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Test and Modelling of Four Rotor
Helicopter Rotors
Anders Hedeager Pedersen
s011258
DTU Ørsted October 2006
Master Thesis
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Abstract
In this thesis two different rotors for small quadrotor helicopters have been
modelled and tested. From this rotor model a fully nonlinear helicopter model is
described. In order to perform sufficient tests of the rotors it was necessary to
build a force and torque measuring instrument.
This report presents the theoretical background for rotor performance and the
derivation of a nonlinear rotor model. This rotor model is calibrated by the use of
tests performed inside and outside a wind tunnel. In this project two different
rotors and performance have been tested with 24 different test configurations.
With use of the model and the tests it is concluded which rotor that has the best
performance for the small quadrotor helicopter
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Index
1 INTRODUCTION .................................................................................................................1
1.1 BRIEF HISTORY OF QUADROTOR HELICOPTER..................................................................2 1.2 PRINCIPLE BEHIND THE QUADROTOR HELICOPTER ..........................................................3
2 QUADROTOR HELICOPTER DYNAMICS ....................................................................5
2.1 ATTITUDE REPRESENTATION...........................................................................................5 2.2 NONLINEAR MODEL ........................................................................................................6
3 QUADROTOR HELICOPTER ROTOR ANALYSIS.......................................................9
3.1 HELICOPTER ROTORS ......................................................................................................9 3.2 METHODS OF ROTOR ANALYSIS.....................................................................................10 3.3 BLADE ELEMENT...........................................................................................................11 3.4 AERODYNAMIC BEHAVIOUR..........................................................................................22
4 MATLAB MODEL..............................................................................................................25
4.1 HELICOPTER MODEL .....................................................................................................25 4.2 ROTOR MODEL ..............................................................................................................26 4.3 SIMULATION .................................................................................................................30
5 TEST SET-UP......................................................................................................................33
5.1 MOTOR AND MOTOR CONTROLLER................................................................................33 5.2 THRUST AND DRAG METER ...........................................................................................34 5.3 SIX DIMENSION TRANSDUCER .......................................................................................39 5.4 WIND TUNNEL...............................................................................................................40 5.5 STROBOSCOPE...............................................................................................................40 5.6 HIGH SPEED CAMERA ...................................................................................................41
6 ESTIMATION OF CONSTANTS .....................................................................................43
6.1 ROTOR CONSTANTS.......................................................................................................43 6.2 HELICOPTER CONSTANTS ..............................................................................................48
7 TEST OF THE ROTOR .....................................................................................................49
7.1 TESTING INSTRUMENTS.................................................................................................49 7.2 FLAPPING ANGLES.........................................................................................................50 7.3 ZERO LIFT ANGLE..........................................................................................................50 7.4 THRUST.........................................................................................................................51 7.5 DRAG ............................................................................................................................51 7.6 THE THRUST/DRAG RELATIONSHIP ................................................................................51 7.7 SMOKE TESTS – AERODYNAMIC BEHAVIOUR .................................................................52
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8 CALIBRATION OF ROTOR MODEL.............................................................................53
8.1 THRUST HOVER .............................................................................................................54 8.2 THRUST HORIZONTAL FLIGHT........................................................................................56 8.3 DRAG HOVER ................................................................................................................63 8.4 DRAG IN HORIZONTAL FLIGHT.......................................................................................64 8.5 OTHER KINDS OF TORQUE IN FLIGHT .............................................................................65 8.6 SUMMERY OF CALIBRATION OF THE SINGLE ROTOR MODEL...........................................66
9 WHITE VS. BLACK ROTOR............................................................................................71
9.1 THRUST AND DRAG .......................................................................................................71 9.2 THRUST/DRAG RELATIONSHIP .......................................................................................72 9.3 EARLIER OBSERVATIONS ...............................................................................................73 9.4 SUMMARY .....................................................................................................................73
10 CONCLUSION ....................................................................................................................74
10.1 PURPOSE BUILT INSTRUMENT ........................................................................................74 10.2 NONLINEAR MODEL.......................................................................................................74 10.3 WHITE VERSUS BLACK ROTOR.......................................................................................75 10.4 FUTURE IMPROVEMENTS ...............................................................................................76
11 BIBLIOGRAPHY................................................................................................................77
A ZERO LIFT ANGLE. .................................................................................................................1
A.1) ZERO LIFT ANGLE ............................................................................................................1
B MICROPROCESSOR DESIGN.................................................................................................1
B.1) INSTRUCTION MANUAL....................................................................................................1
C TEST DATA.................................................................................................................................1
D TEST RESULT............................................................................................................................1
D.1) ESPECIALLY BUILT THRUST AND DRAG METER VS. FS6..................................................1 D.2) WALL EFFECTS................................................................................................................3 D.3) WHITE OR BLACK ROTOR ................................................................................................4 D.4) ZERO LIFT ANGLE ............................................................................................................6 D.5) FLAPPING ANGLES...........................................................................................................7 D.6) THRUST, HOVER ............................................................................................................12 D.7) THRUST, HORIZONTAL FLIGHT.......................................................................................13 D.8) VERTICAL THRUST IN HORIZONTAL FLIGHT ...................................................................14 D.9) HORIZONTAL THRUST IN HORIZONTAL FLIGHT ..............................................................15 D.10) DRAG, HOVER ...............................................................................................................17 D.11) DRAG, IN FORWARD FLIGHT ..........................................................................................18 D.12) THRUST/DRAG RELATIONSHIP .......................................................................................20
E MATLAB MODEL......................................................................................................................1
F TECHNICAL DRAWINGS ........................................................................................................1
G ENCLOSED DVD .......................................................................................................................1
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List of Symbols
A Area of disc [m2]
a Slope lift curve []
B Outer effective radius factor []
b Number of blades []
QC Coefficient of Torque []
TC Coefficient of thrust []
c Blade chord [m]
dc Coefficient of drag []
lc Coefficient of lift []
F Forces acting at the quadrotor [N]
g Acceleration due to gravity [m/s2]]
H Horizontal force induced by the drag [N]
Ih Moment of inertia of the quadrotor [kg/m2]
Ib Moment of inertia of the rotor blade [kg/m2]
M Torque induced from the thrust [Nm]
mh Mass of the quadrotor [kg]
Q Torque (Drag) [Nm]
R Rotor radius [m]
R.N. Reynolds Number
T Thrust [N]
UP Velocity perpendicular to the blade
UT Velocity tangential to the blade
V Velocity of the quadrotor [m/s]
1v Induced velocity [m/s]
x0 Inner effective radius factor []
R Rotation matrix of the quadrotor
α Angle of attack [rad]
0Lα Zero lift angle [rad]
sα Angle of the disc plane [rad]
β Flapping angle [rad]
θ Blade pitch [rad]
µ Inflow ratio []
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ρ Density of air [kg/m3]
σ Solidity of rotor []
φ Inflow angle [rad]
Γ Torques acting on the quadrotor [Nm]
ψ Azimuth position of blade, zero is at the negative x axis [rad]
Ω Rotational velocity of the quadrotor [rad/s]
ω Rotational velocity of rotor [rad/s]
ζ Position of the quadrotor [m]
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1 Introduction
This thesis sets up and calibrates a nonlinear model of a quadrotor helicopter and
compares the use of two different rotor types.
The small quadrotor helicopter principle is in it self very simple, it consists of a
frame with four rotors. The speed of each of the four rotors is used to control
velocity and direction of the quadrotor helicopter.
The background for this thesis is a Master Thesis “Modelling and Control of a 4-
Rotor Helicopter” by A. Bertelsen and S. Magnússon spring 2004 [1] and their
later work with quadrotor helicopters. In their thesis they succeeded in creating,
designing, and controlling a small quadrotor helicopter.
The present control is based on a model of the quadrotor helicopter linearized
about hover. The linearzed model is based on some simple assumptions of rotor
dynamics. These assumptions are:
• The rotor is rigid, no feathering of the blades.
• Ideal aerodynamic flow around the rotor blade.
• No interference between the rotors.
These assumptions may or may not be valid in other conditions than in hovering
flight; this will be examined in this thesis.
The earlier flight tests are all done in hover or at low velocities (0-3 m/s). Thus
there is a desire of Bertelsen and Magnússon to investigate how the quadrotor will
react in horizontal or vertical flight above 3 m/s. In this thesis the performance of
the quadrotor rotors are tested and it is described how high horizontal velocity
influences the quadrotor. From traditional helicopters it is known that there are
unwanted horizontal forces and torque in horizontal flights.
Furthermore a new model of rotor blade to the quadrotor helicopter has just
entered the market. The new rotor is black and the hitherto used rotor is white,
therefore in this project the rotors will be denoted the white and the black rotor.
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This new black rotor should according to the manufacture perform better
performance than the old white one, thus the black rotor model is included in the
research and it is determined which rotor provide the best performance of the
quadrotor.
Summarized the purpose of this thesis is to:
• Describe and by test calibrate a nonlinear model of the quadrotor
helicopter including all relevant factors.
• With tests and calculations on the white and the black rotors describe the
advantages and disadvantages of the new rotors compared with the
hitherto used rotors.
• Build and calibrate the necessary test equipment.
1.1 Brief history of quadrotor helicopter
The quadrotor idea is not new; actually the first quadrotor helicopter was built
earlier than the cyclic/collective pitch helicopter we know today. One of the first
was built in France by the brothers Louis and Jacques Bréguet [3]. It was called
“gyroplane nr 1” and consisted of four rotors in a square. The tests flights did not
end up very well; the helicopter was simply too difficult to control for the pilot.
Figure 1-1: In 1907 the brothers Lous and Jacques Bréguet tested the first four rotor
helicopter, called "gyroplane nr 1"[3].
Even though the concept of the quadrotor helicopter is simpler than the traditional
cyclic/collective pitch helicopter; it was not further developed until the
introduction of a control system to control the helicopter.
In the beginning of the 21st century a number of research projects of the quadrotor
helicopter were done, some of the first were Hamel 2002 [4], Pounds 2002 [5] and
Suter 2001 [6]. All these papers were concerned with designing and controlling a
small unmanned four rotor helicopter.
The question is why it is interesting now and has not been for 100 years. This is
because the idea behind it is very good, as the basic control is simple (just
increasing the lift on some of the rotors) and there are no unwanted horizontal
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forces during forward flight. This is in contrast to the normal helicopter today,
where the pitch angle must be adjusted over one revolution of the rotor which
makes the control more complicated and gives a lot of wear on the system.
Today the quadrotor helicopter configuration is found mostly in radio controlled
toy helicopters. One of the most known manufactures is RC-toys [8] with the
model Draganflyer, from which the is the model used in this project.
The future prospect of the quadrotor is the development of a small foldable model
helicopter which can be used for inspection where no human can go. This could
be aerial electricity cables, into mines after accidents or into unsafe buildings. The
quadrotor may also be used for future TV surveillance from above by replacing
the use of a full size helicopter or a lift.
1.2 Principle behind the quadrotor helicopter
The quadrotor helicopter consists of a frame with four rotors mounted in the
corners, see Figure 1-2. The two rotors r1 and r3 have positive or counter
clockwise rotation while the two others have negative or clockwise rotation.
Figure 1-2: The quadrotor helicopter, top view, with the body frame drawn.
Altitude and velocity control is obtained by regulating the thrust and drag of the
four rotors. The trust and drag can be controlled by the pitch angle or the angular
velocity. In small quadrotor helicopters with electric motors the simplest method
is to use the angular velocity, thus is it also used in this project.
When all rotors have same angular velocity, the quadrotor is in hover or is in
vertical flight. Increased horizontal velocity in the x direction is obtained by
increasing the angular velocity of r2 and r3, while increase of r3 and r4 increases
velocity in the horizontal y direction. Increased rotational velocity around the
vertical z axis is obtained by increasing the angular velocity of r1 and r3, while
negative angular velocity is obtained by increasing the velocity of r2 and r4.
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2 Quadrotor Helicopter Dynamics
This chapter describes a mathematical nonlinear model of the quadrotor
dynamics. This setup of a model is important to determine which factors that have
influence of the dynamics of the system. It is also important with a detailed
mathematical model if a proper controller has to be dimensioned.
2.1 Attitude representation
The quadrotor helicopter is configured with four rotors placed in a square as
shown in Figure 2-1. These four rotors are named r1, r2, r3 and r4, according to
the quadrant in the body-fixed coordinate system in which they are placed. The
two rotors r1 and r3 run with counter-clockwise angular velocity while the two
other rotors r2 and r4 run with clockwise angular velocity.
Figure 2-1: Quadrotor helicopter configuration
In order to describe the motion of the quadrotor it is necessary to have a reference
frame. The reference frame is rigid and right hand where the z axis has the same
direction as the acceleration due to gravity. The x-y plane is given by the
horizontal plane and the position in the reference frame is given by:
[ ], ,r r r rP x y z=
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A body frame in the quadrotor body is defined to describe the position of the
forces acting on the quadrotor. The initial rotation of the body frame is the same
as the reference frame where zb is in the positive downwards direction. The
position in the body frame is given by:
[ ], ,b b b bP x y z=
The two frames and their initial rotation are seen in Figure 2-1.
The relation between the reference and body frame is the rotation matrix
b r: P P→R . The rotation matrix is constructed from pitch, yaw, roll, and is
described by the three Euler angles ( ), ,η φ θ ψ= .
c c s s c c s c s c s s
c s s s s c c c s s s s
s s c c c
θ φ ψ θ φ ψ φ ψ θ φ ψ φ
θ φ ψ θ φ ψ φ ψ θ φ ψ φ
θ ψ θ ψ θ
− −
= + − −
R (2.1)
where cα = cos(α) and sα = sin(α)
Thus a position in the body frame is in the reference frame given by:
r r b
P Pζ= +R (2.2)
where r
ζ is the position of the origin of body frame in the reference frame.
And a position in the reference frame is in the body frame given by:
p r rP Pζ= − + TR (2.3)
The linear and angular velocity vectors of the quadrotor in the body frame are
denoted , ,x y z
V V V V= and , ,φ θ ψΩ = Ω Ω Ω respectively.
2.2 Nonlinear model
To obtain Newton’s laws the linear acceleration and angular acceleration are
given by:
h
m V F= (2.4)
h
I Ω = Γ (2.5)
where F is the force and Γ is the torque acting on the helicopter, mh is the mass of
the helicopter, and Ih is the moment of inertia of the helicopter.
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The derivative of the position r
ζ and rotation R in the reference frame of the
helicopter is given from the derivative of the velocity and angular velocity in the
body frame.
Vζ = R (2.6)
( )sk= ΩR R (2.7)
where the notation ( )sk a is the skew-symmetric matrix leading to ( )sk a b a b= ×
for vectors in 3v ∈ℜ .
This force and torque are the sum of a number of components; these components
are described in the following subsections.
2.2.1 Force components
Fgravity
The force due to gravity is given by the mass of the helicopter, mh, the
acceleration due to gravity, g, and the rotation matrix:
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T
gravity hF m g e= R (2.8)
Fthrust
The total force from the thrust is given by the sum of the thrust from each of the
four rotors:
1, 2, 3, 4
thrust i
r r r r
F T= ∑ (2.9)
Frotation
The cross product of the angular velocity of the helicopter and the helicopter
velocity results in a force perpendicular to both of them.
rotation h
F m V= − Ω× (2.10)
FH
The H force is the horizontal forced induced by the drag of the rotor blades. It is
derived in chapter 3. The sum of the forces is given by:
1, 2, 3, 4
H i
r r r r
F H= ∑ (2.11)
Fdrag
When the helicopter flies through the air there is a negative drag force in the flight
direction. The drag is given by [2]:
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2drag d
F V Acρ= − (2.12)
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where ρ is the density of the fluid, A is the area of the quadrotor towards the
wind and d
c is a drag coefficient. The drag coefficient depends on the shape of
the quadrotor.
