d32linb (V3 /ta )
Piezoelectric element strain caused by electrical field in upper piezoelectric
element:
p1 V3 d32
p1 V3 d31
sin2 (θa1 ) +
cos2 (θa1 )
Λ311 =
ta
ta
A.5. Actuation
114
Λ321 =
Λ361
p1 V3 d32
p1 V3 d31
sin2 (θa1 ) +
cos2 (θa1 )
ta
ta
1 ³ p1 V3 d31 p1 V3 d32 ´
=
−
sin(2θa1 )
2
ta
ta
Total electrical strain vector upper element:


Λ311
 Λ321 


 Λ361 


Λa31 = 

 0 
 0 
0
Piezoelectric element strain caused by electrical field in lower piezoelectric
element:
p2 V3 d31
p2 V3 d32
Λ312 =
sin2 (θa2 ) +
cos2 (θa2 )
ta
ta
Λ322 =
Λ362
p2 V3 d31
p2 V3 d32
sin2 (θa2 ) +
cos2 (θa2 )
ta
ta
1 ³ p2 V3 d31 p2 V3 d32 ´
−
=
sin(2θa2 )
2
ta
ta
Piezoelectric element strain vector lower element


Λ312
 Λ322 


 Λ362 


Λa32 = 

0


 0 
0
Resultant Strain
·
Al Bl
Bl Dl
¸µ
²
κ
¶
=
l
·
·
¸
·
¸
Aa1 Ba1
Aa2 Ba2
Λa31 +
Λa32 +
Ba1 Da1
Ba2 Da2
¸
·
¸
·
¸
Aa1 Ba1
Aa2 Ba2
As Bs
Ta1 +
Ta2 +
Ts
Ba1 Da1
Ba2 Da2
Bs Ds
Appendix B
PBP Simulation
In this appendix an exact copy of the Matlab code that was used to solve
equation 3.47 is presented [3].
clc clear close all
format short
% Everything in this m-file is based on equation 18 of SPIE paper no 5762-16
% ’Post-buckled precompressed (PBP) benders: a new class of grid-fin
% actuators enhancing high speed autonomous VTOL MAV’s’
PZT_5A
% actuator dimensions:
% Lay-up:
ta = 0.191e-3
ts = 0.0762e-3
tb = 0.0635e-5
% PBP dimensions
L0 = 0.0724
L0tot = 0.0889
b=0.010
;% piezo actuator thickness (m)
;% substrate thickness (m)
;% bonding layer thickness (m)
;% length of piezo actuator (m)
;% length of flex spar (m)
;% width of PBP laminate element [m]
for V3=[20 40 60 70 80 90 100]; %
E3=V3/(ta+tb);
%
d31lin= -(d31lina + d31linb*E3);%
Lambda =d31lin*E3;
%
% Pre-compressive force
Fcr=6.18;
Input Voltage
Electric field (V/m)
Piezoelectric constant (m/V)
Free strain actuator
% buckling load [N]
115
116
Fcrgmf=Fcr/0.009808;
for i=0.001:0.1587/2:1
Fagmf=i*Fcrgmf;
Fa=i*Fcr;
% buckling load [gmf]
% pre-compression force [gmf]
% pre-compression force [N]
% Correction factors
C1=(-3.2e-3)*V3+0.76; C2=((4e-3)*V3+1.25)*C1;
% Definition of the D and B components:
B= C1*(Ea1*(ts*ta+2*tb*ta+ta^2));
D= C2*((Es1*ts^3)/12+Ea1*((ta*(ts+2*tb)^2)/2+(ts+2*tb)*ta^2+ (2/3)*ta^3));
% Initial Curvature [rad/m] (without pre-compression):
kappa=(B/D)*Lambda;
% Left side of equation 18:
sol1=(L0/2)*sqrt(Fa/(D*b));
% Initial values:
sol2=0;
delta0=0;
% [rad]
% Finding delta0 which compares to the given pre-compressive force
for counter=1:10000 if sol1>sol2;
delta0=delta0+.0005;
% Right side of equation 18
dxi=pi/200; xi=(0:dxi:pi/2);
f=(sin(delta0./2)).*(cos(xi))./(sqrt(((sin(delta0./2)).^2).*(cos(xi)).^2+...
((kappa.^2).*D.*b)/(4.*Fa)).*sqrt(1-((sin(delta0./2)).^2).*(sin(xi)).^2));
ff=f*dxi; sol2=sum(ff); end end
delta0=delta0*180/pi;
end end
plot(delta0,Fagmf,’.’)
xlabel(’Rotation, delta0 (1/2 peak to peak) (deg)’)
ylabel(’Axial force (gmf)’)
title(’Axial force vs. Rotation’) axis([0 5 0 600]) hold on
Appendix C
Leading Edge Geometry
Before Xfoil can be used, the shape of the leading edge must be mathematically determined. Figure C.1 shows how the leading edge contour is
divided into three different parts. The top-front part is being represented
by a quarter part of an ellipse:
Figure C.1: Definition of the leading edge geometry of the airfoil.
x1 = a cos θ1 ,
π/2 < θ1 < π
y1 = b sin θ1
a = x0 + r2 ,
b = yp
A quarter circle marks the bottom side of the leading edge contour:
x2 = r2 cos θ2 − x0 ,
π < θ2 < 3π/2
y2 = r2 sin θ2
The last line is the aft part of the leading edge. This is represented as an
inverse cosine shape:
x + x0
y3 = d + A cos(
π)
xp + xo
117
118
A=
yp − t + r1
,
2
d = A − r1
These three curves together determine the shape of the leading edge. In
order to change the shape, essentially four variables can be altered: x0 , xp ,
r2 , and yp . As stated in the first paragraph of this section, the leading
