CALIFORNIA STATE UNIVERSITY, NORTHRIDGE
STATIC PITCH STABILITY OF CANARD CONFIGURED AIRCRAFT
A thesis submitted in partial fulfillment requirements
For the degree of Master of Science in
Mechanical Engineering
By
Mahdi Ghalami
December 2013
The thesis of Mahdi Ghalami is approved:
Stewart Prince, Ph.D.
Date
George Youssef, Ph.D.
Date
Timothy Fox, Chair
Date
California State University, Northridge
ii
Acknowledgement
Dedicated to Sahar
My love, my life, and my wife
For all her love, supports, and understanding
iii
Table of Contents
Signature Page .................................................................................................................... ii
Acknowledgement ............................................................................................................. iii
List of Figures .................................................................................................................. viii
List of Symbols ................................................................................................................ xiii
Abstract ............................................................................................................................. xv
1.
2.
Introduction ................................................................................................................. 1
1.1
Stability ................................................................................................................ 1
1.2
Stable Flight ......................................................................................................... 4
1.3
Canard Concept .................................................................................................... 7
Design and Analysis .................................................................................................. 11
2.1
Canard Configuration ......................................................................................... 11
2.2
Airfoil ................................................................................................................. 13
2.3
Reynolds Number ............................................................................................... 18
2.4
Finite Wing......................................................................................................... 24
2.5
Finite Wing Moment Coefficient (CM) .............................................................. 26
2.6
Finite Wing Lift Coefficient (CL) ....................................................................... 27
2.7
Finite Wing Drag Coefficient (CD) .................................................................... 31
2.8
Parasite Drag of Non-Aerodynamic Parts .......................................................... 33
2.9
Aircraft Actual Lift............................................................................................. 36
iv
3.
2.10
Aircraft Actual Drag........................................................................................... 38
2.11
Drag Polar .......................................................................................................... 39
2.12
Pitch Static Stability Criteria .............................................................................. 42
2.13
Equilibrium Flight .............................................................................................. 46
2.14
Aircraft Moment Coefficient (CM) .................................................................... 47
2.15
Finding CG Location .......................................................................................... 49
2.16
Pitch Static Stability ........................................................................................... 51
2.17
Aerodynamic Center or Neutral Point (NP) ....................................................... 52
2.18
Static Margin(SM).............................................................................................. 54
Excel Spreadsheets .................................................................................................... 57
3.1
Inputs .................................................................................................................. 57
3.2
Aerodynamic Characteristics of Airfoils............................................................ 60
3.3
3D Lift Curve Requirements .............................................................................. 63
3.4
AR Impact on Lift Curves .................................................................................. 65
3.5
3D Aerodynamic Characteristics ....................................................................... 66
3.6
Parasite Drag ...................................................................................................... 69
3.7
Drag Polar and Aerodynamic Efficiency Plots .................................................. 71
3.8
Cruise Optimization ........................................................................................... 71
3.9
Center of Gravity................................................................................................ 72
3.10
Static Margin ...................................................................................................... 73
v
3.11
4.
5.
Optimization ....................................................................................................... 73
Results ....................................................................................................................... 74
4.1
Airfoil Selection ................................................................................................. 74
4.2
Canard and Wing Airfoil Comparison ............................................................... 80
4.3
Canard 2D vs. 3D models .................................................................................. 82
4.4
Wing 2D vs. 3D models ..................................................................................... 84
4.5
Aspect Ratio Impact on Lift Slope ..................................................................... 86
4.6
Wing and Canard Comparison ........................................................................... 87
4.7
Aircraft Lift and Drag Coefficients .................................................................... 89
4.8
Aircraft Drag Polar ............................................................................................. 91
4.9
Aircraft Cruise Condition ................................................................................... 92
4.10
Cruise Optimization ........................................................................................... 94
4.11
Angle of Incidence Results ................................................................................ 95
4.12
CG Location ....................................................................................................... 99
4.13
Static Margin .................................................................................................... 100
4.14
Optimization ..................................................................................................... 100
Conclusion ............................................................................................................... 113
Bibliography ................................................................................................................... 115
Appendices ...................................................................................................................... 117
Appendix A. Excel Spreadsheets ................................................................................ 117
vi
Appendix B. Aircraft Lift Coefficient Equation ......................................................... 138
Appendix C. Aircraft Drag Coefficient Equation ....................................................... 139
Appendix D. Aircraft Moment Coefficient Equation.................................................. 141
vii
List of Figures
Figure 1-1 Stable stability [3] ............................................................................................... 1
Figure 1-2 Unstable stability [3]........................................................................................... 2
Figure 1-3 Neutral stability [3] ............................................................................................. 2
Figure 1-4 Aperiodic dynamic stability [3] .......................................................................... 3
Figure 1-5 Damped oscillation Dynamic Stability [3].......................................................... 3
Figure 1-6 Increasing oscillation dynamic stability [3] ........................................................ 3
Figure 1-7 Aircraft’s axes with translational and rotational motions ................................. 4
Figure 1-8 Moments and forces .......................................................................................... 6
Figure 1-9 Flyer one (National Air and Space Museum) .................................................. 8
Figure 1-10 Rutan’s Voyager design (National Air and Space Museum) .......................... 9
Figure 1-11 Rutan’s VariEz design (Courtesy of National Air and Space Museum) ......... 9
Figure 1-12 Canard Design of CSUN 2011 Aeronautics Team ....................................... 10
Figure 2-1 Earlier stall; for canard (a), and wing (b) ........................................................ 11
Figure 2-2 Earlier hitting zero lift; for wing (a), and canard (b)....................................... 12
Figure 2-3 Airfoil section [3] ............................................................................................. 13
Figure 2-4 Airfoil terms [3] ................................................................................................ 14
Figure 2-5 Low speed airfoil types (Drawn by Profili Software [10]) ............................... 14
Figure 2-6 Definition of forces, moment, and relative wind ............................................ 15
Figure 2-7 NACA 2412 airfoil plots [7] ............................................................................. 16
Figure 2-8 Cambered airfoil vs. symmetric airfoil [9] ....................................................... 17
Figure 2-9 Transition from laminar to turbulent [3] ........................................................... 19
viii
Figure 2-10 cl vs. α curves at different Re number of airfoil E210 (Drawn by Profili
Software [10]) ..................................................................................................................... 20
Figure 2-11 cl vs. AOA curve of some airfoils at same Re number of 250,000 (Drawn by
Profili Software [10]) .......................................................................................................... 21
Figure 2-12 cl vs. AOA curve at different Re number for wing and canard airfoils (Drawn
by Profili Software [10]) ..................................................................................................... 23
Figure 2-13 Infinite wing [3] .............................................................................................. 24
Figure 2-14 Wing vortices on a finite wing ...................................................................... 25
Figure 2-15 Origin of induced drag [3] .............................................................................. 26
Figure 2-16 AR impact on cl vs. AOA curve .................................................................... 28
Figure 2-17 Airfoil LE sharpness parameter [11] ............................................................... 30
Figure 2-18 Angle of attack increment for maximum lift coefficient [11] ......................... 30
Figure 2-19 Drag polar...................................................................................................... 40
Figure 2-20 Aerodynamic efficiency (E or L/D) vs. AOA ............................................... 41
Figure 2-21 Aircraft in steady, equilibrium flight at its trim angle .................................. 42
Figure 2-22 Nose up disturbance ...................................................................................... 43
Figure 2-23 Nose down disturbance ................................................................................. 43
Figure 2-24 Moment coefficient vs. AOA curve with a negative slope ........................... 44
Figure 2-25 Moment coefficient curve with a positive slope ........................................... 45
Figure 2-26 Statically unstable flight ................................................................................ 45
Figure 2-27 Forces and moments ...................................................................................... 46
Figure 2-28 Moments, forces and their distances from CG .............................................. 47
Figure 2-29 Impact of NP location on CM curve slope ..................................................... 53
ix
Figure 2-30 Importance of lift center location, behind the CG [3] .................................... 54
Figure 3-1 cl vs. α curves of airfoil FX63-137(Re=130,000), and SD7062 (Re=350,000)
........................................................................................................................................... 61
Figure 3-2 Airfoil SD7062 lift characters ......................................................................... 62
Figure 3-3 AR impact on the lift slope ............................................................................. 65
Figure 3-4 Angle of incidence impact .............................................................................. 67
Figure 4-1 cl vs. α curve of airfoil FX63-137 at Re number of 130,000 .......................... 74
Figure 4-2 cd vs. α curve of airfoil FX63-137 at Re number of 130,000 .......................... 75
Figure 4-3 cl/cd vs. α curve of airfoil FX63-137 at Re number of 130,000 ...................... 75
Figure 4-4 cl vs. cd curve of airfoil FX63-137 at Re number of 130,000 ......................... 76
Figure 4-5 cm vs. α curve of airfoil FX63-137 at Re number of 130,000 ......................... 76
Figure 4-6 cl vs. α curve of airfoil SD7062 at Re number of 350,000.............................. 77
Figure 4-7 cd vs. α curve of airfoil SD7062 at Re number of 350,000 ............................. 78
Figure 4-8 cl/cd vs. α curve of airfoil SD7062 at Re number of 350,000 ......................... 78
Figure 4-9 cl vs. cd curve of airfoil SD7062 at Re number of 350,000............................. 79
Figure 4-10 cm vs. α curve of airfoil SD7062 at Re number of 350,000 .......................... 79
Figure 4-11 cl vs. α curves of airfoil FX63-137(Re=130,000), and SD7062 (Re=350,000)
........................................................................................................................................... 80
Figure 4-12 cd vs. α curves of airfoil FX63-137(Re=130,000), and SD7062 (Re=350,000)
........................................................................................................................................... 81
Figure 4-13 cl/cd vs. α curves of FX63-137(Re=130,000), and SD7062 (Re=350,000) .. 81
Figure 4-14 cm vs. α curves of airfoil FX63-137(Re=130,000), and SD7062 (Re=350,000)
........................................................................................................................................... 82
x
Figure 4-15 Lift coefficient vs. α- Airfoil FX63-137, and 3D canard (Re=130,000) ....... 82
Figure 4-16 Drag coefficient vs. α- Airfoil FX63-137, and 3D canard (Re=130,000) ..... 83
Figure 4-17 E vs. α -Airfoil FX63-137, and 3D canard (Re=130,000) ............................ 83
Figure 4-18 Lift coefficient vs. α- Airfoil SD7062, and 3D wing (Re=350,000) ............. 84
Figure 4-19 Drag coefficient vs. α- Airfoil SD7062, and 3D wing (Re=350,000) ........... 85
Figure 4-20 E vs. α- Airfoil SD7062, and 3D wing (Re=350,000) .................................. 85
Figure 4-21 AR impact on the lift slope (same surface ratio, different chord length) ...... 86
Figure 4-22 AR impact on the lift slope (same chord length, different surface ratio) ...... 86
Figure 4-23 CL vs. α curves of canard (Re=130,000), and wing (Re=350,000) ............... 87
Figure 4-24 CD vs. α curves of canard (Re=130,000), and wing (Re=350,000)............... 88
Figure 4-25 CL/CD vs. α curves of canard (Re=130,000), and wing (Re=350,000) ......... 88
Figure 4-26 CL vs. α curves of canard, wing, and aircraft (V∞=50ft/s) ............................ 89
Figure 4-27 CD vs. α curves of canard, wing, and aircraft (V∞=50ft/s) ............................ 90
Figure 4-28 CL vs. CD curve of aircraft (V∞=50ft/s) ......................................................... 91
Figure 4-29 Aerodynamic efficiency of aircraft vs. α (V∞=50ft/s) ................................... 91
Figure 4-30 Aircraft lift coefficient vs. velocity at wing loading of 1.5 ........................... 92
Figure 4-31 Aircraft drag coefficient vs. velocity at wing loading of 1.5 ........................ 93
Figure 4-32 Aircraft aerodynamic efficiency vs. velocity at wing loading of 1.5 ............ 93
Figure 4-33 VCruise vs. AOA at different angle of incidence for wing and canard............ 94
Figure 4-34 Canard lift coefficient vs. AOA, with 0 and 2 degree angle of incidence .... 95
Figure 4-35 Canard drag coefficient vs. AOA, with 0 and 2 degree angle of incidence .. 96
Figure 4-36 Canard CL/CD vs. AOA, with 0 and 2 degree angle of incidence ................. 96
Figure 4-37 CL vs. AOA for aircraft, wing, and canard, with angle of incidence ............ 97
xi
Figure 4-38 CD vs. AOA for aircraft, wing, and canard, with angle of incidence ............ 98
Figure 4-39 Aircraft aerodynamic efficiency vs. AOA with angle of incidence .............. 98
Figure 4-40 CG location vs. different angle of incidence for wing and canard................ 99
Figure 4-41 Moment coefficient vs. absolute angle of attack ......................................... 100
Figure 4-42 Stall margin vs. canard angle of incidence, same SW/SRef .......................... 101
Figure 4-43 Cruise AOA vs. canard angle of incidence, same SW/SRef .......................... 102
Figure 4-44 Cruise E vs. canard angle of incidence, same SW/SRef ................................ 103
Figure 4-45 Cruise Aerodynamic efficiency in different angle of incidence ................. 103
Figure 4-46 Cruise increment thrust vs. canard angle of incidence, same SW/SRef ........ 104
Figure 4-47 CG location vs. canard angle of incidence, same SW/SRef ........................... 104
Figure 4-48 Static margin vs. canard angle of incidence, same SW/SRef ......................... 105
Figure 4-49 Moment coefficient curve slope in different chord lengths ........................ 106
Figure 4-50 Stall margin vs. canard angle of incidence, same chords ............................ 107
Figure 4-51 Cruise AOA vs. canard angle of incidence, same chords ........................... 107
Figure 4-52 Cruise E vs. canard angle of incidence, same chords ................................. 108
Figure 4-53 CG location vs. canard angle of incidence, same chords ............................ 109
Figure 4-54 Static margin vs. canard angle of incidence, same chords .......................... 109
Figure 4-55 Stall margin vs. wing angle of incidence, different AR .............................. 110
Figure 4-56 Cruise angle of attack vs. wing angle of incidence, different AR............... 111
Figure 4-57 CG location vs. wing angle of incidence, different AR .............................. 111
Figure 4-58 Static margin vs. wing angle of incidence, different AR ............................ 112
xii
List of Symbols
L
Lift
LW
Wing Lift
LC
Canard Lift
D
Drag
DW
Wing Drag
DC
Canard Drag
DLG
Landing Gear Drag
DF
Fuselage Drag
DVS
Vertical Stabilizer Drag
W
Weight
TNet
Net Thrust
MCG
Moment about Center of Gravity
MW
Wing Moment
MC
Canard Moment
Re
Reynolds Number
V
Flight Velocity

