flight and orbital mechanics - tu delft opencourseware · semester 1 - 2012 challenge the future...
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1Challenge the future
Flight and Orbital Mechanics
Lecture slides
Semester 1 - 2012
Challenge the future
DelftUniversity ofTechnology
Flight and Orbital MechanicsLecture 7 – Equations of motion
Mark Voskuijl
2AE2104 Flight and Orbital Mechanics |
Time scheduleDate Time Hours Topic
4 Sep 10.45 – 12.30 1, 2 Unsteady climb
6 Sep 10.45 – 12.30 3, 4 Minimum time to climb
11 Sep 10.45 – 12.30 5, 6 Turning performance
13 Sep 10.45 – 12.30 7, 8 Take – off
18 Sep 10.45 – 12.30 9, 10 Landing
20 Sep 10.45 – 12.30 11, 12 Cruise
25 Sep 10.45 – 12.30 13, 14 Equations of motion (wind gradient)
27 Sep 10.45 – 12.30 15, 16 Kepler orbits, gravity, Earth-repeat orbits, sun-
synchronous orbits, geostationary satellites
2 Oct 10.45 – 12.30 17, 18 Third-body perturbation, atmospheric drag, solar
radiation, thrust
4 Oct 10.45 – 12.30 19, 20 Eclipse, maneuvers
9 Oct 10.45 – 12.30 21, 22 Interplanetary flight
11 Oct 10.45 – 12.30 23, 24 Interplanetary flight
16 Oct 10.45 – 12.30 25, 26 Launcher, ideal vs. real flight, staging, design
18 Oct 10.45 – 12.30 27, 28 Exam practice
3AE2104 Flight and Orbital Mechanics |
4AE2104 Flight and Orbital Mechanics |
5AE2104 Flight and Orbital Mechanics |
6AE2104 Flight and Orbital Mechanics |
Content
• Introduction
• Axis systems and Euler angles
• Vector / matrix notation
• Accelerations
• Forces
• General equations of motion 3D flight
• Effect of a wind gradient
7AE2104 Flight and Orbital Mechanics |
Introduction
Newton’s laws only hold with respect to a frame of reference which is in absolute rest. This is called an inertial frame of reference
Coordinate systems translating uniformly to the frame of reference in absolute rest are also inertial frames of reference
A rotating frame of reference is not an inertial frame of reference
Newton’s laws
8AE2104 Flight and Orbital Mechanics |
Introduction
• Derivation of equations of motion
• General 3 dimensional flight
• 2 dimensional flight with a wind gradient
• General approach!
Objective
9AE2104 Flight and Orbital Mechanics |
Content
• Introduction
• Axis systems and Euler angles
• Vector / matrix notation
• Accelerations
• Forces
• General equations of motion 3D flight
• Effect of a wind gradient
10AE2104 Flight and Orbital Mechanics |
Axis systems and Euler angles
Earth axis system: {Eg}
1. Xg axis in the horizontal plane, orientation is arbitrary
2. Yg axis in the horizontal plane, orientation: perpendicular to Xg
3. Zg axis points downwards
Earth axis system
11AE2104 Flight and Orbital Mechanics |
Axis systems and Euler angles
2
2
'Centrifugal force'
e
e
W VC
g R h
C V
W R h g
Assumption 1: the earth is flat
Valid assumption
2
2
Example
100 [m/s]
6371 [km]
9.80665 [m/s ]
0 [m]
1000.00016
6371000 0 9.80665
(0.016%)