2.2.2 Torque components
dragΓ
The total torque from the drag is given by the sum of the drag from each of the
four rotors:
1, 2, 3, 4
drag i
r r r r
QΓ = ∑ (2.13)
thrustΓ
The torque induced from the thrust consists of two parts. The first one is the
torque each of the four rotors gives to the helicopter because the lift is not at the
centre of mass of the helicopter. The other one is the torque from the non-uniform
lift distribution on the rotor. The subdivision is further described in chapter 3 and
for now the total torque from the thrust is denoted Mi:
1, 2, 3, 4
thrust i
r r r r
MΓ = ∑ (2.14)
HrotationΓ
The resulting torque from the helicopter rotation is given by:
Hrotation h
IΓ = −Ω× Ω (2.15)
RrotationΓ
Likewise there is a resulting torque from the rotor rotation:
1, 2, 3, 4
Hrotation r i
r r r r
I ωΓ = − Ω×∑ (2.16)
2.2.3 Final nonlinear model
The total force is given by ((2.8)-(2.12)):
gravity thrust drag rotation H
F F F F F F= + + + + (2.17)
And the total torque is given by ((2.13)-(2.16)):
drag Thrust Hrotation Rrotation
Γ = Γ + Γ + Γ + Γ (2.18)
Then the differential equations describing the quadrotor helicopter become:
Vξ = R (2.19)
[ ]3
1, 2, 3, 4
T
h h h i i
r r r r
m V m V m g e T H= − Ω× + + +∑ R (2.20)
( )sk= ΩR R (2.21)
[ ]1, 2, 3, 4 1, 2, 3, 4
h h r i i i
r r r r r r r r
I I I Q MωΩ = −Ω× Ω − Ω× + +∑ ∑ (2.22)
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3 Quadrotor Helicopter Rotor Analysis
The previous chapter described a dynamic model of a quadrotor helicopter, with
the assumption that each rotor gives a well defined thrust, thrust-torque and drag
torque. This chapter provides an introduction to the rotors used in the project and
derives a rotor model to be used to find these three elements.
3.1 Helicopter rotors
The main factor influencing helicopter performance is the rotor. Thus if the
purpose is to increase the performance of for example speed, stability, lift, of a
helicopter, then rotor optimization is the best place to start. During this project
two different rotors have been analysed, the white and the black rotor. This
subsection provides an introduction to the two rotor types.
Figure 3-1: The two rotors used in the project, the black rotor and the white rotor.
The two rotors (shown in Figure 3-1) are both sold by RC-toys [8], and they both
fit the Draganflyer V quadrotor helicopter, which is one of the most known toy
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quadrotor helicopters. The Draganflyer V frame is widely used in quadrotor
construction and it is also used on the quadrotor at DTU.
3.1.1 White rotor
The white rotor is the original one, sold with the earliest models. It is made of a
thin (1.3 mm) plastic sheet which has been curved using high pressure and heat.
An area of 5 mm along the advancing edge is roughened, which should give it a
better aerodynamic behaviour, ss described in section 3.4. The length of the blade
is: Rw=0.14 m. The shape of the airfoil of the blade is shown in Figure 3-2
3.1.2 Black rotor
The black rotor is the new model from Draganfly and according to their website
the new rotor should be a major improvement. Phrases like “Indestructible”,
”greatly increase flight performance” and ”resistant to damage” are widely used in
the description of the rotor [8].
The improvement should be that it is made of injected nylon. This process has the
advantage that if the mould is made adequately accurate, a better and more precise
airfoil can be made. Compared to the white rotor the black one is less flexible but
it is longer, with Rb=0.16 m.
The surface of the black rotor is smoother than the white one, and there is no
roughened area. This may result in worse performance (section 3.4.).
Figure 3-2: The white and the black rotor seen from the tip.
3.2 Methods of rotor analysis
This analysis of the rotor takes into account all factors that have influence on the
rotor performance. There are fundamentally two methods to analyse the rotor
performance, the energy method and the blade element method.
The energy method uses the principles of energy conservation and Newton’s laws
of motion. It makes the assumption that the rotating blades can be seen as a
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uniform disc, and therefore it does not take factors like; flapping, dynamic twist
and tip loss into account; these factors will be described later. The very good thing
about the energy model is that it gives a fast and simple estimate of the rotor
performance. This is used to obtain an initial idea of if the rotor satisfies the
conditions of maximum thrust or minimum drag.
Figure 3-3: Orientation of blade element and local velocities, source Prouty [2] figure 1.6.
The blade element method is more computationally heavy. It looks at the blade as
an infinite number of lift elements. (see Figure 3-3). For each of these elements
the lift and drag vectors are calculated and integrated to get a model of the whole
rotor. It is possible for each of these elements to take into account all the factors
that influence rotor performance. It is difficult to set up the blade model with all
relevant factors, but this provides an exact estimate of how the rotor performs.
Because this project intends to find the most exact model of the rotor, the blade
element model is chosen here.
3.3 Blade element
As mention above, the blade element method divides the blade up into a number
of blade elements and calculates the lift and torque vector for each element. The
lift of the element is given by [2]:
( )2
2l
L r c c rρ
ω∆ = ∆ (3.1)
where ρ is the density of air, ω is the angular velocity of the rotor, c is the
chord, cl is the lift coefficient, and r is the radius from the blade element to the
centre of the rotor, see Figure 3-3, middle figure.
The torque of the blade element, Q∆ , is divided into two parts; the profile drag
times the radius arm to the element, and the induced drag times the radius arm to
the element.
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The induced drag comes from the lift [2]:
i
D Lφ∆ = ∆ (3.2)
where φ is the inflow angle.
And the profile drag is given by [2]:
( )2
02
dD r c c r
ρω∆ = ∆ (3.3)
This gives a horizontal drag force:
( ) ( )2 2
2 2l d
H r c c r r c c rρ ρ
ω φ ω∆ = ∆ + ∆ (3.4)
and a drag torque at the motor shaft:
( ) ( )2 2
2 2l dQ r r c c r r c c r
ρ ρω φ ω
∆ = ∆ + ∆
(3.5)
3.3.1 Lift on a blade element in hover
The lift coefficient for fixed wing with a thin cambered airfoil, is given by [9]
( )0l Lc a α α= − (3.6)
where a is the slope of the lift curve, α is the angle of attack to the air (or here
also the geometric pitch of the chord), and 0Lα is the zero lift angle of the chord.
For a wing with a symmetrical airfoil the zero lift angle is equal to zero, but for
the thin cambered airfoil of the quadrotor helicopter, the zero lift angle is not
necessary equal to zero.
For a helicopter the relative angle of attack to the air is not equal to the
geometrical pitch. For a rotor in hover the rotor disc induces a velocity, v1, that
gives a different angle of attack (as shown in Figure 3-3 left and right). This angle
uses the geometric pitch θ , and the inflow angle φ :
α θ φ= + (3.7)
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The inflow angle is given by:
1arctanv
rφ
ω
=
(3.8)
Because most rotors have an inflow angle less than 0.2 rad, the small angle
assumption will be used here:
1v
rφ =
Ω (3.9)
Therefore it gives a lift coefficient in hover:
10l L
vc a
rθ α
= − − Ω
(3.10)
Thus the increment of the lift of the blade element in hover is:
( )2 1
02
L
vL r a c r
r
ρθ α
∆ = Ω − − ∆ Ω
(3.11)
3.3.2 Lift on a blade element in flight
As shown above, it is the local angle of attack that is the important factor in
determining the lift of the blade element. Figure 3-4 shows the angle of attack and
the pitch angle and the inflow velocities for the thin cambered airfoil. From this a
general expression for the angle of attack can be derived:
0 arctan PL
T
U
Uα θ α
= − +
(3.12)
where Up is a velocity that is perpendicular to the blade, and UT is a velocity
tangential to the blade.
14
Figure 3-4: Angle of attack of a blade element
Again it is expected that the inflow-angle is small, therefore the small angle
assumption is used.
arctan P P
T T
U U
U Uφ
=
(3.13)
This new angle of attack can be inserted in the incremental lift on a blade element
(3.11):
( )2
02
PT L
T
UL U a c r
U
ρθ α
∆ = − + ∆
(3.14)
The rotor thrust can be computed from the lift of each blade element. First the
integration is done along the radius of a blade (0 to R). The lift is found to vary
along the azimuth angle, ψ, which is defined with ψ = 0 over the negative xb axis,
see Figure 3-5. Therefore the resulting lift is found as the integration of the lift
along the azimuth angle, divided by one revolution (2π).
2
0 0
1
2
R LT drd
r
π
ψπ
∆=
∆∫ ∫ (3.15)
This calculation of thrust is done as the lift per blade. To find the total thrust of
the helicopter rotor the double integral has to be multiplied by the number of
blades (b) to find the total thrust.
2
0 02
Rb LT drd
r
π
ψπ
∆=
∆∫ ∫ (3.16)
This is the basic calculation of the thrust of a rotor. The following describes first
the factors t influence on the inflow angle, and how to include these factors in the
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calculation. Thereafter other factors like zero lift angle, tip loss, root cutout and
slope of the lift curve are discussed.
Inflow angle
The inflow angle is the angle between the geometric pitch angle and local the
angle of attack; this is shown ((3.13) and Figure 3-4) to be a combination of two
velocities, UP and UT. These velocities are a combination of contributions from
five effects of the helicopter blade in flight. The five contributions are described
in the subsections below.
I: Angular velocity of the rotor
The main contribution to the tangential induced velocity is the rotor angular
velocity. Therefore:
T
U rω ω= (3.17)
II: Helicopter velocity
The helicopter velocity in the surrounding air also has an influence on both the
tangential and perpendicular induced velocities. A couple of new definitions need
to be introduced to describe this. The velocity is expected to be a vector
, ,b b bx y z
V V V V= (3.18)
In spherical coordinates are used ( ), ,ρ φ θ . This gives:
, ,t sV V α ε= (3.19)
where ε is the angle of the horizontal flying direction, s
α is the angle of the rotor
disc towards the wind, and t
V is the absolute value of the velocity.
The horizontal component of the helicopter velocity contributes to the tangential
velocity, UT, of the blade element:
( )2 2 sinTV x y
U V V ψ ε= + + (3.20)
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Likewise the helicopter velocity has an influence on the perpendicular velocity,
UP:
PV Z
U V= (3.21)
III: Local induced velocity
In flight the local induced velocity, vL, is not constant over the rotor disc plane. It
is seen at Figure 3-5 that in flight the local induced velocity is zero at the front
end of the blade plane and body, and high at the rear end. The local induced
velocity is given by Prouty [2]:
1 1 cosL
rv v K
Rψ
= +
(3.22)
where v1 is the induced velocity in vertical flight:
( )
22 22 2
12 2 2
Tt tC RV V
vω
= − + + (3.23)
Here T
C is a thrust coefficient and K is a coefficient that is 1 in high horizontal
speed and 0 in hover.
Figure 3-5: Assumed induced velocity distribution in flight.
Source Prouty [2] Figure 3.4
The equation for v1 takes both vertical and horizontal velocity into account. The
only factor in the calculation of thrust and drag that has influence in vertical flight
is the induced velocity.
The local induced velocity is parallel but opposite to the direction of UP:
LPv L
U v= − (3.24)
17
IV: Flapping
When a helicopter flies forward, there is a difference in lift on the advancing and
retreating blades. This is simply because the tangential inflow velocity is higher
for the advancing blade than for the retreating blade. It is possible to reduce that
undesirable effect by anchoring the blades to a feathering hinge. The up and
downwards motion of the blades is called flapping ( β ). This flapping is used to
decrease the tilt torque in horizontal flight, because it increases the lift on the
retreating blade, while it decreases the lift on the advancing blade [2]
The blade flapping can be described as a Fourier series, where it is only the first
harmonic that has an important impact on the thrust and torque [10]:
( ) ( )0 1 1cos sins sa a bβ ψ ε ψ ε= − + − + (3.25)
Here a0 is the coning angle, a1s is the longitudinal flapping with respect to
direction of the helicopter velocity, and b1s is lateral flapping.
This blade flapping has two types of impact on UP:
1. The angle to the wind speed changes. It can be described by:
( )2 2
1 tan cosP x y
U V Vβ β ψ ε= − + + (3.26)
β is expected to be small, so the small angle assumption may be used.
( )2 2
1 cosP x y
U V Vβ β ψ ε≈ − + + (3.27)
2. The vertical velocity of the flapping blade contributes to the perpendicular
inflow velocity:
2PU rβ β= − (3.28)
where:
( ) ( )( )1 1sin coss s
a bβ ψ ε ψ ε= Ω + − + (3.29)
Therefore the flapping influence on UP can be stated as:
( )2 2 cosP x y
U r V Vβ β β ψ ε= − − + + (3.30)
18
V: Geometric and dynamic pitch, θ
As seen on Figure 3-1 and Figure 3-2, the pitch varies significantly over the radius
of the blade. Here it is reasonable to use the assumption that the pitch is linear in
the active area of the rotor blade:
0 1
r
Rθ θ θ= + (3.31)
As seen with flapping, the rotor blade is flexible, so the pitch changes with the
rotor speed. When the helicopter flies with a certain speed, it is also expected that
the pitch changes over one revolution of the rotor blade. It has not been possible
to find a mathematical description of the dynamic twist, but it will normally result
in an effectively lower pitch angle for higher inflow velocities. Therefore a
reasonable guess is that the dynamic twist will be:
0 1dyn dyn dyn
r
Rθ ω θ θ
= − +
(3.32)
If the blade only was flexible at the soft part near the hub, it would give the same
dynamic twist over the blade, and then 1dynθ would be zero. In contrast, if the blade
had the same flexibility over the length of the blade, 0dynθ would be zero. But with
both the black and white rotor used on the quadrotor helicopter, it is a
combination of these two effects. Because there is no simple way of calculating
and measuring the dynamic twist, it will in the first case be assumed to be zero,
but during the tests this point will be discussed further.
Zero lift angle
A wing with a symmetrical airfoil will because of its symmetry properties, have
no lift if the pitch angle is zero. The rotor blade used in the quadrotor has a thin
cambered airfoil and therefore it may have a lift even when the pitch is zero.
Kuethe [9] treats this characteristic; the zero lift angle is then given by:
( )00
1cos 1
L
dzd
dx
π
α θ θπ
= − −∫ (3.33)
where z and x are shown in the coordinate system in Figure 3-6 - on the left. The
angle θ denotes the integration angle shown by Figure 3-6 on the right.
19
Figure 3-6: Left: Velocity components establishing boundary conditions.
Right: The integration parameters. Source Kuethe [9] figure 5.5 and 5.6
From geometric measurements of the quadrotor rotor blade airfoil, it is seen that
the best way to describe the mean camber line is as a circle arc. A more detailed
description and calculation of zero lift angle is included in appendix A.
Like the pitch angle, the zero lift angle is not constant over the entire length of the
blade. But the change in zero lift angle is expected to be constant over the blade.
The aero lift angle can be expressed as:
0 10 0 0L L L
r
Rα α α= + (3.34)
Tip loss and root cutout
Up to this point each blade element has been seen as an element of an infinitely
long rotor, but in reality there are some special boundary conditions around the
centre and at the tip, which must be taken into account. Near the centre the lift
goes to zero because there is no lift at the hub and for the first section of the rotor.
This effect is called root cutout.
Around the tip of the rotor there is also a tip boundary condition, which results in
a lower lift near the tip. It is normally the difference in pressure between the upper
and lower side of the rotor that produces the lift. At the tip this difference in
pressure produces turbulence around the end of the blade, instead of lift.
To compensate for this boundary condition the integrations of the thrust and the
drag are only carried out in the area which has lift:
0
2
02
BR
x R
b LT drd
r
π
ψπ
∆=
∆∫ ∫ (3.35)
where x0 is the position of the start of the lift of the rotor and B is where the tip
loss starts [2].
20
Reverse flow region
In horizontal flight the tangential velocity UT is negative in an area of the disc.
This region is called the reverse flow region and is shown in Figure 3-7. The
integral of the thrust (3.16) does not take the reverse flow region into account,
because it takes the integral of the square of UT and this is positive over the disc.
Therefore the integration has to be separated over the rotor into three parts:
2 2 sin
0 0 sin 02 2 2
R R R
R
b L b L b LT drd drd drd
r r r
π π π µ ψ
π µ ψ πψ ψ ψ
π π π
−
−
∆ ∆ ∆= + −
∆ ∆ ∆∫ ∫ ∫ ∫ ∫ ∫ (3.36)
where µ is the tip speed ratio.
Figure 3-7: Tangential velocities in forward flight, source Prouty [2] figure 3.14.
3.3.3 Three dimensional thrust
Because of the flexibility of the rotor blades, it is not all thrust that is directly
upward from the rotor. There is actually a three dimensional thrust from every
blade element. In Figure 3-8 it is shown that when the blade is flapping an
element of the thrust is vertical and an element of the thrust is horizontal pointing
towards the rotor.
21
Figure 3-8: The two components of lift on a blade element.
The thrust vector is given by the sine and cosine of the flapping angle
respectively:
( ) ( )
( ) ( )
( )
sin cos
sin sin
cos
L
L L
L
β ψ
β ψ
β
−∆
∆ = −∆ −∆
(3.37)
3.3.4 Torque from the thrust
The main contribution to the torque on the helicopter is the torque induced
directly from the thrust:
tri i i
M T P= × (3.38)
Where Ti is the thrust of the i’te rotor and Pi is the position of the rotor on the
helicopter.