Air Density,

Air Viscosity
cd
Airfoil Drag Coefficient
CD
Wing Drag Coefficient
CD, min
Aircraft Min. Drag Coefficient in Drag Polar
CD,0L
Aircraft Drag Coefficient at Zero Lift Angle of Attack
CL
Wing Lift Coefficient
cl
Airfoil Lift Coefficient
cm
Airfoil Moment Coefficient
CM
Moment Coefficient
xiii
CM,0L
Aircraft Moment Coefficient at Zero Lift Angle of Attack
Α
Angle of Attack
αe
Trim Angle of Attack
αeff
Effective Angle of Attack
αi
Induced Angle of Attack
α0L
Zero Lift Angle of Attack
αStall
Stall Angle of Attack
clα
Airfoil Lift vs. Angle of Attack Curve Sole
CLα
Wing Lift vs. Angle of Attack Curve Sole
AR
Aspect Ratio
E
Aerodynamic Efficiency
𝑐̅Ref
Reference Chord
𝑐̅W
Wing Chord
𝑐̅C
Canard Chord
xiv
Abstract
Static Pitch Stability of Canard Configured Aircraft
By
Mahdi Ghalami
Master of Science in Mechanical Engineering
This graduate study is about analysis of the static pitch stability of a canard
configured aircraft which flies at low Reynolds number. Design assumptions of this
project are numbers for the canard aircraft which was designed and built by CSUN 2011
aeronautics team (winner of 6th place in 2011 AUVSI competition). After checking and
simulating of several airfoils based on canard configuration requirements FX63-137 and
SD7062 were chosen to use for canard and wing. To meet the canard configuration
requirements and gain a proper stall margin between wing and canard stall points, wing
and canard chord lengths were chosen 13” and 5”. The expected location of CG which
was 3.6” in front of the wing leading edge was achieved by applying the surface ratio of
85% for wing and 15% for canard (the sum of wing and canard surface areas was 10ft 2).
Pitch static stability of the aircraft were confirmed by gaining the static margin of 5.7%
for the expected cruise flight at 50ft/s which occurred at zero angle of attack for the
aircraft. In order to decrease the cruise angle of attack to zero and move the CG location
to expected point, angle of incidence of 1 and 4 were utilized to wing and canard.
xv
1. Introduction
From many years before successful flight of Wright brothers, flight was like a dream for
human. Thousands of people lost their lives in order to acquire this dream. One of the
most famous people in aviation history who lost his life was Otto Lilienthal, considered
as one of the first aeronautical engineers in history
[1]
. He died just because his glider
wasn't stable enough. Wright brothers were not the first fliers of the aviation history, but
they invented the first stable aircraft and they succeeded to control it
[1]
. Now by more
than a century after their first stable flight, control and stability is still one of the most
significant steps in aircraft design.
1.1
Stability
Stability is the ability of aircraft to return to its previous condition if disturbed by a gust
of air or turbulence. There are two kinds of stability; static stability and dynamic stability.
As illustrated in Figures 1-1 thru 1-3, there are three kinds of static stability states:
statically stable, statically unstable, and statically neutral. In Figure 1-1, the ball will tend
to return to previous condition after any displacing from the bottom. The ball is statically
stable.
Figure 1-1 Stable stability [3]
In Figure 1-2, the ball, after any movement from its initial position, will continue to move
and it won't return back to its previous place. This system is statically unstable. In Figure
1
1-3 if the ball is moved from its place, it will stay in the new point. The ball is in a
statically neutral state now.
Figure 1-2 Unstable stability [3]
Figure 1-3 Neutral stability [3]
It can be concluded after these observations:
-The body is statically stable if it moves back to its original position after being disturbed
by any external forces. The body has positive static stability.
-The body is statically unstable if it does not move back to its original position after
being disturbed by any external forces, and continues to move. The body has negative
static stability.
-After any disturbance, the body is statically neutral if it stays in its new equilibrium
position.
Dynamic stability concept deals with how fast an aircraft returns to its previous statically
equilibrium position. There are two types of dynamic stability; aperiodic in which aircraft
does not oscillate before returning to its initial equilibrium, as shown in Figure 1-4.
2
Figure 1-4 Aperiodic dynamic stability [3]
While in damped oscillation, a series of oscillations with decreasing amplitude occurs
while the body returns back to its initial position, Figure 1-5.
Figure 1-5 Damped oscillation Dynamic Stability [3]
The system is dynamically unstable if it never returns to its original state, as illustrated in
Figure 1-6.
Figure 1-6 Increasing oscillation dynamic stability [3]
3
A dynamically stable aircraft is always statically stable [3]. The dynamic stability is not
discussed in further as the focus of this project is on the static type of stability in the
symmetric set of longitudinal motions, called static pitch stability. In particular, the static
pitch stability is studied at low Reynolds number.
1.2
Stable Flight
The term stability is defined as the description of the flying qualities of an aircraft. To
examine the stability of an aircraft, first we need to define an equilibrium flight. With the
assumption of the aircraft as a rigid body, the aircraft has six degrees of freedom; three of
them are translational and the other three are rotational. Forces applied to aircraft govern
the aircraft performance, by their impact on translational motions of the aircraft as
aircraft responds to them. On the other hand, moments about the center of gravity govern
the stability of the aircraft. Moments affect the rotational motions of the aircraft as
aircraft responds to them. An aircraft is illustrated in Figure 1-7. The total weight of
aircraft effectively is exerted to the center of gravity, indicated as CG in the Figure 1-7.
Figure 1-7 Aircraft’s axes with translational and rotational motions
4
All three x, y, and z axes of the aircraft pass through the CG. The x axis along the
fuselage, drawn from nose to tail in the direction of the flight, is called Longitudinal axis
of the aircraft. The y axis, parallel to the wing and perpendicular to the Longitudinal axis,
is called Lateral axis of the aircraft. The z axis, parallel to the fuselage station and
perpendicular to xy plane, is called Vertical axis. Due to moments about each axis, there
is a rotational velocity about each axis. The motion about the Longitudinal axis is called
roll, while pitch is the motion about the Lateral axis, and yaw is the motion about the
Vertical axis. To have an equilibrium flight, sum of all forces and moment must be equal
to zero (∑Fx=0, ∑Fy=0, ∑Fz=0, ∑Mx=0, ∑My=0, ∑Mz=0). These six degrees of
freedom are divided to two groups; symmetric and asymmetric. The symmetric degrees
of freedom describe the longitudinal motion of the aircraft: x and z force equations and
pitching moment equation (∑Fx, ∑Fz, and ∑My). The asymmetric degrees of freedom
are about lateral motion of the aircraft: y force equation and the yawing and rolling
moment equations (∑Fy, ∑Mz, and ∑Mx). Here we discuss about the longitudinal
motion, the symmetric set. Therefore, to have an equilibrium flight the sum of x and z
forces and sum of pitching moments must be equal to zero (∑Fx=0, ∑Fz=0, and ∑My=0)
[2]
.
5
In order to study the equilibrium flight, first the fundamental forces and moments should
be introduced. Forces and moments applied to an aircraft when it flies, shown in Figure
1-8.
Figure 1-8 Moments and forces
Lift (L) is perpendicular to the direction of flight path, drag force (D) is parallel to the
direction of flight path, Weight (W) acts toward the earth center through the aircraft
center of gravity, and thrust (T) propels the aircraft in flight path direction. Pitching
moment of the wing is produced by pressure and shear stress distribution over the wing
surface. Wing pitching moment can be applied to any arbitrary point such as wing
leading edge, or wing trailing edge. The best choice is aerodynamic center of the wing,
indicated as ac, about which theoretically the moments are not dependent to the angle of
attack of the wing. The aerodynamic center plays a significant role in stability
discussions. For a low speed subsonic wing, aerodynamic center is located near the
quarter chord of the wing. Forces on a wing, lift and drag are applied through the
aerodynamic center of the wing (or wings); moreover, the moment acting about this point
is the moment of the system. The pitching moment about the center of gravity of the
6
aircraft, CG, is denoted by MCG. As it can be observed in the Figure 1-8, the moment
about the center of gravity, MCG, comprises from four groups of forces: (1) lift, drag, and
moment applied on the wing; (2) lift, drag, and moment applied on the canard; (3) thrust;
and (4) aerodynamic forces and moments on other parts of the aircraft such as fuselage,
landing gears, and the vertical stabilizer. As weight is applied through the center of
gravity, it is not considered in moment calculation about the CG. In order to understand
how these moments and forces interact to affect the stability of an aircraft, it is essential
to study the concept of stability in the following section.
1.3
Canard Concept
In a canard configuration, in contrast to a conventional configuration, the horizontal
stabilizer instead of rear (in tail) is in the front of the aircraft (in nose), as shown in
Figure 1-8. Higher safety is the major advantage of a canard configuration than a
conventional type. In a canard aircraft, canard stalls before the main wing. Thus bringing
the nose down, it controls the wing angle of attack. Consequently, wing doesn't stall. This
performance hikes the safety and avoids crashes due to stall spin at low altitude. The
other advantage of canard is its contribution in making lift. Canard shares the load;
whereas, a tail type horizontal stabilizer is slightly loaded. This advantage of canard has a
great outcome, decreasing the wing area, and with a lower wing loading, aircraft will be
lighter. In addition, by decreasing the wing area, wing induced drag decreases.
7
The first successful manned powered heavier than air aircraft was a canard configured
aircraft, Flyer One, Wright Brother's great invention, shown in Figure 1-9. They chose to
place the horizontal stabilizer in nose to keep the nose up in order to have a more stable
flight.
Figure 1-9 Flyer one (National Air and Space Museum)
Canard was utilized in most designs of Burt Rutan, the legendary aerospace engineer and
designer
[6]
. Figure 1-10 shows the Rutan's Voyager design, the first aircraft that
circumnavigated the earth with no refueling and stopping. To circumnavigate the globe
an aircraft needs to fly at a high aerodynamic efficiency, also called lift to drag ratio, and
canard by increasing lift and decreasing drag was a very good option.
8
Figure 1-10 Rutan’s Voyager design (National Air and Space Museum)
Figure 1-11 shows the Rutan's VariEz design. It is very efficient in fuel consumption;
lighter weight, lower drag due to canard configuration. Rutan was also interested in
designing an aircraft with a high resistance against the stall and spin; it’s the major
characteristic of canard configuration. Due to high amount of stability, canard may limit
the aerobatic capability and wouldn't be a good option for an aerobatic aircraft.
Figure 1-11 Rutan’s VariEz design (Courtesy of National Air and Space Museum)
9
Figure 1-12 shows canard configured UAV which was designed and built by CSUN 2011
aeronautics team which was awarded the 6th place of 2011 AUVSI competition. This
project is about design steps, analysis, examining the pitch stability, and design
optimization of this successful canard plane.
Figure 1-12 Canard Design of CSUN 2011 Aeronautics Team
During this chapter some fundamental definitions were introduced. In next chapter,
design and analysis of a canard configured aircraft and its pitch stability is discussed.
10
2.
2.1
Design and Analysis
Canard Configuration
This chapter begins by introducing the details of canard configuration, how it works, and
what are canard configuration requirements. Based on requirements, proper airfoil
selection and wing aerodynamic characteristic calculations will be applied. The final goal
is calculation of the pitch stability of the aircraft in an efficient flight condition.
In canard configuration both forward and main wing contribute to create lift; therefore,
there are two significant and extremely crucial requirements on airfoil selection of canard
and wing in order to have a successful design and stable flight:
1. Canard must stall before the main wing. When canard stalls, wing still is able to
make lift and it returns aircraft to the stable orientation, shown in Figure 2-1a.
Where, if wing had stalled first, nose-up pitch would have occurred, and aircraft
would not have returned to stable condition, as sketched in Figure 2-1b.
Figure 2-1 Earlier stall; for canard (a), and wing (b)
11
2. Wing must reach its zero lift angle of attack before canard. As aircraft dives and
wing reaches its zero lift angle of attack first, canard is still able to make lift and
returns the aircraft to its stable condition, shown in Figure 2-2a. If canard had hit
its zero lift angle of attack first, wing would still have made lift and would have
pushed the aircraft to a steeper dive, shown in Figure 2-2b.
Figure 2-2 Earlier hitting zero lift; for wing (a), and canard (b)
Therefore, the first step in designing a canard configured aircraft is designing a proper
wing and a proper canard based on discussed requirements. First step in wing and canard
design and analysis is selecting of a proper airfoil. Next discussion of the project
introduces the airfoil.
12
2.2
Airfoil
One of the most significant steps in aircraft design is the selecting of a proper airfoil. The
cross section of a wing is called airfoil. Figure 2-3 shows the airfoil obtained by
intersection of the wing with a perpendicular plane.
Figure 2-3 Airfoil section [3]
Airfoil is a slender shape body, also called streamlined shape body, with a good ability of
making lift. In contrast to a blunt shape body, it has a low drag due to late flow separation
over it. One of the major characteristics of airfoil shape is its mean camber line. As
illustrated in Figure 2-4, mean camber line is located in halfway between the top and
bottom surfaces of airfoil. Leading edge (LE) and trailing edge (TE) of an airfoil are the
most forward and the most rearward points of the mean camber line. The straight line
13
between LE and TE is the airfoil chord. The highest distance between the chord and the
mean camber line indicates the airfoil camber [7].
Figure 2-4 Airfoil terms [3]
There are three major types of low speed airfoils
[8]
, as showed in Figure 2-5: heavily
cambered airfoil such as FX63-137, moderately cambered airfoil such as SD7062 and
E197, and no camber airfoil or symmetric airfoil such as E168. A symmetrical airfoil has
a zero camber; mean camber line and chord are same. Aerodynamic characteristics of an
airfoil such as lift, drag, and moment are controlled by its camber, mean camber line, and
its thickness.
Figure 2-5 Low speed airfoil types (Drawn by Profili Software [10])
14
As well as airfoil shape, flight conditions such as flight velocity and angle of flight
direction have impact on aerodynamic characteristics. As illustrated in Figure 2-6, V∞ is
the velocity of upstream air flow. Its direction is called relative wind. The angle between
the airfoil chord and relative wind is defined as geometric angle of attack or angle of
attack. Drag force (D) is described as the force parallel to the relative wind and the lift
(L) is perpendicular to it, these forces are the result of pressure and shear stress
distribution over the wing surface. Besides, a moment (M) is created by pressure and
shear stress distribution which is able to rotate the wing. Changing of angle of attack
affects lift, drag, and moments.
Figure 2-6 Definition of forces, moment, and relative wind
15
Airfoil aerodynamic characteristics are described in “airfoil plots”; Figure 2-7 shows a
sample of airfoil plots
[7]
. Note that the data are shown in terms of coefficients, except
angle of attack. cl is lift coefficient, cd is drag coefficient, and cm is moment coefficient
about the aerodynamic center of the airfoil (around the quarter chord point of the airfoil).
Figure 2-7 NACA 2412 airfoil plots [7]
16
Lift, drag, and moment coefficients are defined from following equations:
cl 
L
q   S Re f
cd 
D
q   S Re f
cm 
M
q   S Re f  c Re f
cl is defined as lift force divided by the dynamic pressure and some reference area. c d is
defined as drag force divided by the dynamic pressure and some reference area. c m is
defined as pitch moment divided by the dynamic pressure, some reference area and
reference chord.
Figure 2-8 shows lift coefficient (cl) vs. angle of attack for a symmetric and a cambered
airfoil.
Figure 2-8 Cambered airfoil vs. symmetric airfoil [9]
17
Based on experimental data cl varies linearly with angle of attack (denoted by α) over a
considerable range except near the stall angle. At some point before stall (at stall angle of
attack cl is maximum) the curve becomes non-linear. The difference between the amount
of stall angle of attack and the angle of attack in which curves become non-linear is a
function of airfoil leading edge sharpness and it’s denoted as ∆αcl,max. [9].
There are two significant points on these curves:

Zero lift angle of attack (α0L) in which airfoil doesn't make any lift. For angles
less than this point, airfoil makes downward lift. For a symmetrical airfoil, zero
lift angle of attack is equal to zero degree. As we can see an asymmetrical airfoil
makes lift at zero degree angle of attack and it’s due to the positive camber of the
airfoil.

Stall angle of attack (αStall) in which cl reaches its maximum (clmax) value and then
drops as angle of attack increases. In this situation, stall phenomena happens,
where the lift slashes.
The slope of the linear part of lift curve is called lift slope (clα) and ideally is equal to 2π
(1/Radian).
Airfoils are tested and examined in the wind tunnels, and theses parameters are measured.
For any configuration of airfoil and flow a Re is defined. Re will introduce in next part.
2.3
Reynolds Number
Reynolds number is a significant dimensionless number with consequential impacts on
aerodynamic studies, named after Osborne Reynolds, the scientist who demonstrated the
concept of transition from laminar to turbulent flow [4]. Reynolds number, denoted by Re,
18
is a function of air density, air viscosity, air velocity, and the length of surface through
which flow travels. Re is the ratio of inertia forces to viscous forces in a fluid flow, and it
is a governing parameter for viscous flow. Re is a key parameter to calculate skin friction
drag, boundary layer thickness, and transition point from laminar to turbulent flow.
Figure 2-9 sketches the transition point from laminar flow to turbulent flow which
depends on Re. The discussion of this project is about the flow at low Re number, less
than 500,000. It means flow over the wing is laminar and it doesn't transit to turbulent [5].
Laminar flow has a lower skin friction drag than turbulent flow; therefore, drag is lower.
Figure 2-9 Transition from laminar to turbulent [3]
Re is also used to find the aerodynamic characteristics of the airfoils. As mentioned,
airfoils are tested and examined in the wind tunnels, and aerodynamic characteristics of
the any airfoil are measured at a specific Re. Consequently, each curve in airfoil plots is
created at a specific Re.
19
Figure 2-10 shows how lift curves of airfoil E210 varies by Re. The discussion of this
project revolves around low Re numbers such as 130,000, and 350,000.
Figure 2-10 cl vs. α curves at different Re number of airfoil E210 (Drawn by Profili
Software [10])
Shortly, information given by airfoil lift plots are required to select proper airfoils and
these plots are create for different airfoils and in different Re numbers. In Figure 2-11 lift
20
coefficient vs. angle of attack (cl vs. α) curves of some low speed airfoils are plotted (at
same Re number equal to 250,000).
Figure 2-11 cl vs. AOA curve of some airfoils at same Re number of 250,000 (Drawn by
Profili Software [10])
Re number is defined as:
Re 
 V l

Where   is air density,   is air viscosity, V is flow velocity, and l is length of area
over which air flows. For case of finding the Re number for wing l is equal to wing chord
21
length ( l  cW ). Same way is used to find Re number for canard ( l  c C ), and to find the
Re number over the fuselage, fuselage length is considered ( l  l Fuselage ).
Re number used in plotting the wing curve is higher than used Re number for canard, due
to longer chord length. Therefore, curves shouldn’t be plotted at the same Re number.
After simulating several airfoils, FX63-137 was designated for canard. Re number to plot
the airfoil aerodynamic characteristics is equal to 130,000. This Re number is calculated
based on flight mission at sea level altitude and cruise velocity of 50 ft/s and also canard
chord length. , SD7062 was designated for wing in Re number of 300,000, due to longer
chord length than canard chord length. Figure 2-12 shows how selected airfoils meet the
canard requirements. Zero lift angle of attack of FX 63-137 at Re of 130,000 is less than
zero lift angle of attack of SD 7062 at Re of 350,000. Therefore, designated airfoil for
wing (SD 7062) reaches zero lift angle of attack before designated airfoil for canard (FX
63-137). FX-63-137 stalls before SD 7062; consequently, selected airfoils meet both
critical requirements of the canard configuration.
22
Figure 2-12 cl vs. AOA curve at different Re number for wing and canard airfoils (Drawn
by Profili Software [10])
Based on canard requirements proper airfoils for wing and canard are selected. Now due
to the difference between airfoil and wing model, aerodynamic characteristics must be
converted.
23
2.4
Finite Wing
The discussion so far is revolved around the airfoil. Now it’s the time to translate our
knowledge about airfoil to finite wing. Airfoil can be considered an infinite wing versus
an airplane wing which is a finite wing. In wind tunnels, airfoil is attached to both
sidewalls; therefore, there are no wing tips on the sides of airfoil. This makes the airfoil a
wing with an infinite span, illustrated in Figure 2-13. That's why it's called infinite wing.
Figure 2-13 Infinite wing [3]
The person who established the first practical finite wing theory was German engineer
Ludwig Prandtl [4], who is given the title “father of aerodynamics”. Flow over an infinite
wing, doesn't vary span wise, that's why it’s called two-dimensional flow (2D flow) and
airfoil is called 2D wing. However, a three-dimensional flow (3D flow) streams over a
finite wing, and it causes the wing tip vortices. Wing tip vortices induce a small
downward component of the velocity and it’s called downwash. Therefore, wing tip
24
vortices are the major difference between flow over a 2D wing (airfoil) and flow over a
3D wing (actual wing), shown in Figure 2-14. Due to wing tip vortices in a 3D flow, drag
and lift coefficients for the airfoil are different from the drag and lift coefficients for the
wing with a same shape of airfoil and at same angle of attack. Aerodynamic coefficients
of a wing are shown by capital letters; CL, and CD, and aerodynamic coefficients of an
airfoil are shown by small letters; cl, and cd.
Figure 2-14 Wing vortices on a finite wing
Wing tip vortices have two important impacts on wing aerodynamic characteristics: (1)
Wing tip vortices decrease the geometric angle of attack. It means the considered angle of
attack for wing is smaller than airfoil geometric angle of attack. The decrement angle is
called induced angle of attack (αi). It is the difference between the local flow direction
over the wing and the relative wind. The considered angle of attack for wing is called the
effective angle of attack (αeff), shown in Figure 2-15. When wing is at angle of attack of
25
α, local airfoil sections of the wing see an angle of attack lower than α because of induced
angle of attack. Clearly, wing have lower lift coefficient than airfoil lift coefficient. Since
the lift of the wing is an integration of the lift from each local segment, we can state that
CL<cl
[3]
, (2) Wing tip vortices raise the drag, this increment is called induced drag or
drag due to lift. As a result, in contrast to the lift coefficient, wing drag coefficient is
larger than airfoil drag coefficient (CD>cd).
Figure 2-15 Origin of induced drag [3]
As a result, 2D and 3D coefficients are different. 2D coefficients are acquired from airfoil
plots and to find the actual lift, drag, and moment we need to find the 3D coefficients.
2.5
Finite Wing Moment Coefficient (CM)
The wing pitching moment coefficient about aerodynamic center is largely determined by
airfoil pitching moment. The following equation provides an adjustment for wing aspect
ratio and sweep angle at low subsonic speed [11].
 AR  cos2  

C M ,Wing  C m,airfoil  
 AR  2  cos  
26
AR is the wing aspect ratio and Λ is the wing swept back angle.
2.6
Finite Wing Lift Coefficient (CL)
Lift coefficient vs. angle of attack curves of airfoils are available from airfoil plots. First,
curves of a 2D wing (airfoil) should be converted to a 3D wing (actual wing) to find the
lift and drag of aircraft wing and canard. Then, by having the wing and canard
aerodynamic characteristic curves, finding the actual lift and drag of aircraft would be
possible. The zero lift angle of attack of the airfoil ( 0 L ,airfoil ) and the zero lift angle of
attack of the wing ( 0 L , wing ) with same airfoil are very similar and they can be assumed as
a same number
[11]
. For a straight wing, maximum lift coefficient of a 3D wing drops
about 10% [12]; therefore, CL, max is about 90% of the cl, max.
 0 L , wing   0 L ,airfoil
C L ,max  0.9  cl ,max
In order to plot the lift coefficient vs. angle of attack curve, two significant points are
required. Zero lift angle of attack which is the same as airfoil zero lift angle of attack
( 0 L , airfoil ) and stall point. Wing maximum lift coefficient is found from airfoil maximum
lift coefficient. The slope of the wing lift curve is used to calculate the angle of attack in
which wing reaches its maximum lift (stall angle of attack).
Several key parameters affect the lift curve slope of the wing such as: wing aspect ratio
(the ratio of wing span to wing chord), flight Mach number (ratio of flight velocity to the
acoustic speed), airfoil efficiency (relates to lift distribution over wing), fuselage lift
factor (fuselage contribution in making lift in wing-body concept), and the exposed
surface of the wing. The major parameter in finding the wing lift slope is wing aspect
ratio (AR). As AR increases, lift slope rises. An infinite span is assumed for an airfoil;
27
therefore, an airfoil by having an infinite AR has a higher lift slope than a wing. Francis
Wenham who designed and made what was most likely the world's first wind tunnel was
the first person in history to perceive the importance of aspect ratio in subsonic flight
and then Wright Brothers examined the detail of aspect ratio
[4]
[4]
,
. They used the wind
tunnel which they constructed to find out that a skinny wing with a high aspect ratio has
greater lift coefficient at the same angle of attack than a short, fat wing with a low aspect
ratio. Figure 2-16 illustrates the effect of aspect ratio on lift curve; by increasing the AR
slope of CL vs. α curve increases.
Figure 2-16 AR impact on cl vs. AOA curve
Consequently, the most important part of plotting the curve of CL vs. α is finding the
slope (CLα) of the curve which is a function of several parameters [11].
C L 
2  AReff
 S exp osed

2
2 2 

 S
tan
(

)
AR  
max,t

2 4 2
1

 airfoil 
2

28

( FF )

AR 
b
c
Endplate  AReff  AR (1  1.9h / b)
Winglet  AReff  1.2 AR
By adding endplate or winglets the effective aspect ratio will be more than the geometric
aspect ratio. In aspect ratio equations b is wing span, c is wing chord, and h endplate
height. As we can see, there are other parameters than aspect ratio in lift slope equation. β
is the impact of Mach number (Mach) and here is about one (due to a very small Mach).
 2  1  Mach 2
 airfoil is airfoil efficiency that is a function of cl vs. α curve slope (clα) and Mach number
impact.
 airfoil 
c l
2 / 
 max,t is the sweep angle of the wing at thickest point of the airfoil. Here, we discuss
about a straight wing; therefore, this part has no impact. Sexposed is part of the wing surface
which is not covered by the fuselage. In a high wing configuration, Sexposed is equal to
wing planform area and this part has no impact to slope. In this project, the wing-body
model is discussed. In a wing-body model, fuselage makes some lift due to the “spillover” of lift from the wing
and b, wing span.
[11]
. FF is fuselage lift factor related to d, fuselage diameter,
FF  1.07(1  d / b)
When wing lift coefficient reaches its maximum amount, it stays constant for a while
even by increasing the angle of attack. This range of angle of attack in which maximum
lift coefficient stays constant is called angle of attack increment for maximum lift (
29
 CL,max ) and it’s a function of the leading edge sharpness parameter, y [9].As illustrated
in Figure 2-17, y is the vertical separation between the points on the upper surface of the
airfoil, which are 0.15% and 6% of the airfoil chord back from the leading edge.
Figure 2-17 Airfoil LE sharpness parameter [11]
By referring to Figure 2-18 based on y and wing leading edge sweep back angle (ɅLE)
we can find the angle of attack increment for maximum lift (  CL,max ) to plot the CL vs. α
curve.
Figure 2-18 Angle of attack increment for maximum lift coefficient [11]
30
Therefore we can find the angle of attack at the stall point by using zero lift angle of
attack ( 0 L , wing ) , slope of the curve (CLα), and angle of attack increment for maximum lift
(  CL,max ) :
C L 
C L max  0
 CL,max   0 L
 CL,max   0 L 
 Stall   0 L 
C L max
C L
C L max
  CL,max
C L
Now we are able to plot the CL vs. α for both wing and canard. By plotting the lift vs.
angle of attack curve of wing and canard, we are able to find the lift coefficients of wing
and canard at any angle of attack of the flight which are helpful to calculate actual lift of
wing and canard. After calculating lift coefficients, drag coefficients have to be
calculated to find actual drag.
2.7
Finite Wing Drag Coefficient (CD)
As discussed in section 2.2, drag coefficients for a finite wing with a given shape of
airfoil is larger than the drag coefficients for the wing airfoil at same angle of attack due
to wing tip vortices. This difference of drag is called drag due to lift or induced drag.
C D  cd  C Di
Therefore, wing drag coefficient is the sum of profile drag coefficient and the induced
drag coefficient. Profile drag coefficient is the airfoil drag coefficient due to pressure and
shear stress distribution over the wing, 2D drag coefficient or cd. Induced drag
31
coefficient, sometimes called as drag due to lift, is produced by the wing tip vortices and
is a function of lift coefficient for the wing, aspect ratio, and the wing span efficiency.
Wing span efficiency is a function of lift distribution over wing.
C Di  KC L
2
K is induced drag coefficient of the wing and is calculated as:
K
1
AR    espan
AR is the wing aspect ratio, and espan is the wing span efficiency factor, and for a straight
wing, it is given by [11]:
espan  1.78  (1  0.045 AR 0.68 )  0.64
Consequently, drag coefficient of a wing is a function of airfoil drag coefficient at any
angle of attack, wing lift coefficient, and wing span efficiency factor as:
C D  c d  KC L
C D  cd 
2
1
AR    espan
CL
2
Therefore, to find the wing and canard drag coefficients at any angle of attack we need
their airfoil drag coefficients (cd) and their 3D lift coefficients (CL).
C D ,W  c d ,W  C Di ,W
C D ,W  c d ,W  K W C L ,W
C D ,W  c d ,W 
2
1
2
C L ,W
ARW    e span,W
32
and for canard
C D ,C  c d ,C  C Di ,C
C D ,C  c d ,C  K C C L ,C
C D ,C  c d ,C 
2
1
2
C L ,C
ARC    e span,W
Wing and canard are major parts to make lift and fuselage has a minor impact in lift
slope. On the other hand, in drag calculations, other parts such as fuselage and landing
gears affect the total drag.
2.8
Parasite Drag of Non-Aerodynamic Parts
In addition to wing and canard as main major aerodynamic parts of an aircraft, other nonaerodynamic parts produce drag which is called parasite drag. Parts such as fuselage,
landing gears, and vertical stabilizers are considered as non-aerodynamic parts in pitch
stability discussions. Parasite drag calculation of slender body objects like fuselage is
different from blunt body objects like landing gears. Fuselage drag is mostly skin friction
drag plus a small separation pressure drag (form drag). The skin friction drag is estimated
by calculating the flat-plate skin friction coefficient. Landing gears drag is estimated by
calculating the blunt body drag coefficient. The subsonic parasite drag of slender body
components of the aircraft is estimated by using a calculated flat-plate skin friction drag
coefficient (Cf) and a component form factor (FF) that estimates the pressure drag due to
viscous separation (form drag). The interference effects on the components drag are
estimated as a factor Q and the total component drag is determined as the product of the
wetted area (Swet), Cf, FF, Q [11].
CD0,Component  C f , Part  FFPart  QPart  SWet
33
Like wing and canard, drag of components depends on Re. Skin friction drag is
significantly affected by the extent to which the aircraft has laminar flow over its
surfaces. Laminar flow is maintained if the local Re number is below roughly half a
million (Re=500,000) [11].
For portion of aircraft that has laminar flow, flat plate skin friction coefficient is
expressed as [5]:
C f , La min ar 
1.328
Re
For turbulent flow (Re>500,000), the flat plate skin friction coefficient for a low speed
aircraft is determined by following equation [5]:
C f ,Turbulent 
0.074
Re 0.2
Form factor (FF) of fuselage [11] is calculated as:
FFFuselage  1 
(l / d )
60
 F F
3
400
(l F / d F )
df is the fuselage diameter and l f is length of the fuselage, and form factor (FF) of tail [14]
is calculated as:


FFTail  1  L  (t / c )  100 (t / c ) 4 R
L is airfoil thickness location parameter, t is thickness, and c is chord. L is equal to 1.2
when (t / c ) max  0.3c and L is equal to 2.0 when (t / c ) max  0.3c . R is the lifting surface
correlation parameter. For a low speed unswept wing R is approximately 1.05 [14].
34
For a high-wing, mid-wing or well-filleted low-wing, the interference is negligible so the
Q factor is approximately one. Q factor of an unfilleted low-wing could be between 1.1
and 1.4
[11]
. The fuselage has a negligible interference factor (Q=1). For tail surfaces,
interference ranges from three percent (Q=1.03) for a clean V-tail to eight percent
(Q=1.08) for H-tail. For a conventional tail, four to five percent may be assumed [15].
To find the vertical stabilizer drag coefficient, hinge leakage should be considered as a
parameter in drag coefficient equation.
CD,VS  C f ,VS  FFVS  QVS  S wet  I
I is hinge leakage of the vertical stabilizer.
Fuselage and vertical stabilizer were assumed as slender body objects. Landing gears are
not slender body objects. They are assumed as blunt body objects and flow over blunt
body objects is not similar to slender body objects. To find drag coefficient of a blunt
body object, blunt body drag coefficient ( C D ,blunt ) is used instead of finding flat plate skin
friction. Blunt body drag coefficient is equal to 1.01 [15].
CD, LG  n  CD,blunt  S Frontal  QLG
LG is used to denote landing gears in equations, n is number of landing gears, S is frontal
area of landing gears. Landing interference (Q) changes based on the type of landing
gear, for an open wheel landing gear interference is equal to 1.5 [11].
By using discussed methods, lift and drag coefficients of wing and canard are calculated.
Parasite drag that is the drag produced by other components is also calculated. Now
requirements are prepared to calculate the aircraft actual lift and drag.
35
2.9
Aircraft Actual Lift
Aircraft lift is sum of lift produced by wing and canard (as discussed fuselage has a minor
impact in producing lift which is considered in calculating the lift slope of wing and
canard):
L AC  LW  LC
AC is used to indicate aircraft’s aerodynamic characteristics, W indicates wing’s
aerodynamic characteristics, and C indicates canard’s aerodynamic characteristics.
Aircraft lift also is a function of airstream density (ρ∞), airstream velocity (V∞), reference
surface area (SRef), and lift coefficient of aircraft (CL).
L AC  0.5     V  S Ref  C L , AC
2
Airstream density and airstream velocity are given by assumed flight condition.
Reference surface area is assumed as the sum of wing surface area and canard surface
area.
S Ref  SW  S C
Now to find the aircraft actual lift, aircraft lift coefficient must be found.
C L , AC 
L AC
0.5     V  S Ref
2
LAC  LW  LC
LW  0.5     VW  SW  CL ,W
2
LC  0.5     VC  SC  CL ,C
2
Airstream density is related to flight altitude and air temperature and pressure. Air flow
velocity over wing (VW) is not the same as the airstream velocity (V∞) due to canard
downwash. Since canard is in front of the wing, the airstream velocity (V∞) is applied for
36
airflow velocity over canard. 1% drop in air velocity is assumed over wing due to canard
VW  0.99  V
downwash.
VC  V
LW  0.5     (0.99  V ) 2  SW  C L ,W
LC  0.5     V  S C  C L ,C
2
Therefore, to find the aircraft lift coefficient we have:
C L , AC 
C L , AC  W  (
2
LW  LC
0.5     V  S Ref
2
SW
S
2
)  C L ,W   C  ( C )  C L ,C
S Ref
S Ref
(Sea appendix B)
Aircraft lift coefficient is a function of wing and canard lift confidents (found from their
plots, 3D lift vs. angel of attack curve). Selected airfoil plots give 2D lift coefficients
(angle of attack, airfoil shape, and Re number govern 2D plots). Based on wing planform,
aspect ratio, and other discussed parameters we are able to find the 3D coefficient plots.
Based on above equation, aircraft lift coefficient also is a function of wing and canard
surface ratio, and wing and canard efficiencies ( ) .
W 
VW 0.99  V

 0.99
V
V
C 
VC V

1
V V
Calculated aircraft lift coefficient completes requirements of calculating the aircraft
actual lift:
L AC  0.5     V  S Ref  C L
2
37
 2

S
S
2
2
L AC  0.5     V  S Ref  W  ( W )  C L ,W   C  ( C )  C L ,C 
S Ref
S Ref


 2

SW
SC
2
2
L AC  0.5     V  ( SW  S C )  W  (
)  C L ,W   C  (
)  C L ,C 
SW  S C
SW  S C


2.10 Aircraft Actual Drag
Aircraft Drag is sum of produced drag by aerodynamic components such as wing and
canard and parasite drag which produced by non-aerodynamic components such as
fuselage, landing gears and vertical stabilizers.
D AC  DW  DC  DParasite
DAC  DW  DC  DLG  DF  DVS
AC is used to indicate aircraft’s aerodynamic characteristics. W is used for wing, C for
canard, LG for landing gears, F for fuselage, and VS for vertical stabilizers. Aircraft drag
also is a function of airstream density (ρ∞), airstream velocity (V∞), reference surface area
(SRef), and drag coefficient of aircraft (CD).
D AC  0.5     V  S Ref  C D , AC
2
Now to find the aircraft actual drag, aircraft drag coefficient must be found.
C D , AC 
C D , AC  W 
2
C 2 
D AC
0.5     V  S Ref
2
SW
1
2
 (c d ,W 
C L ,W ) 
S Ref
ARW    e span,W
SC
1
2
 ( c d ,C 
C L ,C ) 
S Ref
ARC    e span,C
38
S LG
 (n  C D ,blunt  Q LG ) 
S Ref
SF
 (C f , F  FFF  Q F ) 
S Ref
SVS
 (C f ,VS  FFVS  QVS  I )
S Ref
(Sea appendix C)
D AC  0.5     V  S Ref 
2
( W 
2
C 2 
SW
1
2
 (c d ,W 
C L ,W ) 
S Ref
ARW    e span,W
SC
1
2
 (c d ,C 
C L ,C ) 
S Ref
ARC    e span,C
S LG
 (n  C D ,blunt  QLG ) 
S Ref
SF
 (C f , F  FFF  QF ) 
S Ref
SVS
 (C f ,VS  FFVS  QVS  I ))
S Ref
2.11 Drag Polar
Drag polar is culminating of our aerodynamic discussions and basically aerodynamics of
the complete airplane. Drag polar expresses the relation between lift and drag in a plot (L
vs. D) or in an equation (D=D0+ f (L)). It has to be noted that each point on the drag polar
plot corresponds to a different angle of attack for the airplane. Also, note that a plot of L
versus D yields the same curve as a plot of CL, AC vs. CD, AC.
D AC  0.5     V  S Re f  C D , AC  C D , AC 
2
L AC  0.5     V  S Re f  C L , AC  C L , AC 
2
39
D AC
0.5     V  S Re f
2
L AC
0.5     V  S Re f
2
Figure 2-19 Drag polar
A polar drag, plot of CL vs. CD, is sketched in Figure 2-19. The slope of the dashed line
drawn from origin to point 1, 2 and 3 on the drag polar is equal to CL/CD which is lift to
drag ratio. (Points 1, 2, 3, or any other points on the drag polar correspond to a certain
angle of attack of the aircraft). Lift to drag ratio is also called aerodynamic efficiency (E).
As it can be observed from Figure 2-19, the slope of the straight dashed lines first
increase, reach the maximum at point 2, and then decrease to point 3. Therefore, the line
of origin-2 is tangent to the drag polar and locates the point of maximum lift to drag ratio
for the airplane, Emax or (L/D)max . Moreover, the angle of attack associated with point 2 is
the angle of attack for the airplane when it is flying at its maximum aerodynamic
efficiency
[13]
. Point 2 in which aerodynamic efficiency reaches maximum is called
design point for aircraft, and corresponding value of CL is called the design lift
coefficient for the airplane. The design point clearly does not correspond to the point of
minimum drag (CD,min), and also when airplane is pitched to its zero lift angle of attack,
40
the parasite drag (CD,0L) may be slightly higher than the minimum drag value (CD,min)
which occurs at some small angle of attack slightly above the zero lift angle of attack
[13]
.
Figure 2-20 also shows the aerodynamic efficiency (lift to drag ratio) vs. angle of attack.
The maximum point of the curve shows the angle of attack of the design point.
Figure 2-20 Aerodynamic efficiency (E or L/D) vs. AOA
By examining the drag polar of the aircraft, it’s possible to choose an appropriate angle of
attack for cruise flight. This is based on aircraft design objectives, its mission, and its
expected performance at cruise flight. In short, understanding of drag polar is essential to
good aircraft design and design optimization depends on an accurate drag polar.
Drag polar is critically helpful to calculate the aerodynamic characteristics of aircraft (L
and D) at cruise flight. Now the stability of aircraft at selected cruise condition by
calculated lift and drag must be examined and checked.
41
2.12 Pitch Static Stability Criteria
In the design of an aircraft, static stability about all three axes (shown in Figure 1-7) are
essential. Here, we provide only the pitch static stability details about the y axis. It's the
most important static stability type. In the aircraft design, the main focus is on pitch static
stability rather than lateral and directional stability because it's significantly sensitive to
the location of the center of gravity for the aircraft. Figures 2-21 thru 2-23 show the
variation of pitch moment with angle of attack.
Figure 2-21 Aircraft in steady, equilibrium flight at its trim angle
In Figure 2-21, aircraft flight is in equilibrium and steady condition. The angle of attack
in which aircraft flight is equilibrium and steady is called trim angle of attack or
equilibrium angle of attack (αe). If the aircraft gets disturbed by a gust wind, there will be
two foreseeable prospects: increasing in angle of attack (α) or decreasing.
42
In Figure 2-22, the aircraft pitches upward (α > αe). To bring back the aircraft to the
equilibrium flight, a negative moment about the center of gravity is required, in a
counterclockwise direction.
Figure 2-22 Nose up disturbance
As shown in Figure 2-23, the aircraft pitches downward (α < αe). To bring back the
aircraft to the equilibrium flight, a positive moment about the center of gravity is
required, in a clockwise direction.
Figure 2-23 Nose down disturbance
43
Curve shown in Figure 2-24 illustrates above situations that could happen to the pitch
stability.
C M ,cg

Figure 2-24 Moment coefficient vs. AOA curve with a negative slope
The curve is nearly linear. As it could be observed, about equilibrium angle of attack the
moment coefficient about the center of the gravity is equal to zero. For angles larger than
equilibrium angle of attack, moment coefficient is negative. It must be negative to
decrease the angle of attack in order to push back the nose downward to its equilibrium
position. In the lower angle than equilibrium angel of attack, moment coefficient gets
positive. It must be positive to increase the angle of attack in order to push back the nose
upward to its equilibrium position. This is precisely the definition of pitch static stability.
CM,0L is the value of moment coefficient at the zero lift angle of attack. CM,0L must be a
positive number and the slope (CMα) of the curve (CM vs. α) must be negative. More
negative slope results in more stable flight. Less negative slope, as the absolute value of
the slope decreases, results in the less stable flight. It must be considered that discussed
angle of attacks in the curve are smaller than the stall angle of attack.
44
Figure 2-25 Moment coefficient curve with a positive slope
Now let's consider a different condition for the aircraft in which moment coefficient
curve has a positive slope, as shown in Figure 2-25. As the disturbing wind increases the
angle of attack and pushes the nose upward (α > αe), a positive moment coefficient is
applied. The direction of this moment is clockwise pushing the nose more upward, farther
from the equilibrium position. Similar result happens when a gust wind disturbs the
aircraft to nose down position. Negative moment pitches the nose more downward and
farther from the equilibrium position. Therefore, this curve proves that the flight
condition is not stable. Figures 2-26a and b show a statically unstable flight.
a
b
Figure 2-26 Statically unstable flight
45
The curve of moment coefficient vs. angle of attack shows the pitch stability of the
aircraft. To plot this curve at an equilibrium flight, the location of the center of gravity of
the aircraft and the moment about it at different angle of attacks must be found. Then the
slope shows to what extend aircraft is stable.
2.13 Equilibrium Flight
A longitudinal equilibrium flight occurs when sum of all forces and moments become
equal to zero (in three equations of symmetric degrees of freedom).
F
F
M
X
Z
 0   FX  TNet  D AC  0  TNet  D AC
 0   FZ  L AC  W AC  0  L AC  W AC
cg
0
All applied forces on the aircraft are illustrated in Figure 2-27.
Figure 2-27 Forces and moments
TNet  TEngine  D Engine
D AC  DW  DC  D LG  DF  DVS
D AC  0.5     V  S Ref  C D
2
L AC  LW  LC
46
L AC  0.5     V  S Ref  C L
2
S Ref  S W  S C
Aircraft weight is applied about aircraft center of gravity (CG). TNet is net thrust of the
engine, available thrust made by engine minus the drag of the engine. In addition to
forces, sum of moments about the CG must be equal to zero. To find the moment about
the CG and the CG location, calculating of all lift and drag forces of all parts such as
aerodynamic parts(wing and canard), and non-aerodynamic parts(fuselage, landing gear,
and vertical stabilizer) is required.
2.14 Aircraft Moment Coefficient (CM)
As discussed before, to fly in longitudinal static equilibrium sum of all moment about the
CG must be equal to zero. In moment calculations, clockwise rotation about the CG is
assumed positive direction for the rotation. Forces, moments, and distances from the CG
are shown in Figure 2-28. Moments about the wing and canard aerodynamic centers are
denoted as Mac,W and Mac,C.
Figure 2-28 Moments, forces and their distances from CG
47
The vertical and horizontal locations of components such as landing gears, mean chord of
vertical stabilizer, engine installation location, and fuselage center are calculated based
on the distance between them and CG. ZLG is vertical distance between the landing gears
and CG, ZVS is vertical distance between the mean chord of vertical stabilizer and CG,
ZEngine is vertical distance between the engine installation location and CG, and ZF is
vertical distance between the fuselage center and CG. The vertical and horizontal
distances between wing aerodynamic center and canard aerodynamic center also are the
project assumptions (XC+XW and ZC+ZW).
M cg  M C  M W  M Engine  M F  M LG  M VS
M cg  0.5  V  S Ref  c Ref  C M
2
CM 
M cg
0.5  V  S Ref  c Ref
2
(Sea appendix D)
CM  C 
SC
2
 W 
S Ref  c Ref
SW
2
S Ref  c Ref


S LG
S Ref  c Ref


1
2
  C M ,C  c C  C L ,C  X C  (c d ,C 
C L ,C )  Z C 


ARC    e span,C



1
2
  C M ,W  c W  C L ,W  X W  (c d ,W 
C L ,W )  Z W

ARW    e span,W

TNet  Z Engine
0.5  V  SRef  c Ref
2

SF
SRef  c Ref
 (n  C D ,blunt  QLG )  Z LG 
 (C f , F  FFF  QF )  Z F
SVS
S Ref  c Ref
48
 (C f ,VS  FFVS  QVS  I )  Z VS




2.15 Finding CG Location
In moment equation discussed in the last sections, the location of components such as
landing gears, mean chord of vertical stabilizer, engine installation location, and fuselage
center are calculated based on the distance between them and CG. The vertical and
horizontal distances between wing aerodynamic center and canard aerodynamic center
also are assumptions here (XC+XW and ZC+ZW). Therefore we need to find the CG
location between them. We need to find XC, XW, ZC, and ZW. Now to solve these 4
unknowns, 4 equations are required. Distances between them are assumed and known
(XC+XW and ZC+ZW). By writing moment equations about two different points, two more
equations would be taken. Moment equation about CG and moment equation about wing
aerodynamic center (Wac).
(1) : X W  X C  A
(2) : Z W  Z C  B
(3) : M CG  0
(4) : M Wac  0
 2

SC
(3) : M CG  0   C 
 C L ,C   X C
S Ref  c Ref


 2

SC
1
2
  C 
 (c d ,C 
C L ,C )   Z C


ARC    e span,C
S Ref  c Ref


 2
SW
 W 
 C L ,W
S Ref  c Ref


  XW


 2

SW
1
2
 W 
 (c d ,W 
C L ,W )   Z W


ARW    e span,W
S Ref  c Ref


49




TNet  Z Engine
0.5  V  S Ref  c Ref
2
SF
S Ref  c Ref
S LG
S Ref  c Ref
SVS
S Ref  c Ref
  C 
2
 (C f , F  FFF  QF )  Z F
 (n  C D ,blunt  QLG )  Z LG
 (C f ,VS  FFVS  QVS  I )  Z VS
SC
 C m ,C  c C   W 
2
S Ref  c Ref
SW
S Ref  c Ref
 C m,W  cW
 2

SC
(4) : M Wac  0   C 
 C L ,C   ( X C  X W )
S Ref  c Ref


 2

SC
1
2
  C 
 (c d ,C 
C L ,C )   ( Z C  Z W )