edge can be maximally 30% of the entire chord (c) of the airfoil. If it was
to be made larger, there would not be much left of the morphing wing
concept. The design would then degenerate to an ordinary wing with a
fancy flapping mechanism for roll and direct lift control. That is not the
goal of this investigation. Consequently one of the constraints is:
r2 + x0 + xp < 0.30 · c
(C.1)
As established in the preliminary design, the length of the flat part of the
airfoil amounts to 82 mm. Consequently, the maximum length of the leading
edge amounts to 34 mm. The maximum thickness of the leading edge,
determined by adding the value of yp and r2 , is unconstrained. However, for
practical purposes the investigation is limited to a maximum of 15% of the
chord length:
r2 + yp < 0.15 · c
(C.2)
In practice this means that the maximum thickness of the leading edge is
17 mm. All variables represent distances, so a final set of constraints can
be added:
x0 , xp , r2 , tp > 0
(C.3)
Appendix D
Trailing Edge Analysis
The function of the trailing edges of the thin airfoil wing is threefold. It has
to provide an aerodynamically favorable end of the airfoil. It has to transfer
the precompression force to the adaptive laminates. And, it provides a frame
for the flexible aluminum skin. The first of these functions dictates a thin,
sharp trailing edge. This is not much of a problem, since the aft part of the
wing is not thicker as 0.5 mm. The load introduction of the precompression
force on the adaptive laminates does require some more thought.
The buckling load for a 2 cm wide element amounts to 11.5 N [8]. If a
precompression of 90% of the buckling load is applied, that means that each
rubber band must apply a force of 10.4 N . The load on the laminates is
being introduced, using thin steel rods (diameter: 0.5 mm). When only this
rod would be applied, the PZT laminates would be asymmetrically loaded
(compare figure D.1). To counteract this undesirable effect, a broader beam
is required.
(a) Deflection
(b) Force
Figure D.1: Asymmetric loading of the adaptive laminate.
119
120
In the preliminary design a trailing edge with a width of 1 cm is proposed.
A thickness of 0.5 mm would make the trailing edge as thick as the adaptive
laminate. A glass fibre reinforced polymer is chosen as trailing edge material.
The calculations in this section will show that such a trailing edge will be
stiff enough to assume it to be rigid. Figure D.2 shows a side view on the
adaptive part of the airfoil. The airfoil is deflected over a certain angle, δ.
The angle θ is larger than δ, causing an out of plane force on the trailing
edge, P 0 . The difference between both angles, ∆, determines the value of
P 0 . For large tip deflections (6.4mm is assumed to be the maximum), ∆
amounts to 3.2 degrees. The in-plane force will consequently almost be
equal to P , while the out of plane force, P 0 = P cos ∆ = 0.66 [N ].
Figure D.2: Pre-compression force on the trailing edge.
The force on the trailing edge will cause a moment in plane of the trailing
edge (in the direction of P 0 ). A model of the two laminates and the two
rubber bands is depicted in figure D.3. The trailing edge is represented as a
simply supported beam. The laminates are modeled as two simple supports.
The distance, l, is the width of one panel (95 mm). The distance to the
point of application of the force and the laminate is denoted by d. The cross
section of the trailing edge is drawn to the right of the beam.
(a) Simply supported beam
(b) Cross section
Figure D.3: Model of trailing edge loading.
The maximum moment is constant between the two supports: M = P d.
The curvature, κ,is given by:
M
κ=
(D.1)
EI
121
D Trailing Edge Analysis
The E-modulus for glass fibre reinforced polymer is assumed to be 25 GP a.
The moment of inertia about the vertical axis through P 0 ,I1 , is calculated
as follows:
1
I1 = tw3
(D.2)
3
