Download Chapter 4- The Equations of Motion Aircraft

Chapter 4- The Equations of Motion
Aircraft Performance
Aircraft performance is defined as how the aircraft responds (its motion) to the
four forces of flight.
It is considered to be a branch of the Flight Mechanics discipline.
We have already reviewed aerodynamics and propulsion. We use the
following information in performance:
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aerodynamics
drag polar
propulsion
thrust or power, SFC
Dr. Danielle Soban
Georgia Institute of Technology
Chapter 4- The Equations of Motion
The Four Forces of Flight
Lift, Drag, Weight, Thrust
Lift and Drag are for complete airplane
L
perpendicular to V
by definition
not necessarily in the
flight direction
T
ε
D
parallel to V by
definition
W
always acts towards the
center of the earth
Steady, Level Flight
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V
always in the direction
of the local flight of
the aircraft. Shows
flow velocity relative
to the airplane
Dr. Danielle Soban
Georgia Institute of Technology
Chapter 4- The Equations of Motion
Four Forces in Climbing Flight
flight path
T
L
V
ε
θ
local climb
angle
D
θ
W
earth
AE 3310 Performance
Dr. Danielle Soban
Georgia Institute of Technology
Chapter 4- The Equations of Motion
Now, Bank the Aircraft
φ Bank (roll) angle
φ
L
T sin ε
φ
Wcosθ
AE 3310 Performance
Dr. Danielle Soban
Georgia Institute of Technology
Chapter 4- The Equations of Motion
The Equations of Motion
Based on Newton’s Second Law:
F=ma
note this is vector form
In scalar form, for arbitrary direction in space, s
Fs = m as
General, Formal Derivation
rotating spherical earth
acceleration of gravity with
distance from center of the earth
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Less Formal, more Physical Derivation
flat, stationary earth
Dr. Danielle Soban
Georgia Institute of Technology
Chapter 4- The Equations of Motion
Climbing, Banking Flight
Replace aircraft with point mass at its center of gravity (because we are only
concerned with translational motion).
+
radius of curve
r1
Flight Path
s
L cos φ
V
T sin ε cos φ
θ
instananeous flight
path direction
T cos ε
D
θ
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W
center of gravity
of the airplane
Dr. Danielle Soban
Georgia Institute of Technology
Chapter 4- The Equations of Motion
First Equation of Motion
Take components parallel to the flight path
The force is
F
The acceleration is
= T cos ε - D - W sin θ
dV
a =
dt
Therefore, Newton’s Second Law
parallel to the flight path is
dV
m
dt
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First Equation of Motion
= T cos ε - D - W sin θ
ma = F
Dr. Danielle Soban
Georgia Institute of Technology
Chapter 4- The Equations of Motion
Second Equation of Motion
Take components perpendicular to the flight path
The force is
F
= L cos φ + T sin ε cos φ - W cos θ
The radial acceleration is
V
a =
r1
Therefore, Newton’s Second Law
perpendicular to the flight path is
m
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V
r1
2
Second Equation of Motion
2
= L cos φ + T sin ε cos φ - W cos θ
ma = F
Dr. Danielle Soban
Georgia Institute of Technology
Chapter 4- The Equations of Motion
Forces on Horizontal Plane
Now look at flight path from a “top” view
D cos θ
T cos ε cos θ
V
cos θ
T sin ε sin φ
L sin φ
projection of
flight path
r2
+
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Dr. Danielle Soban
Georgia Institute of Technology
Chapter 4- The Equations of Motion
Third Equation of Motion
Take components perpendicular to the flight path in
the horizontal plane (2)
The force is
F2 = L sin φ + T sin ε sin φ
The radial acceleration is
(V cos θ)
a2 =
r2
Therefore, Newton’s Second Law
perpendicular to the horizontal flight path is
2
Third Equation of Motion
2
m
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(V cos θ)
= L sin φ + T sin ε sin φ
r2
ma = F
Dr. Danielle Soban
Georgia Institute of Technology
Chapter 4- The Equations of Motion
Summary
The three Equations of Motion are simply statements of Newton’s Second
Law.
The three Equations of Motion describe the translational motion of an
airplane through three-dimensional space over a flat earth.
There are three additional equations of motion that describe the rotational
motion of the aircraft about its three axes.
Final note: the three equations of motion here do not assume a yaw
component. The free stream velocity vector is assumed always parallel
to the symmetry plane of the aircraft.
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Dr. Danielle Soban
Georgia Institute of Technology