ARC    e span,C
S Ref  c Ref


TNet  ( Z Engine  Z W )
W  XW 
2
0.5  V  S Ref  c Ref



SF
S Ref  c Ref
S LG
S Ref  c Ref
SVS
S Ref  c Ref
  C 
2
 (C f , F  FFF  Q F )  ( Z W  Z F )
 (n  C D ,blunt  Q LG )  ( Z LG  Z W )
 (C f ,VS  FFVS  QVS  I )  ( Z VS  Z W )
SC
S Ref  c Ref
 C m ,C  c C   W 
2
50
SW
S Ref  c Ref
 C m ,W  c W
2.16 Pitch Static Stability
As discussed before, an aircraft is stable when moments about the CG would be able to
return the aircraft to trim angle after any disturbance. This will happen if some conditions
on aircraft moment coefficient vs. angle of attack curve are met. CM,0L which is the value
of moment coefficient at the zero lift angle of attack must be a positive number, and the
slope (CMα) of the curve (CM vs. α) must be negative. The moment coefficient at the trim
angle of attack is equal to zero. Trim angle is selected by examining the drag polar and
aircraft expected mission. The equation of aircraft moment coefficient is a function of
lift, drag, and moment coefficients of aircraft components, which are related to angle of
attack, and CG location. Based on the aircraft trim angle of attack and zero moment
coefficients at this angle of attack, CG location is found. Now to plot the aircraft moment
coefficient vs. angle of attack curve, calculating the zero lift moment coefficient (CM,0L)
is required. Zero lift angle of attack of aircraft was found from calculating aircraft lift
coefficient. By having the angle of attack, lift, drag, and moment coefficient of aircraft
components could be calculated.
 trim : C M  0
 0 L : C M  C M ,0 L
To plot the aircraft moment coefficient vs. angle of attack curve, absolute angle of attack
is used. Absolute zero lift angle of attack is equal to zero. Therefore, absolute trim angle
of attack is equal to sum of trim angle and absolute value of zero lift angle of attack.
 0 L ,abs  0
 trim,abs   trim   0 L
51
C M 
C M 
C M 
C M ,trim  C M , 0 L
 trim,abs   0 L ,abs
0  C M ,0 L
 trim,abs  0
 C M ,0 L
 trim,abs
Negative slope is required to get pitch static stability; therefore, moment coefficient of
the aircraft at zero lift angel of attack must be positive.
The static pitch stability of the aircraft was examined. Now it has to be shown to what
extend the aircraft is stable. In order to gain this goal, finding the neutral point of the
aircraft and static margin is required.
2.17 Aerodynamic Center or Neutral Point (NP)
For any aircraft, there is a CG location that provides no change in pitching moment as
angle of attack varies. This location of CG is called aircraft aerodynamic center. From
consideration of pitch stability, the aerodynamic center of the aircraft must lie behind the
aircraft’s CG. The aerodynamic center of the aircraft is also called neutral point for the
aircraft
[13]
. Neutral point represents neutral stability and is the most-aft CG location
before the aircraft becomes unstable
[11]
. Figure 2-24 shows to have a stable flight the
pitching moment vs. angle of attack curve must have a negative slope. Figure 2-25 shows
that the flight is unstable whenever the slope is positive.
52
In addition to these situations, Figure 2-29 shows a new situation. Pitch moment curve
with a slope of zero. This situation results in a statically neutral flight. That particular
center of gravity about which the slope for pitching moment vs. angle of attack curve is
equal to zero is called neutral point.
Figure 2-29 Impact of NP location on CM curve slope
In an equilibrium and balanced flight, the center of lift (the point about which the sum of
produced lift by both wing and canard is applied), must be behind the center of gravity of
the aircraft. By changing the angle of attack to have a stable flight, lift center must be at
or forward of the neutral point. Neutral point is located in the farthest point (toward the
back) that we can consider for the center of gravity by which flight still is stable.
Therefore, to have a stable flight, there is a limited distance in which lift can be applied,
from center of gravity to neutral point that is situated behind the center of gravity.
53
Figure 2-30 shows the flight is statically stable when the new lift center is placed behind
the center of gravity, between the center of gravity and the neutral point.
Figure 2-30 Importance of lift center location, behind the CG [3]
2.18 Static Margin(SM)
The distance between the center of gravity and the neutral point is the margin in which
lift center of the aircraft (total lift of the whole aircraft) can be placed. The ratio of this
distance (the distance between center of gravity and the neutral point) to the reference
chord is called static margin(SM). Reference chord was used to find the aircraft pitch
moment coefficient (𝑐̅Ref). Therefore, SM shows the distance between CG and NP as a
percentage of the reference chord.
SM 
X NP  X CG
c Re f
54
Static margin is considered as a measure of the pitch stability of the aircraft since it
indicates how far the center of gravity can be moved behind the designed location of CG
before a statically stable flight changes to a neutral and then unstable flight. However,
this equation has two unknown parameters: SM and XNP. There is also the other way to
find the SM. Static margin is the ratio of slopes for moment coefficient vs. AOA and lift
coefficient vs. AOA. It is numerically equal to the magnitude of -CMα/CLα. For a statically
stable flight, SM must be positive. SM of 5% to 15% gives a good stability to a canard
aircraft [16]. Low SM gives less static stability; on the other hand, a large SM makes the
aircraft nose heavy which may result in elevator (control surface that controls pitch) stall
at take-off and landing. Too large SM also gives an excessive stability to the aircraft that
decreases the aircraft maneuverability.
SM  
C M
C L
C
 C M ,0 L   C L ,Trim  C L ,0 L 
 /

SM   M ,Trim
 






Trim
,
abs
0
L
,
abs
Trim
,
abs
0
L
,
abs

 

C M ,Trim  0
C L,0 L  0
 0 L ,abs  0
  CM ,0 L   C L,Trim 
 /

SM  
 

  Trim,abs    Trim,abs 
 C M ,0 L 

SM  

C
 L ,Trim 
Now calculation the location of the neutral point is possible:
55
SM 
X NP  X CG
c Re f
X NP  ( SM  c Re f )  X CG
 C M ,0 L 
  c Re f  X CG
X NP  