There is in addition also a component of torque around the centre of the rotor. If
the torque on each blade element is calculated and integrated, the total torque is
found from each blade element:
m
M L p∆ = ∆ × (3.39)
where p is the position of the blade element:
( ) ( )
( ) ( )
( )
sin cos
sin sin
cos
r
p r
r
β ψ
β ψ
β
−
= −
(3.40)
22
Thus the total torque the rotor thrust gives to the quadrotor is given by:
2
0 02
R
i i i
b LM T P p drd
r
π
ψπ
∆ = × + ×
∆ ∫ ∫ (3.41)
3.3.5 Drag
As described in the beginning of this chapter the drag is given by the two drag
components, induced drag and the profile drag. The total drag force and torque is
the sum of these:
( ) ( )2 2
2 2l d
H r c c r r c c rρ ρ
φ∆ = Ω ∆ + Ω ∆ (3.42)
( ) ( )2 2
2 2l dQ r r c c r r c c r
ρ ρφ
∆ = Ω ∆ + Ω ∆
(3.43)
The first term in the parentheses is the induced drag given by the equations above.
The second term in the parentheses is the profile drag. The profile drag coefficient
cd is in simple approximations a constant, but it is more correct to define it as a
power series [2]:
0 1 2
2
d d d dc c c cα α= + + (3.44)
The drag coefficient cannot be found mathematically; it has to be determined from
tests on the physical rotor.
3.4 Aerodynamic behaviour
The above considerations are based on the assumption that all aerodynamic
behaviour is ideal with no stall, no boundary effects and no other turbulence than
the one that results in tip losses. These effects are especially seen on airfoils with
low Reynolds numbers. The Reynolds number is an important non-dimensional
number in fluid dynamics. It describes the transitions from laminar to turbulent
flow. At low Reynolds numbers the flow is laminar, while it is turbulent at higher
Reynolds numbers [2].
. . 68000D
VLR N VL
ρ
µ= ≈
Where 21.25 kg
mρ = is the density of air, 3
51.84 10 kg
D mµ −= ⋅ is the dynamic
viscosity, V is the velocity in ms
and L is the chord in m . The Reynolds number is
under suspicion to be low, therefore the maximum Reynolds number for the
23
quadrotor rotors is found. Following maximum values for the rotor blade are
chosen:
V Rω= , 0.15R m= , 350 rads
ω = , 0.025L m=
This results in a Reynolds number:
4. . 8 10R N = ⋅
For full size helicopters and flights the Reynolds numbers are in normally
between 104 and 109. The transition point from laminar to turbulent flow is
commonly given by 5. . 10 .R N = Thus the flow around a wing with a Reynolds
number lower than 105 is laminar, and the flow around a wing with Reynolds
number higher than 105 is turbulent. There is not a sharp step from laminar to
turbulent flow; the natural transition is closer to 106, but it is possible to have
turbulence, also at Reynolds number under 105.
Therefore the flow around the rotors of the quadrotor can be expected to be
laminar. Laminar flow results in lower skin friction, which decreases the drag, but
it can result in the air flow not following the curve of the airfoil. When the flow
does not follow the airfoil the lift efficiency decreases. To counteract this
behaviour a rough surface of the wing can be used. This increases the skin fiction
which deflects the airflow around the airfoil. It is very difficult to predict this
behaviour and estimate it occurs
The white and black rotors used in the project both have a very smooth surface.
This has apparently been a problem at the white rotor, because the manufacturer
has roughened the first 5 mm of the inflow side of the blade; see section 3.1.1.
3.4.1 Slope of the lift curve
The slope of the lift curve is introduced earlier in this section as a constant a. The
lift coefficient cl is a function of the local angle of attack. Figure 3-9 shows the
steady lift coefficient as a function of the angle of attack (the dotted line). At
small angles the lift coefficient is linearly proportional to the angle of attack. For
the airfoil documented in the figure there is at about 10 degrees a maximum lift
coefficient and it decreases for larger angles. From this point of view it is best to
set the angle of attack to be around the maximum of the lift curve.
But this curve only holds for the static case, where the flow has time to finds its
own equilibrium. The angle of attack varies significantly in one revolution of the
rotor blade. Then inertia in the airflow results in another dynamic effect, which is
shown by the dynamic curve around the lift curve. This curve is unique for each
airfoil and for each mean angle of attack. The one shown in Figure 3-9 is a
simplified approximation, a more exact estimation of the dynamic lift curve is
24
more complex. Therefore this dynamic lift curve is not easily measured. Thus a
simple assumption that the slope of the lift curve is constant is used in this project.
Later it will be discussed if this assumption is valid.
Figure 3-9: An example of the dynamic lift slope curve, source Prouty [2] figure 6.22.
3.4.2 Distortion from other rotors
When rotors operate close to each other there is interference between the rotors.
The quad rotor helicopter has not been studied in traditional helicopter theory but
the configuration can be seen as two tandem rotors placed side-by-side. Both the
tandem and the side by side configuration have been studied and the results are
useable on the quadrotor.
The side-by-side rotor is studied by Tong & Sun 2000 [11] and the conclusion is
that the two rotors do not influence each other noticeably. For the tandem rotor
the front rotor will act like a single rotor but the rear rotor will experience
significant interference. There have been different estimates of the rear rotor
performance but this has always been based on a specific helicopter and rotor.
Dingeldein 1964 [12] concludes from his rotor setup that over a specific tip speed
ratio, the rear rotor performance can be predicted fairly well with the theory by
“considering the rear rotor to be operating in the fully developed downwash of the
front rotor” Dingeldein 1964 [12]. Tong & Sun 2000 [11] treat the same problem
and end up concluding that the performance of the rear rotor is significantly worse
than the front rotor. Heyson 1954 [13] concludes that with at tip speed ratio of
0.15, the rear rotor requires about three times as much power.
Therefore if this is included in the model, the performance must be derived
through test results and due to limited time in the project this has not been
possible.
Dynamic lift curve
Static lift curve
25
4 Matlab Model
To design a good feedback system for the helicopter, it is important to have a
good mathematical model. The model presented has been designed based on the
equations from chapter 2 and 3. As in these chapters this review will start with the
helicopter model, and then afterwards go into details with the elements.
4.1 Helicopter model
The theoretical model is based on the dynamic helicopter model derived in
equations (2.19) to (2.22):
Vξ = R (4.1)
3
1, 2, 3, 4
T
h h h i i
r r r r
m V m V m g e T H= − Ω× + + +∑ R (4.2)
( )sk= ΩR R (4.3)
[ ]1, 2, 3, 4 1, 2, 3, 4
h h r i i i
r r r r r r r r
I I I Q MωΩ = −Ω× Ω − Ω× + +∑ ∑ (4.4)
4
OMEGA
3
R
2
XI
1
V
V
RXI
XI (position)
T
OMEGA
R
V
V (velocity)
OMEGA R
R (rotation)
Q
omegaOMEGA
OMEGA
(angular velocity)
omega r1
omega r2
omega r3
omega r4
Vx
Vy
Vz
Tx
Ty
Tz
Qx
Qy
Qz
Four rotors
force and torque contribution
Demux
4
omega r4
3
omega r3
2
omega r2
1
omega r1
Figure 4-1: Matlab model of the quadrotor. Input is the angular velocity of the four rotors,
and outputs are the position, velocity, angular position and angular velocity of the
quadrotor.
26
The helicopter model built with these four differential equations is shown in
Figure 4-1. The four differential equations are shown as four subsystems. In the
model the thrust, Txyz, vector is used to describe all forces both from thrust and
from horizontal drag force.
1, 2, 3, 4
xyz i i
r r r r
T T H= +∑ (4.5)
The same is done for the torque from the rotors. The drag, Qxyz, consists of torque
induced by the drag (3.43) and torque induced by the thrust (3.41):
[ ]1, 2, 3, 4
xyz i i
r r r r
Q M Q= +∑ (4.6)
4.2 Rotor model
The rotor model is given by the equations for thrust, T, torque from thrust, M,
drag torque, Q and drag force H. In chapter 3 the equations for these factors are
derived.
Thrust is a combination of (3.35) and (3.36):
2 2 sin
0 0 sin 02 2 2
BR BR BR
R
b L b L b LT drd drd drd
r r r
π π π µ ψ
π µ ψ πψ ψ ψ
π π π
−
−
∆ ∆ ∆= + −
∆ ∆ ∆∫ ∫ ∫ ∫ ∫ ∫ (4.7)
Where L∆ is derived in section 3.3.2.
It has shown to be far from simple to take into account both the root cutout and
reverse flow region in the integration. Thus it is chosen only to take the reverse
flow region into account. The thrust is very low near the centre, so the error is
assumed negligible.
Torque from thrust is given by (3.41) where the tip loss and reverse flow region is
taken into account:
2
0 0 sin
2 sin
0
2 2
2
BR BR
i i iR
BR
b L b LM T P p drd p drd
r r
b Lp drd
r
π π
π µ ψ
π µ ψ
π
ψ ψπ π
ψπ
−
−
∆ ∆ = × + × + ×
∆ ∆
∆ − ×
∆
∫ ∫ ∫ ∫
∫ ∫(4.8)
27
Drag force, H, and drag torque, Q, are given by the integral of the incremental
force H∆ and torque Q∆
2
0 0 sin
2 sin
0
2 2
2
BR BR
R
BR
b H b HH drd drd
r r
b Hdrd
r
π π
π µ ψ
π µ ψ
π
ψ ψπ π
ψπ
−
−
∆ ∆= +
∆ ∆
∆−
∆
∫ ∫ ∫ ∫
∫ ∫ (4.9)
2
0 0 sin
2 sin
0
2 2
2
BR BR
R
BR
b Q b QQ drd drd
r r
b Qdrd
r
π π
π µ ψ
π µ ψ
π
ψ ψπ π
ψπ
−
−
∆ ∆= +
∆ ∆
∆−
∆
∫ ∫ ∫ ∫
∫ ∫ (4.10)
where H∆ and Q∆ are given by (3.42) and (3.43):
4.2.1 Implementation
The above equations were written into Matlab using the Symbolic Math Toolbox
[14]. The symbolic math toolbox is a Matlab entry to Maple [15] so the equations
are actually written into Maple. But the simulations are carried out in Matlab thus
it is obvious to use the Symbolic Math Toolbox.
The symbolic Matlab code is found on the enclosed DVD (appendix G), here it is
seen that first the equations for the perpendicular velocity (UP) then the tangential
velocity (UT) are defined. From these, the pitch and the zero lift angle the angle of
attack and the incremental forces are derived. All these equations are the ones
derived in chapter 3.
Figure 4-2 shows the lift distribution over a black rotor in horizontal flight, Vt=10
m/s. The coefficients applied are derived in chapter 6 or measured in chapter 7. In
this figure the reverse flow region is also seen as the small positive peak near the
centre.
28
Figure 4-2: Lift distribution over a black rotor in flight. Vt = 5 ms, ω = rad/s, αs = 0°
The lift and drag forces are integrated to the forces T and H and the torques M and
Q are derived ((2.19) (2.20) (2.21) (2.22)). The derived equations are loaded into
Simulink. Figure 4-3 shows the Simulink implementation of the four rotors, where
the blocks for four rotors are implemented using the “embedded MATLAB
function”.
4.2.2 Implementation status
The final implementation of the model in Matlab showed up to be very calculation
heavy. Thus the calculations were very slow, it took over 3 hours for Matlab to
calculate the integrations, and it took at least 4 hour to compile the Simulink
simulation. And before some simple simplifications were made the error
“symbolic expression too long” came often. The simplifications
were to widely use the small angle assumption and exclude the torque induced
from the blade elements, these are described hereafter
First simplifications
The first simplification is made because one of the equations that is the most
difficult for Matlab to calculate symbolically are the trigonometric equations.
Thus the assumption of small angles is used for the angles β ,φ ,θ ,s
α and 0Lα :
( )
( )
( )
sin
tan
cos 1
ϕ ϕ
ϕ ϕ
ϕ
=
=
=
(4.11)
29
Second simplification
The torque induced from the lift on each blade element, Mi, is too complex an
equation. In flight this torque does not have the major effect, because it is
negligible compared to the torque induced from the trust of each rotor, i i
T P× .
From the final Matlab model of the quadrotor it is known that in forward flight, Vh
= 5 m/s, then i i
T P× is in the range of 0.5 Nm to 1.2 Nm, while Mi is in the range
0.05 Nm to 0.10 Nm.
The model was compiled in Simulink and it was tested that the sign and equations
were correct. At the point where it worked properly it was decided to stop the
work with deriving a better and simpler model and at the same time stop the
adjustment of constants. It is possible that the constants can be turned more
accurately; this is done in chapter 8.
6
Qz
5
Qy
4
Qx
3
Tz
2
Ty
1
Tx
Vx
Vy
Vz
omega
Tx
Ty
Tz
Qx
Qy
Qz
rotor4
Embedded
MATLAB Function
Vx
Vy
Vz
omega
Tx
Ty
Tz
Qx
Qy
Qz
rotor3
(Embedded
MATLAB Function)
Vx
Vy
Vz
omega
Tx
Ty
Tz
Qx
Qy
Qz
rotor2
(Embedded
MATLAB Function)
Vx
Vy
Vz
omega
Tx
Ty
Tz
Qx
Qy
Qz
rotor1
(Embedded
MATLAB Function)
r1
r2
r3
r4
x
y
Out3
Spli t up and sum2
r1
r2
r3
r4
x
y
Out3
Spli t up and sum1
r1
r2
r3
r4
x
y
Out3
Split up and sum
In1
In2Out1
Cross product3
In1
In2Out1
Cross product2
In1
In2Out1
Cross product1
In1
In2Out1
Cross product
-C-
Constant3
-C-
Constant2
-C-
Constant1
-C-
Constant
7
Vz
6
Vy
5
Vx
4
omega r4
3
omega r3
2
omega r2
1
omega r1
Figure 4-3: Simulink implementation of the four rotors and how they affect the quadrotor.
This is a subsystem of Figure 4-1.
30
4.3 Simulation
Even if the model is not fully calibrated; it is relevant to simulate it to see the
dynamic properties of the model. The nonlinear model is simulated in two stages.
The first simulation is performed with a simple model where the helicopter
velocity is not taken into account. This means all rotors act like they are in hover,
with no wind influence on the performance. This simulation is carried out because
it is simple to examine if the model is implemented correctly.
The second simulation is done with the nonlinear model as it is at the end of the
project. Here all relevant factors are taken into account, but it is not calibrated (see
above)
4.3.1 Simple simulation
To verify the basic performance of the Matlab helicopter model, the first tests is
done at the simplified one, where all rotors acts like in hover
1. With no initial velocity, and same velocity at each of the rotor, the flight
must be only vertical, and there must be no rotation around the zb axis.
2. When increasing angular velocity at rotors 2 and 3, the helicopter must fly
in the positive xb direction.
3. When increasing angular velocity at rotor 3 and 4, the helicopter must fly
in the positive yb direction.
4. When increasing angular velocity at rotor 2 and 4, the helicopter must
rotate in the positive direction around the zb axis.
The tests showed the model was set up correctly. And it showed the angular
velocity of the rotor if the helicopter shall hover has to be 196 rad/s. The last three
of the four tests described above is shown in Figure 4-4.
31
Figure 4-4: Tests of the simple model.
2: Black: [ω1 = 198 ω2 = 198.005 ω3 = 198.005 ω4 = 198] rad/s – positive xb velocity.
3: Red: [ω1 = 198 ω2 = 198 ω3 = 198.005 ω4 = 198.005] rad/s – positive xb velocity.
4: Blue: [ω1 = 197 ω2 = 199 ω3 = 197 ω4 = 199] rad/s – positive angular velocity.
4.3.2 Simulation of the final nonlinear model
For the final model it is more complex to imagine the responses from the
helicopter model. It must still follow the same conditions as the simple setup
(Figure 4-4) but furthermore the following conditions can be tested:
• For the same conditions for horizontal flight, the final model must have a
lower horizontal velocity than the simple one. This is because in forward
flight the x component of the force is negative (reverse flight direction).
• In vertical climb the vertical upwards velocity must be lower in the final
model because the thrust is lower in climb.
• In vertical decent the thrust is higher in the final model, thus the
downwards vertical velocity must be lower when the helicopter velocity is
taken into account.
• The horizontal forces perpendicular to the flight direction must cancel
each other.
32
Figure 4-5: Simulation of the Matlab model. They both starts in [0 0 0] and flies upwards
(negative z). Left: simple model where all rotors acts link in hover, Right: Final model where
helicopter velocity is taken into account.
