Nomenclature
1. Nomenclature
Principal Axes
Principal Axes
Coordinate Systems
Wing Geometry
2. Four Forces of Flight
AIRCRAFT STABILITY & CONTROL (AE401)
Weight, Thrust, Drag
Lift
Chapter1: Introduction
3. Control Surfaces
Aileron
Elevator
Rudder
4. References
Axis, as applied to aviation, is defined as "an imaginary line about which
a body rotates". An aircraft is free to revolve or move around three axes
as shown in Figure 1. These axes, referred to as longitudinal, lateral
and vertical, are each perpendicular to the others and intersect at the
aircraft centre of gravity. The longitudinal axis of an aircraft is a straight
line through the aircraft’s nose cone and the endpoint of the fuselage.
The lateral axis is parallel to the wings and extends throughout wingtip
to wingtip. The vertical axis is "normal" of the geometric plane formed
by the longitudinal and lateral axes. Motion around the longitudinal axis,
the lateral axis and the vertical axis are referred to as roll, pitch and yaw
respectively.
Asst.Prof. M. Orkun Öǧücü
+90 (342) 360 1200 | 3526
oogucu@gantep.edu.tr
http://www1.gantep.edu.tr/ oogucu/
GAZIANTEP UNIVERSITY
Faculty of Aeronautics and Astronautics
Department of Aircraft and Aerospace Engineering
Figure 1: The three axes of rotation
September 16, 2019
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1. Nomenclature
Principal Axes
Nomenclature
Coordinate Systems
Coordinate Systems
Wing Geometry
2. Four Forces of Flight
Weight, Thrust, Drag
Lift
3. Control Surfaces
Aileron
Elevator
Rudder
4. References
Nomenclature
1. Nomenclature
Coordinate Systems
Principal Axes
Coordinate Systems
To describe the motion of an airplane it is necessary to define a suitable
coordinate system as shown in Figure 2 to formulate the equations of
motion. For most problems dealing with aircraft motion, two right-handed
coordinate systems are used. The first one is denoted by E and fixed to
the Earth. It may be considered for the purpose of aircraft motion analysis
to be an inertial coordinate system. The second one is denoted by B and
referred to body coordinate system. It is fixed to the center of gravity and
translates/rotates with the aircraft. Note that, the x- and z-axes are in the
plane of symmetry, with the x-axis pointing along the fuselage and the
positive y-axis along the right wing.
Wing Geometry
2. Four Forces of Flight
Weight, Thrust, Drag
Lift
3. Control Surfaces
Here the force components are denoted by X, Y , Z and moments components are indicated by L, M , N along and about the x, y, and z axes.
They can be defined in terms of dimensionless coefficients (Cx , Cy , Cz ,
Cl , Cm , Cn ), the flight dynamic pressure (Q), a reference area (S) and a
characteristic length (l) as follows;
Aileron
Elevator
Rudder
4. References
Axial Force
Side Force
Normal Force
Rolling Moment
Pitching Moment
Yawing Moment
X = Cx QS
Y = Cy QS
Z = Cz QS
L = Cl QSl
M = Cm QSl
N = Cn QSl
Here, the reference area S is taken as the wing planform area and the
characteristic length l is taken as the wing span for the rolling and yawing
moment and the mean chord for the pitching moment.
Figure 2: Earth-fixed and body-fixed coordinate systems [1]
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Nomenclature
Nomenclature
Coordinate Systems
Wing Geometry
The angle of attack (α) and sideslip (β), which are illustrated in Figure 3,
can be defined in terms of the velocity components as follows;
The wings can be mounted to the fuselage in many different ways as shown in Figure 4. Also, at
each configuration by angling the wings up or down, dihedral or anhedral designs can be obtained.
1. Nomenclature
Principal Axes
Coordinate Systems
Wing Geometry
2. Four Forces of Flight
Weight, Thrust, Drag
LOW-WING
Lift
MID-WING
HIGH-WING
Positive
3. Control Surfaces
Aileron
Elevator
Rudder
4. References
w
u
(1)
β = sin−1
v
V
(2)
u2 + v 2 + w 2 .
Negative
where, V =
√
α = tan−1
Zero
Figure 3: Definition of a) angle-of-attack b) angle-of-sideslip [1]
If the angle of attack and sideslip are small, that is < 150 , then Equations 1 and 2 can be approximated by;
α = w/u
(3)
β = v/u
(4)
Figure 4: Low, mid and high mounting configurations of the wing with dihedral/anhedral designs
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Nomenclature
Wing Geometry
1. Nomenclature
Principal Axes
Nomenclature
Wing Geometry
Coordinate Systems
In some situations arising from performance requirements, stability or physical constraints such as
visibility, dihedral angle may be varied along the span of the wing as in the case of gull-wing or
inverted gull-wing designs. For example, gull-wing has sharp dihedral on the root section, little or
none on the main section. Besides, inverted gull-wing has anhedral on the root section and dihedral
on the main section.
LOW-WING
MID-WING
HIGH-WING
Wing Geometry
2. Four Forces of Flight
The shape of the wing, when viewed from above or below, is called planform as shown in Figure 6.