e
V
R
g
h
C
W
Assumption
12AE2104 Flight and Orbital Mechanics |
Axis systems and Euler angles
Assumption 2: the earth is non-rotating
Assumptions
13AE2104 Flight and Orbital Mechanics |
14AE2104 Flight and Orbital Mechanics |
Axis systems and Euler angles
Moving earth axis system: {Ee}
1. Xe parallel to Xg axis but attached to c.g. of aircraft
2. Ye parallel to Yg axis but attached to c.g. of aircraft
3. Ze axis points downwards
Moving earth axis system
15AE2104 Flight and Orbital Mechanics |
Axis systems and Euler angles
Body axis system: {Eb}
1. Origin is fixed to the aircraft c.g.
2. Xb lies in plane of symmetry and points towards the nose
3. Yb is perpendicular to the plane of symmetry and is directed to the right wing
4. Zb is perpendicular to Xb and Yb
Body axis system
16AE2104 Flight and Orbital Mechanics |
Axis systems and Euler anglesYaw angle (body axis)
17AE2104 Flight and Orbital Mechanics |
Axis systems and Euler anglesPitch angle (body axis)
18AE2104 Flight and Orbital Mechanics |
Axis systems and Euler anglesRoll angle (body axis)
19AE2104 Flight and Orbital Mechanics |
Axis systems and Euler angles
Air path axis system: {Ea}
1. Origin is fixed to the aircraft c.g.
2. Xa lies along the velocity vector3. Za taken in the plane of
symmetry of the airplane4. Ya is positive starboard
Air path axis system
20AE2104 Flight and Orbital Mechanics |
Axis systems and Euler anglesAzimuth angle (air path axis)
21AE2104 Flight and Orbital Mechanics |
Axis systems and Euler anglesFlight path angle (air path axis system)
22AE2104 Flight and Orbital Mechanics |
Axis systems and Euler anglesAerodynamic angle of roll (air path axis system)
23AE2104 Flight and Orbital Mechanics |
Axis systems and Euler angles
• Four axes systems can be defined
• Earth
• Moving Earth
• Body axes
• Air path axes
• Three Euler angles define the orientation of the aircraft (body
axes) Yaw Pitch Roll
• Three Euler angles define the orientation of the aircraft (body
axes) Azimuth Flight path Aerodynamic roll
• The sequence of the Euler angles is very important!!!
Summary
24AE2104 Flight and Orbital Mechanics |
25AE2104 Flight and Orbital Mechanics |
Content
• Introduction
• Axis systems and Euler angles
• Vector / matrix notation
• Accelerations
• Forces
• General equations of motion 3D flight
• Effect of a wind gradient
26AE2104 Flight and Orbital Mechanics |
Vector / matrix notation
X
Z
Y
P
r
i
kz
x
r xi yj zk
i, j and k are unit vectors along the axes
27AE2104 Flight and Orbital Mechanics |
Vector / matrix notation
: Row
: Column
: Square matrix
i
r x i y j z k x y z j
k
x y z E
28AE2104 Flight and Orbital Mechanics |
Content
• Introduction
• Axes systems and Euler angles
• Vector / matrix notation
• Accelerations
• Forces
• General equations of motion 3D flight
• Effect of a wind gradient
29AE2104 Flight and Orbital Mechanics |
Accelerations
dVa
dt
0 0 aV V E
0 0 0 0a a
dVV E V E
dt
?
What is the time derivative of the air path axis system?
30AE2104 Flight and Orbital Mechanics |
Accelerations
x
yz
Xa
Ya
Za
i
j
k
-ky
jz
kx
-iz-jx
iy
0
0
0
z y
a z x a
y x
E E
0 z y
dii j k
dt
0 y x
dki j k
dt
0 z x
dji j k
dt
Time derivative of the air path axis system
31AE2104 Flight and Orbital Mechanics |
Accelerations
0 0 0 0a a
dVV E V E
dt
0
0 0 0 0 0
0
z y
a z x a
y x
dVV E V E
dt
z y a
dVV V V E
dt
32AE2104 Flight and Orbital Mechanics |
Content
• Introduction
• Axes systems and Euler angles
• Vector / matrix notation
• Accelerations
• Forces
• General equations of motion 3D flight
• Effect of a wind gradient
33AE2104 Flight and Orbital Mechanics |
Forces
F L D T W
0 0 0 1a eF T D L E W E