The tests have been performed and the model behaviour verified. Figure 4-5
shows the simple and the final model. The angular velocities of the four rotors are
[ω1 = 210 rad/s ω2 = 198 rad/s ω2 = 210.05 rad/s ω4 = 198 rad/s] which should
give:
• A negative vertical velocity because the rotor velocity is higher than 196
rad/s which is hover.
• A negative angular velocity around the zb axis, because rotor 1 and 3 has
higher angular velocity than the two others.
• A positive vertical velocity in the xb yb direction ( [ , ,0]A A
x yV E E= )
because the angular velocity of rotor 3 is higher.
This is also what is seen in Figure 4-5 and it is seen that the final model has lower
speed than the simple one. This is also expected because in forward flight there is
a negative xb force, in climb the thrust decreases and in decent the thrust increases.
33
5 Test Set-up
The preceding chapters have been concerned with how to derive a good model of
the rotor and estimate its parameters. Now the rotor has to be tested to see if the
calculations and assumptions are valid. Most of the tests have been conducted in
a wind tunnel and this chapter describes the test setup in the wind tunnel.
Rotor
Motor
Thrust meter
Drag meter
Hinge link
Wind tunnel walls
FS6
Wind
Angular velocity
Smoke pipeWind tunnel
rotor
Figure 5-1: The wind tunnel setup.
The final setup in the wind tunnel is shown in Figure 5-1. Here a number of
instruments are shown; these instruments are described in the following
subsections.
5.1 Motor and motor controller
It is desirable to do the tests with a motor that is more powerful than the one used
on the original quadrotor model. Then it is possible to conduct tests over a larger
range than the normal operating area and thereby achieve a better understanding
of the rotor performance. For this reason the Speed Gear 600 plus from Graupner
GmbH [16] was chosen, the datasheet is found in appendix G. It is a low-cost
34
geared motor with a controller, so it is easy to control for example directly from a
microprocessor. The chosen motor has a maximum output power of 74W (8.4V)
[17] the one used in the quadrotor to day has a maximum output power of 8.61W
(7.2W) [1]. It is a huge step in power, but it is necessary because there is a high
increase in drag torque for high velocities.
The built-in motor controller for the rotors is controlled by a 5 v signal. For every
2 ms a pulse is received and the angular speed of the motor depends on the length
of this pulse. The length of this pulse, tp, must be in the interval:
1 µs < tp < 2 µs
where 1 µs is zero speed and 2 µs is high forward angular velocity. The actual
motor can not run with negative angular velocities. When a negative velocity is
required, the polarity of the motor must be reversed in hardware.
5.2 Thrust and Drag meter
In earlier work reported by Bertelsen [1] it is stated that there is no measurement
equipment available at DTU which can be used to measure the thrust and drag of
the rotor with good resolution. It was therefore necessary to build new
measurement equipment which would be able to measure three parameters: the
vertical thrust, Tz, the horizontality drag Dz and the velocity. The technical
drawings of the instrument are included in appendix F.
The in the project the Thrust/Drag instrument (T/D) constructed functions as
shown in Figure 5-1 as the support of the rotor. It positions the rotor in the middle
of the test section of the wind tunnel.
The height of the wind tunnel is 0.50 m thus the part of the T/D instrument that is
inside the wind tunnel must be 0.25 m high. This 0.25 m includes a hinge link,
drag meter, thrust meter and the drive motor. Furthermore it is prepared for a twin
rotor configuration. Figure 5-2 shows the final instrument with one rotor
mounted. In the following subsections, first the mechanical design is described
and then the analog circuit and the microcontroller design are described. Finally
the calibration is described.
35
Figure 5-2: Thrust and drag measuring device
5.2.1 Link:
Figure 5-2 shows the T/D instrument which was constructed from a Ø30 mm H7
hard metal cylinder (on the left in the figure). The cylinder is used to mount the
instrument in the wind tunnel. To the right of that there is a link. This is used to
test the rotor for a varying angle of the helicopter in the wind. This simulates the
helicopter flying up and down inside the wind tunnel.
5.2.2 Drag meter
The drag meter consists of a hub so the top of the T/D instrument is able to rotate
compared to its base. A force sensor (Honeywell FSG15N1A, see appendix G) is
mounted between the two parts, to measure the rotational torque of the rotor. The
force sensor is not accurate at very small forces; therefore the drag meter is
prestressed with a tension spring.
The Honeywell FSG15N1A force sensor has a measuring range from 0 to 15 N.
The force sensor is placed 15 mm from the centre, thus it gives a range for the
drag force measurements:
0 0.225Q Nm≤ ≤
These values will in reality be less because the system is prestressed. The exact
values depend on the spring loading and therefore vary from test to test.
Compared with the measurements reported by Bertelsen [1], these values of drag
should be reasonable.
5.2.3 Thrust meter
The same principle as for the drag meter is used for the thrust meter. A detailed
photo of the thrust meter is seen in Figure 5-3, where the two mutual vertical
moving parts, the force sensor, and the pressure spring are seen. Both for the drag
and the thrust meter the ball bearings are used to keep the static friction low.
36
The same force sensor as for the drag meter is used
here (Honeywell FSG15N1A). It has a measuring
range from 0 to 15 N, so the range for the thrust is:
0 15T N≤ ≤
The actual range is lower in the tests because of the
prestressing of the spring. In the measurements
reported by Bertelsen [1] the thrust is less than 3 N,
therefore the measuring range for thrust is
acceptable
5.2.4 Motor mounting
The motor is mounted on the T/D instrument with
an aluminium profile. It is prepared for twin rotor
configuration but unfortunately the time frame of this project has made it
impossible to perform twin rotor tests. Figure 5-4 shows the twin rotor
configuration, it is done by mounting an aluminium profile on the T/D instrument,
and mounting a motor at each end. The motors are mounted with the same
aluminium profile as for the single rotor configuration.
In the twin rotor configuration the rotors will run with opposite angular velocities.
Motors
Rotors
Reflective object sensor
Thrust
meter
Figure 5-4: Rotor mounting for twin rotor tests
5.2.5 Velocity measurements
The rotational velocity is measured with an optical sensor at the rotor shaft. The
sensor measures the difference in the reflected light from the rotor blades and
therefore there are three black bands (no reflection) painted on the shaft. This
gives three pulses per revolution. To log the velocity the microcontroller and a
frequency to voltage converter are used.
Pressure spring
Ball bearings
Force sensor
Vertical movement
Figure 5-3: The thrust meter
37
The optical sensor used is a reflective object sensor (OPB701 appendix G) from
Optec.
5.2.6 Analog electronics
Three different analog circuits are used in the system; operational amplifiers,
frequency to voltage converter, and power supplies. These analog circuits are
based on earlier layouts designed by Elbert Hendricks.
Amplifier
The amplifier used in the project is a fully differential instrumentation amplifier.
The circuit the amplifier is based on had a good dynamic performance, thus it is
only the gain that has been adjusted. The force sensor has a differential output of
0-0.36V, and the ADC has an input of 0-5V. This gives an amplification gain of:
14a
K =
Frequency to voltage converter
The frequency to voltage converter is based on a LM2907 and the circuit is built
as the “Frequency to voltage converter with 2 pole Butterworth filter to reduce
ripple” described in the LM2907 datasheet in appendix G. The purpose of using a
frequency to voltage converter instead of just measuring the output of the optical
sensor with an oscilloscope was an aim of making a feedback loop with the
microprocessor to control the angular velocity. The feedback loop did not work
very well, so the velocity was for the most part measured with the oscilloscope.
Power supply
The print layout for the power supply was also from an earlier project. It is a 230
V AC to +/0/- 15 V DC stabilised power supply. A transformer and a bridge
connection are used to step down the voltage and a 7815 and 7915 (appendix G) is
used to stabilize the voltage. The +/- 5 V for the force sensors and the optical
sensor are made by a voltage divider and Zener diodes.
5.2.7 Microprocessor design
As it is mentioned above a microcontroller is used to control the motor, and to log
the thrust, drag and velocity. It was decided to use an Ethernut 2.1b [18]
development board. The Ethernut 2 series is a low cost series of development
boards, based on an Atmel ATmega 128 Microprocessor. On the board is, among
other things: 512 kb extra memory and an Ethernet controller. All the features of
the Atmel are easy to access. Together with the Ethernut platform there is open
source real time operating system, Ethernut OS, which supports the features of the
Ethernut board.
38
There are many other ways to control the motor and log data, but this was the first
found with both real time thread handling and build-in 10 bit ADC. An instruction
manual is included in appendix B.
5.2.8 Calibration
Four parts of the system needed to be calibrated. The calibration of the thrust
meter, the drag meter, the frequency to voltage converter and ADC are described
in the following subsections:
Thrust meter
The thrust meter calibration is a calibration of
both the force sensor and its amplifier. The thrust
meter is calibrated by fixing the top of the
instrument on a table and then suspending
accurate weights from the bottom of the
instrument, see Figure 5-5. The forces on the
instrument will be the same as when the rotor
affects it with a vertical upwards force.
The calibration was done with weights from 0.05
kg to 1 kg. This is with an interval of 0.025 kg for
mass under 0.5 kg and an interval of 0.05 kg for
mass from 0.5 to 1 kg.
Drag meter
The drag meter is calibrated in the same way as the thrust meter. The top is still
mounted on the table but on the bottom a torque meter is mounted and the weights
are suspended from the torque meter. Then an accurate measurement can be made
if the weights are accurate, and the torque is given by the torque meter.
Frequency to voltage converter
The frequency to voltage converter is first tested with a function generator as
input and a voltmeter as output. Then it is calibrated so with an input range from 0
to 60 Hz the output is 0 to 5V which is the range of the ADC. After that the
optical sensor was connected and the tests done directly on the rotor. The input
frequency is measured with an oscilloscope and the output voltage is measured
with a voltmeter. From that a transmission factor is derived.
ADC in the microprocessor
The built in 10 bit ADC in the microprocessor is tested with a stabilised power
supply at the input and the output is read out from the microprocessor through a
Thrust
meter
Table
Weight
Figure 5-5: Calibration of thrust
meter
39
RS232 cable. A transmission factor for the ADC is derived and the final
transmission factors for the devices are:
ADC to Thrust: 0.0222 N
ADC to Drag: 0.000288 Nm
ADC to angular velocity: 1.0044 rad/s
This means the measuring range and the accuracy for the three instruments are:
Range Accuracy
Thrust: 0 to 23 N ± 0.0222 N
Drag: 0 to 0.3 Nm ± 0.000288 Nm
ADC to angular velocity: 0 to 1028 rad/s ± 1.0044 rad/s
The thrust drag system is then compared with the six dimensional force transducer
FS6, see appendix D.
5.3 Six dimension transducer
The Department of Mechanical Engineering at DTU has a six dimensional
force/torque sensor; this is a FS6-250 from AMTI [19]. To amplify the signal
from the force sensor an AMIT Mini-AMP serial amplifier is used. The Mini-
AMP has a serial interface that can be accessed with the AMTI Netforce software.
The FS6 has an operational range:
• Fx, Fy: 550 N with an accuracy of 0.13 N
• Fz: 1100 N with an accuracy of 0.27 N
• Mx, My: 28 Nm with an accuracy of 0.007 Nm
• Mz: 14 Nm with an accuracy of 0.003 Nm
From earlier tests reported by Bertelsen [1], the thrust and drag of the white rotor
are known to be in the range:
• Thrust: 0-2 N
• Drag: 0-0.1 Nm
This shows that especially for the thrust measurement the FS6 may not be
sufficiently accurate. The transducer is installed in the wind tunnel and it is
natural to use it. Thus all tests will be carried out with both the especially built
thrust and drag meter and the AMTI FS6-250. After the tests are documented it
will be discussed which sensor that has the sufficient range and sensitivity.
40
5.4 Wind tunnel
The wind tunnel used is located at the Department of Mechanical Engineering,
building 414, DTU. The tunnel is a so called vacuum type wind tunnel; this means
the fan in the tunnel is placed downstream of the test section, resulting in a more
uniform flow than if the fan injected the air to the tunnel. The wind tunnel has an
active air speed from 0 to 60 m/s but at very low velocities it can be hard to
control precisely. The cross sectional area in the test section of the tunnel with
homogeneous wind is 0.5 x 0.5 m. The rotor diameter is about 0.3 m, so the wind
tunnel is slightly too small to avoid wall and ground effects. Therefore it has to be
tested how the tunnel walls influence the rotor
Figure 5-6: Photo of the setup in the wind tunnel. The rotor is difficult to see because it is
rotating but it is at the top of the pylon.
During the period of the project the wind tunnel was equipped with a smoke pipe.
This makes it possible to see the turbulence around the rotor. It can also help to
find the effects of for example tip losses and wall effects. Figure 5-6 shows the
wind tunnel and how it is equipped with the FS6 force transducer under its floor
and the thrust and drag meter mounted on the FS5.
5.5 Stroboscope
In horizontal flight there is some kind of flapping of the blades. This is because
the rotor blades are flexible and the lift varies over one revolution (see chapter 3).
It was attempted to measure the flapping in two ways. The one is with a Brüel &
Kjær stroboscope and a normal digital camera, the other one is using a high speed
camera. To test with the stroboscope the frequency of the stroboscope is set to the
same as the rotor frequency and thus it seems as the rotor is not rotating, here
41
photos of the rotor are taken, see photos in appendix G. Unfortunately the velocity
of the rotor is not constant; therefore it is difficult to get the photos in the right
rotor position.
5.6 High Speed Camera
The other way to determine the flapping is to use a High Speed Camera from
Dantec Dynamics. The camera is able to take photos with a frequency of 4000 s-1
and it gives minimum 70 photos per revolution of the rotor, which is sufficiently
fast. A strong light is needed when the exposure time is short and the light has to
be constant. Thus two 500 W halogen lamps were used to light the rotor with a
constant light.
42
43
6 Estimation of Constants
Chapter 2 and 3 have shown which parameters have an influence on the
quadrotor performance. Some of the constants have to be determined
geometrically, some have to be found in handbooks and finally some have to be
determined by tests.
6.1 Rotor constants
One of the aims of this thesis is to compare two different rotors for best
performance and find which parameters that must be adjusted to increase the
rotor’s performance. Thus in the following subsections the rotor constants for the
two rotors are determined.
6.1.1 Geometric size
It is shown in appendix A that the best way to describe the airfoil of the two rotors
is as a circle arc.
Figure 6-1: Geometric pitch θ. RR is the radius of the circle the airfoil is curved after and c is
the chord of the airfoil.
44
Figure 6-1 shows the airfoil as a circle arc with a horizontal tangent at the leading
edge. Thus the geometric pitch can be estimated to be:
arcsin2
R
c
Rθ = (6.1)
where RR is the radius of the circle the airfoil is curved after and c is the chord of
the airfoil.
An estimate of the white rotor’s chord is:
r
0.0357- 0.0148wc mR
=
(6.2)
This equation for the chord does not fit the chord near the centre of the rotor, see
chapter 3. Near the centre the blade is narrower than the equation says but in the
same area the pitch is zero and thus there is no lift. This is taken care of by the
compensation for tip loss and the root cutout see section 6.1.3.
The black rotor has the following equation for the chord:
r
0.0348- 0.0158bc mR
=
(6.3)
The radius RR is determined by comparing the airfoil with different shapes
(appendix A):
0.04Rbw
R m= (6.4)
0.035Rb
R m= (6.5)
From (6.1) and (3.33) the pitch and the zero lift angle are calculated:
0.4614 - 0.1974w
rrad
Rθ
=
(6.6)
0.5164 - 0.2451b
rrad
Rθ
=
(6.7)
0 0.1341- 0.0826L w
rrad
Rα
=
(6.8)
0 0.1641- 0.1124L b
rrad
Rα
=
(6.9)
45
The black rotor has a larger value both for the pitch and the zero lift angle. This is
expected because the black rotor is more curved than the white one.
The radii of the two rotors are measured to be:
0.14w
R m= (6.10)
0.16b
R m= (6.11)
6.1.2 Slope of the lift curve
As discussed in section 3.4.1, the lift curve is complex, but for normal helicopters
with small angles of attack it is known to be 12a radπ −= . For rotors with small
Reynolds numbers a lower estimate is more correct. Therefore a widely used
estimate is [2]:
15.73a rad−= (6.12)
The Reynolds number for the rotors used in the project is known to be small, see
section 3.4.
6.1.3 Tip loss and root cutout, B and x0
The tip loss and root cutout are determined from a number of measurements [1]
made in the autumn of 2004. Figure 6-2 shows the wind speed, v2, just beneath the
rotor. It is seen that at the inner part of the rotor the velocity goes to zero, but as
Figure 3-1 shows the first four centimetres of the rotor have no airfoil, it will not
induce any air velocity. At the tip of the rotor, the velocity becomes negative; this
is a result of turbulence around the tip as described in section 3.4 and further
demonstrated with smoke experiments, section 7.7.