Inserting this in equation D.1 gives κ = 9.53 · 10−12 [1/mm]. The amount
of bending is so small that it can be assumed to be zero. Consequently the
laminates will be loaded uniformly.[11]
For the force P 0 an analogous approach can be applied. The only difference
is that the moment of inertia must now be taken around the cross section’s
axis of symmetry:
1
I2 = wt3
(D.3)
12
The moment, M = P 0 d. Substituting this in equation D.1 yields κ = 9.78 ·
10−10 [1/mm]. Although this is a factor 100 larger than the value found for
the in-plane bending, it still is very small. Therefore, also this curvature is
neglected.
Finally, the force P 0 will cause a moment around the center of mass of the
cross section. To prevent torsional deformation the fibres in the laminate
could be placed at a relative angle of 45 degrees. Figure D.4 shows how the
fibres should be oriented to counteract the torsional moment.[11]
Figure D.4: Fibre orientation in the trailing edge (top view).
122
Acknowledgements
The past year has been an adventure for me, trying to design and build the
first morphing wing, actuated by piezoelectric materials. I was not alone in
this adventure and there are a lot of people that I would like to thank for
helping me bringing it to a good ending. First of all my esteemed mentor
in Delft, ir. Lars Krakers and my supervisor, prof. dr. ir. Michel van
Tooren. Also at the faculty of aerospace engineering, I would like to thank
dr. Mostafa Abdallah (MSc) and dr. ir. Leo Veldhuis.
But what is an engineer if there are no people who can tell him how to build
his design? Therefore I would like the following people at Delft Aerospace
for their practical help in the field of composites, electronics and wind tunnel
tests: ir. Maarten Labordus, Hans Weerheim and Leo Molenwijk. My fellow
students with whom I did this project with, ir. Roeland de Breuker, Ross
McMurtry and Domenico Casella, thank you for all your efforts that made
this project to a success.
A special thank you is reserved for the person that inspired me, motivated
me, and laid the basis for this project. That is my supervisor at Auburn
University, and my dear friend, dr. Ron Barrett. His creativity really inspired me in my project work. For technical advise, also at AU, I thank
Christoph Burger, without whom I would never have been able to make
a flight worthy morphing wing. At the aerospace department there are a
number of people I would like to thank for helping me, either with small
technical adjustments, or with big building projects: Lori Prothero, Shannon Wheatley, Mike Brennison, Amanda Gilliand, Ryan Leurck, Richard
Bramlette, Robert Love and Brooke Hill.
123
124
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Dame, 117 Hessert Center
[25] Drela, Mark and Youngren, Harold ”XFOIL 6.94 User Guide” MIT and Aerocraft, Inc. 10 Dec 2001. http://raphael.mit.edu/xfoil/xfoil doc.pdf
[26] www.defense-aerospace.com; found on September 16th , 2004.
[27] www.newscientist.com/news; found on September 16th , 2004.
[28] www.aae.uiuc.edu/m-selig/uiuc-lsat.html; found September 23rd , 2004
[29] www.nd.edu/ mav/competition.htm; found September 21st , 2004
127
BIBLIOGRAPHY
[30] www.aae.uiuc.edu/m-selig/uiuc-lsat.html; found September 23rd , 2004
[31] www.nd.edu/ mav/competition.htm; found September 21st , 2004
[32] www.epson.co.jp/e/newsroom; found September 16th , 2004
[33] www.ae.msstate.edu/ masoud/Teaching/SA2; found October 10th , 2004
[34] www.picoelectronics.com/dcdclow/pe62 63.htm; found December 13th , 2004
[35] www.engineering.gr/design/home; found January 10th 2005
[36] www.darpa.mil/dso/thrust/matdev/mas.htm; found August 9th 2005
[37] www.afrl.af.mil/accomprpt/dec02/images/dec7 a.jpg; f oundAugust9th 2005
[38] www.libraries.wright.edu/icons/special/flyer.gif; found August 9th 2005
[39] www.texasbeyondhistory.net/forts/images/eagle-sm.jpg; found August 9th
2005
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