No sideslip
Assume thrust in direction of airspeed vector
34AE2104 Flight and Orbital Mechanics |
ForcesSideslip angle
35AE2104 Flight and Orbital Mechanics |
ForcesSideslip angle
36AE2104 Flight and Orbital Mechanics |
Forces Sideslip angle
37AE2104 Flight and Orbital Mechanics |
Forces
F L D T W
0 0 0 1a eF T D L E W E
Problem: different axis systems Express all forces in 1 axis system
38AE2104 Flight and Orbital Mechanics |
Forces
{Ee}
{Ea}
Transformation matrices
39AE2104 Flight and Orbital Mechanics |
Forces
Xe X1
Ye
Y1
1j
ej
1iei
1
1
1
cos sin 0
sin cos 0
0 0 1
e
e
e
i i
j j
k k
1
1
1
cos sin
sin cos
e e
e e
e
i i j
j i j
k k
1 eE T E
Rotation over azimuth angle ()
40AE2104 Flight and Orbital Mechanics |
Forces
2i
1i
2k
X1
Z1
X2
Z2
1k 2 1E T E
cos 0 sin
0 1 0
sin 0 cos
T
2 1 1
2 1
2 1 1
cos sin
sin cos
i i k
j j
k i k
Rotation over flight path angle ()
41AE2104 Flight and Orbital Mechanics |
2j
aj
2k
Y2
Ya
Z2
Ya
ak
2aE T E
1 0 0
0 cos sin
0 sin cos
T
2
2 2
2 2
cos sin
sin cos
a
a
a
i i
j j k
k j k
Rotation over aerodynamic angle of roll ()
Forces
42AE2104 Flight and Orbital Mechanics |
Forces
{Ee}
{Ea}
Transformation matrices
43AE2104 Flight and Orbital Mechanics |
Forces
a eE T T T E
1
2
e
e
a e
E T E
E T T E
E T T T E
1 1 1
e aE T T T E
Transformation matrices
44AE2104 Flight and Orbital Mechanics |
Forces
1
1 1
. . . . . .
. . . . . .
. . . . . .
T
I
Properties of transformation matrices
45AE2104 Flight and Orbital Mechanics |
Forces
0 0 0 1a e
F L D T W
T D L E W E
0 0 0 1T T T
a aF T D L E W T T T E
sin cos sin cos cos aF T D W W L W E
cos sin 0 cos 0 sin 1 0 0
0 0 1 0 0 1 sin cos 0 0 1 0 0 cos sin
0 0 1 sin 0 cos 0 sin cos
T T T
e aW E W E
0 0 1 sin cos sin cos cose aW E W W W E
All results combined
46AE2104 Flight and Orbital Mechanics |
Equations of motion
F m a
sin cos sin cos cos aF T D W W L W E
z y a
dVa V V V E
dt
sin
cos sin
cos cos
z
y
T D W mV
W mV
L W mV
3 equations of motion!
47AE2104 Flight and Orbital Mechanics |
Equations of motion
sin
cos sin
cos cos
sin
cos sin
cos cos
z
y
T D W mV
W mV
L W mV
mV T D W
mV L
mV L W
Using transformation matricesto convert x, y and z
Rewrite in traditional form
48AE2104 Flight and Orbital Mechanics |
Equations of motion
1 1
1 aE T T E
cos 0 sin 1 0 0
0 0 0 1 0 0 cos sin
sin 0 cos 0 sin cos
aE
10 0 E
Conversion x, y, z to d/dt, d/dt, d/dt
sin cos sin cos cos aE
49AE2104 Flight and Orbital Mechanics |
Equations of motion
1
2 aE T E
1 0 0
0 0 0 cos sin
0 sin cos
aE
20 0 E
Conversion x, y, z to d/dt, d/dt, d/dt
0 cos sin aE
50AE2104 Flight and Orbital Mechanics |
Equations of motion
3 aE E
30 0 E
Conversion x, y, z to d/dt, d/dt, d/dt
0 0 aE
51AE2104 Flight and Orbital Mechanics |
Equations of motion
sin cos sin cos cos cos sin
x y z a
a
E
E
Conversion x, y, z to d/dt, d/dt, d/dt
52AE2104 Flight and Orbital Mechanics |
Content
• Introduction
• Axes systems and Euler angles
• Vector / matrix notation
• Accelerations
• Forces
• General equations of motion 3D flight
• Effect of a wind gradient
53AE2104 Flight and Orbital Mechanics |
Equations of motion
sin
cos sin
cos cos
mV T D W
mV L
mV L W
Final result
54AE2104 Flight and Orbital Mechanics |
Content
• Introduction
• Axes systems and Euler angles
• Vector / matrix notation
• Accelerations
• Forces
• General equations of motion 3D flight
• Effect of a wind gradient
55AE2104 Flight and Orbital Mechanics |
Effect of a wind gradient
An aircraft is flying from A to B over a distance of 1000 nautical miles. The True Airspeed of this aircraft is 120 knots (1kt = 1 nautical mile per hour). The aircraft is experiencing a constant headwind of 20 kts.