C
 L ,Trim 
56
3.
Excel Spreadsheets
All calculations and optimizations of the design of this project are applied in an Excel
tool. During this chapter the spreadsheets of the used Excel tool will be introduced.
Spreadsheets are including: the required inputs, 2D and 3D calculations in order to plot
the drag polar, selecting an efficient AOA, finding the CG and SM in order to examining
the pitch stability, and optimizing to acquire the design expectations such as canard
requirements, CG, and SM.
3.1
Inputs
The first spreadsheet is the input spreadsheet where the value of given parameters and
design assumptions should be inserted. Inputs are categorized to:
1-flight ambient condition parameters which are important to find the Re number and
other important parameters:
Ambient Conditions
Name
Value
Temperature(20°C)
T
68
Air Kinematic Viscosity(@20°C)
ν 1.63E-04
Acoustic Speed(@20°C)
a
1,126
Cruise Velocity
V
50
Air Density
ρ 0.00237
Dynamic Pressure
q
57
2.96
Unit
°F
ft^2/s
ft/s
ft/s
slug/ft^3
lb/ft^2
2-Airplane reference parameters which are the design assumptions such as wing loading
and size of planform area, and etc.:
Airplane References
Name
Reference Planform Area
Sref.
Reference Chord
cref
Weight
W
Horizontal Distance b/w Cac and Wac Xc+Xw
Vertical Distance b/w Cac and Wac
Zw+Zc
Value Unit
10
ft^2
1.21
Ft
15
Lb
3
Ft
0.54
Ft
3-Size and other parameters of aerodynamic parts such as wing:
Wing
Name
Wing Chord
Wing Surface Ratio
Wing Sweep Back Angle
Wing Efficiency
Chord
Sw/Sref
Λ
Ƞ
Value
1.21
0.85
0
0.99
Unit
Ft
%
degree
-
And canard:
Canard
Name
Canard Chord
Canard Surface Ratio
Canard Sweep Back Angle
Canard Efficiency
Chord
Sc/Sref
Λ
Ƞ
58
Value
0.42
0.15
0
1
Unit
Ft
%
degree
-
4- Size and other parameters of non-aerodynamic parts such as fuselage:
Fuselage
Name
Fuselage Diameter
Fuselage Length
Fuselage Nose Area
Fuselage Wetted Area
Fuselage Center below Wac (above CG)
Value Unit
D
0.46
ft
l
4
ft
Sn 0.11 ft^2
Sw 5.12 ft^2
Zf 0.23
ft
To find fuselage wetted area, fuselage area should be subtracted by fuselage area covered
by wing and canard. At the beginning, CG location is unknown; that’s why, wing and
canard aerodynamic centers are selected as references to indicate the other part’s
locations.
Vertical stabilizers:
Vertical Stabilizer
Name
Value Unit
VS Mean Chord
Chord
0.5
ft
VS Span
Span
0.5
ft
VS Wetted Area
Surface 0.25 ft^2
VS Mean Chord above Wac
Zvs
0.25
ft
Engine:
Engine
Name
Engine Installation Point above Wac
Engine Installation Point behind Wac
59
Ze
Xe
Value
0
1.2
Unit
ft
ft
And landing gears:
Landing Gears
Name
Wheel Diameter
Wheel Thickness
Wheel Frontal Area
Strut Thickness
Strut Height
Strut Frontal Area
# of Main Landing Gears
Main Wheels bellow Cac
Main Struts bellow Cac
Nose Wheel bellow Cac
Nose Strut bellow Cac
3.2
Value
d
2
tw
0.7
Sw 0.0097
ts
0.5
hs
5
Ss 0.017
n
2
ZG
0.5
ZS
0.21
ZG
0.5
ZS
0.21
Unit
Inch
Inch
ft^2
Inch
Inch
ft^2
ft
ft
ft
ft
Aerodynamic Characteristics of Airfoils
Choosing suitable airfoils for canard and wing is a very crucial step in designing a canard
aircraft. In two next spreadsheets, aerodynamic characteristics of wing and canard airfoils
should be introduced to the Excel tool of this project. In order to find these characteristics
from sources of airfoil plots (Profili software
[17]
is used as a source of airfoil plots)
calculating the Re is required. Re is calculated based on sea level cruise flight at velocity
of 50ft/s and chord length of wing and canard.
FX63-137 at Re=130000
SD7062 (14%) - Re = 370000
Alfa
Cl
Cd
Cl/Cd
Cm
Alfa
-6.5
-0.0864
0.0812
-1.064
-0.0902
-7.5
-5
-0.0246
0.0437
-0.5629
-0.1387
-7
-4.5
0.0915
0.0317
2.8864
-0.1569
-6.5
….
……
…
……
……
….
60
Cl
0.3035
0.2613
0.2172
……
Cd
-0.0973
0.0228
Cl/Cd
11.9488
11.4605
0.02
-10.86
-0.0934
0.0254
……
……
Cm
-0.0952
……
0
0.7836
0.0209
37.4928
-0.1885
-3
0.1422
0.0105
13.5429
-0.0866
0.5
0.8183
0.0214
38.2383
-0.1847
-2.5
0.1964
0.01
19.64
-0.0862
1
0.8903
0.0209
42.5981
-0.1869
-2
0.2505
0.0095
26.3684
-0.0858
1.5
0.9405
0.0213
44.1549
-0.186
-1.5
0.3047
0.0092
33.1196
-0.0851
2
0.9976
0.0211
47.2796
-0.1856
-1
0.3591
0.0091
39.4615
-0.0843
……
…
……
……
……
……
……
….
……
….
12.5
1.653
0.0526
31.4259
-0.1235
9.5
1.3771
0.0174
79.1437
-0.0569
13
1.6717
0.0572
29.2255
-0.1214
10
1.4031
0.0185
75.8432
-0.0527
12
1.6493
0.0471
35.017
-0.1268
10.5
1.4253
0.0198
71.9848
-0.0485
12.5
1.6519
0.0525
31.4648
-0.124
11
1.4522
0.021
69.1524
-0.0453
13
1.6691
0.0569
29.3339
-0.122
11.5
1.4668
0.0229
64.0524
-0.0411
In Figure 3-1 lift curve of the airfoils are compared. In result section all aerodynamic
characteristics of the airfoils will be discussed.
Cl vs. α (Canard & Wing Airfoils)
1.8
1.6
1.4
1.2
1
0.8
Wing
Cl
0.6
Canard
0.4
0.2
-10
-8
-6
-4
0
-2
0
-0.2
2
4
6
8
10
12
14
16
-0.4
-0.6
Angle of Attack
Figure 3-1 cl vs. α curves of airfoil FX63-137(Re=130,000), and SD7062 (Re=350,000)
61
2D Lift Slope, 2D clmax, and 2D Zero Lift AOA are very significant and useful values
known from airfoil lift vs. AOA curve, shown in Figure 3-2. These values are used to
graph the 3D wing lift vs. AOA curve.
Canard Airfoil Characteristics
Name
Value
Reynolds Number
Re
128,149
2D Lift Slope
Clα
0.11
2D clmax
Clmax
1.69
2D Zero Lift AOA
α0L
-4.89
Unit
1/degree
Degree
Wing Airfoil Characteristics
Name
Value
Reynolds Number
Re
371,634
2D Lift Slope
clα
0.10
2D clmax
clmax
1.51
2D Zero Lift AOA
α0L
-4.33
Unit
1/degree
Degree
Figure 3-2 Airfoil SD7062 lift characters
2D aerodynamic characteristics of airfoils are known now next step is converting them to
3D values.
62
3.3
3D Lift Curve Requirements
To calculate the lift of canard and wing, 3D lift curve must be plotted. To plot the 3D lift
curve of canard and wing, some significant values are required: 3D zero lift angle of
attack, 3D lift slope, 3D CLmax, angle of attack increment, and 3D stall AOA. In next
spreadsheets, theses parameters are calculated in order to convert 2D aerodynamic
characteristics to 3D. (Only canard spreadsheet is shown here)
Canard Calculated Parameters
Name
Canard Surface Ratio
Canard Exposed Planform Area
Canard Effective(Exposed) Span
Canard Chord
Canard Effective Aspect Ratio
Sc/Sref
Area
b
c
AR
Value
0.15
1.5
3.6
0.42
8.64
Unit
%
ft^2
ft
Ft
-
Canard 2D Slope
Canard 2D Slope
clα
clα
0.107
6.11
1/degree
1/Radian
Canard Span Efficiency Factor
Canard Induced Drag Factor
e
K
0.79
0.046
-
M
β
Ƞ
Λ
d
S
Sex/Spf
F
0
1
0.97
0
0.46
0.191
1
1.09
Radian
Ft
ft^2
ft^2
-
CLα
CLα
5.32
0.093
1/Radian
1/degree
Mach Number
Mach Number Effect
Airfoil Efficiency
Canard Sweep Back Angle
Fuselage Width
Fuselage Planform Area
Exposed to Reference Canard Area Ratio
Fuselage Lift Factor
Canard 3D Slope
63
Canard 3D Lift Plot Parameters
Name
Value
Canard 2D max. Lift Coefficient
clmax
1.69
Canard 3D max. Lift Coefficient
CLmax
1.52
Canard 2D Zero Lift AOA
α0L
-4.89
Canard 3D Zero Lift AOA
α0L
-4.89
0.15% Chord Back From LE
0.15%c
0.03
6% Chord Back From LE
6%c
0.063
Leading Edge Sharpness Parameter
∆y-c
0.033
Angle of Attack Increment
∆αCLmax
1.3
Canard 3D Stall Angle
αmax
12.75
αmax∆αCLmax
11.45
Unit
Degree
Degree
Chord%
Chord%
Chord%
Degree
Degree
Degree
After finding 3D required parameters, checking the canard aircraft requirements for zero
lift and stall conditions is a very crucial and important step. 3D values, like airfoils, must
meet the canard requirements.
Wing and Canard Stall and Zero Lift Points Comparison
Canard 3D Stall Angle
αmax-∆αCLmax
Wing 3D Stall Angle
αmax-∆αCLmax
Stall Margin
αmax
αmax
α0L
α0L
Canard 3D Zero Lift AOA
Wing 3D Zero Lift AOA
Zero Lift Margin
12.75
11.45
13.94
12.59
1.19
degree
-4.89
-4.33
0.56
degree
degree
degree
degree
degree
degree
Canard stalls before wing and wing reaches its zero lift angle of attack before canard.
Canard configuration crucial requirements are now met.
64
3.4
AR Impact on Lift Curves
AR is the major parameter in calculating the 3D lift slope. Impact of different ARs is
shown in the following spreadsheets. Two model are used to change the AR: 1) by
utilizing different surface ratios of wing and chord (Sw/Sref and SC/Sref) when chord for
wing and canard are same, 2) by utilizing same surface ratios and changing chords for
wing and canard. Impact of AR on lift curves in different conditions is plotted in
spreadsheet. Maximum lift coefficient doesn’t change and by changing the AR, only
curve slope changes resulting in changing of stall angle of attack, shown in Figure 3-3.
AR impact on Lift Slope (Same Surface Ratio, Diff. Chord)
1.6
1.4
1.2
AR=4.41
CL
1
0.8
AR=6
0.6
AR=11.52
0.4
0.2
0
-5
0
5
AOA
10
Figure 3-3 AR impact on the lift slope
Sc/Sref =15% ,C=5", AR=8.64
AOA
CL
-4.89
0
11.45
1.52
12.75
1.52
Sc/Sref =15% ,C=6" , AR=6
AOA
CL
-4.89
0
12.97
1.52
14.27
1.52
65
15
Sc/Sref =15% ,C=7" , AR=4.41
AOA
CL
-4.89
0
14.87
1.52
16.17
1.52
Sc/Sref =20% ,C=5", AR=11.52
AOA
CL
-4.89
0
10.68
1.52
11.98
1.52
Sc/Sref =25% ,C=5", AR=14.40
AOA
CL
-4.89
0
10.22
1.52
11.52
1.52
3.5
3D Aerodynamic Characteristics
In these spreadsheets, 3D aerodynamic characteristics of wing and canard such as lift,
drag coefficients, and other aerodynamic characteristics are calculated and 3D plots are
graphed by using calculated 3D zero lift angle of attack, 3D lift slope, 3D C Lmax, angle of
attack increment, and 3D stall AOA. Calculations are based on aircraft angle of attack.
Without utilizing any angle of incidence for wing and canard (angle of incidence is the
angle between the wing chord, where the wing is mounted to the fuselage, and reference
axis along the fuselage, the longitudinal axis), airstream flows at the same angle of attack
over aircraft, wing and canard; therefore, same angle of attack should be applied for
them. However, by utilizing angle of incidence for wing or canard, flowing airstream
over them wouldn’t be at the same angle of attack any more. As discussed before, zero
lift angle of attack and stall angle of attack for canard are very crucial in canard
66
requirements. Therefore, canard angle of attack is assumed as a primary angle of attack,
and then aircraft and wing angle of attacks are calculated based on canard angle of attack
and the angle of incidence. For example in the following spreadsheet angle of incidence
of 4 degree is applied for canard. For angle of incidence of 4 degree, the angle between
the canard chord and longitudinal axis is 4 degree. Therefore, when aircraft AOA is 8.89, canard AOA is at its zero lift angle of attack (-4.89 degree). Due to angle of
incidence of 4 degree for canard, it reaches 12.75 degree (its stall angle of attack), when
aircraft is at 8.75 angle of attack, shown in Figure 3-4.
Canard CL with αi vs. without αi
Lift Coefficient
1.60
1.40
1.20
1.00
0.80
0.60
0.40
0.20
0.00
-10.00
-5.00
No αi
With αi
0.00
5.00
Angle of Attack
10.00
15.00
Figure 3-4 Angle of incidence impact
Canard CL &cd vs. α
AOA
-4.89
….
9.17
9.67
10.17
10.67
11.17
11.45
12.75
CL
0.00
…..
1.30
1.35
1.40
1.44
1.49
1.52
1.52
Cd
0.041
….
0.026
0.028
0.030
0.033
0.038
0.041
0.055
Canard(With Angle of Incidence)
Aircraft
AOA
-8.89
….
5.17
5.67
6.17
6.67
7.17
7.45
8.75
CL KCL^2
0.00 0.000
…
…
1.30 0.079
1.35 0.085
1.40 0.091
1.44 0.097
1.49 0.103
1.52 0.107
1.52 0.107
67
cd
0.041
…
0.026
0.028
0.030
0.033
0.038
0.041
0.055
CD CL/CD
0.041
0.00
…
…
0.105 12.44
0.112 12.02
0.121 11.57
0.130 11.08
0.141 10.54
0.148 10.23
0.162
9.37
In the following spreadsheet, angle of incidence of 1 degree is assumed for wing. Canard
angle of attack is assumed as a primary angle of attack. With 4 degree angle of incidence
for canard, aircraft angle of attack is -8.89 when canard is at its zero lift angle of attack.
Wing angle of attack at this point is -7.89 due to 1 degree angle of incidence. Angle of
attack angles start with canard zero lift angle of attack (AOAC =-4.89, AOAw =-7.89, and
AOAAircraft = -8.89) and ends with canard stall angle of attack (AOAC =12.75, AOAw
9.75, and AOAAircraft = 8.75), however, wing hasn’t reached its stall angle of attack at this
point yet.
Wing CL vs. α(Delta i)
AOA CL
cd
-7.89 -0.29
0.027
-7.33 -0.24
0.025
7.67 0.97
0.015
…
….
….
8.45 1.03
0.016
9.59 1.12
0.018
9.75 1.13
0.018
A/C AOA
-8.89
-8.33
6.67
….
7.45
8.59
8.75
Wing(With angle of incidence)
CL
KCL^2
cd
CD
-0.29
0.005 0.027 0.033
-0.24
0.004 0.025 0.028
0.97
0.058 0.015 0.073
….
…..
…..
….
1.03
0.066 0.016 0.082
1.12
0.078 0.018 0.096
1.13
0.080 0.018 0.098
CL/CD
-8.79
-8.57
13.25
…..
12.58
11.66
11.54
Without using angle of incidence for canard and wing, same angle of attack would be
used, as shown below.
Canard CL & cd vs. α
AOA
-4.89
-4.33
…
11.45
12.59
12.75
CL
0.00
0.05
…
1.52
1.52
1.52
cd
0.041
0.031
….
0.041
0.053
0.055
Canard(With no angle of incidence)
A/C AOA
-4.89
-4.33
….
11.45
12.59
12.75
68
CL
0.00
0.05
….
1.52
1.52
1.52
KCL^2
0.000
0.000
…..
0.107
0.107
0.107
cd
0.041
0.031
…..
0.041
0.053
0.055
CD CL/CD
0.041
0.00
0.031
1.69
…..
…..
0.148 10.23
0.161
9.45
0.162
9.37
Wing CL vs. α(Delta i)
AOA
CL
cd
-4.89 -0.05
0.014
-4.33 0.00
0.013
…
….
…
12.59 1.36
0.027
12.75 1.37
0.028
Wing(With no angle of incidence)
A/C AOA
CL KCL^2
cd
CD
-4.89
-0.05 0.000 0.014 0.014
-4.33
0.00 0.000 0.013 0.013
….
….
….
….
….
12.59
1.36 0.116 0.027 0.143
12.75
1.37 0.118 0.028 0.146
CL/CD
-3.19
0.00
….
9.51
9.39
Aerodynamic characteristics of aerodynamic parts, wing and canard, are found. Plotting
the whole aircraft lift vs. angle of attack curve now is possible. Now computing the drag
of non-aerodynamic parts are required to find the total drag of aircraft on different angle
of attacks during the flight.
3.6
Parasite Drag
In this spreadsheet drag of fuselage, vertical stabilizer, and landing gears would be
calculated. Note that the engine drag is included in net thrust of the engine.
Transition
Turbulant
Form Factor
Interference Factor
Transition Location
Turbulant
Swet/Sref
Fuselage
Re
Re
cf
CD0
69
500000
114351
1.11
1
5.332
0.0072
0.51
ft
-
0.0041
-
Vertical Stabilizer
Flat Plate Form Factor
Mean Chord
Mean Chord
Re
Cf
Swet/Sref
H-Tail Interference
Hinge Leakage
CD0
1
0.5
0.152
14294
0.0111
0.1
1.08
1
Ft
M
-
0.00108
-
Landing Gears
Wheel Frontal Area
Swheel
0.0097
ft^2
Strut Frontal Area
Sstrut
0.0174
ft^2
Blunt Object
CDo
1.01
-
# of Back LG's
n
2
-
1.5
-
Open Wheel Interference
Wheel
CD0
0.0029
-
Strut
CD0
0.0052
-
Back Gears
CD0
0.0081
-
Nose Gear
CD0
0.0041
-
Total Cdmin of Non-Aerodynamic Parts
CD0
0.0174
-
Now by finding the drag of all parts, plotting the whole aircraft drag vs. angle of attack
curve is possible. After calculating lift and drag of all parts such as aero and non-aero
parts, and calculating the total lift and drag of whole aircraft, drag polar of the aircraft
and its aerodynamic efficiency plot would be feasible.
70
3.7
Drag Polar and Aerodynamic Efficiency Plots
In drag polar spreadsheet, lift and drag of aircraft will be calculated in order to plot the
drag polar and aerodynamic efficiency. Plotting the drag polar is significant to find a
proper angle of attack for an efficient cruise flight, high aerodynamic efficiency and less
required thrust.
A/CAOA CAOA
-8.89
-4.89
…
…
0.17
4.17
0.67
4.67
1.17
5.17
1.67
5.67
2.17
6.17
2.67
6.67
3.17
7.17
…
…..
7.17
11.17
7.45
11.45
3.8
CCL
0.00
…
0.84
0.89
0.93
0.98
1.03
1.07
1.12
….
1.49
1.52
CCD
0.041
…
0.054
0.057
0.061
0.066
0.070
0.075
0.080
….
0.141
0.148
Aircraft
WAOA WCL
-8.89 -0.37
…
….
0.17 0.36
0.67 0.40
1.17 0.44
1.67 0.48
2.17 0.52
2.67 0.56
3.17 0.60
….
….
7.17 0.93
7.45 0.95
WCD
0.041
….
0.017
0.019
0.021
0.024
0.026
0.030
0.033
….
0.068
0.070
A/CCL
-0.31
….
0.43
0.47
0.51
0.55
0.59
0.63
0.68
….
1.00
1.03
A/CCD
0.058
….
0.040
0.042
0.044
0.047
0.050
0.053
0.057
….
0.095
0.099
A/CE
-5.31
…
10.80
11.22
11.53
11.74
11.84
11.88
11.86
….
10.50
10.37
Cruise Optimization
In cruise optimization spreadsheet, an appropriate flight angle of attack would be selected
based on flight aerodynamic efficiency (to have an efficient flight and less required
thrust) and aircraft mission (cruise velocity). This angle of attack should be close to
optimum aerodynamic efficiency. By adding angle of incidence, we would be able to find
a proper angle of attack.
71
3.9
Center of Gravity
After selecting an appropriate angle of attack for cruise flight, calculating the exact
amount of lift, drag, and moment of aircraft and other parts is possible. Now finding the
aircraft center of gravity based on lift, drag and moment of all parts is possible.
A/C AOA
1.1
Aircraft at Stable Cruise Condition
Canard
Canard
AOA
Canard CL
CD
5.1
0.93
0.061
Wing AOA
Wing CL
Wing CD
1.1
0.44
0.021
A/C CL
A/C CD
0.51
0.0441
Xc+Xw
ft
3
A/C E
11.4827
Vcruise
50.00
Thrust
1.3064
Lift
15
Zc+Zw
Ft
0.542
Xe
ft
1.2
Ze
ft
0
Zf
ft
0.229
ZLG
Wheel
0.5
Xc
Xw
Canard Cm
-0.178
Wing Cm
-0.078
A/C Cm
0
Strut
0.208
2.35
0.65
ZLG(Nose)
Wheel
Strut
0.5
0.208
ZVS
ft
0.25
CG Location infront of Wing LE
0.34
ft
4.12
inch
72
3.10 Static Margin
In static margin spreadsheet, absolute zero lift angle of attack and selected cruise angle of
attack for aircraft in absolute form are being used to plot the moment vs. angle of attack
curve.
Aircraft at Zero Lift Condition
A/C
AOA
-5.11
A/C Cm
0.046
0
Canard AOA
-1.1
Wing AOA
-5.1
Abs. AOA
0
6.20
Canard CL Canard CD
0.35
0.027
Wing CL
Wing CD
-0.06
0.015
A/C CL
A/C CD
0.01
0.034
Slope
-0.01
CLα
0.08
Canard Cm
-0.184
Wing Cm
-0.090
A/C Cm
0.046
SM
0.092
Negative slope of moment vs. angle of attack curve shows the stability of the aircraft.
Static margin of aircraft will be found by slope of moment vs. angle of attack curve and
lift vs. angle of attack curve.
3.11 Optimization
At the last spreadsheet, based on design parameters such as wing chord, canard chord,
and wing surface ratio, expecting parameters such as stall margin (the difference between
stall points of wing and canard), cruise AOA, cruise E, center of gravity, and static
margin at different angle of incidence are analyzed to get proper and efficient results.
73
4.
4.1
Results
Airfoil Selection
In this project airfoil FX63-137 was chosen to use for canard. Canard Re number is equal
to 130,000 (based on assumed chord length for canard, 6”, and cruise velocity of 50 ft/s
at standard condition of atmosphere). Figures 4-1 thru 4-5 show the aerodynamic
characteristics of canard airfoil (FX63-137) at Re number of 130,000.
Cl vs. α (Canard Airfoil)
1.8
1.6
1.4
1.2
1
Cl
0.8
0.6
0.4
0.2
0
-8
-6
-4
-2
-0.2
0
2
4
6
8
10
12
14
Angle of Attack
Figure 4-1 cl vs. α curve of airfoil FX63-137 at Re number of 130,000
As illustrated in Figure 4-1,canard airfoil has a zero lift angle of attack of -4.89 degree,
clmax of 1.69 at 10 degree, and the slope for linear part of curve is 0.107(1/degree).
74
Figure 4-2 shows that drag coefficient of FX63-137 is almost constant from 1 to 3 degree
of angle of attack, which is a perfect range for cruise angle of attack.
Cd vs. α (Canard Airfoil)
0.09
0.08
0.07
0.06
Cd
0.05
0.04
0.03
0.02
0.01
0
-8 -7 -6 -5 -4 -3 -2 -1 0
1
2 3 4 5 6
Angle of Attack
7
8
9 10 11 12 13 14
Figure 4-2 cd vs. α curve of airfoil FX63-137 at Re number of 130,000
Figure 4-3 shows aerodynamic efficiency (cl/cd) of the canard airfoil.
Cl/Cd vs. α (Canard Airfoil)
80
70
60
Cl/Cd
50
40
30
20
10
0
-8 -7 -6 -5 -4 -3 -2 -10
-1 0
1
2
3
4
5
6
7
8
9 10 11 12 13 14
Angle of Attack
Figure 4-3 cl/cd vs. α curve of airfoil FX63-137 at Re number of 130,000
75
Figure 4-4 shows that minimum drag coefficient happens when the lift coefficient is not
equal to zero. This figure shows the range in which drag coefficient is constant.
Cl vs. Cd (Canard Airfoil)
1.8
1.6
1.4
1.2
1
Cl
0.8
0.6
0.4
0.2
0
-0.2 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Cd
Figure 4-4 cl vs. cd curve of airfoil FX63-137 at Re number of 130,000
FX63-137 has a constant moment coefficient at lower angle of attacks, helpful for
stability, Figure 4-5.
Cm vs. α (Canard Airfoil)
0
-8 -7 -6 -5 -4 -3 -2 -0.02
-1 0
1
2
3
4
5
6
7
8
9 10 11 12 13 14
-0.04
-0.06
Cm
-0.08
-0.1
-0.12
-0.14
-0.16
-0.18
-0.2
Angle of Attack
Figure 4-5 cm vs. α curve of airfoil FX63-137 at Re number of 130,000
76
The airfoil SD7062 was chosen to be used for wing. Wing Re number is equal to 350,000
(based on assumed chord length for canard, 14.5”, and cruise velocity of 50 ft/s at
standard condition of atmosphere). Figures 4-6 thru 4-10 show the aerodynamic
characteristics of wing airfoil (SD7062) at Re number of 350,000.
Cl vs. α (Wing Airfoil)
1.8
1.6
1.4
1.2
1
Cl
0.8
0.6
0.4
0.2
0
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
14
16
-0.2
-0.4
Angle of Attack
Figure 4-6 cl vs. α curve of airfoil SD7062 at Re number of 350,000
As illustrated in Figure 4-6, wing airfoil has a zero lift angle of attack of -4.33 degree,
clmax of 1.51 at 13 degree, and the slope for linear part of curve is 0.103(1/degree).
77
Figure 4-7 shows that drag coefficient of SD7062 does not change linearly but it is low at
small angle of attack, from 1 to 3 degree of angle of attack, which is a perfect range for
cruise angle of attack.
Cd vs. α (Wing Airfoil)
0.035
0.03
0.025
Cd
0.02
0.015
0.01
0.005
0
-9 -8 -7 -6 -5 -4 -3 -2 -1 0
1 2 3 4 5
Angle of Attack
6
7
8
9 10 11 12 13 14 15
Figure 4-7 cd vs. α curve of airfoil SD7062 at Re number of 350,000
Figure 4-8 shows aerodynamic efficiency (cl/cd) of the wing airfoil.
Cl/Cd vs. α (Wing Airfoil)
100
80
Cl/Cd
60
40
20
0
-9 -8 -7 -6 -5 -4 -3 -2 -1 0
-20
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
Angle of Attack
Figure 4-8 cl/cd vs. α curve of airfoil SD7062 at Re number of 350,000
78
Figure 4-9 shows that minimum drag coefficient doesn’t happen at zero lift angle of
attack.
Cl
Cl vs. Cd (Wing Airfoil)
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-0.2 0
-0.4
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Cd
Figure 4-9 cl vs. cd curve of airfoil SD7062 at Re number of 350,000
Cm vs. α (Wing Airfoil)
0
-9 -8 -7 -6 -5 -4 -3 -2 -1 0
-0.02
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
Cm
-0.04
-0.06
-0.08
-0.1
-0.12
Angle of Attack
Figure 4-10 cm vs. α curve of airfoil SD7062 at Re number of 350,000
Moment coefficient for SD7062 increases at angle of attack of one. It gives us a nose up
impact for stability, shown in Figure 4-10.
79
4.2
Canard and Wing Airfoil Comparison
As illustrated in Figure 4-11, the selected airfoils meet all canard configuration
requirements. Wing airfoil has a less negative zero lift angle than canard airfoil.
Therefore, wing airfoil reaches its zero lift angle of attack before canard airfoil. On the
other hand, canard airfoil stalls at lower angle of attack than wing airfoil, canard airfoil
stalls first. Canard airfoil has higher lift coefficient than wing airfoil at any angle of
attack, and it has higher maximum lift coefficient also.
Cl vs. α (Canard & Wing Airfoils)
1.8
1.6
1.4
1.2
1
0.8
Wing
Cl
0.6
Canard
0.4
0.2
-10
-8
-6
-4
0
-2
0
-0.2
2
4
6
8
10
12
14
16
-0.4
-0.6
Angle of Attack
Figure 4-11 cl vs. α curves of airfoil FX63-137(Re=130,000), and SD7062 (Re=350,000)
80
Canard airfoil is a moderately cambered airfoil and that’s the reason for its higher lift
coefficient. For same reason it has a higher drag coefficient, as illustrated in Figure 4-12.
Despite having a higher lift coefficient, canard airfoil has a lower aerodynamic efficiency
(cl/cd) than wing airfoil due to having higher drag coefficient. It is shown in Figure 4-13.
Cd vs. α (Canard & Wing Airfoils)
0.1
0.09
0.08
0.07
Wing
Cd
0.06
0.05
Canard
0.04
0.03
0.02
0.01
0
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
14
16
Angle of Attack
Figure 4-12 cd vs. α curves of airfoil FX63-137(Re=130,000), and SD7062 (Re=350,000)
Cl/Cd
Cl/Cd vs. α (Canard & Wing Airfoils)
-10
-8
-6
-4
100
90
80
70
60
50
40
30
20
10
0
-10 0
-2
-20
Wing
Canard
2
4
6
8
10
12
14
16
Angle of Attack
Figure 4-13 cl/cd vs. α curves of FX63-137(Re=130,000), and SD7062 (Re=350,000)
81
Cm vs. α (Canard & Wing Airfoils)
-8
-6
0
-2 0
-0.05
-4
Cm
-10
2
4
6
8
10
12
14
16
Wing
-0.1
Canard
-0.15
-0.2
Angle of Attack
Figure 4-14 cm vs. α curves of airfoil FX63-137(Re=130,000), and SD7062 (Re=350,000)
Figure 4-14 shows that canard airfoil has a much higher moment coefficient, a proper
option for a tailless aircraft.
4.3
Canard 2D vs. 3D models
Airfoil has higher lift and drag coefficient than 3D model of wing. In Figure 4-15 the lift
coefficient for FX63-137, canard airfoil, and lift coefficient for 3D model of the canard
are compared. Zero lift angles are same, 3D model due to its aspect ratio has lower slope
and lower maximum lift coefficient. On the other hand, 3D model of canard has a higher
stall angle of attack than its airfoil.
Lift Coefficient
Lift Coefficient vs. α (Canard 2D vs. 3D models)
-8
-6
-4
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-0.2 0
-2
2D,Airfoil
3D,Wing
2
4
6
8
Angle of Attack
10
12
14
Figure 4-15 Lift coefficient vs. α- Airfoil FX63-137, and 3D canard (Re=130,000)
82
As illustrated in Figure 4-16, FX63-137, canard airfoil, 3D model of the canard has same
drag coefficient at zero lift angle of attack. Because the induced drag is zero at zero lift
angle of attack. As angle of attack increases, due to induced drag, 3D model of the canard
has higher drag coefficient than the airfoil. Due to having lower lift coefficient and higher
drag coefficient, 3D model of the canard has a much lower aerodynamic efficiency (c l/cd)
than canard airfoil, shown in Figure 4-17
Drag Coefficient vs. α (Canard 2D vs. 3D models)
0.16
0.14
0.12
Drag Coefficient
0.1
2D,Airfoil
0.08
0.06
0.04
3D,Wing
0.02
0
-8
-6
-4
-2
0
2
4
6
8
10
12
