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1 Challenge the future
Introduction to Aerospace Engineering
Lecture slides
15-12-2012
Challenge the future
Delft University of Technology
Flight mechanics
Introduction Aerospace Engineering Flight Mechanics
Dr. Ir. Mark Voskuijl
2 Flight mechanics
1.& 2. Flight Mechanics
3 Flight mechanics
Question
A Boeing 747 runs out of fuel at 10 km altitude
Suddenly all engines stop…
How far will this aircraft be able to glide?
a) 180 [m]
b) 1800 [m]
c) 18000 [m]
d) 180000 [m]
4 Flight mechanics
5 Flight mechanics
Flight mechanics
• What is the performance of a given aircraft; i.e. how far, high,
fast, slow can it fly?
• How long can an aircraft remain airborne following an engine
failure and how far can it glide?
• How is aircraft morphology related to aircraft performance?
Key questions
6 Flight mechanics
Flight Mechanics
The performance of the complete vehicle is analyzed
High altitude Orbits
Launchers
Steep climbs
Gliding flight
Low speed
7 Flight mechanics
Flight Mechanics
Equations of motion
Aerodynamics
Aircraft Structure
Pilot (flying
strategy)
Propulsion system
Atmosphere
8 Flight mechanics
Contents
1. What is Flight mechanics?
2. Practical matters
3. General equations of motion
4. Propulsion
5. Aerodynamics
6. Summary
7. Additional material (background information for AE1100 project)
9 Flight mechanics
Contents
1. What is Flight mechanics?
2. Practical matters
3. General equations of motion
4. Propulsion
5. Aerodynamics
6. Summary
7. Additional material (background information for AE1100 project)
10 Flight mechanics
Flight mechanics
Hours 1 & 2 : Introduction, general equations of motion
Hours 3 & 4 : Horizontal flight performance
Hours 5 & 6 : Climbing and descending flight
Hours 7 & 8 : Flight envelope
Hours 9 & 10: Example questions and solutions
11 Flight mechanics
What do you need to learn?
• Most important!!! Lecture sheets (blackboard)
• Introduction to flight: paragraphs; 6.1, 6.2, 9.1 9.2, 9.4, 9.6
• Better book to consult for more information: Ruijgrok, “Elements of Airplane Performance” (only 14 euro’s at VSV)
• Practice questions (blackboard)
12 Flight mechanics
Contents
1. What is Flight and Orbital Mechanics?
2. Practical matters
3. General equations of motion
4. Propulsion
5. Aerodynamics
6. Summary
7. Additional material (background information for AE1100 project)
13 Flight mechanics
Equations of motion
• Newton’s laws
• Coordinate systems
• Assumptions
• Equations of motion
Overview
14 Flight mechanics
Equations of motion Newton’s laws
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
15 Flight mechanics
Equations of motion
V
Zb
Ze
Xe
Xb
Xg
Zg
Xa
Za
G: Earth axis system
E: Moving Earth axis system
Coordinate systems
A: Air path axis system
B: Body axis system
Angles: - pitch attitude - angle of attack - flight path angle
16 Flight mechanics
Assumptions
Assumption 1: the earth is flat
2
2
Centrifugal force
e
e
W VC
g R h
C V
W R h g
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
17 Flight mechanics
Assumptions
Assumption 2: the earth is non-rotating
18 Flight mechanics
Foucault - story
19 Flight mechanics
Assumptions
Assumption 3: Gravity is constant
1 2
2
M MF
R
(Newton’s law of gravitation)
At maximum altitude for atmospheric flight (60 – 80 km), g is very close to g at sea level
Note: these assumptions are fine for flight mechanics but not for orbital mechanics
20 Flight mechanics
Equations of motion Free Body Diagram - Forces
L
D
W
T V
T
Zb
Ze
Xe
Xb
Xa
Important: • Lift vector perpendicular
to airspeed • Drag parallel to airspeed • Thrust not necessarily in
direction of Xb
21 Flight mechanics
Xa
Equations of motion Kinetic diagram - accelerations
V
Zb
Ze
Xe
dV
dt
dV
dt
Xb
22 Flight mechanics
Equations of motion
// : cos sin
: cos sin
V T
V T
W dVF T D W
g dt
W dF V L W T
g dt
23 Flight mechanics
Contents
1. What is Flight and Orbital Mechanics?
2. Practical matters
3. General equations of motion
4. Propulsion
5. Aerodynamics
6. Summary
7. Additional material (background information for AE1100 project)
24 Flight mechanics
Propulsion
1. Fundamental equations and definitions
2. Propeller or Jet engine?
3. Pure Jet engine
4. Propeller
Overview
25 Flight mechanics
Propulsion
0
0
out in
f j
j
F I I I
T m m V mV
T m V V
Propulsion System
Momentum equation
Force V0 Vj
Fuel is added to the mass flow
Conclusion A force is created by acceleration of mass. Two fundamental options:
1. Give a small amount of mass a large acceleration
2. Give a large amount of mass a small acceleration
Fundamental equations
26 Flight mechanics
Fundamental equations
• A small acceleration to a large mass:
• A large acceleration to a small mass:
• The mass can be taken from the surrounding
air (airbreathing engine) but it can also be
taken along
27 Flight mechanics
Fundamental equations
28 Flight mechanics
Fundamental equations
• Which propulsion type is the best?