46
Figure 6-2: Air velocity 1.1 cm below the white rotor. Source Bertelsen [1]
From Figure 6-2 the tip loss and root cutout constants are determined to be:
0 0.32x = (6.13)
0.82B = (6.14)
These experiments are unfortunately only made with the white rotor, but because
of the similar design of the rotors, it is assumed that the same parameters describe
the black rotor.
6.1.4 Flapping and coning angle
For full size helicopters flapping is normally used to decrease the negative effects
of sideways torques in forward flight. Flapping increases the lift on the retreating
blade, while it decreases the lift of the advancing blade. To increase this effect the
blade is often fasten to the hub with a hinge. The quadrotor blade is made in one
piece and has thereby no hinge. But the inner part of the rotor, where it is not
curved is quite soft, so it is assumed to act like a hinge. For very small flapping
angles this is a good assumption, but at larger angles this is no longer valid (see
Appendix D for evaluating this problem).
From full size helicopter theory [2] an equation for the flapping for a helicopter in
forward flight with only first harmonic oscillations is known to be:
( ) ( )0 1 1cos sins sa a bβ ψ ε ψ ε= − + − + (6.15)
47
And the constants are given by:
2
0
12
13 12
T
e
C Ra
ea
R
σγ
−
=
+
(6.16)
0 1
1 2 3 4
8 22 2 123 2 4 3
33 211 1 122 4
s T
s T
eC
R aa CeeRR
σ µγθ µ θ µ µ µα σµ σ
σµ µµ
γ
+ + − = + + +− − −
(6.17)
1 0 12 3 4
4 2 1283 3 2 2
3 2 3 211 1 122 4
T
s s T
eC
Rab Ce eR R
µγσ
σ σθ µ θ µ µ µα σ
µ µ µµγ
= + + + + − ++ − −
(6.18)
Where γ is the lock number
4
b
c aR
I
ργ = (6.19)
CT is the non-dimensional thrust coefficient:
( )
2T
TC
A Rρ=
Ω (6.20)
σ is the solidity ratio:
bc
Rσ
π= (6.21)
And e is the hinge offset, which is the distance from the centre of the hub to the
hinge. Ib is the moment of inertia of the rotor blade.
The lock number, solidity ratio, and hinge offset are calculated by use of
geometric measurements of the two rotors directly (by use of the measurements in
section 6.1.1 and following constants):
31.25 kg
mρ = , 15.73 rada = , 0.005
bbm kg= , 0.004
bwm kg=
48
Then the rotor parameters become:
3.31w
γ = 0.129w
σ = 0.02 mw
e = (6.22)
3.17b
γ = 0.1139b
σ = 0.02 mb
e = (6.23)
The thrust coefficient is estimated from the first tests made with the rotor to be:
40.1871+1.24*10Tw
C ω−= (6.24)
50.2310+9.57*10Tb
C ω−= (6.25)
In section 7.2 and appendix D it is discussed if the calculated flapping angles are
appropriate.
6.2 Helicopter constants
This project does not work directly with the physical helicopter, therefore all the
helicopter constants are taken from the Bertelsen 2004 [1]:
Mass of the quadrotor: mH = 0.544 kg.
Acceleration of gravity: 29.82 m
sg =
Position of the four rotors:
[ ]1 0.162 0.162 0T
P m= (6.26)
[ ]2 0.162 0.162 0T
P m= − (6.27)
[ ]1 0.162 0.162 0T
P m= − − (6.28)
[ ]1 0.162 0.162 0T
P m= − (6.29)
Moment of inertia matrix:
2
0.0058 0 0
0 0.0058 0
0 0 0.0117
HI kgm
=
(6.30)
49
7 Test of the Rotor
This report has up to now only been concerned with theory, but now it is time to
test if it works on the real rotors. This chapter contains a short introduction to
which tests that has been performed, followed by the relevant results. If the results
deserve a longer discussion; this has been put in the appendix D.
In the test period 8 tests of the white rotor and 16 tests of the black rotor have
been performed. The configuration and how to read the results is described in
appendix C and the results are enclosed on the DVD.
This review of the tests is done from the control point of view. This means that
the thrust forces or drag torques are plotted directly. This gives a good idea of the
dynamics of the rotor, which is necessary to design a feedback system, but does
not illuminate its aerodynamic behaviour and the possible errors in the model. In
chapter 8 the non-dimensional thrust and drag coefficients are discussed, thus the
appropriate discussion of aerodynamic behaviour is found there.
All tests are performed in the coordinate system described in chapter 2. And the
rotor position in this coordinate system is:
0
0
0.38
P m
= −
(7.1)
7.1 Testing instruments
In the tests two different instruments are used to measure the force and torque, the
Thrust/Drag meter designed in chapter 5.2 (T/D) and the 6 dimensional force
sensor (FS6). After they had been calibrated, a number of tests were executed.
The tests showed that both instruments had a reasonable precision for static
forces. The FS6 instrument did not have a sufficient accuracy to measure the force
at low velocities in hover. But when the wind was turned on and the tests of
horizontal flight were performed, the T/D instrument did not have a sufficiently
large measuring range. The vibrations from flapping and rotations were large in
50
the wind tunnel. The noise was often over 5 times the actual values, especially in
high wind velocities. Thus in high wind and rotor velocities it was only the FS6,
which were used. But the measurements are taken as the average over one minute
with 200 measurements per second; this is done to reduce the influence of noise
and poor accuracy.
The wind tunnel also has influence on the results, when the rotor is placed closed
to the tunnel floor; the floor has a large influence on the rotor performance. This
effect is called ground effect and it was observed by Bertelsen and Magnüsson [1]
that the quadrotor did not need very much power to lift the first half meter from
the ground. From Bramwell [10] it is known that for traditional helicopters in
hover the ground effect is pronounced until the helicopter is three times the rotor
radius over the ground.
In the wind tunnel the rotor is placed less than two times the distance of its radius
over the tunnel floor. Tests performed on the rotor inside and outside the wind
tunnel show that in hover the thrust is at least 6% higher inside the wind tunnel,
while the drag is 15% lower inside the wind tunnel. In horizontal flight the test
results indicate that the ground effect is less pronounced, which fits very well with
the theory, because the effective distance to the ground along the wake is longer
in horizontal flight.
Likewise the test has shown that the turbulence around the pylon where the motor
is mounted is not negligible. It has a great influence on the rotor performance
because it is a quite a large pylon placed under the rotor.
7.2 Flapping angles
The flapping angles are measured by photos taken of the rotating rotor with the
high speed camera. On the photos the angles are measured in varying air velocity
and rotor angular velocity. The measured flapping angles were compared with the
those derived mathematically in section 6.1.4. This showed that the theoretical
estimation could not be used. The main reason is that the blades do not have a real
hinge but the flexible part of the blade near the centre acts as a hinge. This flexible
part is not as flexible as a hinge; therefore the flapping is in reality much lower
than that determined theoretically. See appendix D.
7.3 Zero lift angle
To measure the zero lift angle one blade is placed in the wind tunnel and the lift is
measured for varying angles of attack. The measured zero lift angle is compared
with that calculated from appendix A. This shows the zero lift angle is less than
that calculated. One reason for this could be that there is some kind of dynamic
twist of the blade or the aerodynamic calculations do not hold for these small
airfoils.
51
7.4 Thrust
The thrust is measured in hover and in forward flight. In forward flight the angle
of the rotor disc plane is varied from -30° to 5° in respect to horizontal. From
these tests, shown in appendix D, it can be concluded that:
• The thrust is only vertical in hover. This is expected and indicates that the
test setup is well calibrated.
• The thrust is higher in hover, than in forward flight. This is not expected,
thus it can be a special aerodynamic behaviour in the wind tunnel [20].
• In wind the thrust decreases when the disc plane angle, αs, becomes
negative. This is because the rotor in that situation acts like in climb.
• The horizontal force in the flight direction is highest when the wind is
purely horizontal. This means the horizontal, xb, force is only a function of
the horizontal component of the wind.
• The horizontal force perpendicular to the flight direction in wind is seen to
be negative at low angular velocities but at higher angular velocities it
reverses its direction. Here it is the positive effects of flapping that
dominate at high velocities.
7.5 Drag
The drag is measured in the same conditions as the thrust and the conclusions of
the tests are:
• In hover the slope of the drag curves are comparable to the slope of the
thrust curves. This indicates that the there is a constant thrust/drag
relationship.
• In flight the drag is higher than in hover. This is in contrast to the thrust
but it fits in very well with the fact that the ground effect is less in flight.
The ground effect results in a smaller induced velocity, which decrease the
inflow angle. When the inflow angle decreases, the thrust increases and
the drag decreases.
• The drag does not vary with the helicopter horizontal velocity. It is only
the change from hover to flight that has influence on the drag.
7.6 The thrust/drag relationship
The thrust versus drag relationship has also been studied. From appendix D
following are apparent:
• In higher wind velocities the drag is higher for the same thrust.
• The drag decreases as the quad rotor flies towards the wind (negative αs)
compared to when it flies in the same direction as the wind. This is not
expected, but is a result of the oscillations in the measurements being too
high to use the thrust/drag meter. And because the FS6 uses a different
52
basis (the ground reference freme) there is interference from the other
torques from the rotor.
7.7 Smoke tests – aerodynamic behaviour
To get an idea of the aerodynamic behaviour around the rotor smoke tests were
performed. Figure 7-1 shows a photo of the smoke taken with the high speed
camera. Red dotted lines are used to emphasise the smoke turbulence.
Figure 7-1: Smoke test in the wind tunnel, the turbulence is emphasized by red lines.
ω=290 rad/s, Vt=5 m/s, αs=15 deg
The smoke tests showed that there is unwanted turbulences around the rotor. The
turbulence is especially large in the area of the rotor disc towards the wind; this is
the seen in Figure 7-1. On the opposite side of the disc the same turbulence was
not seen.
With the simple setup with smoke and high speed camera it was not possible to
see the exact flow around the rotor blade. Thus it was not possible to get a better
estimation of the aerodynamic boundary conditions described in section 3.4 and
the tip losses described in section 3.3.2.
53
8 Calibration of Rotor Model
After a theoretical model has been constructed it is now time to calibrate the
model and test if the assumptions made during the derivation of the model are
correct. This chapter will by tests discuss which parameters that need to be
calibrated if the theoretical model is to be sufficiently calibrated.
In the following, it is the non-dimensional thrust coefficient T
C σ and drag
coefficient Q
C σ , which are of greatest interest (section 6.1.4 and Prouty [2]).
( )
2T
b
TC
A Rσ
ρ=
Ω (8.1)
( )
2Q
b
QC
A R Rσ
ρ=
Ω (8.2)
where A is the blade area.
The advantage of using the non-dimensional coefficients is that it is possible to
compare the performance of rotors with different sizes, angular velocities or in
different media. They is used here to compare the white rotor and the black rotor
to see which has the best thrust and drag efficiency.
Another reason to use the thrust coefficients is that it emphasizes the errors of the
system. When a third or higher order equation with a dominating second order
term and no constant and first order terms are plotted, it is difficult to see the third
or higher order behaviours. Thus if the second order behaviour is divided out the
second order term is the constant and the higher order terms are the slope of the
curve. Thereby both the size of the second order term and the higher order terms
are emphasized. The dominant second order term of the thrust and drag equations
are seen in (3.14).
For clarity all thrust coefficients are plotted positive. The drag coefficient is
plotted positive as for the counter clockwise rotating rotor.
54
In some figures the graph from the Matlab model is not plotted with the right
numerical value. Sometimes it is multiplied by a factor so the slope of the
calculated curves becomes more comparable with the measured results. If this is
done it is clearly stated in the figure commentary.
The single rotor model is calibrated by setting up the same tests as in chapter 7 but
here the output is compared with the calculated model. The rotor used in these
experiments is a rotor with positive or counter clockwise angular velocity. It
would be r1 or r3 on the quadrotor.
8.1 Thrust hover
In hover, the thrust is only vertical and the drag is only around the zb-axis.
Figure 8-1: Thrust coefficient in hover versus rotor angular velocity.
Figure 8-1 shows the thrust coefficient as function of rotor angular velocity. In the
figure the measured results from white rotor inside (red) and outside (magenta)
the wind tunnel and the black rotor inside the wind tunnel (black) are shown. The
results from the model calculations are for the white rotor shown as the blue line
and for the black rotor showed as the cyan line.
It is seen that both the black rotor and the white rotor have a significantly higher
thrust than the calculated model but have the correct tendencies. Thus to be able
to compare the slope of the curves the two dotted lines are included, the blue
dotted line is the calculated model for the white rotor plus 39 %, and the cyan
55
dotted line is the calculated model for the black rotor plus 44 %. Some of the
difference is because the results are measured inside a wind tunnel thus there is
some ground effect, see section 7.1. But still for the white rotor there is a
difference of 31% between the calculated and the measured results. The ground
effect is probably higher for the black rotor; this is because the radius is larger and
thereby the relative distance z/R is shorter. This means that the thrust coefficient
that of the black rotor is higher than of the white rotor. But is only the white rotor
that is tested outside the wind tunnel.
The slope of the measured and the calculated curves are both negative, which
means the thrust coefficient decreases for higher angular velocities. In the model it
is a result of coning of the blades. For the measured results the same effect of
coning is seen but here there is also an effect of dynamic twist, which must be
taken into account. Since the slopes of the curves for the calculated results are
lower than the slopes of the curves for the measured, this can indicate that the
coning angle in the model has been overstated.
56
8.2 Thrust horizontal flight
For normal helicopters today the most of the flight is horizontal, therefore this
case is the most important, but it is actually also the one that is most difficult to
model. Unfortunately the white rotor had a tendency to break when it encountered
high horizontal wind velocities. At air velocities around 5 m/s the white rotor
broke two times. The wind tunnel is very hard to control precisely at velocities
less than 5m/s. Thus it is chosen to do the main analysis on the black rotor and
additionally some test results from the white rotor are discussed.
The first setup is a situation where the helicopter has fixed rotor velocity but with
different horizontal velocities. Figure 8-2 shows the vertical thrust coefficient a as
function of the helicopter velocity. The curves of the calculated model are
multiplied by a factor of 1.20. In no wind (hover) the measured results are still
much higher than the model but for the two cases αs = 0° and αs = 5° in wind the
measured results follows the calculated plus 20%. The measured results in hover
are significantly higher than the ones in flight. That could indicate that the ground
effect in the wind tunnel is of a significant size.
Figure 8-2: Vertical thrust coefficient in horizontal flight
57
From Figure 8-2 it can be seen that at higher wind velocities the thrust coefficient
increases, which is a result of the equation (section 3.3.2):
( )
( ) ( )22 2
sin
sin sin
T
T
U R V
U R V RV
ω ψ
ω ψ ω ψ
= +
= + + (8.3)
The integral of the second term of 2
TU will increase as the forced speed increases,
while the integral of the first and the last does not change. The thrust coefficient is
a function of 2
TU thus it will increase when the horizontal velocity increases.
The magenta plot in Figure 8-2 shows that at low horizontal velocities the thrust
coefficient decreases and then is constant for higher horizontal velocities. The
magenta plot, with negative αs, is the situation where the quadrotor flies towards
the wind (normal flight condition), this situation would be like a combination of
climb and horizontal flight and the to the blade perpendicular velocity Up
decreases.
Figure 8-3: Effects of Ground Vortex on inflow Patterns, source Prouty [2] figure 3.12
For the cases where αs is zero or positive the dynamic performance of the model
fits the measured results. But the agreement is not good for negative αs; the test
has therefore been carried out a couple of times but with same result. This can be
a result of the aerodynamic behaviour around the pylon where the motor is
mounted, it can be a result of ground effect in forward flight, see (Figure 8-3),
which have not been calculated in this project, or it can be a result of the
aerodynamic behaviours around the rotor [20] described in section 3.4.
Figure 8-3 shows one kind of the turbulence due to the ground effect in forward
flight, in other velocities or distance over the ground, the turbulence is different.
The smoke tests (section 7.7) did not show a clear image of how the turbulence is
in forward flight, thus it is from this not possible to say how the ground effect
influences the performance in forward horizontal velocity.
58
Figure 8-4: Horizontal x thrust coefficient in horizontal flight
Matlab calculations and the measurements for the horizontal thrust coefficient are
shown in Figure 8-4. The horizontal forces should be zero in hover and increase
when the velocity increases too. This is seen both in the backward, xb, force and
the sideward, yb, force. The calculated backwards thrust coefficient is seen to
increase significantly with the wind at low velocities, see the red, blue and
magenta line in Figure 8-4. Here the calculated results are not adjusted by a factor.
