Download - Ae2104 flight-mechanics-slides 3
1Challenge the future
Flight and Orbital Mechanics
Lecture slides
Semester 1 - 2012
Challenge the future
DelftUniversity ofTechnology
Flight and Orbital MechanicsLecture hours 5, 6, 7 – Turning performance
Mark Voskuijl
2AE2104 Flight and Orbital Mechanics |
Question
Lockheed F-104 StarfighterTop speed Mach 2.2 (at 10 km altitude)
How large would the turning radius be if an aircraft performs a horizontal steady turn at this airspeed, with a bank angle of 30 deg?
3AE2104 Flight and Orbital Mechanics |
Turn at Mach 2.2, bank angle 30 deg
Circle with radius 75 km…
4AE2104 Flight and Orbital Mechanics |
Content
• Aim
• How to fly a turn
• Equations of motion
• Load factor and performance
diagrams
• Turning performance
• Example calculations
• Advanced manoeuvres
• Altitude effects
• Summary
• Example exam question
5AE2104 Flight and Orbital Mechanics |
Content
• Aim
• How to fly a turn
• Equations of motion
• Load factor and performance
diagrams
• Turning performance
• Example calculations
• Advanced manoeuvres
• Altitude effects
• Summary
• Example exam question
6AE2104 Flight and Orbital Mechanics |
Aim
Calculate turning performance of a given aircraft
• Minimum turn radius (tightest turn)
• Minimum time to turn (fastest turn)
• Maximum load factor (steepest turn)
• Rate one/two/three turn (civil operations)
• Assumption: sustained turns
7AE2104 Flight and Orbital Mechanics |
Content
• Aim
• How to fly a turn
• Equations of motion
• Load factor and performance
diagrams
• Turning performance
• Example calculations
• Advanced manoeuvres
• Altitude effects
• Summary
• Example exam question
8AE2104 Flight and Orbital Mechanics |
How to fly a turn?
9AE2104 Flight and Orbital Mechanics |
How to fly a turn
L
W
L W
T D
Steady horizontal flight
10AE2104 Flight and Orbital Mechanics |
How to fly a turn
W
Lcos
Only roll:Vertical component of the lift is smaller than the weight
cosL W Roll the aircraft
11AE2104 Flight and Orbital Mechanics |
How to fly a turn
cos
L W
L W
Lcos
Lcos
D
D T
Roll the aircraft(increase angle of attack)
12AE2104 Flight and Orbital Mechanics |
How to fly a turnRoll the aircraft and increase the pitch
13AE2104 Flight and Orbital Mechanics |
How to fly a turn
1. Roll the aircraft to start turning
2. Nose up to maintain altitude
3. Increase the thrust to maintain airspeed
14AE2104 Flight and Orbital Mechanics |
Content
• Aim
• How to fly a turn
• Equations of motion
• Load factor and performance
diagrams
• Turning performance
• Example calculations
• Advanced manoeuvres
• Altitude effects
• Summary
• Example exam question
15AE2104 Flight and Orbital Mechanics |
Equations of motion
0 T D
2
sin WV
LgR
T
Lcos
D
L
Lsin
W
V
Lsin
=d / dt
R
Free Body Diagram
0
mV2 / R
mV2 / R
0
Kinetic Diagram
F ma
Eq. of motion
2
sin
cos
T D
WVL
gR
L W
Horizontal sustained turn
0 cosL W
16AE2104 Flight and Orbital Mechanics |
Content
• Aim
• How to fly a turn
• Equations of motion
• Load factor and performance
diagrams
• Turning performance
• Example calculations
• Advanced manoeuvres
• Altitude effects
• Summary
• Example exam question
17AE2104 Flight and Orbital Mechanics |
Load factor and performance diagrams
1
cos costurn
Ln
L
cosW L
Ln
W
L nW
Load factor:
Apparent weight(C is a virtual / fictitious force)
Load factor during steady horizontal turn:
,30
,45
,60
1.15
1.41
2
turn
turn
turn
n
n
n
Load factor
Video 1 F22 high g turnhttp://www.youtube.com/watch?v=n34RwIUlnAo&feature=related
Video 2 (pilot’s perspective)http://www.youtube.com/watch?v=5LMVVey5bCM&feature=related
18AE2104 Flight and Orbital Mechanics |
Load factor and performance diagrams
Drag
V
Performance diagram
What will happen to this diagram in a turn?
