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Flight and Orbital Mechanics
Lecture slides

Challenge the future

1
Flight and Orbital Mechanics
Lecture hours 5, 6, 7 – Turning performance
Mark Voskuijl
Semester 1 - 2012

Delft
University...
Question

Lockheed F-104 Starfighter
Top speed Mach 2.2 (at 10 km altitude)

How large would the turning radius be if an a...
Turn at Mach 2.2, bank angle 30 deg
Circle with radius 75 km…

AE2104 Flight and Orbital Mechanics

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Content
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Aim
How to fly a turn
Equations of motion
Load factor and performance
diagrams
Turning perfor...
Content
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Aim
How to fly a turn
Equations of motion
Load factor and performance
diagrams
Turning perfor...
Aim
Calculate turning performance of a given aircraft
• Minimum turn radius (tightest turn)

• Minimum time to turn (faste...
Content
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Aim
How to fly a turn
Equations of motion
Load factor and performance
diagrams
Turning perfor...
How to fly a turn?

AE2104 Flight and Orbital Mechanics

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How to fly a turn
Steady horizontal flight
L

L W
T D

W

AE2104 Flight and Orbital Mechanics

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How to fly a turn
Roll the aircraft

L cos   W
Lcos

Only roll:
Vertical
component of
the lift is
smaller than
the weig...
How to fly a turn
Roll the aircraft
Lcos

L W
L cos   W

D
D T

(increase angle of attack)

Lcos



AE2104 Flight ...
How to fly a turn
Roll the aircraft and increase the pitch

AE2104 Flight and Orbital Mechanics

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How to fly a turn
1. Roll the aircraft to start turning
2. Nose up to maintain altitude
3. Increase the thrust to maintain...
Content
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Aim
How to fly a turn
Equations of motion
Load factor and performance
diagrams
Turning perfor...
Equations of motion
Horizontal sustained turn
Free Body Diagram
Lcos

Kinetic Diagram

F  ma

0

L

mV2 / R

Lsin
W

0 ...
Content
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Aim
How to fly a turn
Equations of motion
Load factor and performance
diagrams
Turning perfor...
Load factor and performance diagrams
Load factor
Load factor:
L
W
L  nW

n

Load factor during steady horizontal turn:

...
Load factor and performance diagrams
Performance diagram
Drag

What will happen to this
diagram in a turn?

V

AE2104 Flig...
Load factor and performance diagrams
Performance diagram
Airspeed

L  nW

CL 1 V 2 S  nW
2
nW 2 1
V
S  CL

V  functi...
Load factor and performance diagrams
Performance diagram
Effect of load factor (n) on airspeed (V)
(at constant angle of a...
Load factor and performance diagrams
Performance diagram
Drag

Each point relates to 1
angle of attack (CL)

(note: this g...
Load factor and performance diagrams

AE2104 Flight and Orbital Mechanics

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Load factor and performance diagrams
Performance diagram

1.

T, D

2.

3.

Tmax

4.

Vnmax

Vmin increases when n
increas...
Load factor and performance diagrams
Performance diagram
T, D

2 speed ranges

Tmax

V
n limited by CLmax

n limited by en...
Content
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Aim
How to fly a turn
Equations of motion
Load factor and performance
diagrams
Turning perfor...
Turning performance
• Maximum load factor (steepest turn)
• Minimum turn radius (tightest turn)

• Minimum time to turn (f...
Turning performance
• Maximum load factor (steepest turn)
• Minimum turn radius (tightest turn)

• Minimum time to turn (f...
Steepest turn
T, D

Aim: For each airspeed nmax

Tmax

V1V34 V5 V6 V7 V8 V9
VV
2

V

AE2104 Flight and Orbital Mechanics

...
Steepest turn
Achievable load factor as function of airspeed
n

1

V
AE2104 Flight and Orbital Mechanics

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Steepest turn

T, D

Calculation of nmax(V)
Tmax

• Range A:

CL  CL max
L  nW  n 

• Range B:

CL max
V 2
W 2 1
S  ...
Steepest turn
Solution for jet aircraft
• Vnmax at Dmin

D, T

D at nmax

CD
nW
• D
CL
 CL 


 CD max

Tmax

 CL ...
Steepest turn
Solution for jet aircraft
CD
T D
nW
CL
nmax

Tmax

W

(A numerical example will follow later)

 CL 

...
Turning performance
• Maximum load factor (steepest turn)
• Minimum turn radius (tightest turn)

• Minimum time to turn (f...
Do not confuse turn radius with time to turn!