Weight, Thrust, Drag
Lift
3. Control Surfaces
Aileron
Elevator
Rudder
Variable
4. References
Figure 6: Rectangular, tapered, elliptical, swept and delta wing planform geometries
Figure 5: Low, mid and high mounting configurations of the wing with dihedral/anhedral designs
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The ends of the wing are called the wing tips, and the distance from one
wing tip to the other is called the span, given by the symbol b. The leading
edge is the part of the wing that first contacts the air; alternatively it is the
foremost edge of the wing. On the other hand, the trailing edge is rear of
the wing, where the airflow separated by the leading edge rejoins. The
wing area, S, is the projected area of the planform and is bounded by the
leading and trailing edges and the wing tips. Aspect ratio is a measure of
how long and narrow a wing is from tip to tip. The Aspect Ratio of a wing
is defined to be the square of the span divided by the wing area and is
given the symbol AR. There is a component of the drag, which is called
induced drag, depends inversely on the aspect ratio. A higher aspect ratio
wing has a lower drag and a slightly higher lift than a lower aspect ratio
wing.
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Nomenclature
1. Nomenclature
Wing Geometry
Principal Axes
Coordinate Systems
Wing Geometry
2. Four Forces of Flight
Weight, Thrust, Drag
Lift
3. Control Surfaces
Aileron
Elevator
Rudder
4. References
Nomenclature
1. Nomenclature
Wing Geometry
Principal Axes
Coordinate Systems
The distance from the leading edge to the trailing edge is called the chord,
denoted by the symbol c. The chord length can be constant as in the case
of rectangular wing, or may be varied along the span as in the case of tapered, elliptical, swept and delta wings 1 . For example, as shown in the
detailed view of the swept wing planform, root and tip chord lengths are
represented by cr and ct , respectively. To give a characteristic parameter
that can be compared among various wing shapes, the mean aerodynamic chord, or MAC, is used. It is defined as;
2
M AC =
S
Wing Geometry
2. Four Forces of Flight
Weight, Thrust, Drag
Lift
3. Control Surfaces
Aileron
Elevator
Rudder
4. References
b/2
M GC = S/b
Apart from the wing, the chord or chord length is also used to describe
width of the horizontal/vertical stabilizer, flap, aileron and rudder.
Taper ratio can be either in planform or thickness, or both. In its simplest
terms, it is a decrease from wing root to wing tip in wing chord or wing
thickness.
c(y)2 dy
0
The angle between the lateral axis and the quarter-chord line is called
sweep angle as shown in Figure 6 indicated by ΛLE leading edge sweep
angle and ΛT E trailing edge sweep angle.
where, S is the wing area, b is the wing span, y is the coordinate along
the wing span and c is the chord at the coordinate y. The position of
center of mass (CoM) of an aircraft is usually measured relative to the
MAC, so not only the length but also the position of it is often important.
The aerodynamic center lies on the mean aerodynamic chord.
1
Another characteristic parameter related to the chord length that is rarely
used in aerodynamics is mean geometric chord (or standard mean chord).
It is the chord of a rectangular wing with the same area and span as those
of the given wing and defined as;
For further reading please refer to https://en.wikipedia.org/wiki/Wing_configuration
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Nomenclature
Wing Geometry
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1. Nomenclature
Principal Axes
Four Forces of Flight
Weight, Thrust, Drag
Coordinate Systems
A cut through the wing perpendicular to the leading and trailing edges shows the cross-section of
the wing which is called airfoil as shown in Figure 7. The straight line drawn from the leading to
trailing edges of the airfoil is called the chord line. The chord line cuts the airfoil into an upper
surface and a lower surface. If we plot the points that lie halfway between the upper and lower
surfaces, we obtain a curve called the mean camber line. For a symmetric airfoil (upper surface
the same shape as the lower surface) the mean camber line will fall on top of the chord line. But in
most cases, these are two separate lines. The maximum distance between the two lines is called
the camber, which is a measure of the curvature of the airfoil (high camber means high curvature).
The maximum distance between the upper and lower surfaces is called the thickness.
Wing Geometry
2. Four Forces of Flight
Weight, Thrust, Drag
Lift
3. Control Surfaces
Aileron
Elevator
Rudder
4. References
There are four basic forces that allow an airplane to fly; weight, lift, drag
and thrust.
Weight is a force that is always directed toward the center of the earth.
The magnitude of the weight depends on the total mass of all airplane
parts, amount of fuel, and any payload on board. It is distributed throughout the airplane, but we assume that it collected and acting through a
single point called the center of gravity.
Flying encompasses two major problems; overcoming the weight of an
object by some opposing force, and controlling the object in flight. Both of
these problems are related to the object’s weight and the location of the
center of gravity. During a flight, an airplane’s weight constantly changes
as the aircraft consumes fuel. The distribution of the weight and the center
of gravity also changes. So the pilot must constantly adjust the controls
to keep the airplane balanced, or trimmed.