14
Angle of Attack
Figure 4-16 Drag coefficient vs. α- Airfoil FX63-137, and 3D canard (Re=130,000)
CL/CD
CL/CD vs. α (Canard 2D vs. 3D models)
-8
-6
-4
80
70
60
50
40
30
20
10
0
-2-10 0
2D,Airfoil
3D,Wing
2
4
6
8
Angle of Attack
10
12
14
Figure 4-17 E vs. α -Airfoil FX63-137, and 3D canard (Re=130,000)
83
4.4
Wing 2D vs. 3D models
In Figure 4-18, lift coefficient for SD7062, wing airfoil, and lift coefficient for 3D model
of the wing are compared. Like canard; zero lift angles are same, 3D model of wing has a
lower slope, lower maximum lift coefficient, and higher stall angle of attack than its
airfoil. It can be seen the difference between lift coefficients of wing airfoil and 3D
model of wing at the same angle of attack is smaller than the difference between lift
coefficients of canard airfoil and 3D model of canard at the same angle of attack. In our
design assumptions canard has a higher aspect ratio and it increases its slope.
Lift Coefficient vs. α (Wing 2D vs. 3D models)
1.8
1.6
1.4
1.2
Lift Coefficient
1
2D,Airfoil
0.8
3D,Wing
0.6
0.4
0.2
-10
-8
-6
-4
0
-2
0
-0.2
2
4
6
8
10
12
14
16
-0.4
Angle of Attack
Figure 4-18 Lift coefficient vs. α- Airfoil SD7062, and 3D wing (Re=350,000)
84
For same reason as discussed for canard and its airfoil, after zero lift angle of attack, by
increasing the angle of attack 3D model of the wing has higher drag coefficient than the
airfoil, illustrated in Figure 4-19.Also 3D model of the wing has a much lower
aerodynamic efficiency (cl/cd) than wing airfoil, shown in Figure 4-20.
Drag Coefficient vs. α (Wing 2D vs. 3D models)
0.14
Drag Coefficient
0.12
0.1
0.08
2D,Airfoil
0.06
3D,Wing
0.04
0.02
0
-10
-8
-6
-4
-2
0
2
4
6
Angle of Attack
8
10
12
14
16
Figure 4-19 Drag coefficient vs. α- Airfoil SD7062, and 3D wing (Re=350,000)
Lift Coefficient/Drag Coefficient vs. α (Wing 2D vs. 3D models)
100
Lift Coefficient/Drag Coefficient
90
80
70
60
2D,Airfoil
50
40
3D,Wing
30
20
10
-10
-8
-6
-4
0
-2-10 0
2
4
6
8
10
12
14
16
-20
Angle of Attack
Figure 4-20 E vs. α- Airfoil SD7062, and 3D wing (Re=350,000)
85
4.5
Aspect Ratio Impact on Lift Slope
The major parameter in finding the wing lift slope is wing aspect ratio (AR). As shown in
Figures 4-21 and 4-22 by increasing the AR, the lift slope rises. Two different method is
utilized to show the impact of AR on the lift slope:
1) Surface ratio remains same (S/SRef) and chord length is changed, Figure 4-21.
AR impact on Lift Slope (Same Surface Ratio, Diff. Chord)
1.6
1.4
1.2
AR=4.41
CL
1
0.8
AR=6
0.6
AR=11.52
0.4
0.2
0
-5
0
5
AOA
10
15
Figure 4-21 AR impact on the lift slope (same surface ratio, different chord length)
2) Chord length remains same and Surface ratio(S/SRef) is changed, Figure 4-22.
AR impact on Lift Slope (Same Chord, Diff. Surface Ratio)
1.6
1.4
1.2
AR=8.64
CL
1
0.8
AR=11.52
0.6
AR=14.40
0.4
0.2
0
-5
0
5
AOA
10
15
Figure 4-22 AR impact on the lift slope (same chord length, different surface ratio)
86
4.6
Wing and Canard Comparison
For next step 3D models of wing and canard for canard configuration requirements
should be checked. As illustrated in Figure 4-23, 3D models of wing and canard like the
selected airfoils meet all canard configuration requirements. Wing has a lower negative
angle of zero lift; therefore, wing arrives at its zero lift angle of attack before canard.
Also, canard stalls at a lower angle of attack and this satisfies the other requirement
which canard must stall first. Canard has higher lift coefficient than wing at any angle of
attack, and also it has higher maximum lift coefficient.
CL vs. α (Canard & Wing 3D Models)
1.8
1.6
1.4
1.2
CL
1.0
0.8
Wing
0.6
Canard
0.4
0.2
-8
-6
-4
0.0
-2
0
-0.2
2
4
6
8
10
12
14
-0.4
Angle of Attack
Figure 4-23 CL vs. α curves of canard (Re=130,000), and wing (Re=350,000)
All curves which are used to compare 3D models of canard and wing are plotted up to
canard stall angle of attack since the reference point for stall is stall angle of attack for
canard.
87
Canard has higher drag coefficient than wing, illustrated in Figure 4-24.
CD vs. α (Canard & Wing 3D Models)
0.2
0.1
0.1
CD
0.1
Wing
0.1
Canard
0.1
0.0
0.0
0.0
-8
-6
-4
-2
0
2
4
6
8
10
12
14
Angle of Attack
Figure 4-24 CD vs. α curves of canard (Re=130,000), and wing (Re=350,000)
CL/CD vs. α (Canard & Wing 3D Models)
25
20
15
CL/CD
10
Wing
Canard
5
0
-8
-6
-4
-2
0
2
4
6
8
10
12
14
-5
-10
Angle of Attack
Figure 4-25 CL/CD vs. α curves of canard (Re=130,000), and wing (Re=350,000)
As Figure 4-25 shows, at some angle of attacks that are proper for cruise flight, wing has
higher aerodynamic efficiency than canard. It proves that besides satisfying canard
88
requirements, a good cruise condition for the aircraft is provided by selecting proper
aspect ratios for wing and canard. Wing makes most of the lift and its surface is larger
than canard, it’s the reason that its higher aerodynamic efficiency is significantly helpful
for cruise flight.
4.7
Aircraft Lift and Drag Coefficients
Figure 4-26 illustrates lift coefficient of the aircraft, wing, and canard. The aircraft total
lift is sum of lift produced by wing and canard. Wing has a larger surface than canard;
therefore, wing makes larger portion of the total lift. It can be seen that the aircraft lift
curve is closer to the wing curve than the canard curve.
CL vs. AOA
1.6
1.4
Canard
CL
1.2
1.0
Wing
0.8
0.6
A/C
0.4
0.2
0.0
-10
-5
-0.2
0
5
10
15
-0.4
Angle of Attack
Figure 4-26 CL vs. α curves of canard, wing, and aircraft (V∞=50ft/s)
89
CD vs. AOA
0.16
0.14
CD
0.12
0.10
Canard
0.08
0.06
Wing
0.04
A/C
0.02
0.00
-10
-5
0
5
10
15
Angle of Attack
Figure 4-27 CD vs. α curves of canard, wing, and aircraft (V∞=50ft/s)
Drag coefficient of aircraft, wing, and canard is illustrated in Figure 4-27. Aircraft drag is
sum of wing drag, canard drag, and drag for other components (parasite drag). Like lift,
produced drag by wing and canard is based on how large is their portion from the total
surface (SW/SRef and SC/SRef).
90
4.8
Aircraft Drag Polar
Figure 4-28 illustrates the aircraft drag polar. It shows the relationship between aircraft
drag and lift coefficients. It can be seen that the aircraft minimum drag coefficient
doesn’t happen when aircraft lift is equal to zero. Also, the maximum aerodynamic
efficiency (Emax or (L/D) max) doesn’t happen at the minimum drag.
CL vs. CD
1.2
1.0
0.8
CL
0.6
0.4
0.2
0.0
-0.2 0.00
0.02
0.04
0.06
0.08
0.10
0.12
-0.4
CD
Figure 4-28 CL vs. CD curve of aircraft (V∞=50ft/s)
As it is shown in Figure 4-29 the maximum aerodynamic efficiency of the aircraft
doesn’t occur at the angle of attack in which drag coefficient is minimum.
Aero. Efficiency
EAircraft vs. AOA
-10
-5
14
12
10
8
6
4
2
0
-2 0
-4
-6
Angle of Attack
5
10
Figure 4-29 Aerodynamic efficiency of aircraft vs. α (V∞=50ft/s)
91
The aerodynamic efficiency reaches its maximum amount when CL/CD reaches its
maximum and it’s not only a function of drag coefficient. Therefore, as observed on
EAircraft vs. angle of attack curve, the AOA of 0 to 5 degree for cruise flight is a very
efficient range.
4.9
Aircraft Cruise Condition
The aircraft has an acceptable and efficient aerodynamic efficiency when the cruise flight
occurs at 0 to 5 degree of AOA; however, by changing the angle of attack, lift coefficient
changes. Besides cruise angle of attack and cruise lift coefficient, cruise velocity is
another significant parameter to be considered. When aircraft wing loading is constant,
velocity varies by lift coefficient. Figure 4-30 shows how lift coefficient varies by
velocity.
Velocity vs. Aircraft CL
1.0
0.9
0.8
Aircraft CL
0.7
0.6
0.5
0.4
0.3
0.2
30
35
40
45
50
55
60
Velocity(ft/s)
Figure 4-30 Aircraft lift coefficient vs. velocity at wing loading of 1.5
92
65
Drag coefficient varies by lift coefficient due to induced drag; consequently, by changing
the angle of attack, drag coefficient changes too. Also, drag coefficient varies by velocity,
illustrated in Figure 4-31.
Velocity vs. Aircraft CD
0.09
0.08
Aircraft CD
0.07
0.06
0.05
0.04
0.03
0.02
30
35
40
45
50
55
60
65
Velocity(ft/s)
Figure 4-31 Aircraft drag coefficient vs. velocity at wing loading of 1.5
Figure 4-32 shows how aircraft aerodynamic efficiency varies by velocity.
Velocity vs. Aerodynamic E
13.00
12.50
Aerodynamic E
12.00
11.50
11.00
10.50
10.00
30
35
40
45
50
55
60
65
70
Velocity(ft)
Figure 4-32 Aircraft aerodynamic efficiency vs. velocity at wing loading of 1.5
As a result, cruise velocity has impacts on cruise angle of attack, lift, drag, and
aerodynamic efficiency. The cruise velocity is one of the design assumptions. Therefore,
93
to fly at the assumed cruise velocity and reaching an optimum aerodynamic efficiency for
aircraft, the flight angle of attack during the cruise has to be optimized.
4.10 Cruise Optimization
The cruise flight is optimum when: 1) aircraft flies at its assumed cruise velocity, 2) at a
high aerodynamic efficiency ,and 3) cruise angle of attack of zero or very close to zero.
Utilizing the angle of incidence (angle of incidence is the angle between the chord, where
the wing or canard are mounted to the fuselage, and reference axis along the fuselage,
the longitudinal axis) would be very helpful in order to decrease the cruise AOA. In
Figure 4-33 some different conditions of wing and canard angle of incidence are
compared.
V cruise(ft/s)
Vcruise vs. AOA
-4
-3
-2
80
75
70
65
60
55
50
45
40
35
30
-1
0
Wi=0,Ci=0
Wi=0,Ci=2
Wi=1,Ci=2
Wi=2,Ci=3
1
2
3
4
Angle of Attack
5
6
7
8
9
Figure 4-33 VCruise vs. AOA at different angle of incidence for wing and canard
When no angle of incidence is used for wing and canard, cruise velocity of 50ft/s is
reached at AOA of about 2 degree. Flight at AOA of close to zero is the goal. Therefore,
to fly at AOA of zero, adding angle of incidence to wing and canard would be helpful.
94
Angle of attack vs. cruise velocity are compared for: angle of incidence of 2 degree for
canard and zero degree for wing, angle of incidence of 2 degree for canard and 1 degree
for wing, and angle of incidence of 3 degree for canard and 2 degree for wing.
4.11 Angle of Incidence Results
To fly at an angle of attack of zero degree or very close to zero degree for aircraft, angle
of incidence of 1 degree for wing and 2 degree for canard are added and analyzed.
Figures 4-34 thru 4-36 show how adding the angle of incidence affects canard
aerodynamic characteristic curves. (Same thing happens to wing curves). By adding the
angle of incidence to wing or canard, they will be able to produce same lift and drag in
less angle of attack. This decrement of AOA is equal to the angle of incidence. By adding
the angle of incidence, the angle of attack increases; therefore, canard and wing produce
same lift and drag at the less angle of attack.
Lift Coefficient
Canard CL with αi=2 vs. without αi
-10 -8
-6
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-4 -0.2
-2 0 2 4 6
Angle of Attack
With Incident
Angle
No Incident
Angle
8
10 12 14
Figure 4-34 Canard lift coefficient vs. AOA, with 0 and 2 degree angle of incidence
As illustrated in Figure 4-34, canard lift coefficient curve is shifted to left after adding 2
degree angle of incidence. For example, after adding 2 degree of angle of incidence, zero
95
lift AOA of canard from -4.89 shifts to -6.89. The important point is that the curve just
shifts to the left and curve’s slope doesn’t change and remains same. Consequently,
adding angle of incidence has no impact on the slope of the curve. Same shifting to the
left happens to the drag coefficient curve as illustrated in Figure 4-35.
Canard CD with αi=2 vs. without αi
0.16
0.14
0.12
With Incident
Angle
Drag Coefficient
0.10
0.08
No Incident
Angle
0.06
0.04
0.02
-10
-8
-6
-4
0.00
-2 0
2
4
6
Angle of Attack
8
10
12
14
Figure 4-35 Canard drag coefficient vs. AOA, with 0 and 2 degree angle of incidence
Canard CL/CD with αi=2 vs. without αi
20
15
CL/CD
10
With Incident
Angle
5
0
-10
-8
-6
-4
-2 0
-5
2
4
6
8
10
12
14
No Incident
Angle
Angle of Attack
Figure 4-36 Canard CL/CD vs. AOA, with 0 and 2 degree angle of incidence
Consequently, aerodynamic efficiency curve shifts to left like CL and CD, as illustrated in
Figure 4-36.
96
After comparing the aircraft lift coefficient curves before adding angle of incidence
(Figure4-26) and after adding them (Figure4-37), it can be seen that the lift coefficient at
zero degree of AOA is increased. Therefore, by adding angle of incidence now more lift
is produced at the same angle of attack for cruise.
CL vs. AOA
1.8
1.6
CL
1.4
1.2
1.0
Canard
0.8
0.6
Wing
0.4
0.2
A/C
0.0
-10
-5
0
5
10
15
-0.2
-0.4
Angle of Attack
Figure 4-37 CL vs. AOA for aircraft, wing, and canard, with angle of incidence
97
CD vs. AOA
0.16
0.14
CD
0.12
0.10
Canard
0.08
0.06
Wing
0.04
A/C
0.02
0.00
-10
-5
0
5
Angle of Attack
10
15
Figure 4-38 CD vs. AOA for aircraft, wing, and canard, with angle of incidence
After adding angle of incidence, drag coefficient at zero degree AOA increases due to lift
growing (comparison of Figure 4-27 and Figure 4-38), but this increment is smaller than
the lift coefficient increment. Therefore, the aerodynamic efficiency of zero degree AOA
increases (comparison of Figure 4-29 and Figure 4-39). Now by adding angle of
incidence, the design goals were achieved. Cruise velocity is 50ft/s at an AOA of very
close to zero degree with a better aerodynamic efficiency.
Aero. Efficiency
EAircraft vs. AOA
-10
-5
14
12
10
8
6
4
2
0
5
-2 0
-4
-6
Angle of Attack
10
15
Figure 4-39 Aircraft aerodynamic efficiency vs. AOA with angle of incidence
98
4.12 CG Location
To find a proper CG location, impacts of angle of incidence on the CG location and the
static margin will be analyzed. Figure 4-40 shows how different selected angle of
incidence affect the CG location. Proper location of the CG is the other goal that should
be achieved by choosing right angle of incidence. It can be observed that by increasing
the wing angle of incidence CG location comes closer to wing LE and it goes back (aft
CG). By in increasing the canard angle of incidence, CG location moves forward and it
becomes further than wing LE (front CG). The reason is creating more lift at same cruise
angle of attack. As lift increases, moment arm decreases.
CG Location vs. Angle Of Incidence
3.5
CG(inch) From Wing LE
3
2.5
2
1.5
1
0.5
0
0,0
0,2
1,2
1,4
Wi,Ci
Figure 4-40 CG location vs. different angle of incidence for wing and canard
99
4.13 Static Margin
Now finding the static margin is the last step to prove the pitch static stability of the
aircraft. To find the static margin, slopes of lift vs. angle of attack curve and moment vs.
angle of attack curve are required. Figure 4-41 shows the moment coefficient curve vs.
absolute angle of attack. Negative slope of this curve is a significant proof of a positive
SM and pitch stability of the aircraft.
Moment Coefficient vs. AOA
0.035
0.030
0.025
0.020
CM
0.015
0.010
0.005
0.000
0
1
2
3
4
Absolute AOA(degree)
5
6
7
Figure 4-41 Moment coefficient vs. absolute angle of attack
4.14 Optimization
To get an acceptable stall margin (difference between stall points of wing and canard) for
meeting the canard requirements, proper pitch stability for aircraft (SM), expected
position for center of gravity (CG), desired cruise flight angle of attack (AOA), and an
efficient cruise flight (E or CL/CD), different conditions are examined:
100
a) Different chord length for wing and canard with constant surface ratios of wing (85%)
and canard (15%); different wing and canard AR with constant surface areas.
Stall Margin vs. Canard Angle of Incidence
6
Stall Margin(degree)
5
Wc=13"
,Cc=4"
4
3
Wc=14.5"
,Cc=5"
2
1
Wc=15"
,Cc=6"
0
0
1
2
3
4
5
Canard Angle of Incidence(degree)
Figure 4-42 Stall margin vs. canard angle of incidence, same SW/SRef
In this situation, the surface of wing and canard are kept constant and their planform
shape change as their chords change. By increasing chords, the stall margin that is the
difference between stall points of wing and canard decreases, shown in Figure 4-42.
Because as chord increases, the slope of the lift curve decreases. It means that smaller
chords are more suitable for canard configuration. As illustrated, by adding a higher
angle of incidence to canard, stall margin increases. When chord of equal 15” for wing
and 6” for canard are chosen, with no angle of incidence, wing and canard will stall at
same angle and it won’t meet canard requirements. Therefore, by adding angle of
incidence, canard configuration stall requirement will meet. As shown, better stall margin
101
will be obtained when wing chord is equal to 13”, canard chord is equal to 4”, and canard
angle of incidence is equal to 4 degree. By this design, canard stall angle of attack is 5.4
degree smaller than wing stall angle of attack. The stall margin obtained by these design
selections appears a very good amount to meet the canard stall requirement. On the other
hand, there are also other design objectives that have to be examined. In next parts,
impacts of choosing different chords and different angle of incidence on other significant
design parameters will be discussed.
As Figure 4-43 illustrates, by decreasing chords and increasing angle of incidence, cruise
angle of attack goes to zero degree that would be a desired angle of attack. As chord
decreases, it raises lift curve slope; therefore, more lift will be produced at smaller AOA.
Also angle of incidence shifts the lift curve to the left toward smaller angle of attacks.
Cruise AOA(degree)
Cruise AOA vs. Canard Angle of Incidence
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
Wc=13"
,Cc=4
Wc=14.5"
,Cc=5
Wc=15"
,Cc=6
1
2
3
4
5
Canard Angle of Incidence (degree)
Figure 4-43 Cruise AOA vs. canard angle of incidence, same SW/SRef
102
Cruise E
Cruise E vs. Canard Angle of Incidence
12.6
12.4
12.2
12
11.8
11.6
11.4
11.2
11
10.8
Wc=13"
,Cc=4"
Wc=14.5"
,Cc=5"
Wc=15"
,Cc=6"
0
1
2
3
4
5
Canard Angle of Incidence(degree)
Figure 4-44 Cruise E vs. canard angle of incidence, same SW/SRef
Higher aerodynamic efficiency is acquired by a lower chord at cruise, same reason of
higher AR and higher lift slope, shown is Figure 4-44. Also, by increasing the angle of
incidence, aerodynamic efficiency decreases. Because by utilizing higher angle of
incidence, the required lift is created at lower angle of attack in which aerodynamic
efficiency becomes lower, shown in Figure 4-45 (Wc=15” and Cc=6”).
E vs. angle of attack
12
10
8
E
6
ic=2
4
ic=3
2
ic=4
0
-8
-6
-4
-2
-2
0
2
4
6
8
10
-4
Angle of atack(degree)
Figure 4-45 Cruise Aerodynamic efficiency in different angle of incidence
103
Due to decreasing the aerodynamic efficiency, the cruise required thrust increases
slightly, shown in Figure 4-46. This result is not considered beneficial, but it’s very small
and ignorable. Also as chords decrease, cruise flight becomes more effective by
decreasing the cruise required thrust.
Cruise T vs. Canard Angle of Incidence
1.4
Wc=13"
,Cc=4"
Cruise T
1.35
1.3
Wc=14.5"
,Cc=5"
1.25
1.2
1
2
3
4
5
Wc=15"
,Cc=6"
Canard Angle of Incidence(degree)
Figure 4-46 Cruise increment thrust vs. canard angle of incidence, same SW/SRef
As illustrated in Figure 4-47, as chords increase, the center of gravity location moves
backward, closer to wing leading edge. Therefore, back CG will be acquired by utilizing
the longer chord lengths. By using of longer chords, canard lift slope decreases more than
wing’s; consequently, wing lift decrease less than canard’s and wing needs smaller
CG from WLE(inch)
moment arm to achieve stability requirements and CG moves backward.
CG from WLE vs. Canard Angle of Incidence
6
5
4
3
2
1
0
Wc=13"
,Cc=4"
Wc=14.5"
,Cc=5"
1.5
2
2.5
3
3.5
4
Canard Angle of Incidence(degree)
4.5
Wc=15"
,Cc=6"
Figure 4-47 CG location vs. canard angle of incidence, same SW/SRef
104
Adding the angle of incidence for canard pushes the CG forward because it increases
canard lift. Now, canard needs smaller moment arm and this moves the CG forward,
further from wing leading edge.
Static margin is ratio of the lift curve slope to moment curve slope, and higher static
margin results in higher pitch stability. As Figure 4-48 illustrates, by using shorter
chords, static margin increases.
SM(%)
SM vs. Canard Angle of Incidence
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
-1 0
-2
-3
-4
-5
Wc=13"
,Cc=4"
Wc=14.5"
,Cc=5"
1
2
3
4
5
Wc=15"
,Cc=6"
Canard Angle of Incidence(degree)
Figure 4-48 Static margin vs. canard angle of incidence, same SW/SRef
By decreasing the chord lengths, moment curve slope increases more than the lift curve
slope, shown in Figure 4-49. Because by decreasing chord lengths, cruise angle of attack
becomes smaller and zero lift moment coefficient increases; consequently, the slope of
the curve increases.
105
Moment Coefficient vs. AOA
0.08
0.07
0.06
0.05
CM 0.04
0.03
0.02
0.01
0
Cw=13,Cc=4
Cw=14.5,Cc=5
0
2
4
6
8
Absolute AOA(degree)
Figure 4-49 Moment coefficient curve slope in different chord lengths
Analysis of the static margin is the last step of the project and is very crucial. In Figure 448, there are some points in which the static margin is negative. The goal of analysis is
aircraft pitch stability and it will be obtained by a positive static margin. Adding angle of
incidence increases the pitch stability.
b) Different surface ratios for wing and canard with constant chord lengths of wing
(14.5”) and canard (5”); different wing and canard AR with constant chord lengths:
By assuming different surface ratios for wing and canard, there will be a direct
correlation between AR’s of wing and canard. By increasing the surface ratio of wing,
surface of the canard decreases. Chords are assumed constant; consequently, wing AR
increases and canard AR decreases. In last section, there was not such a correlation. By
decreasing wing surface, canard surface increases and the stall margin increases, because
canard AR increases and wing AR decreases, shown in Figure 4-50. It means that smaller
surface for wing and bigger for canard is highly suitable for stall requirements of canard
configuration.
106
Stall Margin(degree)
Stall Margin vs. Canard Angle of Incidence
8
7
6
5
4
3
2
1
0
Sw.ratio
=0.85
Sw.ratio
=0.8
Sw.ratio
=0.75
0
1
2
3
4
5
Canard Angle of Incidence(degree)
Figure 4-50 Stall margin vs. canard angle of incidence, same chords
As Figure 4-51 illustrates, by decreasing wing surface and increasing angle of incidence,
cruise angle of attack goes to zero degree angle of attack.
Cruise AOA(degree)
Cruise AOA vs. Canard Angle of Incidence
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Sw.ratio
=0.85
Sw.ratio
=0.8
Sw.ratio
=0.75
0
1
2
3
4
5
Canard Angle of Incidence(degree)
Figure 4-51 Cruise AOA vs. canard angle of incidence, same chords
As wing surface decreases, canard surface increases and it results growing of aircraft lift
curve slope; therefore, more lift will be produced at smaller AOA. Angle of incidence
also shifts the curve to the left towards a smaller angle of attack. Higher aerodynamic
107
efficiency is acquired by a smaller wing surface at cruise, same reason of higher canard
AR and higher aircraft lift slope, shown is Figure 4-52.
Cruise E
Cruise E vs. Canard Angle of Incidence
11.66
11.64
11.62
11.60
11.58
11.56
11.54
11.52
11.50
11.48
11.46
Sw.ratio=
0.85
Sw.ratio=
0.8
0
1
2
3
4
5
Sw.ratio=
0.75
Canard Angle of Incidence(degree)
Figure 4-52 Cruise E vs. canard angle of incidence, same chords
At higher wing surface ratios (80% and 85%) by increasing the angle of incidence,
aerodynamic efficiency decreases, illustrated in Figure 4-52. The AOA in which required
lift is created decreases. When the cruise angle of attack is less than the AOA in which
Emax occurs, by decreasing the AOA, aerodynamic efficiency decreases. But
aerodynamic efficiency changes differently by utilizing the canard angle of incidence
when wing surface ratio is 75%. As Figure 4-52 shows, by increasing the angle of
incidence, aerodynamic efficiency first rises (for wing surface ratio of 75%) and then
decreases. It demonstrates that the cruise AOA initially is higher than the AOA in which
Emax occurs. By decreasing the AOA, aerodynamic efficiency first increases and
becomes closer to Emax (by increasing the canard angle of incidence from two to three
degree), and then decreases (by increasing the canard angle of incidence from three to
four).
108
As illustrated in Figure 4-53, as wing surface increases, the center of gravity location
moves backward, closer to wing leading edge. Therefore, back CG will be acquired by
utilizing the bigger wing surface ratio. Using the bigger wing, increases wing lift;
consequently, wing needs smaller moment arm to achieve stability requirements and CG
moves backward. Adding the angle of incidence for canard pushes the CG forward
because it increases canard lift. Now, canard needs smaller moment arm and this moves
the CG forward, further from wing leading edge.
CG from WLE vs. Canard Angle of Incidence
CG from WLE(inch)
12
10
Sw.rati
o=0.85
8
6
Sw.rati
o=0.8
4
2
Sw.rati
o=0.75
0
0
1
2
3
4
5
Canard Angle of Incidence(degree)
Figure 4-53 CG location vs. canard angle of incidence, same chords
SM vs. Canard Angle of Incidence
25
Sw.rati
o=0.85
SM(%)
20
15
Sw.rati
o=0.8
10
5
0
-5 0
1
2
3
4
5
Sw.rati
o=0.75
Canard Angle of Incidence(degree)
Figure 4-54 Static margin vs. canard angle of incidence, same chords
As Figure 4-54 illustrates, by using bigger wing surface, static margin decreases. It