The decrease at high velocities is an error in the model; this is because the
modelling of flapping does not fit the velocity range exceeding 9 m/s, see
appendix D.
The measured results seem to be much higher than the theoretical model. A part
of the reason is that the calculated model as described earlier has 20 % lower lift
than the real rotor but there is more to this. After the test phase ended and the
wind tunnel setup was removed it was discovered that the measurements have not
been adjusted for the influence of wind pressure on the pylon where the motor is
mounted. Thus all the measurements have the same error. The following provides
an estimate of the effect of the wind pressure on the pylon.
If the pylon is assumed to be square shaped with a drag coefficient cdp = 2, width
wp = 0.04 m and height hp = 0.25 m, then the drag force and drag coefficient from
the pylon is given by [2]:
2
2
1
2
0.0125
d dp p p
d
F v c h w
F v
ρ= −
= −
(8.4)
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where v is the air velocity in the wind tunnel.
If this correction factor is used in Figure 8-4 the direction of the drag is reversed,
showed in Figure 8-5. This is because the wind around the pylon is influenced by
the wake of the rotor. The aerodynamic pressure on pylon has only influence on
the horizontal xb force, thus it is only the xb force that due to that reason is not
possible to measure correct.
Figure 8-5: Horizontal xb thrust coefficient calibrated for the pressure on the pylon
Figure 8-6: Horizontal yb thrust coefficient in horizontal flight
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Sideways along the yb-axis, the force is expected to increase when the helicopter
velocity increases, see the red, blue and magenta lines in Figure 8-6. It is
interesting that the measured results in contrast to the earlier results lie under the
graph for the model, even though the model has not been multiplied by a factor.
This can be due to the fact that the model doesn’t take root cutout into account. In
the part of the rotor disc, which should have been excluded with root cutout the
horizontal yb force is high. This is because the advancing blade with positive
thrust and the retreating blade with negative thrust both have the same direction of
the horizontal yb component of force.
Another reason could be that flapping is lower than expected, and thereby the
horizontal forces are lower. As discussed in appendix D the flapping in the yb
direction was the most difficult parameter to estimate accurately.
Figure 8-7: Thrust coefficients in horizontal flight.
Top: Horizontal x. Left: Horizontal y. Right: Vertical z.
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It was possible to finish one test of thrust as function of wind speed. The results
are shown in Figure 8-7, and will be compared to the results from the black rotor.
The test is carried out in normal forward flight αs = -5°. Thus it is compared to the
magenta curves above. The behaviours of the three plots in Figure 8-7 are in
accordance with the above plots for the black rotor. This is discussed further
hereafter.
The size of the xb component of the thrust is about the same for the two rotors. It
indicates that the aerodynamic pressure on the pylon as described earlier, has a
great influence on the two measurements. Therefore it is only the dissimilarities of
the slopes of the two curves which can be compared. The white rotor has a steeper
slope; this indicates that there are higher effects of flapping. This fits with the
observation in section 3.1: the white rotor is more flexible than the black one.
For the yb component (Figure 8-7 left and Figure 8-6) it is clearly seen that the
horizontal force of the white rotor increases faster than the one of the black rotor.
This is partly because the tip speed ratio increases faster on the white rotor
because the radius is smaller. It is also due to higher flexibility of the white rotor.
In the vertical zb-direction the thrust coefficient is seen to be lower for the white
rotor (Figure 8-7-right) than for the black rotor (Figure 8-2). These results show
that the white rotor has higher sideways force coefficients than the black rotor.
This effect is often undesirable in flight, so this could indicate that in this case the
black rotor is better. Data shows that both the black and the white rotors do not fit
the calculated model for αs = -5°. This means that it is not just an error in the
measurements but some aerodynamic forces that need to be taken into account.
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Figure 8-8: Thrust forces versus angular velocity of the rotor.
Top: Horizontal x. Left: Horizontal y. Right: Vertical z.
All forces are in N.
Finally the thrust is plotted as function of the angular velocity, see Figure 8-8. The
xb component is as before much larger than the theoretical model. The yb
component has a maximum at a specific frequency and decreases afterwards. The
same trend is found in the measured results but as expected it is lower. This is
discussed in appendix D.7). The measured vertical thrust is as seen before 20 %
higher than that calculated.
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8.3 Drag hover
As described in section 4.2 the modelled data from the drag is expected to be
much lower than measured. This is because the model only takes induced drag
and not profile drag into account. Therefore it is expected that the measured
results for the drag are more than 40% higher than the model. 40% is what that
was the case for the thrust. Also the fact that near the ground the drag decreases
indicates that the modelled drag should be lower compared to the measured
results.
Figure 8-9: Drag coefficient in hover.
The actual tests show that the drag is much higher than the model predicts, see
Figure 8-9. For the white rotor the drag is 95 % higher in the wind tunnel and 120
% higher outside the wind tunnel. The tests outside the wind tunnel are the ones
that should be comparable with the model. It means that the profile drag must be
taken into account. More interesting is the slope of the curve. In free air the theory
predicts that the drag coefficient should be close to constant, and only increases
very little. This is also what is seen at the two measured curves for the white rotor
(the red and the magenta graph).
For the black rotor the measured results again are much higher than the model
(50%), but the slope of the measured results is not the same. Here it has low drag
coefficient for low velocities with a maximum at 275 rad/s and starts to decrease
for higher angular velocities. From the simple theory used in this thesis it is not
possible to say why this characteristic appears but the aerodynamic behaviour
when the flow around the blade changes can result in behaviour like that [20].
These effects are described in section 3.4.
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8.4 Drag in horizontal flight
When the helicopter flies horizontally the increasing tangential velocity UT reduce
the inflow angle, which will result in a decrease in the induced drag coefficient.
This is visualized by the red, blue, magenta and black lines in Figure 8-10. Note
that none of them are multiplied by a factor. The profile drag (not calculated) is
expected to increase with the horizontal velocity because the integral of the square
of the inflow velocity increases. Because these two components of the drag have
opposite influences on the drag in horizontal flight it is not immediately possible
to predict the slope of the measured drag coefficient as a function of wind
velocity.
Figure 8-10: Drag coefficient in horizontal flight.
The slope of the measured drag coefficient for varying helicopter velocity is
constant for the black rotor (red, blue and magenta dots) but increases for the
white rotor (black dots) as shown in Figure 8-10. On that basis it can be concluded
that for the black rotor the positive influence from induced drag cancel the
negative influence from profile drag in horizontal flight. For the white rotor the
drag increases for higher horizontal flight. This is because the profile drag is
higher compared to the induced drag. It is seen that the results with αs = 0° have
significantly higher measured drag than the other cases. This is because of
influence from other torques in other directions when αs ≠ 0°.
65
The drag coefficient as a function of angular velocity is plotted in Figure 8-11. In
this test both of the rotors have an angle αs = 0° to the wind. From Figure 8-10 it
is expected that the results with αs = 0 should have significant higher measured
drag than modelled. It is also what seen both for the white and the black rotor that
the measured drag is much higher than the calculated (blue and red lines)
Figure 8-11: Drag coefficient versus angular velocity
8.5 Other kinds of torque in flight
In flight there are also torqueses around the horizontal axis besides those from
horizontal forces. Unfortunately one of the simplifications done in section 4.2 was
to exclude these forces but they are discussed here because they give a good idea
of the rotor parameters. The torque around the yb axis corresponds to the force xb.
This torque will not be further discussed here because the other tests showed large
effects from the aerodynamic pressure on the pylon where the rotor is mounted.
The torque around the xb axis is more interesting. This is the torque that flapping
should help to cancel. Figure 8-12 shows this torque versus angular velocity. The
torque is combined from two parts, the one is the sideways thrust at the rotor hub
and the other is torque around the rotor hub. The contributions from the horizontal
forces (with h = 0.38 m) are negligible, see Figure 8-6.
In Figure 8-12 it is seen that the torque coefficient decreases with angular
velocity. One of the reasons is because the tip speed ratio decreases for higher
velocities. Another reason is because of flapping: then the more flexible white
66
rotor should decrease most. This is not what is seen in the figure; the torque
coefficient for the white rotor is higher than for the black rotor. Thus it can be
concluded that the more flexible white rotor does not have better performance
from flapping. The assumption from section 3.3.2 where it is assumed that the
flexibility of the rotors can be calculated as flapping is thus not valid. It is also
seen that the angle αs has influence on the slope of the curves but not the actual
values in the interval 250 < ω < 350 rad/s, which is the area where the maximum
torque is found. This means the maximum torque around the horizontal xb axis is
not influenced by the angle αs.
Figure 8-12: Torque coefficient xb, versus angular velocity.
8.6 Summery of calibration of the single rotor model
The test series shows that this first roughly calibrated model needs some
adjustment before it fits the quadrotor rotors. In this subsection it will be
discussed which parameters that need to be adjusted. If possible the estimation of
a new value is presented.
In hover the thrust is seen to be about 40% higher than the model predicts. Some
of this is because the tests are done inside a wind tunnel but it is not the entire
explanation. In horizontal flight the vertical thrust coefficient is 20% higher than
the model. Here the ground effect is less pronounced. Thus it would probably be
the best to calibrate the model to fit that situation. To increase the thrust in hover
the following parameters in the model can be adjusted:
67
• The integration area, X0 and B. From measurements of the induced
velocity it is determined that 2.5 cm of the tip does not have any lift. By
reducing that area to 2 cm the lift will increase 8 % in the model for the
black rotor and 9 % for the white. From Figure 6-2 it is seen that 2 cm tip
loss could be probable.
• The angle of attack, α, can be increased. This could be done by increasing
the pitch or zero lift angle, but tests of the zero lift angle indicate that these
are already too high (Appendix D)
• Lift coefficient, cl. In the model the slope of lift curve is assumed to be
constant. The slope is in the model chosen to be 5.73, which is lower than
the assumption normally used in large helicopters, which is 2π. If 2π are
used the lift coefficient will increase with 10 %. An implementation of a
dynamic lift coefficient could also have en impact on the thrust [20].
From these considerations it is chosen to decrease the tip loss area to be the end 2
cm of the rotor blades (B = 0.86) and the slope of the lift curve is chosen to be a =
2π. Thereby the lift forces are increased by 20 %.
The same 20 % increment of the lift forces is not seen for the horizontal forces.
The force along the flight direction, xb, is much higher in the tests than in the
calculated model, but this is probably an error because a part of the force
originates from the wind pressure on the pylon where the motor is mounted. In the
component of the force perpendicular to the helicopter forward direction, yb, the
measured results is 20% lower than the calculated model. This could indicate that:
• The flapping angle is too high. The size of the yb force is highly influenced
by the size of the flapping factor, b1s. Thus by reducing b1s the y
component is reduced. In the estimation of the flapping angles in appendix
D, it is concluded that b1s is the least accurately estimated flapping angle.
The force in the calculated model is in previous subsection increased by 20 %.
Thus to obtain the correct horizontal force this must be decreased by 33 %, this
means the angle b1s must be decreased by more than 33%.
Just before the project was closed it was possible to do one simulation of the
model with the calibrated parameters. Following conditions were hard coded in
the simulation:
• ω = 290 rad/s
• αs = 0°
• ε = 0°
Figure 8-13 shows the new modified model for the horizontal yb force compared
with the measured results and Figure 8-14 shows the values for the vertical thrust.
Both of those figures show that for wind velocities over 2 m/s the new calibrated
68
model fits the measured results. At low velocities less than 2m/s it is seen that the
measured thrust is higher than the modelled thrust but this is because of ground
effect in the measured results. In Figure 8-14 it is seen that the slope of the curve
of the model is too low but this is because the root cutout is not taken into account
in the model. Thus the reverse flow region has too high an impact on the results.
Figure 8-13: Horizontal yb thrust coefficient versus wind velocity.
Red: Measured results. Blue: Calibrated model
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Figure 8-14: Vertical zb thrust coefficient versus wind velocity.
Red: Measured results. Blue: Calibrated model
The tests of the models ability to estimate the drag shows that the assumption that
the drag only consists of the induced drag is not valid. The profile drag has to be
taken into account for a correct model. It is not possible from the tests in this
thesis to give a good estimation of the profile drag. To do so, further tests needs to
be performed and ideas for these tests are described in subsection 10.4.
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71
9 White vs. Black Rotor
One of the goals in the project was to find out which of the white and black rotor
that has the best performance. In the two previous chapters where the test results
are described, some factors have been discussed that influence which of the two
rotors should have the best performance. In this chapter these factors will be
discussed and in the end it will be concluded which rotor that has the best
performance in the quadrotor configuration.
9.1 Thrust and drag
First the values for the thrust and drag is discussed, the non-dimensional thrust
coefficient, T
C σ , is higher for the black rotor. A maximum thrust is in high
relationship to the angular velocity, two factors have influence on the maximum
angular velocity: the one is the power of the motor and the other is strengthens of
the blade. The white rotor broke two times at 290 rad/s, thus it is not desirable to
choose a normal velocity to more than 200 rad/s. The same maximum is not found
for black rotor, thus the comparison of the thrust and drag is done at 200 rad/s.
For an angular velocity of 200 rad/s the thrust coefficients and thrust for the two
rotors are:
White: 0.179T
C σ = 1.32T N=
Black: 0.185T
C σ = 1.99T N=
This means that even if the two rotors had the same size the black rotor would still
have a higher lift than the white. The non-dimensional drag coefficient was found
to be lower for the black rotor than for the white rotor but the drag is higher for
the black rotor. At 200 rad/s the values are:
White: 0.0297Q
C σ = 0.0255Q Nm=
Black: 0.0231Q
C σ = 0.0414Q Nm=
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9.2 Thrust/drag relationship
One new presentation of the measured results will be taken into account; this is
the relationship between the non-dimensional thrust and drag coefficients. This
relationship defines the efficiency of the rotors. Because the non-dimensional
coefficients are used, it is possible to compare the different rotors in different
conditions.
Figure 9-1: The relationship between the thrust and the drag coefficients.
Figure 9-1 shows the relationship between the non-dimensional thrust and drag
coefficients. The efficiency is defined as:
T
Q
C
C
ση
σ= (9.1)
From the graph this efficiency is higher (better) in the lower right corner and
lower (worse) in the upper left corner.
It is seen that for both the black and white rotor the efficiency is better in hover,
this is expected because of the ground effect. The black rotor has a significantly
better efficiency; this is seen in the results for the black rotor which has a lower
drag coefficient, while the thrust coefficient has a tendency to be higher than for
the white rotor.
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9.3 Earlier observations
Most of the observations concerning the difference between the two rotors in
chapter 7 and 8 are included in Figure 9-1. The observations that are not included
are:
• The horizontal thrust forces and torques are relatively higher for the white
rotor than for the black rotor. In the quadrotor configuration this does not
directly have a negative influence of the performance but it is never
desirable to have unwanted forces in the system. The reason for the
increased forces is that the white rotor blades more flexible than the black
rotor blades. From the theory of flapping it is expected that flapping
should give lower horizontal forces and torque but it is not what seen here.
Thus it can be concluded that the flapping theory of can not be directly
used on the quadrotor.
• The white rotor had a tendency to break if the horizontal velocity was too
high. If the rotor is used in a light quadrotor designed for indoor use,
where the horizontal velocity is low, this will not be a problem. But else
the risk of break down will be undesirable in most quadrotor helicopters.
9.4 Summary
Overall it is concluded that the black rotor has the best performance. The two
main reasons for the black rotor being better are:
1. The attempt from the manufacturer to make a rotor with a better airfoil
than the white rotor has succeeded.
2. The black rotor is less flexible than the white. This results in lower
horizontal forces than for the white rotor.
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10 Conclusion
The goal of this thesis was to:
• By tests and calculations of the white and the black rotors describe the
advantages and disadvantages of the new rotors compared with the
hitherto used rotors.
• Build and calibrate the necessary test equipments.
• Describe and by test calibrate a nonlinear model of the quadrotor
helicopter including all relevant factors.
The conclusion is divided into three sections: one is the purpose built instrument,
second is the nonlinear model and the third is a comparison of the white and the
black rotor. At the end of this chapter some ideas for future work with the
quadrotor helicopter is presented.
10.1 Purpose built instrument
Earlier tests previous to this thesis have shown that the measurement instruments
at DTU did not have the accuracy to test the rotors; thus it was decided to build an
instrument to measure vertical thrust and drag torque around the vertical axis.
This instrument was built and calibrated during this thesis and the final instrument
had better accuracy than the force and torque sensors available at DTU. In the test
phase it was realized that the instrument was disturbed too much by vibrations
from the motor and rotor. This exceeded the measuring range and led to wrong
measurements. Thus a commercial six dimensional force and torque sensor was
used in some of the tests.