How long does it take to fly from A to B?
A. Less than 10 hours
B. 10 hours
C. More than 10 hours
Question
56AE2104 Flight and Orbital Mechanics |
Effect of a wind gradient
57AE2104 Flight and Orbital Mechanics |
Effect of a wind gradient
58AE2104 Flight and Orbital Mechanics |
Effect of a wind gradient
g wV V V
(wind speed)wV
(ground speed)gr V
(airspeed)V
Xa
Xg
Zg
r
59AE2104 Flight and Orbital Mechanics |
Effect of a wind gradient
0WdV
dt
Wind is not constant:
This lecture: only horizontal wind
( )WdVf H
dt
60AE2104 Flight and Orbital Mechanics |
Effect of a wind gradient
g wV V V
ga V
g wa V V V
Absolute acceleration
So we need to define the velocity first
0 0 0 0g a w gV V E V E
The acceleration can be determined by taking the time derivative
0 0 0 0a w g
d da V E V E
dt dt
0 0 0 0 0 0 0 0a a w g w ga V E V E V E V E
61AE2104 Flight and Orbital Mechanics |
0 0 0 0 0 0 0 0a a w g w ga V E V E V E V E
What are and ???a gE E
The ground axis system is at rest
0gE
However, the air path axis system is rotating and translating
62AE2104 Flight and Orbital Mechanics |
Effect of a wind gradient
a
di
dt
dE j
dt
dk
dt
0 0d
k i j kdt
0 0d
i i j kdt
0 0 0d
j i j kdt
0 0 0 0
0 0 0 0 0 0
0 0 0 0
a a
di
dt id
E j j Edt
kd
kdt
63AE2104 Flight and Orbital Mechanics |
Effect of a wind gradient
0 0 0 0 0 0 0 0a a w g w ga V E V E V E V E
0 0
0 0 0 0 0 0 0 0 0
0 0
a a w ga V E V E V E
0 0 0 0 0 0a a w ga V E V E V E
0 0 0a w ga V V E V E
Fill in the results
Write out
Simplify
Two axis systems…
64AE2104 Flight and Orbital Mechanics |
Effect of a wind gradient
cos 0 sin
0 1 0
sin 0 cos
g aE E
cos 0 sing a a ai i j k
0 1 0g a a aj i j k
sin 0 cosg a a ak i j k
0 0 0a w ga V V E V E
cos 0 sin
0 0 0 0 1 0
sin 0 cos
a w aa V V E V E
65AE2104 Flight and Orbital Mechanics |
Effect of a wind gradient
cos 0 sin
0 0 0 0 1 0
sin 0 cos
a w ga V V E V E
0 cos 0 sina w w aa V V E V V E
cos 0 sinw w aa V V V V E
66AE2104 Flight and Orbital Mechanics |
Effect of a wind gradient
sin 0 cosa a aF T D W i j L W k
V
Zb
Ze
Xe
Xb
Xg
Zg
Xa
Za
D
W
T
Assumption: T // V
sin 0 cos aF T D W L W E
L
67AE2104 Flight and Orbital Mechanics |
Effect of a wind gradient
sin 0 cos
cos 0 sin
g
a
w w a
F m V
F T D W W L E
a V V V V E
sin cos
0 0
cos sin
w
w
WT D W V V
g
WL W V V
g
68AE2104 Flight and Orbital Mechanics |
Effect of a wind gradient
sin
cos w
H RC V
s V V
69AE2104 Flight and Orbital Mechanics |
Effect of a wind gradient
70AE2104 Flight and Orbital Mechanics |
71AE2104 Flight and Orbital Mechanics |
Questions?
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