proves using bigger canard is helpful to get a better pitch stability. Analysis of the static
109
margin is the last step of the project and is very crucial. In Figure 4-54, there are some
points in which the static margin is negative. The goal of analysis is aircraft pitch stability
and it will be obtained by a positive static margin. Adding angle of incidence increases
the pitch stability.
Figures 4-55 to 4-58 show the impact of wing angle of incidence in different aspect ratios
on stall margin, cruise angle of attack, CG location and static margin. The impact of
different aspect ratios is same as discussed for canard. As illustrated in Figure 4-55, when
wing angle of incidence increases, stall margin decrease. Because wing angle of
incidence shifts the lift curve to the left and to the smaller angle of attack.
Stall Margin vs. Wing Angle of Incidence
6
Stall Margin
5
ARw=7.24
,ARc=8.64
4
3
ARw=5.82
,ARc=8.64
2
1
ARw=5.44
,ARc=6
0
0
0.5
1
1.5
2
2.5
Wing Angle of Incidence (degree)
Figure 4-55 Stall margin vs. wing angle of incidence, different AR
110
Due to same reason, wing angle of incidence decreases the cruise angle of attack, as
shown in Figure 4-56.
Cruise AOA vs. Wing Angle of Incidence
1.5
ARw=7.24
,ARc=8.64
Cruise AOA
1
0.5
ARw=5.82
,ARc=8.64
0
0
0.5
1
1.5
2
2.5
ARw=5.44
,ARc=6
-0.5
-1
Wing Angle of Incidence(degree)
Figure 4-56 Cruise angle of attack vs. wing angle of incidence, different AR
As illustrated in Figure 4-57, CG location moves backward by using wing incidence
angle. Wing angle of incidence increases produced lift by wing; consequently, wing
moment arm decreases and CG becomes closer to wing leading edge, as opposed to
canard incidence angle impact.
CG from WLE(inch)
CG from WLE vs. Wing Angle of Incidence
5
4
ARw=7.24,
ARc=8.64
3
ARw=5.82,
ARc=8.64
2
1
ARw=5.44,
ARc=6
0
0
0.5
1
1.5
2
Wing Angle of Incidence(degree)
2.5
Figure 4-57 CG location vs. wing angle of incidence, different AR
111
By utilizing the wing incidence angle, CG moves back and it decreases the static margin,
as shown in Figure 4-58.
SM vs. Canard Angle of Incidence
14
12
10
ARw=7
.24,AR
c=8.64
SM(%)
8
6
4
ARw=5
.82,AR
c=8.64
2
0
-2 0
0.5
1
1.5
2
2.5
-4
-6
ARw=5
.44,AR
c=6
Canard Angle of Incidence(degree)
Figure 4-58 Static margin vs. wing angle of incidence, different AR
112
5.
Conclusion
In canard configuration, horizontal stabilizer is installed in the nose of aircraft instead of
the tail. In a conventional aircraft with a tailed horizontal stabilizer, the aircraft whole
required lift is made by wing. But in canard configuration aircraft, canard produces a
portion of required lift. Consequently, wing becomes smaller. Smaller wing makes less
lift; therefore, it produces less induced drag. Also smaller wing makes aircraft lighter. For
canard configuration there are two significant requirements: 1) Canard must stall first (the
benefit is high resistance against the stall and spin). 2) Wing must reach its zero lift angle
of attack before canard. These requirements gives canard a high resistance against stall
and spin. Based on canard requirements, selecting the appropriate airfoils for wing and
canard is very crucial. Aerodynamic characteristics of chosen airfoils (2D characteristics)
must be converted to 3D wing model characteristics in order to find the lift, drag and
moment of wing and canard. Aircraft lift is sum of produced lift by wing and canard
based on their surface portions from the total reference surface. Aircraft drag is sum of
wing drag, canard drag, and drag of other parts like landing gears, vertical stabilizer(s),
and fuselage. In a low Re number (less than 500,000), boundary layer of the flow is
laminar. Therefore, flow over wing, canard, and a vertical stabilizer(s) is laminar. Only
over the fuselage does the flow go to turbulent. To find the landing gear drag, blunt body
model is applied. After finding the aircraft lift and drag, next step is creating the drag
polar and aerodynamic efficiency curves of the aircraft. Aerodynamic efficiency (C L/CD)
curve shows at which angle of attack for cruise flight a better efficiency could be gained.
Utilizing the angle of incidence for wing and canard would be helpful to fly at an angle of
attack of zero or very close to zero. Also, adding angle of incidence to wing and canard is
113
helpful in order to move the CG location in order to gain the expected location. Moment
of aircraft at cruise angle of attack about its CG must be equal to zero to fly in a stable
equilibrium condition. After utilizing angle of incidence and applying proper surface area
and ARs for wing and canard, the expected location of CG would be acquired. CG
location has a significant impact on the pitch stability of the aircraft. Aircraft pitch
stability would be acquired when the center of lift of the aircraft (neutral point or aircraft
aerodynamic center) was behind the aircraft CG. Aircraft pitch stability causes the
aircraft to return to previous condition after any nose-down or nose-up disturbances.
Pitch stability of the aircraft is revealed by static margin which is found by slopes of CM
vs. AOA and CL vs. AOA curves of the aircraft. To achieve pitch stability, SM must be a
positive number and it shows how stable the aircraft is. SM of 5% to 15% for a canard
aircraft is a perfect range. Analysis to get the acceptable SM for the aircraft is the last
step to check the aircraft static pitch stability.
114
Bibliography
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Published by The Johns Hopkins University Press (April 7, 2004).
2. Introduction to Aircraft Performance, Selection and Design by Francis J Hale,
Published by Wiley & Sons, Incorporated, John (March 20, 1984).
3. Introduction to Flight by John D. Anderson, Published by McGraw-Hill Higher
Education (March 2004).
4. A History of Aerodynamics and Its Impact on Flying Machines by John D. Anderson,
Published by Cambridge University Press (1997).
5. Fundamentals of Aerodynamics by John D. Anderson, Published by McGraw-Hill
Higher Education (2010).
6. Burt Rutan: Aeronautical and Space Legend by Daniel Alef, Published by Titans of
Fortune Publishing (December 28, 2011).
7. Airfoil Design and Data by Richard Eppler, Published by Springer-Verlag (October
1990).
8. Airfoils at Low Speeds by Michael Selig, John Donovan, and David Fraser, Published
by SoarTech Publications (Jan 1, 1995).
9. Design of Aircraft by Thomas C. Corke, Published by Prentice Hall PTR (2002).
10. Profili Software (2011 Version).
11. Aircraft Design: A Conceptual Approach by Daniel P. Raymer, Published by AIAA
(July 1, 2012).
12. Aerodynamics, Aeronautics, and Flight Mechanics by Barnes W. McCormick,
Published by Wiley (September 13, 1994).
115
13. Aircraft Performance and Design by John D. Anderson, Published by McGraw-Hill
Science/Engineering/Math (December 5, 1998).
14. Fundamentals of Aircraft Design by Leland Nicolai, Published by Amer Inst of
Aeronautics (March 15, 2010).
15. Fluid Dynamic Drag by S. Hoerner, Published by Hoerner Fluid Dynamics (June 25,
1993).
16. R/C Model Aircraft Design by Andy Lennon, Published by Air Age (September
1996).
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Design Dimension Press (October 31, 2002).
116
Appendices
Appendix A. Excel Spreadsheets
Calculated Parameters
Canard
Canard Eff.
Planform Area
Canard Chord
Canard Chord
Canard Effective
Span
Canard Effective
Span
Canard Surface
Ratio
Canard Span
Efficiency Factor
0.15% Chord Back
From LE
6% Chord Back
From LE
LE Sharpness
Parameter
AOA Increment
Area
Chor
d
Chor
d
1.5
0.4
2
ft^2
5.0
inch
Span
3.6
ft
Span
Sc/Sr
ef
43
0.1
5
0.9
5
0.0
39
0.0
3
0.0
63
0.0
33
inch
e
K
0.15
%c
6%c
∆y-c
∆αCL
max 1.3
Wing Planform
Area
ft
Wing Span
Wing Span
Efficiency Factor
7
0.9
e
5
0.0
K
58
0.15% Chord Back 0.15 0.0
From LE
%c
3
6% Chord Back
0.0
From LE
6%c 63
LE Sharpness
0.0
Parameter
∆y-c 33
∆αCL 1.3
AOA Increment
max
5
%
Cho
rd%
Cho
rd%
Cho
rd%
degr
ee
Must be less
-1 than
0.
Sref. 93
m^2
∆i
Reference
Planform Area
Wing
8.5
Area
0
ft^2
117
Span
Stall
Margin
Zero Lift
Margin
1.1
9
0.5
6
ft
Cho
rd%
Cho
rd%
Cho
rd%
degr
ee
degr
ee
degr
ee
Canard Lift Curve Slope
Name
Canard Eff. Span
Canard Chord
Canard Eff. Aspect Ratio
Canard Eff. Planform Area
Canard 2D Slope
Canard 2D Slope
Mach Number
Mach Number Effect
Airfoil Efficiency
Canard Sweep Back Angle
Fuselage Width
Fuselage Planform Area
Canard Exposed to Reference Area Ratio
Fuselage Lift Factor
Canard 3D Slope
Slope of 3d Canard W/O fuselage
Value
b
1.10
c
0.13
AR
8.64
Sref
0.14
clα
0.107
clα
6.11
M
0
β
1
Ƞ
0.97
Λ
0
d
0.14
S
0.018
Sex/Spf
1
F
1.09
CLα
5.32
CLα
0.093
CLα
4.94
CLα
0.086
CLα/Clαo 1.08
118
Unit
m
m
m^2
1/degree 0.110 ideal
1/Radian
2π
ideal
Radian
m
m^2
m^2
1/Radian
1/degree
1/Radian
1/degree
Name
Canard 2D max. Lift Coefficient
clmax
Canard 3D max. Lift Coefficient
CLmax
Canard 2D Zero Lift AOA
α0L
Canard 3D Zero Lift AOA
α0L
0.15% Chord Back From LE
0.15%c
6% Chord Back From LE
6%c
Leading Edge Sharpness Parameter
∆y-c
Angle of Attack Increment
∆αCLmax
Canard 3D Stall Angle
αmax
αmax-∆αCLmax
αmax
αmax-∆αCLmax
Wing 3D Zero Lift AOA
α0L
Wing 3D Stall Angle
Stall Margin
Zero Lift Margin
119
Value
Unit
1.69
1.52
-4.89 degree
-4.89 degree
0.03 Chord%
0.063 Chord%
0.033 Chord%
1.3
degree
12.75 degree
11.45 degree
13.94
12.59
-4.33
degree
1.19
0.56
degree
degree
degree
Wing Lift Curve Slope
Name
Wing Span
Wing Chord
Wing Aspect Ratio
Wing Planform Area
Wing 2D Slope
Value
Unit
b
2.14
m
c
0.37
m
AR
5.82
Sref
0.79
m^2
clα
0.103 1/degree 0.11 ideal
clα
5.89 1/Radian 2π ideal
Mach Number
M
0
Mach Number Effect
β
1
Airfoil Efficiency
Ƞ
0.94
Wing Sweep Back Angle
Λ
0
Radian
Fuselage Diameter
d
0.14
m
Fuselage Planform Area
S
0.051
m^2
Wing Exposed to Reference Area Ratio Sex/Spf
1
m^2
Fuselage Lift Factor
F
1.07
Wing 3D Slope
CLα
4.61 1/Radian
CLα
0.08 1/degree
Slope of 3d Wing W/O fuselage
CLα
4.40 1/Radian
CLα
0.08 1/degree
CLα/Clαo 1.05
Name
Wing 2D max. Lift Coefficient
Wing 3D max. Lift Coefficient
Wing 2D Zero Lift AOA
Wing 3D Zero Lift AOA
0.15% Chord Back From LE
6% Chord Back From LE
Leading Edge Sharpness Parameter
Angle of Attack Increment
Wing 3D Stall Angle
Value
Unit
clmax
1.51
CLmax
1.36
α0L
-4.33 degree
α0L
-4.33 degree
0.15%c
0.03 Chord%
6%c
0.063 Chord%
∆y-c
0.033 Chord%
∆αCLmax
1.35
degree
αmax
13.94 degree
αmax-∆αCLmax 12.59
120
Calculated Area
Canard
2D Slope
clα
2D clmax
clmax
2D Zero
Lift AOA
3D Stall
Angle
α0L
αmax
αmax∆αCLmax
Wing
0.1
07
1.6
9
4.8
9
12.
75
11.
45
1/degre
e
at 10
degree
2D Slope
clα
2D clmax
clmax
2D Zero
Lift AOA
3D Stall
Angle
degree
degree
degree
α0L
αmax
αmax∆αCLmax
Air Kinematic Viscousity(20°C) ν 1.51E-05 m^2/s
Cruise Velocity
V
15.24
m/s
Canard Chord
c
0.127
m
Canard Reynolds Number
Re 128149.8
Wing Chord
c
0.37
m
Wing Reynolds Number Re 371634 -
121
0.1
03
1.5
1
4.3
3
13.
94
12.
59
1/degre
e
at 13
degree
degree
degree
Canard Airfoil Aerodynamic Characteristics
FX63-137 at 130000 Re
Alfa
Cl
Cd
Cl/Cd
Cm
-6.5
-0.086
0.081
-1.064
-0.0902
-5
-0.025
0.044
-0.5629
-0.1387
-4.5
0.0915
0.032
2.8864
-0.1569
-4
0.185
0.029
6.446
-0.1646
-2.5
0.4688
0.02
23.094
-0.188
-2
0.5271
0.021
25.341
-0.1864
-1
0.6373
0.022
29.505
-0.1837
-0.5
0.702
0.021
33.113
-0.1842
0
0.7836
0.021
37.493
-0.1885
0.5
0.8183
0.021
38.238
-0.1847
1
0.8903
0.021
42.598
-0.1869
1.5
0.9405
0.021
44.155
-0.186
2
0.9976
0.021
47.28
-0.1856
2.5
1.0552
0.021
50.01
-0.1854
3
1.1035
0.021
52.299
-0.1837
3.5
1.1655
0.021
55.766
-0.1841
4
1.2035
0.021
57.31
-0.1805
4.5
1.253
0.021
60.825
-0.1784
5
1.3015
0.021
62.874
-0.1774
5.5
1.3731
0.021
65.386
-0.181
6
1.4422
0.021
68.676
-0.1834
6.5
1.4898
0.021
70.607
-0.1817
7
1.5338
0.022
71.009
-0.1795
7.5
1.5761
0.022
70.996
-0.1771
8
1.6141
0.023
70.178
-0.174
8.5
1.6472
0.024
68.633
-0.1701
9
1.6693
0.025
66.242
-0.1645
9.5
1.6821
0.027
62.765
-0.1577
10
1.6859
0.029
58.135
-0.1503
122
11
1.682
0.032
52.563
-0.1429
11
1.6697
0.036
45.997
-0.136
12
1.6522
0.042
39.338
-0.1301
12
1.6457
0.048
34.501
-0.1261
13
1.653
0.053
31.426
-0.1235
13
1.6717
0.057
29.226
-0.1214
123
Wing Airfoil Aerodynamic Characteristics
SD7062 (14%)' at 350000 Re
Alfa
Cl
Cd
Cl/Cd
Cm
-7.5
-0.304
0.025
-11.949
-0.0973
-7
-0.261
0.023
-11.461
-0.0952
-6.5
-0.217
0.02
-10.86
-0.0934
-6
-0.172
0.017
-10.287
-0.0919
-5.5
-0.122
0.015
-8.0329
-0.0905
-5
-0.07
0.014
-4.9296
-0.0896
-4
0.035
0.012
2.8455
-0.0877
-3.5
0.0888
0.012
7.7217
-0.0869
-3
0.1422
0.011
13.5429
-0.0866
-2.5
0.1964
0.01
19.64
-0.0862
-2
0.2505
0.01
26.3684
-0.0858
-1.5
0.3047
0.009
33.1196
-0.0851
-1
0.3591
0.009
39.4615
-0.0843
-0.5
0.4131
0.009
45.9
-0.0835
0
0.4656
0.009
51.7333
-0.0822
0.5
0.5164
0.009
57.3778
-0.0804
1
0.5626
0.009
62.5111
-0.0776
1.5
0.6291
0.009
69.9
-0.0791
2
0.6909
0.009
74.2903
-0.0805
2.5
0.7431
0.01
77.4062
-0.0798
3
0.7965
0.01
80.4545
-0.0793
3.5
0.8483
0.01
82.3592
-0.0786
4
0.9009
0.011
84.9906
-0.078
4.5
0.9519
0.011
85.7568
-0.0771
5
1.0021
0.012
87.1391
-0.0762
5.5
1.0523
0.012
87.6917
-0.0753
6
1.0988
0.013
87.2063
-0.0738
6.5
1.148
0.013
87.6336
-0.0728
7
1.1909
0.014
85.6763
-0.0709
124
7.5
1.2372
0.014
86.5175
-0.0694
8
1.2752
0.015
83.8947
-0.0668
8.5
1.3174
0.016
83.9108
-0.0647
9
1.3446
0.017
80.515
-0.0603
9.5
1.3771
0.017
79.1437
-0.0569
10
1.4031
0.019
75.8432
-0.0527
11
1.4253
0.02
71.9848
-0.0485
11
1.4522
0.021
69.1524
-0.0453
12
1.4668
0.023
64.0524
-0.0411
12
1.4891
0.025
60.5325
-0.0381
13
1.5074
0.027
56.4569
-0.0353
13
1.5129
0.03
50.5987
-0.032
125
Canard Aerodynamic Characteristics with angle of incidence
Canard
AOA
-6.00
-5.50
-5.00
-4.89
-4.50
-4.33
-4.00
-3.50
-3.00
-2.50
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
5.50
6.00
6.50
7.00
7.50
8.00
8.50
9.00
9.50
10.00
CL &cd vs. α
CL
cd
-0.10 0.069
-0.06 0.056
-0.01 0.044
0.00 0.041
0.04 0.032
0.05 0.031
0.08 0.029
0.13 0.026
0.18 0.023
0.22 0.020
0.27 0.021
0.32 0.021
0.36 0.022
0.41 0.021
0.45 0.021
0.50 0.021
0.55 0.021
0.59 0.021
0.64 0.021
0.69 0.021
0.73 0.021
0.78 0.021
0.83 0.021
0.87 0.021
0.92 0.021
0.96 0.021
1.01 0.021
1.06 0.021
1.10 0.022
1.15 0.022
1.20 0.023
1.24 0.024
1.29 0.025
1.34 0.027
1.38 0.029
Canard(With angle of incidence)
A/C AOA CL KCL^2
cd
CD CL/CD
-8.00
-0.10 0.0004 0.069 0.069 -1.49
-7.50
-0.06 0.0001 0.056 0.056 -1.00
-7.00
-0.01 0.0000 0.044 0.044 -0.22
-6.89
0.00 0.0000 0.041 0.041
0.00
-6.50
0.04 0.0001 0.032 0.032
1.15
-6.33
0.05 0.0001 0.031 0.031
1.69
-6.00
0.08 0.0003 0.029 0.029
2.86
-5.50
0.13 0.0006 0.026 0.027
4.87
-5.00
0.18 0.0012 0.023 0.024
7.23
-4.50
0.22 0.0019 0.020 0.022 10.00
-4.00
0.27 0.0028 0.021 0.024 11.38
-3.50
0.32 0.0038 0.021 0.025 12.58
-3.00
0.36 0.0051 0.022 0.027 13.55
-2.50
0.41 0.0064 0.021 0.028 14.75
-2.00
0.45 0.0080 0.021 0.029 15.72
-1.50
0.50 0.0097 0.021 0.031 16.09
-1.00
0.55 0.0116 0.021 0.033 16.83
-0.50
0.59 0.0137 0.021 0.035 16.98
0.00
0.64 0.0159 0.021 0.037 17.30
0.50
0.69 0.0183 0.021 0.039 17.43
1.00
0.73 0.0208 0.021 0.042 17.48
1.50
0.78 0.0235 0.021 0.044 17.53
2.00
0.83 0.0264 0.021 0.047 17.41
2.50
0.87 0.0295 0.021 0.050 17.41
3.00
0.92 0.0327 0.021 0.053 17.20
3.50
0.96 0.0361 0.021 0.057 16.90
4.00
1.01 0.0396 0.021 0.061 16.67
4.50
1.06 0.0434 0.021 0.064 16.40
5.00
1.10 0.0473 0.022 0.069 16.03
5.50
1.15 0.0513 0.022 0.074 15.65
6.00
1.20 0.0555 0.023 0.079 15.24
6.50
1.24 0.0599 0.024 0.084 14.81
7.00
1.29 0.0645 0.025 0.090 14.38
7.50
1.34 0.0692 0.027 0.096 13.91
8.00
1.38 0.0741 0.029 0.103 13.41
126
10.50
11.00
11.45
12.00
12.59
12.75
1.43
1.48
1.52
1.52
1.52
1.52
0.032
0.036
0.041
0.048
0.053
0.055
8.50
9.00
9.45
10.00
10.59
10.75
1.43
1.48
1.52
1.52
1.52
1.52
127
0.0792
0.0844
0.0893
0.0893
0.0893
0.0893
0.032
0.036
0.041
0.048
0.053
0.055
0.111
0.121
0.131
0.137
0.143
0.144
12.85
12.22
11.60
11.08
10.63
10.52
Wing Aerodynamic Characteristics with angle of incidence
AOA
-6.00
-5.50
-5.00
-4.89
-4.50
-4.33
-4.00
-3.50
-3.00
-2.50
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
5.50
6.00
6.50
7.00
7.50
8.00
8.50
9.00
9.50
10.00
10.50
Wing
CL
-0.13
-0.09
-0.05
-0.05
-0.01
0.00
0.03
0.07
0.11
0.15
0.19
0.23
0.27
0.31
0.35
0.39
0.43
0.47
0.51
0.55
0.59
0.63
0.67
0.71
0.75
0.79
0.83
0.87
0.91
0.95
0.99
1.03
1.07
1.11
1.15
1.19
CL vs. α(No i)
cd
CD CL/CD
0.017 0.018
-7.6
0.015 0.016
-6.0
0.014 0.014
-3.7
0.014 0.014
-3.2
0.013 0.013
-1.0
0.013 0.013
0.0
0.012 0.012
2.2
0.012 0.012
5.7
0.011 0.011
9.6
0.010 0.011
13.1
0.010 0.012
16.3
0.009 0.012
18.7
0.009 0.013
20.3
0.009 0.014
21.3
0.009 0.016
21.8
0.009 0.018
22.0
0.009 0.020
21.9
0.009 0.022
21.7
0.009 0.024
21.0
0.010 0.027
20.4
0.010 0.030
19.7
0.010 0.033
19.0
0.011 0.036
18.4
0.011 0.040
17.7
0.012 0.044
17.1
0.012 0.048
16.5
0.013 0.052
15.9
0.013 0.057
15.3
0.014 0.062
14.8
0.014 0.066
14.3
0.015 0.072
13.8
0.016 0.077
13.4
0.017 0.083
12.9
0.017 0.089
12.5
0.019 0.095
12.1
0.020 0.102
11.7
128
Wing CL vs. α(Delta i)
AOA
CL
cd
-7.00 -0.21 0.023
-6.50 -0.17 0.020
-6.00 -0.13 0.017
-5.89 -0.13 0.016
-5.50 -0.09 0.015
-5.33 -0.08 0.015
-5.00 -0.05 0.014
-4.50 -0.01 0.013
-4.00 0.03 0.012
-3.50 0.07 0.012
-3.00 0.11 0.011
-2.50 0.15 0.010
-2.00 0.19 0.010
-1.50 0.23 0.009
-1.00 0.27 0.009
-0.50 0.31 0.009
0.00
0.35 0.009
0.50
0.39 0.009
1.00
0.43 0.009
1.50
0.47 0.009
2.00
0.51 0.009
2.50
0.55 0.010
3.00
0.59 0.010
3.50
0.63 0.010
4.00
0.67 0.011
4.50
0.71 0.011
5.00
0.75 0.012
5.50
0.79 0.012
6.00
0.83 0.013
6.50
0.87 0.013
7.00
0.91 0.014
7.50
0.95 0.014
8.00
0.99 0.015
8.50
1.03 0.016
9.00
1.07 0.017
9.50
1.11 0.017
11.00
11.45
12.00
12.59
12.75
1.23
1.27
1.31
1.36
1.36
0.021
0.023
0.025
0.027
0.028
0.109
0.116
0.124
0.134
0.135
11.4
11.0
10.6
10.2
10.1
10.00
10.45
11.00
11.59
11.75
1.15
1.19
1.23
1.28
1.29
Wing(With angle of incidence)
A/C AOA CL KCL^2
cd
CD CL/CD
-8.00
-0.21 0.003
0.023 0.025 -8.43
-7.50
-0.17 0.002
0.020 0.022 -8.01
-7.00
-0.13 0.001
0.017 0.018 -7.56
-6.89
-0.13 0.001
0.016 0.017 -7.26
-6.50
-0.09 0.001
0.015 0.016 -5.98
-6.33
-0.08 0.000
0.015 0.015 -5.28
-6.00
-0.05 0.000
0.014 0.014 -3.73
-5.50
-0.01 0.000
0.013 0.013 -1.01
-5.00
0.03 0.000
0.012 0.012
2.17
-4.50
0.07 0.000
0.012 0.012
5.70
-4.00
0.11 0.001
0.011 0.011
9.61
-3.50
0.15 0.001
0.010 0.011 13.11
-3.00
0.19 0.002
0.010 0.012 16.28
-2.50
0.23 0.003
0.009 0.012 18.70
-2.00
0.27 0.004
0.009 0.013 20.26
-1.50
0.31 0.005
0.009 0.014 21.31
-1.00
0.35 0.007
0.009 0.016 21.80
-0.50
0.39 0.009
0.009 0.018 21.97
0.00
0.43 0.011
0.009 0.020 21.90
0.50
0.47 0.013
0.009 0.022 21.65
1.00
0.51 0.015
0.009 0.024 21.02
1.50
0.55 0.017
0.010 0.027 20.37
2.00
0.59 0.020
0.010 0.030 19.71
2.50
0.63 0.023
0.010 0.033 19.01
3.00
0.67 0.026
0.011 0.036 18.38
3.50
0.71 0.029
0.011 0.040 17.69
4.00
0.75 0.032
0.012 0.044 17.09
4.50
0.79 0.036
0.012 0.048 16.47
5.00
0.83 0.040
0.013 0.052 15.87
5.50
0.87 0.044
0.013 0.057 15.34
6.00
0.91 0.048
0.014 0.062 14.77
129
0.019
0.020
0.021
0.023
0.024
6.50
7.00
7.50
8.00
8.50
9.00
9.45
10.00
10.59
10.75
0.95
0.99
1.03
1.07
1.11
1.15
1.19
1.23
1.28
1.29
0.052
0.057
0.061
0.066
0.071
0.077
0.081
0.088
0.094
0.096
130
0.014
0.015
0.016
0.017
0.017
0.019
0.020
0.021
0.023
0.0238
0.066
0.072
0.077
0.083
0.089
0.095
0.101
0.109
0.118
0.120
14.32
13.81
13.40
12.94
12.55
12.13
11.76
11.36
10.89
10.77
Non-Aerodynamic Components Drag Coefficients
Min. Drag's
Vertical Stabilizer
Flat Plate Form Factor
1
Mean Chord
0.5
ft
Mean Chord
0.152 m
Re
153780 cf
0.0034 Swet/Sref
0.1
Ƞ
0.9
H-Tail Interference
Q
1.08
Hinge Leakage
l
1
CDmin
0.00033 Landing Gears
Wheel Frontal Area
Swheel
Strut Frontal Area
Sstrut
Blunt Object
CDo
# of Back LG's
n
Open Wheel Interference
Ƞ
Wheel
CDmin
Strut
CDmin
Back Gears
CDmin
Nose Gear
CDmin
Fuselage
Transition Re
500000
Turbulant Re
1230238 Form Factor
1.11
Interference Factor
1
Transition Location
0.496
ft
Laminar cf
0.0019
Turbulant cf
0.0045
Total cf
0.0064
Swet/Sref
0.51
Ƞ
1
CDmin
0.0036
Total Cdmin of Non-Aerodynamic Parts
CDmin
0.0161
-
131
0.0097 ft^2
0.0174 ft^2
1.01
2
1.5
0.99
0.0029
0.0052
0.0081
0.0041
Aircraft Drag Polar
Aircraft
A/C AOA Canard AOA Canard CL Canard CD
-8.00
-6.00
-0.10
0.069
-7.50
-5.50
-0.06
0.056
-7.00
-5.00
-0.01
0.044
-6.89
-4.89
0.00
0.041
-6.50
-4.50
0.04
0.032
-6.33
-4.33
0.05
0.031
-6.00
-4.00
0.08
0.029
-5.50
-3.50
0.13
0.027
-5.00
-3.00
0.18
0.024
-4.50
-2.50
0.22
0.022
-4.00
-2.00
0.27
0.024
-3.50
-1.50
0.32
0.025
-3.00
-1.00
0.36
0.027
-2.50
-0.50
0.41
0.028
-2.00
0.00
0.45
0.029
-1.50
0.50
0.50
0.031
-1.00
1.00
0.55
0.033
-0.50
1.50
0.59
0.035
0.00
2.00
0.64
0.037
0.50
2.50
0.69
0.039
1.00
3.00
0.73
0.042
1.50
3.50
0.78
0.044
2.00
4.00
0.83
0.047
2.50
4.50
0.87
0.050
3.00
5.00
0.92
0.053
3.50
5.50
0.96
0.057
4.00
6.00
1.01
0.061
4.50
6.50
1.06
0.064
5.00
7.00
1.10
0.069
5.50
7.50
1.15
0.074
6.00
8.00
1.20
0.079
6.50
8.50
1.24
0.084
7.00
9.00
1.29
0.090
7.50
9.50
1.34
0.096
8.00
10.00
1.38
0.103
8.50
10.50
1.43
0.111
132
9.00
9.45
10.00
10.59
10.75
11.00
11.45
12.00
12.59
12.75
1.48
1.52
1.52
1.52
1.52
0.121
0.131
0.137
0.143
0.144
Aircraft
Wing AOA Wing CL Wing CD A/C CL A/C CD A/C E
-7.00
-0.21
0.025
-0.20
0.048
-4.09
-6.50
-0.17
0.022
-0.16
0.043
-3.62
-6.00
-0.13
0.018
-0.11
0.038
-3.04
-5.89
-0.13
0.017
-0.11
0.037
-2.87
-5.50
-0.09
0.016
-0.07
0.034
-2.15
-5.33
-0.08
0.015
-0.06
0.034
-1.78
-5.00
-0.05
0.014
-0.03
0.033
-1.00
-4.50
-0.01
0.013
0.01
0.031
0.26
-4.00
0.03
0.012
0.05
0.030
1.62
-3.50
0.07
0.012
0.09
0.029
3.06
-3.00
0.11
0.011
0.13
0.029
4.49
-2.50
0.15
0.011
0.17
0.029
5.84
-2.00
0.19
0.012
0.21
0.030
7.11
-1.50
0.23
0.012
0.25
0.031
8.28
-1.00
0.27
0.013
0.29
0.032
9.29
-0.50
0.31
0.014
0.33
0.033
10.14
0.00
0.35
0.016
0.38
0.034
10.89
0.50
0.39
0.018
0.42
0.036
11.47
1.00
0.43
0.020
0.46
0.038
11.97
1.50
0.47
0.022
0.50
0.040
12.36
2.00
0.51
0.024
0.54
0.043
12.58
2.50
0.55
0.027
0.58
0.046
12.73
3.00
0.59
0.030
0.62
0.048
12.80
3.50
0.63
0.033
0.66
0.052
12.82
4.00
0.67
0.036
0.70
0.055
12.80
4.50
0.71
0.040
0.74
0.059
12.70
5.00
0.75
0.044
0.78
0.062
12.59
5.50
0.79
0.048
0.82
0.066
12.45
6.00
0.83
0.052
0.87
0.071
12.26
6.50
0.87
0.057
0.91
0.075
12.08
7.00
0.91
0.062
0.95
0.080
11.85
133
7.50
8.00
8.50
9.00
9.50
10.00
10.45
11.00
11.59
11.75
0.95
0.99
1.03
1.07
1.11
1.15
1.19
1.23
1.28
1.29
0.066
0.072
0.077
0.083
0.089
0.095
0.101
0.109
0.118
0.120
134
0.99
1.03
1.07
1.11
1.15
1.19
1.23
1.27
1.31
1.32
0.085
0.090
0.095
0.101
0.107
0.114
0.121
0.128
0.137
0.139
11.67
11.42
11.21
10.95
10.71
10.43
10.17
9.88
9.56
9.48
Cruise Condition
A/C
AOA
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
Canard
CL
0.45
0.50
0.55
0.59
0.64
0.69
0.73
0.78
0.83
0.87
0.92
0.96
1.01
1.06
1.10
1.15
Optimal Flight Area
Canard
Wing
Wing
CD
CL
CD
0.029
0.27
0.013
0.031
0.31
0.014
0.033
0.35
0.016
0.035
0.39
0.018
0.037
0.43
0.020
0.039
0.47
0.022
0.042
0.51
0.024
0.044
0.55
0.027
0.047
0.59
0.030
0.050
0.63
0.033
0.053
0.67
0.036
0.057
0.71
0.040
0.061
0.75
0.044
0.064
0.79
0.048
0.069
0.83
0.052
0.074
0.87
0.057
Lift Vcruise Thrust
lbf
ft/s
lbf
15
65.6
1.61
15
61.5
1.48
15
58.1
1.38
15
55.1
1.31
15
52.6
1.25
15
50.4
1.21
15
48.5
1.19
15
46.7
1.18
15
45.2
1.17
15
43.8
1.17
15
42.5
1.17
15
41.3
1.18
15
40.2
1.19
15
39.2
1.20
135
A/C
CL
0.29
0.33
0.38
0.42
0.46
0.50
0.54
0.58
0.62
0.66
0.70
0.74
0.78
0.82
0.87
0.91
A/C
CD
0.032
0.033
0.034
0.036
0.038
0.040
0.043
0.046
0.048
0.052
0.055
0.059
0.062
0.066
0.071
0.075
A/C
E
9.29
10.14
10.89
11.47
11.97
12.36
12.58
12.73
12.80
12.82
12.80
12.70
12.59
12.45
12.26
12.08
MaxE
12.82
Cruise Optimization
Wi=0,Ci=0
A/C AOA A/C CL A/C CD A/C E V cruise
degree
ft/s
0.0
0.36
0.034
10.65
59.18
0.5
0.40
0.036
11.27
56.09
1.0
0.44
0.038
11.82
53.45
1.5
0.48
0.040
12.21
51.14
2.0
0.52
0.042
12.47
49.12
2.5
0.57
0.045
12.64
47.31
3.0
0.61
0.048
12.73
45.69
3.5
0.65
0.051
12.76
44.23
4.0
0.69
0.054
12.75
42.89
4.5
0.73
0.057
12.69
41.68
5.0
0.77
0.061
12.59
40.56
5.5
0.81
0.065
12.45
39.52
6.0
0.85
0.069
12.28
38.56
6.5
0.89
0.074
12.12
37.67
7.0
0.93
0.078
11.89
36.84
7.5
0.97
0.083
11.72
36.06
Wi=0,Ci=2
A/C AOA A/C CL A/C CD A/C E V cruise
degree
ft/s
-2.0
0.23
0.030
7.49
74.82
-1.5
0.27
0.031
8.59
68.87
-1.0
0.31
0.032
9.57
64.14
-0.5
0.35
0.034
10.38
60.27
0.0
0.39
0.035
11.08
57.02
0.5
0.43
0.037
11.64
54.25
1.0
0.47
0.039
12.10
51.84
1.5
0.51
0.041
12.47
49.73
2.0
0.55
0.044
12.66
47.86
2.5
0.59
0.046
12.80
46.19
3.0
0.63
0.049
12.85
44.68
3.5
0.67
0.053
12.83
43.30
4.0
0.72
0.056
12.80
42.05
4.5
76.00
0.060
12.69
40.90
5.0
0.80
0.063
12.57
39.84
5.5
0.84
0.068
12.40
38.86
136
Wi=1,Ci=2
A/C AOA A/C CL A/C CD A/C E V cruise
degree
ft/s
-2.0
0.29
0.032
9.29
65.64
-1.5
0.33
0.033
10.14
61.51
-1.0
0.38
0.034
10.89
58.07
-0.5
0.42
0.036
11.47
55.15
0.0
0.46
0.038
11.97
52.63
0.5
0.50
0.040
12.36
50.42
1.0
0.54
0.043
12.58
48.48
1.5
0.58
0.046
12.73
46.74
2.0
0.62
0.048
12.80
45.18
2.5
0.66
0.052
12.82
43.76
3.0
0.70
0.055
12.80
42.47
3.5
0.74
0.059
12.70
41.28
4.0
0.78
0.062
12.59
40.19
4.5
0.82
0.066
12.45
39.19
5.0
0.87
0.071
12.26
38.25
5.5
0.91
0.075
12.08
37.38
Wi=2,Ci=3
A/C AOA A/C CL A/C CD A/C E V cruise
degree
ft/s
-3.0
0.29
0.032
9.29
65.64
-2.5
0.33
0.033
10.14
61.51
-2.0
0.38
0.034
10.89
58.07
-1.5
0.42
0.036
11.47
55.15
-1.0
0.46
0.038
11.97
52.63
-0.5
0.50
0.040
12.36
50.42
0.0
0.54
0.043
12.58
48.48
0.5
0.58
0.046
12.73
46.74
1.0
0.62
0.048
12.80
45.18
1.5
0.66
0.052
12.82
43.76
2.0
0.70
0.055
12.80
42.47
2.5
0.74
0.059
12.70
41.28
3.0
0.78
0.062
12.59
40.19
3.5
0.82
0.066
12.45
39.19
4.0
0.87
0.071
12.26
38.25
4.5
0.91
0.075
12.08
37.38
137
Appendix B. Aircraft Lift Coefficient Equation
C L , AC 
LW
0.5     V  S Ref
2
0.5     VW  SW  C L ,W