• Propeller
• Jet
• Turbofan
• Ramjet
• Rocket
• Before we can answer this question we must define what is efficient
29 Flight mechanics
Useful definitions
• Power Available
• Jet Power
• Thermal Power
• Total efficiency
• Propulsive efficiency
• Thermal efficiency
30 Flight mechanics
Useful definitions
2 1
W F x
W T x x
2 1
WP
t
T x x xP T TV
t t
Power available
Work (energy) Power (energy per second)
V V
x t + t t
T T
aP TV
31 Flight mechanics
Useful definitions
Jet power is defined as the increase in kinetic energy of the flow
Jet power
Propulsion System
Force V0 Vj
Fuel is added to the mass flow
2 21 12 2
j jP mV mV
32 Flight mechanics
Useful definitions
FH
Qg
Thermal power
Thermal power (Q) is the heat energy supplied to the process (burning fuel)
f
in
Q m H
FHQ
g
Q mq
Variables: mf mass flow of fuel [kg/s] F fuel flow [N/s] H heating value [J/kg] (constant) m mass flow of air [kg/s] qin heat per kg air [J/kg]
33 Flight mechanics
Useful definitions
• Total efficiency is defined as the ratio of Power available
(energy for transportation) over Thermal power (fuel required)
• It is also the multiplication of propulsive efficiency and thermal
efficiency
Total Efficiency
atot
jatot j th
j
P
Q
PP
P Q
34 Flight mechanics
Useful definitions
2
1
aj
jj
P
VP
V
Propulsive efficiency
Conclusions: • The jet velocity Vj must be
larger than V in order to create thrust
• Therefore propulsive efficiency must be smaller than 100%!
35 Flight mechanics
Useful definitions
100 [N]
Option 1:
1 200 100 100 [N]
Option 2:
1 300 200 100 [N]
j
T
T m V V
T
T
,1
2 266%
20011
100
jjV
V
Jet or propeller?
• At large airspeeds it becomes more efficient to give a large acceleration to air
• At low airspeeds it is more efficient to give a large amount of air a small acceleration
,2
2 280%
30011
200
jjV
V
36 Flight mechanics
Useful definitions
Jet or propeller?
37 Flight mechanics
Useful definitions
jT m V V
2
1
atot j th
aj
jj
j
th
P
Q
P
VP
V
P
Q
Summary
2 21 12 2
a
j j
P TV
P mV mV
FHQ
g
Thrust Power Efficiency
Try to understand what these equations mean. Then you will be able to derive them. Do not memorize them
38 Flight mechanics
Working principle jet engine
See also, Physics 1 – lecture 7 (Chapter 9 of Cengel & Boles)
39 Flight mechanics
Working principle jet engine
• The compression at the intake depends on the flight velocity (kinetic
energy)
• The compression ratio (p2 / p1) at the intake is relatively low (in the
order of 1 2)
• Compression ratio in the compressor is relatively high (in the order of
30 – 40)
• Jet velocity is very high 0jT m V V
40 Flight mechanics
Propulsive force
Thrust is assumed to be independent of airspeed
Typical analytical assumption – Jet
41 Flight mechanics
Jet engine - simplified
Thrust
Airspeed
For basic flight mechanics applications, thrust of a turbojet can be assumed to be constant with airspeed for a given flight altitude
42 Flight mechanics
Working principle turboprop Schematic
43 Flight mechanics
Working principle turboprop
• Shaft power (Pbr) is provided by gasturbine
• Propeller accelerates air
44 Flight mechanics
Working principle propeller Motion of a propeller blade section
45 Flight mechanics
Working principle turboprop Induced velocity
46 Flight mechanics
Working principle propeller Actuator disk theory – blade element theory
47 Flight mechanics
Working principle propeller
Local velocities at a blade section
48 Flight mechanics
Working principle propeller Variable blade pitch
49 Flight mechanics
Working principle propeller Variable blade pitch
50 Flight mechanics
Propulsive force
Pa is assumed to be constant and independent of airspeed
Typical analytical assumption – Propeller
51 Flight mechanics
Propeller - simplified
Power available
Airspeed
For basic flight mechanics applications, power available of a propeller aircraft can be assumed to be constant with airspeed for a given flight altitude