19AE2104 Flight and Orbital Mechanics |
Load factor and performance diagrams
212LC V S nW
2 1
L
nWV
S C
, LV function n C
L nW L nWD D D
L L rP DV
Airspeed Aerodynamic drag Power required
Note:
a r
T D
P P
Conclusion:V, D, Pr are functions of lift coefficient and load factor(In symmetric flight, they only depend on the lift coefficient)
D
L
CD nW
C
, LD function n C
2 1Dr
L L
C nWP nW
C S C
23 3
3
2 Dr
L
Cn WP
S C
,r LP function n C
Performance diagram
20AE2104 Flight and Orbital Mechanics |
Load factor and performance diagrams
2 1
L
nWV n
S C
D
L
CD nW n
C
Effect of load factor (n) on airspeed (V)(at constant angle of attack)
rP DV n n
Effect of load factor (n) on aerodynamic drag (D)(at constant angle of attack)
Effect of load factor (n) on Power required (Pr)(at constant angle of attack)
Performance diagram
21AE2104 Flight and Orbital Mechanics |
Load factor and performance diagrams
Drag
V
Each point relates to 1 angle of attack (CL)
(note: this graph is purely qualitative)
Performance diagram
22AE2104 Flight and Orbital Mechanics |
Load factor and performance diagrams
23AE2104 Flight and Orbital Mechanics |
Load factor and performance diagrams
Tmax
Vnmax
T, D
V
1. Vmin increases when n increases
2. Vmin first aerodynamically limited then thrust limited
(safety: regulation for stick force/g)
3. Vmax decreases when n increases
4. At nmax Vmin = Vmax
Performance diagram
24AE2104 Flight and Orbital Mechanics |
Load factor and performance diagrams
Tmax
n limited by engine
T, D
V
n limited by CLmax
2 speed ranges
Performance diagram
25AE2104 Flight and Orbital Mechanics |
Content
• Aim
• How to fly a turn
• Equations of motion
• Load factor and performance
diagrams
• Turning performance
• Example calculations
• Advanced manoeuvres
• Altitude effects
• Summary
• Example exam question
26AE2104 Flight and Orbital Mechanics |
Turning performance
• Maximum load factor (steepest turn)
• Minimum turn radius (tightest turn)
• Minimum time to turn (fastest turn)
• Rate one/two/three turn (civil operations)
Assumption: sustained turns
27AE2104 Flight and Orbital Mechanics |
Turning performance
• Maximum load factor (steepest turn)
• Minimum turn radius (tightest turn)
• Minimum time to turn (fastest turn)
• Rate one/two/three turn (civil operations)
Assumption: sustained turns
28AE2104 Flight and Orbital Mechanics |
Steepest turn
Tmax
T, D
V
Tmax
T, D
V
Aim: For each airspeed nmax
V1V2V3V4 V5 V6 V7 V8 V9
29AE2104 Flight and Orbital Mechanics |
Steepest turn
V
n
1
Achievable load factor as function of airspeed
30AE2104 Flight and Orbital Mechanics |
Steepest turn
2max
2
2 1
LCL nW n V
W
S V
D L
L D
C CTT D nW n
C W C
• Range A:
• Range B:
maxL LC C
max Vary LT T C
Tmax
n limited by engine
T, D
Vn limited by CLmax
Calculation of nmax(V)
L
2 1 at every C ,
L
nWV n V
S C
31AE2104 Flight and Orbital Mechanics |
Steepest turn
max
L
D
C
C
Tmax
Vnmax
D at nmax
V
D, T
• Vnmax at Dmin
• D
L
CD nW
C
Solution for jet aircraft
Note: You must be able to derive the final step
0L DC C Ae
32AE2104 Flight and Orbital Mechanics |
Steepest turn
0
max
maxmax
max
max 2 1
opt
opt
D
L
L
D
L D
n
L
CT D nW
C
T Cn
W C
C C Ae
n WV
S C
Solution for jet aircraft
(A numerical example will follow later)
33AE2104 Flight and Orbital Mechanics |
Turning performance
• Maximum load factor (steepest turn)
• Minimum turn radius (tightest turn)
• Minimum time to turn (fastest turn)
• Rate one/two/three turn (civil operations)
• Assumption: sustained turns
34AE2104 Flight and Orbital Mechanics |
Do not confuse turn radius with time to turn!