2
C L , AC 
0.5     V  S Ref
2
VW  SW  C L ,W
C L, AC  (
V  S Ref
2
0.5     V  S Ref
2
0.5     VC  S C  C L ,C
2

2
C L , AC 
LC
0.5     V  S Ref
2
VC  S C  C L ,C
2

V  S Ref
2
VW 2
S
V
S
)  ( W )  C L ,W  ( C ) 2  ( C )  C L ,C
V
S Ref
V
S Ref
138
Appendix C. Aircraft Drag Coefficient Equation
D AC  (0.5     VW  SW  C D ,W ) 
2
(0.5     VC  S C  C D ,C ) 
2
(0.5     VLG  S LG  C D , LG ) 
2
(0.5     VF  S F  C D , F ) 
2
(0.5     VVS  SVS  C D ,VS )
2


1
2
2
D AC   0.5     VW  SW  (c d ,W 
C L ,W )  


ARW    espan,W




1
2
 0.5     VC 2  S C  (c d ,C 
C L ,C )  


ARC    espan,C


0.5  
0.5  
0.5  


 VLG  S LG  (n  C D ,blunt  QLG ) 

 VF  S F  (C f , F  FFF  QF ) 

 VVS  SVS  (C f ,VS  FFVS  QVS  I )
2

2
2
C D , AC 
DW
0.5     V  S Ref
2
DLG
0.5     V  S Ref
2



DC
0.5     V  S Ref
2
DF
0.5     V  S Ref
2
DVS
0.5     V  S Ref
2
139


C D , AC


1
2
 0.5     VW 2  SW  (c d ,W 
C L ,W ) 


ARW    espan,W


2
0.5     V  S Ref


1
2
 0.5     VC 2  S C  (c d ,C 
C L ,C ) 


ARC    e span,C


2
0.5     V  S Ref
0.5  
 VLG  S LG  (n  C D ,blunt  QLG )
2

0.5     V  S Ref
2
0.5  
 VF  S F  (C f , F  FFF  QF )
2

0.5     V  S Ref
2
0.5  

 VVS  SVS  (C f ,VS  FFVS  QVS  I )
2


0.5     V  S Ref
2
140

Appendix D. Aircraft Moment Coefficient Equation
M cg  M ac ,C  LC  X C  DC  Z C
 M ac ,W  LW  X W  DW  Z W  TNet  Z Engine
 DF  Z F  DLG  Z LG  DVS  Z VS
CM 



M ac ,C
0.5  V  S Ref  c Ref
2
M ac ,W

0.5  V  S Ref  c Ref
2
TNet  Z Engine

0.5  V  S Ref  c Ref
2

LC  X C
0.5  V  S Ref  c Ref
2
LW  X W
0.5  V  S Ref  c Ref
2
DF  Z F
0.5  V  S Ref  c Ref
2

DC  Z C
0.5  V  S Ref  c Ref
2
DW  Z W

0.5  V  S Ref  c Ref
2
D LG  Z LG

0.5  V  S Ref  c Ref
2
DVS  Z VS
0.5  V  S Ref  c Ref
2
0.5  VC  S C  c C  C M ,C
2
CM 
0.5  V  S Ref  c Ref
2
0.5  VC  S C  C D ,C  Z C
2

0.5  V  S Ref  c Ref
2
0.5  VW  SW  C L ,W  X W

0.5  V  S Ref  c Ref
2
TNet  Z Engine
0.5  V  S Ref  c Ref
2
2

0.5  V  S Ref  c Ref
2
0.5  VW  SW  C D ,W  Z W
2

0.5  V  S Ref  c Ref
2
0 .5  V F  S F  C D , F  Z F
2

0.5  V  S Ref  c Ref
2
0.5  V LG  S LG  C D , LG  Z LG
0.5  V  S Ref  c Ref
2
2
0.5  VW  SW  cW  C M ,W
2

0.5  V  S Ref  c Ref
2

2

0.5  VC  S C  C L ,C  X C
0.5  VVS  SVS  C D ,VS  Z VS
2

141
0.5  V  S Ref  c Ref
2
SC
cC
 C M ,C 
S Ref
c Ref
S
X
2
  C  C  C L ,C  C
S Ref
c Ref
S
Z
1
2
2
  C  C  ( c d ,C 
C L ,C )  C
S Ref
ARC    e span,C
c Ref
CM  C 
2
SW
cW
 C M ,W 
S Ref
c Ref
S
X
2
 W  W  C L ,W  W
S Ref
c Ref
 W 
2
 W 
2

SW
Z
1
2
 (c d ,W 
C L ,W )  W
S Ref
ARW    espan,W
c Ref
TNet  Z Engine
0.5  V  S Ref  c Ref
2

SF
Z
 (C f , F  FFF  QF )  F
S Ref
c Ref

S LG
Z
 (n  C D ,blunt  Q LG )  LG
S Ref
c Ref

SVS
Z
 (C f ,VS  FFVS  QVS  I )  VS
S Ref
c Ref
142