Equations of Aircraft Motion
Force Diagram Conventions
θ
line
Chord
T α α
V
T
ght path
li
F
θ
M
D
Horizontal
W
Definitions
V≡
θ≡
α≡
W ≡
L≡
D≡
M ≡
T≡
αT
≡
flight speed
angle between horizontal & flight path
angle of attack (angle between flight path and chord line)
aircraft weight
lift, force normal to flight path generated by air acting on aircraft
drag, force along flight path generated by air acting on aircraft
pitching moment
propulsive force supplied by aircraft engine/propeller
angle between thrust and flight path
To derive the equations of motion, we apply
K
K
(1)
∑ F = ma
Note: we will not be including the potential for a yaw force.
Applying (1) in flight path direction:
dV
∑ F& = ma& = m dt
and examining the force diagram
∑F
&
= T cos αT − D − W sin θ
⇒ T cos αT − D − W sin θ = m
dV
dt
(2)
Equations of Aircraft Motion
Now applying (1) in ⊥ − direction to flight path
∑ F⊥ = ma⊥ = m
V2
rc
where rc ≡ radius of curvature of flight path
∑F
⊥
= L + T sin αT − W cos θ
⇒ L + T sin αT − W cos θ = m
V2
rc
(3)
Equations (2) & (3) give the equations of motion for an aircraft (neglecting yawing
motions) and are quite general. One important specific case of these equations
is level, steady flight with the thrust aligned w/ the flight path.
⇒
dV
= 0, rc → ∞, αT = 0, θ = 0
dt
⇒
T =D
L =W
Level, steady flight
Moment definitions
The pitching moment must be defined relative to a specific location. The two
typical locations are:
• leading edge
• 1 c , quarter of mean chord
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Equations of Aircraft Motion
Force & Moment Coefficients
Typically, aerodynamicists use non-dimensional force & Moment coefficients.


1
ρ ∞V∞2 S 

2
 3D Drag/Lift coefficients
D

CD ≡
1
2 
ρ ∞V∞ S 

2
L
CL ≡
where
ρ ∞ is freestream density
V∞ is freestream velocity (flight speed)
S is a reference area (problem dependent)
q∞ ≡
1

ρ ∞V∞2  Freestream dynamic pressure
2

The moment coefficient requires another length scale:
CM ≡
A ref
M
1
ρ ∞V∞2 S A ref
2
≡ reference length scale (problem dependent)
For 2-D problems, such as an airfoil, the forces are actually forces/length. So, for
example
3D force
L
2D force/length
L′
D
D′
Similarly, M → M ′ . The non-dimensional coefficients for 2-D are defined:
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Equations of Aircraft Motion
Cl ≡
L′
1
ρ ∞V∞2cref
2
D′
Cd ≡
1
ρ ∞V∞2cref
2
M′
Cm ≡
1
2
ρ ∞V∞2cref
2
where cref is a reference length such as the chord of an airfoil.
Forces on Airfoils
The forces & moments on airfoils are normalized by the chord length. So,
Cl ≡
L′
1
ρ ∞V∞2c
2
, Cd ≡
D′
1
ρ ∞V∞2c
2
, Cm ≡
M′
1
ρ ∞V∞2c 2
2
Force coefficients data is generally plotted in 2 forms:
Lift curve
Cl
Cl
max
α
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Equations of Aircraft Motion
Drag polar
Cl
Cd
Cd
min
Cl
Forces on Wings
Cd
min
Cd
x
Wing planform
y
c(y)
b
c( y ) ≡
b≡
S≡
chord distribution
wing span
b
2
planform area =
∫ cdy
−b
2
b2
S
We can think of the 3-D or total lift on the wing as being the sum (i.e. integral) of
the 2-D lift acting on the wing.
A ≡ aspect ratio ≡
⇒
L=
b
2
∫ L′ ( y ) dy
where L′( y ) = lift distribution
−b
2
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Equations of Aircraft Motion
The average 2-D lift on the wing L′ can be defined:
b
1 2
L
L′ ≡ ∫ L′dy =
b −b
b
2
Plugging that into CL :
b
2
CL ≡
∫ L′dy
L
1
ρ ∞V∞2 S
2
=
−
b
2
1
ρ ∞V∞2 S
2
=
L′b
1
ρ ∞V∞2 S
2
But, the average chord or mean chord can be defined as:
c≡
b
2
1
S
cdy =
∫
b −b
b
2
⇒ CL ≡
L
L′
=
1
1
ρ ∞V∞2 S
ρ ∞V∞2c
2
2
In other words, we can think of the 3-D lift coefficient as the mean value of the 2D lift coefficient on the wing. The same is true for drag and moment:
D
CD ≡
=
D′
1
1
ρ ∞V∞2 S
ρ ∞V∞2c
2
2
M
M′
=
CM ≡
1
1
ρ ∞V∞2 S A ref
ρ ∞V∞2c 2
2
2
where A ref = c is used.
A Closer Look at Drag
The drag coefficient can be broken into 2 parts:
C2
C D = C D ,e + L
N N
ΠeA
parasite
drag
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induced
drag
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Equations of Aircraft Motion
where e = span efficiency factor (more on this when we get to lifting line).
The parasite drag contains everything except for induced drag including:
• skin friction drag
• wave drag
• pressure drag (due to separation)
It is a function of α , thus, we can also think of CD ,e as being a function of CL .
The parasitic drag can be well-approximated by:
CD ,e = CD0 + rCL2
where CD0 ≡ drag at CL = 0 , r = empirically determined constant.
1  2

⇒ CD = CD0 +  r +
 CL
ΠeA 

Finally, we can re-define e to include r :
⇒ CD = CD0 +
where e →
1
CL2
ΠeA
e
.
1 + rΠeA
This re-defined e is known as the Oswald efficiency factor.
We will refer to CD0 as the parasite drag coefficient from now on.
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