35AE2104 Flight and Orbital Mechanics |
Minimum turn radius (tightest turn)
First we must have an equation for the turn radius
Using the equilibrium equations:
1
cos
Ln
W
2
sin
cos
W VL
g R
W L
2 2
2
1cos sin 1 sin 1x x
n
2 2
2 2
11
1
W V VnW R
n g R g n
36AE2104 Flight and Orbital Mechanics |
Minimum turn radius
For a given n, R decreases when V decreases. So, tighter turns are possible at lower airspeeds
An increase in n for a given V results in a decrease in R
(remember the example of the aircraft turning at Mach 2)
2
2 1
VR
g n
V
R
37AE2104 Flight and Orbital Mechanics |
2
2 1
VR
g n
nmax
R
V
V
Tightest turn
38AE2104 Flight and Orbital Mechanics |
Turn at Mach 2.2, bank angle 30 deg
Circle with radius 75 km…
39AE2104 Flight and Orbital Mechanics |
MMO
VMO
40AE2104 Flight and Orbital Mechanics |
Turning performance
• Maximum load factor (steepest turn)
• Minimum turn radius (tightest turn)
• Minimum time to turn (fastest turn)
• Rate one/two/three turn (civil operations)
• Assumption: sustained turns
41AE2104 Flight and Orbital Mechanics |
Fastest turn
V
n
1
V
R
V
T
2
2 RT
V
Minimize time to achieve fastest turnTime to complete a full circle:
(Circumference of turn: 2R)
42AE2104 Flight and Orbital Mechanics |
Turning performance
• Maximum load factor (steepest turn)
• Minimum turn radius (tightest turn)
• Minimum time to turn (fastest turn)
• Rate one/two/three turn (civil operations)
• Assumption: sustained turns
43AE2104 Flight and Orbital Mechanics |
Rate one turn
• Rate one turn: 180 deg/min
• Rate two turn: 360 deg/min
Standardized turns are useful for
air traffic control
Safety:< 200’ no turns light a/c< 1000’ no turns heavy a/c
44AE2104 Flight and Orbital Mechanics |
Rate one turn
Example:
Approach: V = 80 m/s, T = 60 [s]
801502 [m]
/ 60
V VR
R
22 2
21 1.09
1
V VR n
gRg n
V
R
1arccos 23
n
45AE2104 Flight and Orbital Mechanics |
Rate one turn
46AE2104 Flight and Orbital Mechanics |
Rate one turn
47AE2104 Flight and Orbital Mechanics |
Content
• Aim
• How to fly a turn
• Equations of motion
• Load factor and performance
diagrams
• Turning performance
• Example calculations
• Advanced manoeuvres
• Altitude effects
• Summary
• Example exam question
48AE2104 Flight and Orbital Mechanics |
Example calculations
Numerical example – turning performance at 7000 [m] altitude
Lift drag polar is known for this aircraft
Aircraft weight, maximum thrust and dimensions are known as well
0
2
LD D
CC C
Ae
49AE2104 Flight and Orbital Mechanics |
Example calculations
0
3
max
2
0.59 [kg/m ] (ISA)
8.67 [kN]
0.021
7 [-]
60000 [N]
30 [m ]
D
T
C
Ae
W
S
Aircraft and atmospheric data at 7000 [m]
Calculate:1. Maximum load factor and corresponding airspeed2. Radius of tightest turn (Rmin) and corresponding airspeed3. Time for fastest turn (T2,min) and corresponding airspeed
50AE2104 Flight and Orbital Mechanics |
Example calculations
max 8670 [N]T
maxmax
max
L
D
T Cn
W C
0
max
LL D
D
CC C Ae
C
0.021 7 0.68 [-]LC
0
2
LD D
CC C
Ae
max
8670 0.682.34
60000 0.042 n
maxmax
2 1
2.34 60000 2 1
30 0.59 0.68
152.7 [m/s]
n
L
n WV
S C
Maximum load factor20.68
0.021 0.042 [-]7
DC
51AE2104 Flight and Orbital Mechanics |
Example calculationsMaximum load factor
52AE2104 Flight and Orbital Mechanics |