10.2 Nonlinear model
A nonlinear model of the quadrotor helicopter is described in chapter 2 and
models for the two rotors are described in chapter 3. The model of the rotor takes
factors like coning, flapping, zero lift angle, tip loss, root cutout and velocity of
the helicopter into account. Thereby it should be possible to give an exact estimate
of the rotor performance.
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The described model was written into Matlab using the Symbolic Math Toolbox
and Simulink. The final model was too complex to simulate in Matlab. A
simplified model which would be compiled in Matlab was found and further
development was stopped. Based on the tests it was discussed how to calibrate the
model but it was not recalibrated due to its complexity. It was possible to
calculate a single simulation with the estimated parameters. This showed that the
yb and the zb forces were correct for forward flight.
24 tests with different configurations of the two rotors were performed and from
these tests an estimate of which constants of the rotor model that need to be
adjusted was made. The calibration of the model showed that there are other
effects that must be taken into account if an exact model has to be made. These
effects are aerodynamic boundary conditions and ground effect. The aerodynamic
boundary conditions are too complex to determine for rotors with low Reynolds
numbers like the one used on the quadrotor.
It can be concluded that it is possible to theoretically setup a nonlinear blade
element model of the quadrotor but if the model must be exact, there are more
aerodynamic behaviours that must be taken into account. These behaviours are not
simple to estimate mathematically, thus they must often be estimated by use of
tests. To setup a proper nonlinear model of the rotors, the best way is to set up a
simple mathematical model to get an idea of the dynamic behaviour and then from
tests derive the exact parameters. It has shown up to be too complex to include all
parameters in the mathematical model.
10.3 White versus black rotor
In the tests carried out the white rotor and the black rotor were compared to
determine which rotor that has the best performance. This comparison focused on
thrust coefficient, thrust versus drag relationship and sideways forces in forward
flight. The black rotor showed a significantly better performance than the white in
all the parameters discussed.
The tests showed that the non-dimensional thrust coefficient, T
C σ , is higher for
the black rotor and the non-dimensional drag coefficient, Q
C σ , is higher for the
white rotor. Thus this means the thrust versus drag relationship is better for the
black rotor than for the white.
The increased stiffness of the black rotor results in less sideways forces in
horizontal flight. The full size helicopter theory predicts that a rotor with flapping
blades should have less horizontal forces and torques. Here it is seen that the
flexible blades from the quadrotor can not be seen as a blade mounted with a
hinge, thus the quadrotor does not have the positive effects from flapping. The
76
very flexible white rotor has had a tendency to break when it was tested in
forward flight with high angular velocities.
The conclusion is that it is possible to increase the rotor performance by changing
the airfoil of the rotor blades. Here it is shown that the airfoil of the new black
rotor has a better performance on the quadrotor than the hitherto used white rotor
but it is not studied if it is possible to find a better airfoil to the quadrotor rotors
than the one on the black rotor. This can be an issue for future studies.
10.4 Future improvements
As described in section 4.2 the calibration of the Matlab model had to be stopped
due to a overly complex Matlab model and too long compilation times. Thus the
first and most important work in the future is to tune and compile the model from
the tests performed.
It was discovered that the profile drag also has to be taken into account in the
calculations. To find the profile drag a number of future tests should be
performed. The tests performed until now focus on the rotating blade but a
number of tests on a single blade at different inflow velocities and angles of attack
should be made. Here it is possible to get the mean value of the profile drag over
the rotor. This together with the results from this project should give good
estimate of both the profile and induced drag.
In the tests the ground effect and the physical effects around the pylon where the
motor is mounted were shown to have high impact on the rotor behaviour. To
reduce ground effect in the tests it could be an idea to perform tests for vertical
flight. In these tests the rotor should be turned so the rotor disc is vertical in the
wind tunnel and thereby have no ground effect. Here will still be some wall
effects but they have less influence on the results.
It is the area below the rotor where the aerodynamic effect has highest impact on
the rotor performance. Thus it could be an idea to turn the rotor around and let it
have lift downwards. Then the pylon will not be under the rotor where it is most
critical.
Anders Hedeager Pedersen
s011258
77
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78
[14] Math Works, the creator of Matlab and Simulink product home page,
http://www.mathworks.com/.
[15] Maple soft, the creator of Maple product home page,
http://www.maplesoft.com/.
[16] Graupner GmbH, RC-toy dealer, http://graupner.de/.
[17] Mabuchi motor, Japanese Motor manufacture. http://www.mabuchi-
motor.co.jp/
[18] Egnite software GmbH, Manufacturer of Ethernut development boards.
http://www.ethernut.de/.
[19] AMTI, the creator of AMTI force sensors website. http://www.amti.biz/.
[20] Mikkelsen, Robert F, Private communication, DTU 2006
A 1
A Zero Lift Angle.
A.1) Zero lift angle
Calculation of the zero lift angle takes it origin in the definitions from Kuethe
Chow, p 146 [9]:
( )00
1cos 1
L
dzd
dx
π
α θ θπ
= − −∫ (21.1)
To find the geometric shape of the airfoil used with the quadrotor, each of the two
different rotors used is cut into small elements. The shapes of each of these small
elements are then compared with different geometric shapes. The shape that had
the best fit was an arc of a circle. These radii of this circle were for the two rotors:
RRw=0.04m. (21.2)
RRb=0.035m. (21.3)
Figure A 1 The definitions of the airfoil.
From Figure A 1 and from the circle equation, it is known:
22 2
2 2
2 2R R
c cR x z R
= − + − −
(21.4)
A 2
where c is the chord of the airfoil. To find dzdx
is it is evaluated and gives:
2 2
2 2
2 2R R
c cZ R R x
= − + − −
2
2
2
2R
cx
dz
dx cR x
−= −
− −
(21.5)
Again from Figure A 1 it is shown:
tan
2
R
za
cx
θ
= −
(21.6)
If (21.4) is inserted, and evaluated it gives:
2 2
2 2
12
2 2tan
2
R R
R
cx
c cR x R
cx
θ
−−
− =
− − −
− −
−
(21.7)
Then (21.5) is equal to (21.7) and it gives:
2
2
1
2tan
2
R
R
dz
dx cR
cx
θ
=
−
+
−
(21.8)
Then x also has to be estimated as function of θR. By use of Figure A 1 and the
sinus relationship it gives:
( )
( )( ) ( )
22
2
2 2sin sin sin
RR R
R R
RR r
v v
π
π πθ π θ
+= =
+ − − − (21.9)
A 3
Evaluated with respect to 2cx − gives:
( ) ( )( )
( )2
2 2
22
2
1sin 2 cos cos
2 2 2R
R R R Rc
R
Rc cx R
Rθ θ θ
− = − − − −
(21.10)
Finial (21.10) is set into (21.8):
( )( )
( ) ( ) ( )
( )( )
( ) ( )
22
22
2
22 2
22
2
cos cos sin cos
sin cos sin 1
RR R R R
cR
RR R R
cR
R
Rdz
dx R
R
θ θ θ θ
θ θ θ
− −−
=
− − −−
(21.11)
B 1
B Microprocessor Design
B.1) Instruction manual
When the system is turned on, and a RS-232 cable is connected to the board,
following line is send via the RS-232 cable to the PC:
Indtast h (hastighed), s (start log), p (print log)
Here there are following possibilities:
• Write hxxxx where xxxx is a number between 1350 and 2000. As higher
the number is the motor get more current, and thereby no current at h1350.
• Write s to start the log, 4000 samples is caches with a frequency of 500 s-
1. When the log is done “færdig” is written from the microprocessor.
• Write p to print the log, here the log will be written with the following
syntax:
[sample nr.] [Thrust] [drag] [velocity]
The values that are written are the directly output from the ADC.
C 1
C Test Data
A number of test has been performed in this thesis, here is a list with all the
performed tests, the data from the results in enclosed at the DVD.
Tests with white rotor:
Vt [m/s] αs[deg] Ωr[rad/s] Remarks
0 0 - Outside wind tunnel
0 0 - Inside wind tunnel, With high speed video
5 0 - With high speed video
5 -2.5 - With high speed video
5 -10 - With strengthened rotor
- -5 290 With strengthened rotor
10 Test of zero lift angle
- Test of lift on single blade
Tests with black rotor
Vt [m/s] αs[deg] Ωr[rad/s] Remarks
0 0 - With high speed video
5 5 - With high speed video
5 0 - With high speed video
5 -2.5 -
5 -5 - With high speed video
5 -7.5 -
5 -10 -
5 -20 -
5 -30 -
- 5 290 With high speed video
- 0 290
- -5 290 With high speed video
10 Test of zero lift angle
15 Test of zero lift angle
- Test of lift on single blade
Test with smoke
C 2
In the columns for wind velocity, angle and angular velocity following rules is
given:
• A number means the value has been fixed during the test.
• A “-“ means the test has been varied during the test.
• Nothing means the value is not used for the test.
At each test the forces and torques are measured with two different instruments,
the FS6 force sensor and the specially built thrust and drag meter (T/D). All logs
from the tests are included at the DVD, and they are placed in folders describing
the test conditions. An overview of the files in the folders is:
The thrust drag meter:
Filename: capxxxx.txt where xxxx is the input to the motor.
Encoding:
0 AAA BBB CCC
1 AAA BBB CCC
….
3999 AAA BBB CCC
where
• AAA is the speed,
• BBB is the thrust force and
• CCC is the drag torque.
All given in digital output from the ADC, therefore they must be multiplied by a
factor:
ADC to Thrust: 0.0222 N
ADC to Drag: 0.000288 Nm
ADC to angular velocity: 1.0044 rad/s
The FS6 force and torque sensor:
Filename: transxxxx.txt where xxxx is the input to the motor.
Encoding:
AAA BBB CCC DDD EEE FFF
AAA BBB CCC DDD EEE FFF
…
AAA BBB CCC DDD EEE FFF
where
• AAA is the x component of the force in N.
• BBB is the y component of the force in N.
C 3
• CCC is the z component of the force in N.
• DDD is the torque around the x axis in Nm.
• EEE is the torque around the y axis in Nm.
• FFF is the torque around the z axis in Nm.
Measured speed
Along with the speed measured with the freq to voltage converter the speed is
measured with an oscilloscope.
Filename: speed.txt
Encoding
AAAA BBB CCC
AAAA BBB CCC
Where:
• AAAA is the input to the rotor.
• BBB is angular velocity of the rotor in Hz.
• CCC is velocity of the air in the wind tunnel in m/s.
Load into Matlab
Filename: DATA.m
Load the measured data to the Matlab workspace. Following arrays are created:
• AAms_BBdeg_CCCC_cap
Data from the T/D with angular velocity [rad/s], thrust [N] and drag [Nm].
• AAms_BBdeg_CCCC_trans
Data from the FS6, with x,y,z force and torque around the x,y,z axis.
• AAms_BBdeg_CCCC_r_trans
Same as above but rotated so it is in the rotor frame.
In all of then are:
• AA given by the air velocity in the wind tunnel
• BB given by the angle of the rotor (αs)
• CC given by the input to the rotor in the test.
In the cases where one of the factors is not constant in the test this is excluded
from the name.
D 1
D Test Result
This appendix describes the performed tests and discusses the results. The first
numbers of tests show if the assumptions done earlier in this project are valid; the
last tests show the dynamic of the rotor. The raw data for each test is found on the
enclosed DVD.
D.1) Especially built Thrust and Drag meter vs. FS6
Before the tests started the two measuring instruments T/D and FS6 (see section
5.2) were calibrated and after the calibration some test were done to verify that the
two instruments show the same result. One of the tests was with the white rotor in
the wind tunnel without wind. Figure D 1 and Figure D 2 show the thrust and drag
from the experiment. The results from both the thrust and drag meter (T/D), and
the 6 dimensional force sensor (FS6) are both shown in the figure, and the average
line is also drawn. From these lines it is concluded that the two instruments agree
on the strength of the force. The same was seen the same for most of the test
results.
When looking directly at the log from the two instruments the results are different.
The log from the microcontroller gives a 10 bit integer (0-1023). When the test
with wind is performed the signal oscillates between zero and maximum in one
revolution of the motor.
D 2
Figure D 1 Thrust vs. Angular velocity for the white rotor in the wind tunnel.
Figure D 2 Drag vs. angular velocity for the black rotor in the wind tunnel.
D 3
D.2) Wall effects
The wall and ground effects are probably one of worst problems in the wind
tunnel. From Bramwell [10] it is known that the rotor must be at least 2*R over
the ground to reduce ground effect, and it is in reality not until 3*R it is acceptable
to not use it in the calculations. It is a bit different with the tests carried out in
wind, because distance to the ground must be measured along the wake, and the
wake is longer in wind. When the rotor is installed in the wind tunnel the distance
to the ground is about 1.6*R which based on Bramwell [10] should give an about
10 % increase of the thrust inside the wind tunnel.
Figure D 3: Thrust versus angular velocity inside and outside the wind tunnel
Figure D 4 Drag versus angular velocity inside and outside the wind tunnel
D 4
Figure D 3 and Figure D 4 show that the thrust is about 6% higher inside the wind
tunnel. The drag is about 15% lower inside than outside, it is expected that it
should be lover because the induced drag is a function of the inflow angle which
decreases near the ground. Figure D 5 shows how the drag depends on the thrust.
It is seen that the drag/thrust curve can considered a linear curve, and the
difference in the slope, is due to a faster speed outside the wind tunnel, and
thereby the profile drag becomes higher.
Figure D 5 Drag versus thrust inside and outside the wind tunnel
D.3) White or black rotor
Inside the wind tunnel but with no wind, tests are done to tell the difference of the
white and black rotor. Figure D 6 and Figure D 7 show that the values for both lift
and drag are abut 40 % higher for the black rotor than for the white rotor (exact is
43% for the lift and 39% for the drag). The difference in the two rotors disc area is
given as how much lower the white rotor is than the black:
( ) ( )( ) ( ) ( )( )( )
( ) ( )( )
2 2 2 2
0 0
2 2
0
44%b b w w
b b
R B R x R B R x
R B R x
− − −=
− (24.1)
D 5
Figure D 6: Comparison of the thrust for the white and the black rotor.
Figure D 7: Comparison of the drag for the white and the black rotor
This could conclude that the main difference of the two rotors is the disc area. But
the fact that the difference in drag is lower than the difference in thrust, could
indicate that the black airfoil has a lower profile drag.
D 6
Test setup b w
w
T T
T
− b w
w
Q Q
Q
−
αs=0° Vx=0ms 43% 39%
αs=0° Vx=5ms 23% 11%
αs=-10° Vx=5ms 17% -12%
Table D 1: Difference in white and black rotor.
Table D 1 shows three selected tests performed with both the white and black
rotor. It shows that the assumption that the only difference between the two rotors
is the disc area is only valid in hover. The difference in thrust, in the three tests, is
a symptom of the change in thrust over the rotor, as explained later. But the
difference in drag is actually more interesting, when the helicopter is in forward
and upward flight. The drag is actually higher for the white rotor than at the black
for the same speed. On this basis it is concluded that the black rotor will have a
better performance in flight.
D.4) Zero lift angle
The zero lift angle is until now based on the geometric size, and derived from full
size airfoil design theory. This theory may not fit small helicopter rotors as the
one used for the quadrotor. It is therefore important to test if the calculated angle
is valid. The rotor pitch and chord are not constant over the radius, thus the zero
lift angle neither is constant. In the wind tunnel the tests are done at an entire
blade, therefore the measured zero lift angle is a mean of the value over the blade.
Figure D 8: Zero lift angle for the white and black rotor.
D 7
Table D 2 shows the pitch and zero lift angles of the white and black rotor blade.
Compared with Figure D 8, the calculated values are higher than the values from
the tests. It is estimated to be about 20%, which could be because there is some
kind of dynamic twist (change the pitch at high velocities) in high speed air flow,
or maybe some kind of error in the geometric calculation.
r θw αL0w θw + α L0w θb αL0b θb + α L0b
0.035 17.5 3.87 21.3 19.1 4.55 23.7
0.045 22.8 6.16 29.0 25.4 7.47 32.8
0.055 22.0 5.79 27.8 24.5 7.00 31.5
0.065 21.3 5.44 26.7 23.6 6.54 30.1
0.075 20.5 5.10 25.6 22.5 6.02 28.5
0.085 19.3 4.62 24.0 21.4 5.49 26.9
0.095 18.6 4.31 22.9 20.5 5.10 25.6
0.105 17.8 4.01 21.9 19.6 4.73 24.3
0.115 17.0 3.73 20.8 18.7 4.74 23.1
0.125 16.3 3.46 19.8 17.9 4.04 21.9
0.135 15.6 3.19 18.8 17.0 3.71 20.7
0.145 - - - 16.2 3.40 19.6
0.155 - - - 15.4 3.15 18.5
average 19.0 4.52 23.5 20.2 5.07 25.3 Table D 2: Pitch and zero lift angles over the radius of the blade. All angles are in radians.