52 Flight mechanics
Contents
1. What is Flight and Orbital Mechanics?
2. Practical matters
3. General equations of motion and power equation
4. Propulsion
5. Aerodynamics
6. Summary
7. Additional material (background information for AE1100 project)
53 Flight mechanics
Aerodynamics
212
2 1
L
L
L W
C V S W
WV
S C
Lift
max
min
2 1
L
WV
S C
54 Flight mechanics
Aerodynamic forces Drag polar
0
2
LD D
CC C
Ae
55 Flight mechanics
Aerodynamic forces Drag polar
0
2
LD D
CC C
Ae
56 Flight mechanics
Aerodynamic forces Lift – Drag polar
0
2
LD D
CC C
Ae
57 Flight mechanics
0
2
LD D
CC C
Ae
58 Flight mechanics
Aerodynamic forces
Lift
0
2
LD D
CC C
Ae21
2
2
2 1
L
L
L W
C V S W
WC
S V
Drag
So, one part of the drag decreases (!) with airspeed (1/V2) and one part increases with airspeed (V2)
Drag as a function of airspeed
212DD C V S
0
22 21 1
2 2L
D
CD C V S V S
Ae
0
22 21 1
2 22 2 4
4 1 1D
WD C V S V S
S V Ae
0
221
2 212
D
WD C V S
Ae V S
59 Flight mechanics
Aerodynamic forces Drag as a function of airspeed
Aircraft are quite unique in the sense that drag increases when airspeed decreases!
0
0
210 2
2
212
i
D
i
D D D
D C V S
WD
Ae V S
60 Flight mechanics
Aerodynamic forces
• Di predominant factor at low airspeeds
Large S; low wing loading (W/S) required
• D0 predominant factor at high airspeeds
Small parasite drag CD0 and small S; high wing loading (W/S)
Consequences for aircraft design
Very low speed aircraft (bicycle plane) high speed aircraft (F104 Starfighter)
61 Flight mechanics
Contents
1. What is Flight and Orbital Mechanics?
2. Practical matters
3. General equations of motion
4. Propulsion
5. Aerodynamics
6. Summary
7. Additional material (background information for AE1100 project)
62 Flight mechanics
Summary
• You should be able to derive the
equations of motion for 2
dimensional flight
• The thrust (T) of a jet aircraft can be
assumed independent of airspeed
• The power available (Pa) of a
propeller aircraft can be assumed
independent of airspeed
• The complete aircraft aerodynamics
can be represented by 1 equation;
the lift drag polar
// : cos sin
: cos sin
V T
V T
W dVF T D W
g dt
W dF V L W T
g dt
0
2
LD D
CC C
Ae
63 Flight mechanics
Questions
64 Flight mechanics
Contents
1. What is Flight and Orbital Mechanics?
2. Practical matters
3. General equations of motion
4. Propulsion
5. Aerodynamics
6. Summary
7. Additional material (background information for AE1100
project)
65 Flight mechanics
Additional material
Extra information which may be useful for the propeller practical in
the AE1100 project is included in the next couple of sheets. You do
not have to learn this for the exam.
Actuator disk theory
66 Flight mechanics
Working principle turboprop Actuator disk theory
67 Flight mechanics
Working principle propeller
221 1
0 0 1 02 2 ap V p V V
0jT m V V
132a aV V
Bernoulli
Momentum equation
Actuator disk theory
2 21 1
2 0 0 0 32 2a ap V V p V V
2
2 1T R p p
22 21 10 3 02 2aT R V V V
2 13 0 32a aT R V V V
0 3 0aT m V V V
3aT mV
2
3 0a aT R V V V
68 Flight mechanics
Working principle propeller
2 2102br jP m V V
22 21
0 0 3 02br a aP R V V V V V
22
0 3br a aP R V V V
Actuator disk theory – propulsive efficiency
This efficiency is a theoretical upper limit Assumptions: - No rotational kinetic energy in slipstream - Axial velocity is uniform over the disk
0 0
0
0
1
1j
abr a
TV V
VP V V
V
2 2
0
2
1 1
jT
R V
Shaft power Pbr may be expressed as the increase in kinetic energy of the air mass flow
69 Flight mechanics
Working principle propeller
Actuator disk theory – blade element theory
70 Flight mechanics
Working principle propeller
Actuator disk theory – blade element theory
71 Flight mechanics
Blade element theory How to calculate the thrust and torque of a prop
Step 1: Calculate angle of attack of blade element
Step 2: Calculate Lift and Drag of element
Step 3: Integrate forces over whole blade (thrust and torque)
Step 4: Calculate induced velocity (actuator disk)
Step 0: assume induced velocity is equal to zero