To overcome drag, aircrafts use a propulsion system to generate a force
called thrust. The direction of the thrust force depends on how the engines are attached to the aircraft. On some aircrafts, such as the Harrier,
the thrust direction can be varied to help the airplane take off in a very
short distance.
Drag is the force that acts opposite to the direction of motion. It tends to
slow an object. Drag is caused by friction and differences in air pressure.
Figure 7: The cross-sectional shapes of various wings
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1. Nomenclature
Principal Axes
Four Forces of Flight
Lift: An Application of the Bernoulli’s Principle
Coordinate Systems
Wing Geometry
2. Four Forces of Flight
Weight, Thrust, Drag
Lift
3. Control Surfaces
Aileron
Elevator
Rudder
4. References
1. Nomenclature
Control Surfaces
Principal Axes
Coordinate Systems
In an airplane wing profile, as represented in Figure 8, the upper surface
is somewhat curved, while the lower surface is relatively flat. When the
airplane moves forward, air travels across both the top and the bottom
surfaces concurrently. Due to the shape of the profile, air on the bottom
surface moves slower, which creates more pressure, and air on the top
surface moves faster, which creates less pressure. Since the pressure
below the wing is higher than the pressure above the wing, there is a
net force upwards which creates lift. So, this phenomenon is a direct
application of the Bernoulli’s principle.
Figure 8: High and low pressure areas for various wing profiles emerging in wind
tunnel tests conducted at the NASA Langley Research Center [2].
Wing Geometry
2. Four Forces of Flight
Weight, Thrust, Drag
Lift
Airplanes come in many different shapes and sizes depending on the mission of the aircraft. The airplane shown in Figure 9 is a turbine-powered
airliner which has been chosen as a representative aircraft. Individual
aircraft may be configured quite differently from this airliner.
3. Control Surfaces
Aileron
Elevator
Rudder
4. References
Figure 9: Turbine-powered airliner, its parts and functions [3]
Aircraft lift acts through a single point called the center of pressure. The
center of pressure is defined just like the center of gravity.
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Control Surfaces
Control Surfaces
Aileron
Elevator
Ailerons, which are small hinged sections on the outboard portion of a wing, can be used to generate
a rolling motion for an aircraft. They usually work in opposition; as the right aileron is deflected
upward, the left is deflected downward, and vice versa. Changing the angle of deflection at the rear
of an airfoil will change the amount of lift generated by the foil. With greater downward deflection, the
lift will increase in the upward direction. Notice from the Figure 10 that the aileron on the left wing,
as viewed from the rear of the aircraft, is deflected down. The aileron on the right wing is deflected
up. Therefore, the lift on the left wing is increased, while the lift on the right wing is decreased. For
the conditions shown in the Figure 10, the resulting motion will roll the aircraft to the right (clockwise)
as viewed from the rear.
The horizontal stabilizer, as shown in Figure 11, is a fixed wing section whose job is to provide
stability for the aircraft, to keep it flying straight. The elevator is the small moving section at the rear
of the stabilizer that is attached to the fixed sections by hinges. They work in pairs; when the right
elevator goes up, the left elevator also goes up. The elevators work by changing the effective shape
of the airfoil of the horizontal stabilizer. Changing the angle of deflection at the rear of an airfoil
changes the amount of lift generated by the foil. With greater downward deflection of the trailing
edge, lift increases. With greater upward deflection of the trailing edge, lift decreases and can even
become negative as shown in Figure 11.
Figure 10: a) Resulting motion in the case of upward deflected right aileron and downward deflected left aileron
b) Animated picture showing the working principle of aileron [3]
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Figure 11: a) Resulting motion in the case of upward deflected elevators b) Animated picture showing the
working principle of elevator [3]
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Control Surfaces
Rudder
References
1. Nomenclature
Principal Axes
Coordinate Systems
The vertical stabilizer, as shown in Figure 12, is a fixed wing section whose job is to provide stability
for the aircraft, to keep it flying straight. It prevents side-to-side, or yawing, motion of the aircraft
nose. The rudder is the small moving section at the rear of the vertical stabilizer that is attached to
the fixed sections by hinges. Changing the angle of deflection at the rear of an airfoil will change the
amount of lift generated by the foil. With increased deflection, the lift will increase in the opposite
direction. The rudder and vertical stabilizer are mounted so that they will produce forces from side
to side. The side force (F ) is applied through the center of pressure of the vertical stabilizer which
is some distance (L) from the aircraft center of gravity. This creates a torque on the aircraft and the
aircraft rotates about its center of gravity.
Wing Geometry
2. Four Forces of Flight
[1] B. Etkin and L. Reid, Dynamics of Flight: Stability and Control, 3rd ed.
Wiley, 1995.
[2] N. L. R. Group, “Aerodynamics: Airfoil camber, flaps, and slots-slats smoke lifts,” 1938.
Weight, Thrust, Drag
Lift
[3] N. G. R. Center, “Aerodynamics index,” 2015.
3. Control Surfaces
Aileron
Elevator
Rudder
4. References
Figure 12: a) Resulting motion in the case of left deflected rudder b) Animated picture showing the working
principle of rudder [3]
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