Figure D 8 also shows that the drag is very low at the zero lift angle, this means
that most of the drag is induced drag and that the profile drag is quite small.
D.5) Flapping angles
The flapping angles are measured by use of the Dantec Dynamics high speed
camera. As in the analytics it is only the first order harmonics that are included in
the test measurements, see section 3.3.2. To find the first order harmonics the
photos of the rotor in the four positions ψ=0, π/2, π, 3π/2 are used. From these the
three constants are calculated:
3
2 20
04
aπ ππα α α α+ + +
= (24.2)
01
2s
a πα α−= (24.3)
3
2 2
12
sb
π πα α−= (24.4)
Due to the structure of the wind tunnel it is only possible to take photos from one
side, and as it is seen at Figure D 9, it is much easier to read the angles at the
upper image than at the lower. At the upper image it is reasonable to assume that
D 8
the angle seen at the image is the real angle. An error could be a result of optical
distortion. The optical distortion can be very high in special wide-angle photos,
but the focal length should be adjusted to the camera.
Figure D 9: Photos from the high speed camera. The red lines are used for calculation the
flapping angles.
In Figure D 9 lower photo there is a larger problem with the precision, at first it is
more difficult to see precise where the tip is, especially the tip pointing towards
the camera. Secondly if there is some kind of uncertainty of the vertical position
of the camera, this will likewise give a high uncertainty in the results. The last
main uncertainty is the difference in distance between the two blade tips and the
camera. In this setup the camera is 2.5 m from the motor, so this difference should
not be significant.
To calculate the angle the distance from the hinge to the tip ( )1R e− and the
height hx. Four photos are used pr measurement, so each of the two blades is
calculated individually.
Even with all these uncertainties the flapping angles are measured and calculated.
There are three factors which have influence on the changing in flapping angles
during flight:
• Rotor velocity.
• Helicopter velocity (in the surrounding air).
• Helicopter flight direction.
This project has been on study the rotor velocity and the helicopter horizontal
velocity.
D 9
D.5.1 Rotor velocity, ωωωω
Flapping as a function of rotor velocity is found by setting up a test in the wind
tunnel, with the rotor in horizontal position, and the wind set to 5 m/s. For
different velocities a video sequence of one round with 4000 photos/s is taken.
Figure D 10: Flapping angles as function of angular velocity.
Figure D 10 shows the measured angles and the tendency lines. This shows that
the second order lines fits satisfactory in the active area of the rotor. From full size
helicopter theory the constants are known to be positive. But here the b1s is
negative, because the main damping in full size helicopter theory is the inertia of
the rotor blade, but here the damping is the stiffness of the rotor, while the mass of
the rotor blade is negligible. Therefore the maximum of the flapping angle will be
in almost the same place as the maximum lift (ψ=π/2)
D 10
Figure D 11: Theoretical flapping angles
The calculated theoretical flapping angles (section 3.3.2) are shown in Figure D
11 as function of the angular velocity. As expected because of the higher damping
they are much higher than the measured. The slope of the curve do not follow the
measured, thus it can be concluded that the full size helicopter theory for flapping
angle can not be used at the quadrotor rotors.
D.5.2 Horizontal velocity
The test is set up with the constant angular velocity 290 rad/s, and the wind speed
is varied. Figure D 12 shows the measured results and tendency lines.
D 11
Figure D 12: Flapping angles as function of horizontal velocity.
The coning angle a0 is constant over the spectrum of horizontal velocity as
expected. The flapping angles a1s and b1s are estimated with second order lines.
These two lines fit well in the measured area, but as seen they will decrease if the
speed exceeds 8 m/s. From the measurements it looks like the flapping angles
finds a constant maximum angle for increasing velocity. Therefore in the final
model it must be best to use following equations:
( )( )
0
3 2
1
4 2
1
0.089
0 5 : 1.6 10 0.023
9.3 10 0.011
h s h h
s h h
a rad
V a V V rad
b V V rad
−
−
=
≤ ≤ = − ⋅ +
= ⋅ −
(24.5)
0
1
1
0.089
5 : 0.0743
0.0332
h s
s
a rad
V a rad
b rad
=
≤ = = −
(24.6)
D.5.3 Conclusive equations for flapping
The above equations are combined, the mean values are taken, and it gives
following equations:
D 12
( )( )( )
( )( )
6 2 4
0
6 2 4 3 2 2
1
6 2 3 3 2 2
1
1.08 10 6.24 10
0 5 : 2.58 10 1.78 10 5.8 10 8.3 10
1.94 10 1.2 10 5.4 10 6.55 10
h s h h
s h h
a rad
V a V V rad
b V V rad
ω ω
ω ω
ω ω
− −
− − − −
− − − −
= − ⋅ + ⋅
≤ ≤ = − ⋅ + ⋅ − ⋅ + ⋅
= − − ⋅ + ⋅ − ⋅ + ⋅
(24.7)
( )( )( )
6 2 4
0
7 2 5
1
7 2 4
1
1.08 10 6.24 10
5 : 6.91 10 4.78 10
3.75 10 2.37 10
h s
s
a rad
V a rad
b rad
ω ω
ω ω
ω ω
− − = − ⋅ + ⋅
< = ⋅ + ⋅
= ⋅ − ⋅
(24.8)
These equations will be used in the final model for the helicopter.
D.6) Thrust, hover
In hover there should only be a vertical thrust vector, if there is another
component it is normally because of uncertainties in the test setup, or turbulence
from the fuselage. Therefore Figure D 13 only shows thrust along the negative z-
axis.
Figure D 13: Thrust versus angular velocity in hover
D 13
From the tendency lines the thrust in hover for the two rotors are given:
( )5 2 9 33.24 10 5.85 10zw
T Nω ω− −= − ⋅ + ⋅ (24.9)
( )5 2 9 34.97 10 2.86 10zb
T Nω ω− −= − ⋅ − ⋅ (24.10)
It is seen that for the black rotor the third order term is opposite sign. In the simple
ideal rotor model the thrust coefficient is constant and thereby the thrust only have
second order terms but the third order term become due to coning and dynamic
twist.
Figure D 14: Three dimensional thrust versus angular velocity in hover
Figure D 14 is included to verify if thrust along the horizontal axis has an effect. It
should be noticed that the horizontal thrust is multiplied by a factor 10. Here it is
seen that the horizontal forces are insignificant small, so they are not used in the
model. It is also seen that the white rotor in general has larger horizontal forces. It
could indicate the white rotor gives more noise and is less effective.
D.7) Thrust, horizontal flight
The thrust becomes three dimensional in horizontal flight because there is a
difference in lift on the advancing and the retreating tip. There will also be a force
in the forward/backward direction depending on the exact angle to the wind. The
vertical and horizontal thrust will be discussed separately, as the vertical thrust is
D 14
still much larger than the horizontal. The data measured from the FS6 force sensor
is in the base coordinate system in the wind tunnel. When the angle αs is changed,
the data also have to be rotated. αs is rotated around the yb axis so the rotation
matrix gives:
( ) ( )
( ) ( )
cos 0 sin
0 1 0
sin 0 cos
s s
R
s s
A
α α
α α
−
=
(24.11)
The following results will be shown in the body frame at the quadrotor. One
exception is the vertical thrust it is shown in the negative z-axis because then it is
plotted positive.
D.8) Vertical thrust in horizontal flight
This test examine if the angle towards the wind has influence on the thrust along
the zb-axis on the helicopter. Figure D 15 and Figure D 16 show the force acting
in the negative zb axis at the quadrotor. It is seen that the thrust has a tendency to
become lower as the helicopter flies against the wind. This is very probable
because it will begin to act like a helicopter in climb, and the induced angle
becomes larger. Thereby the angle of attack decreases and then also the thrust will
decreases, for the same angular velocity.
Figure D 15: Thrust versus angular velocity, white rotor, horizontal wind velocity is 5 m/s.
D 15
Figure D 16: Thrust versus angular velocity, black rotor, horizontal wind velocity is 5 m/s.
D.9) Horizontal thrust in horizontal flight
The forward/backward force from thrust is given by the xb component of the
thrust. It is seen in Figure D 17 that the backward force is highest when the angle
αs is zero; when it becomes positive or negative it decreases. It is because when αs
is zero the wind and the force have same direction and thereby highest
relationship.
D 16
Figure D 17: Horizontal x thrust vs. angular velocity, black rotor, horizontal velocity 5ms,
varying angle towards the wind.
Figure D 18: Horizontal y thrust vs. angular velocity, black rotor, horizontal velocity 5ms,
varying angle towards the wind.
D 17
One of the problems with normal full size helicopters today is that there is
sideward force and moment component when the helicopter is in forward flight. It
is because there is higher lift on the advancing blade than the retreating blade, see
section 3.3.2 flapping. This will give a positive force in the y direction.
At lower velocity of up to 300 rad/s it is seen that as expected the yb-component of
the force is positive. But at higher angular velocities it becomes negative. At the
high velocities the difference in lift on the advancing and retreating blades
decreases, thus here the horizontal force decrease because the positive effects of
flapping is seen, see Figure D 18. It is also discussed in chapter 8.
D.10) Drag, hover
The drag both in hover and in flight is just defined as the drag around the zb axis,
Qψ. In this paper the drag contribution to the vertical forces is given in the above
sections. In hover in the wind tunnel the drag is given by: (se Figure D 19)
( )7 2 10 37.54 10 1.16 10w
Q Nmψ ω ω− −= ⋅ − ⋅ (24.12)
( )6 2 10 31.05 10 2.03 10b
Q Nmψ ω ω− −= ⋅ − ⋅ (24.13)
Figure D 19: Drag versus angular velocity for the white and black rotor.
D 18
As expected it follows more or less the same curves as the thrust, except for a
multiplied factor. It is because the most of the drag originates from the thrust, and
only a smaller component comes from the profile drag.
D.11) Drag, in forward flight
The drag is also tested in flight. It is, as known section D.10), expected to follow
the vertical thrust, but some tests are carried out to se if the expected can be
fulfilled.
Figure D 20 shows the change in drag from hover to horizontal flight. It seems as
the drag compared to hover increases in flight.
Figure D 20: Drag versus angular velocity in horizontal flight
To examine if this assumption is right, a test is carried out with constant rotor
velocity, and with change of the wind velocity. These results are shown in Figure
D 21. They show that the drag is constant over a variation of the wind.
D 19
Figure D 21: Drag versus horizontal velocity, black rotor,
constant angular velocity of 295 rad/s
Figure D 22: Drag versus angular velocity black rotor, horizontal velocity is 5 m/s.
D 20
It is also seen in the figure that the drag decreases with the angle αs. It is not
expected, but is a result of the oscillations in the measurements which were too
large to use the thrust/drag meter. Because the FS6 uses a different frame (the
reference) there is additionally interference from the other torques from the rotor.
D.12) Thrust/drag relationship
The thrust versus drag relationship has also been studied. From Figure D 23
following is seen:
• The white and black rotor has the same thrust/drag relationship. The red and blue curve is close to each other.
• In higher wind velocity the drag is higher for the same thrust. This is especially seen for the black curve, here is the drag consists with wind velocity, but the thrust decreases for an increased wind velocity
• The drag decreases as the quadrotor flies towards the wind, negative αs, compared to when it flies in the same direction as the wind.
Figure D 23: The thrust versus drag relationship.
E 1
E Matlab Model
outXI2
To Workspace3
outXI1
To Workspace2
outXI
To Workspace1
outV
To Workspace
omega r1
omega r2
omega r3
omega r4
V
XI
R
OMEGA
Subsystem
210
Constant4
210
Constant3
210
Constant2
210
Constant1
Figure E1: The overall helicopter model, rotor angular velocity as input and
position, velocity, rotation and angular velocity as output
E 2
Figure E2: Subsystem from in figure E1
1
XI
Matrix
Multiply
Product
1/s
1/s
1/s
Demux
2
R
1
V
Figure E3: Subsystem “XI (position)” in figure E2
4
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eg
a r
4
3
om
eg
a r
3
2
om
eg
a r
2
1
om
eg
a r
1
E 3
1
VuT
Transpose
Scope
1/s
1/s
1/s
u*K
-K-
-K-Demux
In1
In2
Out1
Cross product3
R
2
OMEGA
1
T
Figure E4: Subsystem “V (velocity)” in figure E2
1
R
simout6
To Workspace4
simout5
To Workspace3
simout2
To Workspace2
simout1
To Workspace1
simout
To Workspace
UU(R,C)
UU(R,C)
UU(R,C)
UU(R,C)
UU(R,C)
UU(R,C)
UU(R,C)
UU(R,C)
UU(R,C)
Matrix
Mul tiply
Product
Horiz Cat
Matrix
Concatenation1
Horiz Cat
Matrix
Concatenation
1/s
1/s
1/s
1/s
1/s
1/s
1/s
1/s
1/s
-1
-1
-1
Demux
0
Constant
1
OMEGA
Figure E5: Subsystem “R (rotation)” in figure E2
1
OMEGA
1
s
1
s
1
s
K*u
K*u
K*u
K*u
inv(I_h)* u
I_h* u
Demux
Demux
In1
In2Out1
Cross product4In1
In2Out1
Cross product3In1
In2Out1
Cross product2
In1
In2Out1
Cross product
In1
In2Out1
1
2
omega
1
Q
Figure E6: Subsystem “OMEGA (angular velocity)” in figure E2
E 4
6
Qz
5
Qy
4
Qx
3
Tz
2
Ty
1
Tx
Vx
Vy
Vz
omega
Tx
Ty
Tz
Qx
Qy
Qz
rotor4
Embedded
MATLAB Function
Vx
Vy
Vz
omega
Tx
Ty
Tz
Qx
Qy
Qz
rotor3
(Embedded
MATLAB Function)
Vx
Vy
Vz
omega
Tx
Ty
Tz
Qx
Qy
Qz
rotor2
(Embedded
MATLAB Function)
Vx
Vy
Vz
omega
Tx
Ty
Tz
Qx
Qy
Qz
rotor1
(Embedded
MATLAB Function)
r1
r2
r3
r4
x
y
Out3
Split up and sum2
r1
r2
r3
r4
x
y
Out3
Split up and sum1
r1
r2
r3
r4
x
y
Out3
Split up and sum
In1
In2Out1
Cross product3
In1
In2Out1
Cross product2
In1
In2Out1
Cross product1
In1
In2Out1
Cross product
-C-
Constant3
-C-
Constant2
-C-
Constant1
-C-
Constant
7
Vz
6
Vy
5
Vx
4
omega r4
3
omega r3
2
omega r2
1
omega r1
Figure E7: Subsystem “four rotors force and torque contribution” in figure E2
F 1
F Technical Drawings
F 2
F 3
F 4
F 5
F 6
F 7
G 1
G Enclosed DVD
The DVD contains 6 folders:
Report
The report in PDF format.
Microprocessor
C code for the Atmel MEGA128 microprocessor at the Ethernut 2.1b board.
Matlab model:
The Matlab model consists of two parts, the one is the blade element model and
the other is the Simulink model.
The main file in the blade element model is blade_element.m. This file uses
subfunctions in the three files
• calculate_beta.m
• calculate_chord_and_pitch_angles.m
• rotor_constants.m
Four Simulink models is included at the DVD:
• Quadrotor helicopter model with white rotors.
• Quadrotor helicopter model with black rotors.
• Single rotor model of white rotor.
• Single rotor model of black rotor.
To initialise the helicopter model the file simulering.m must be run.
Datasheets:
Following datasheets are included on the DVD
• Ethernut, a collection of datasheets for the Ethernut hardware and
software used in the project.
• AMTIFS6, datasheet for the AMTI FS6 force and torque sensor.
• Graupner speed gear 600 plus, datasheet for the motor
• Honywel FSG15N1A, datasheet for the force sensor used in the especially
built force and torque sensor
• L7815, L7915, datasheet for the plus and minus 15 V voltage regulators.
G 2
• LM2907, datasheet for the frequency to voltage converter.
• OPTEK OPB701, datasheet for the optic sensors.
Articles
A collection of articles used in the project.
Tests
This folder contains all the 2.3 GB test data from the project, the data is
described in appendix C.
![Active Helicopter Rotor Control Using Blade-Mounted Actuators2-1 Block diagram of coupled rotor and inflow dynamics. Adapted from Pitt and Peters [35]. ..... 34 2-2 Helicopter rotor](https://img.pdfslide.us/doc/110x75/5f440e16ded27235c7483c01/active-helicopter-rotor-control-using-blade-mounted-actuators-2-1-block-diagram.jpg)