Example calculations
Vnmax , “steepest turn”
Maximum load factor
53AE2104 Flight and Orbital Mechanics |
Example calculationsMinimum turn radius
Vrmin – “tightest turn”
54AE2104 Flight and Orbital Mechanics |
Example calculations
VTmin – fastest turn
Minimum time to turn (fastest turn)
55AE2104 Flight and Orbital Mechanics |
Example calculations
2
2tan
V
gT
max maxn
2
2 RT
V
All results combined
56AE2104 Flight and Orbital Mechanics |
Content
• Aim
• How to fly a turn
• Equations of motion
• Load factor and performance
diagrams
• Turning performance
• Example calculations
• Advanced manoeuvres
• Altitude effects
• Summary
• Example exam question
57AE2104 Flight and Orbital Mechanics |
Advanced manoeuvresImmelmann
58AE2104 Flight and Orbital Mechanics |
Advanced ManoeuvresThrust vectoring
http://www.youtube.com/watch?v=CGbOs0vgYOA&NR=1(thrust vecotring turn at 2:50)
59AE2104 Flight and Orbital Mechanics |
Content
• Aim
• How to fly a turn
• Equations of motion
• Load factor and performance
diagrams
• Turning performance
• Example calculations
• Advanced manoeuvres
• Altitude effects
• Summary
• Example exam question
60AE2104 Flight and Orbital Mechanics |
Altitude effects
0.75
max
max
L
D
CTn
W C
max
2 1
n
L
nWV
S C
max
2
2 if
1n
VR
g n
Equations
61AE2104 Flight and Orbital Mechanics |
Altitude effectsExample
62AE2104 Flight and Orbital Mechanics |
Altitude effectsExample
63AE2104 Flight and Orbital Mechanics |
Content
• Aim
• How to fly a turn
• Equations of motion
• Load factor and performance
diagrams
• Turning performance
• Example calculations
• Advanced manoeuvres
• Altitude effects
• Summary
• Example exam question
64AE2104 Flight and Orbital Mechanics |
Summary
• Be able to derive the equations of motion for a horizontal sustained
turn (what is the definition of horizontal and sustained?)
• Load factor is defined as n = L/W
• In a horizontal sustained turn, n = 1/cos
2
sin
cos
T D
mVL
R
L W
65AE2104 Flight and Orbital Mechanics |
Summary
• Maximum load factor for jet aircraft:
• Turn radius:
• You must be able to derive the equations mentioned above
• Time to turn 360 deg:
• Rate one turn is defined as 180 deg / min
2
2 RT
V
maxmax
max
L
D
T Cn
W C
2
21
VR
g n
66AE2104 Flight and Orbital Mechanics |
Content
• Aim
• How to fly a turn
• Equations of motion
• Load factor and performance
diagrams
• Turning performance
• Example calculations
• Advanced manoeuvres
• Altitude effects
• Summary
• Example exam question
67AE2104 Flight and Orbital Mechanics |
Example exam question
The following data is known for a small propeller aircraft (Cessna 172)
Pa = 100kW (independent of airspeed)
W = 10 kN
S = 15 m2
The maximum rate of climb in steady symmetric flight of this aircraft equals 6 m/s when
operating at sea level (0 m) in the International Standard Atmosphere.
a.Calculate the maximum load factor and corresponding bank angle of this aircraft in a horizontal
steady turn when operating at the same altitude, power setting and angle of attack
b. The aircraft performs a rate one turn with an airspeed of 180 km/h at sea level in the
international standard atmosphere. Calculate the required bank angle and corresponding turn
radius.
68AE2104 Flight and Orbital Mechanics |
Questions?