AE2104 Flight and Orbital Mechanics

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Minimum turn radius (tightest turn)
First we must have an equation for the turn radius
Using the equilibrium equations:

W...
Minimum turn radius
R

V

2

R

g n2  1

V

For a given n, R decreases when V decreases. So,
tighter turns are possible ...
nmax

V

R 

R

V2
g n2 1

Tightest turn

V
AE2104 Flight and Orbital Mechanics

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Turn at Mach 2.2, bank angle 30 deg
Circle with radius 75 km…

AE2104 Flight and Orbital Mechanics

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MMO

VMO

AE2104 Flight and Orbital Mechanics

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Turning performance
• Maximum load factor (steepest turn)
• Minimum turn radius (tightest turn)

• Minimum time to turn (f...
Fastest turn
Minimize time to achieve fastest turn
Time to complete a full circle:

2 R
T2 
V

n
1

V
R

(Circumference...
Turning performance
• Maximum load factor (steepest turn)
• Minimum turn radius (tightest turn)

• Minimum time to turn (f...
Rate one turn
• Rate one turn: 180 deg/min
• Rate two turn: 360 deg/min
Standardized turns are useful for
air traffic cont...
Rate one turn
Example:
Approach: V = 80 m/s, T = 60 [s]

V
V
80
 R 
 1502 [m]
R
  / 60

V

R

2

V 
R
n 
 ...
Rate one turn

AE2104 Flight and Orbital Mechanics

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Rate one turn

AE2104 Flight and Orbital Mechanics

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Content
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Aim
How to fly a turn
Equations of motion
Load factor and performance
diagrams
Turning perfor...
Example calculations

Numerical example – turning performance at 7000 [m] altitude
Lift drag polar is known for this aircr...
Example calculations
Aircraft and atmospheric data at 7000 [m]

  0.59 [kg/m3 ] (ISA)
Tmax  8.67 [kN]
CD0  0.021
Ae  ...
Example calculations
Maximum load factor
nmax

Tmax  CL 



W  CD max

0.682
CD  0.021 
 0.042 [-]
7

Tmax  86...
Example calculations
Maximum load factor

AE2104 Flight and Orbital Mechanics

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Example calculations
Maximum load factor

Vnmax , “steepest turn”

AE2104 Flight and Orbital Mechanics

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Example calculations
Minimum turn radius

Vrmin – “tightest turn”
AE2104 Flight and Orbital Mechanics

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Example calculations
Minimum time to turn (fastest turn)

VTmin – fastest turn

AE2104 Flight and Orbital Mechanics

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Example calculations
All results combined
nmax  max

tan  

T2 

2 V
gT2

2 R
V
AE2104 Flight and Orbital Mechani...
Content
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Aim
How to fly a turn
Equations of motion
Load factor and performance
diagrams
Turning perfor...
Advanced manoeuvres
Immelmann

AE2104 Flight and Orbital Mechanics

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http://www.youtube.com/watch?v=CGbOs0vgYOA&NR=1
(thrust vecotring turn at 2:50)

Advanced Manoeuvres
Thrust vectoring

AE2...
Content
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Aim
How to fly a turn
Equations of motion
Load factor and performance
diagrams
Turning perfor...
Altitude effects
Equations
 CL 
  0.75


 CD max

nmax

T

W

Vnmax

nW 2 1


S  CL

Rnmax 

V2
g n 1
2

 ...
Altitude effects
Example

AE2104 Flight and Orbital Mechanics

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Altitude effects
Example

AE2104 Flight and Orbital Mechanics

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Content
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Aim
How to fly a turn
Equations of motion
Load factor and performance
diagrams
Turning perfor...
Summary
• Be able to derive the equations of motion for a horizontal sustained
turn (what is the definition of horizontal ...
Summary
• Maximum load factor for jet aircraft:
• Turn radius: R 

nmax

Tmax

W

 CL 


CD max


V2
g 1  n2

• Y...
Content
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Aim
How to fly a turn
Equations of motion
Load factor and performance
diagrams
Turning perfor...
Example exam question
The following data is known for a small propeller aircraft (Cessna 172)
Pa = 100kW (independent of a...
Questions?

AE2104 Flight and Orbital Mechanics

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These are the lecture slides to the third lecture of the course AE2104 Flight and Orbital mechanics.

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Transcript of "Ae2104 flight-mechanics-slides 3"

  1. 1. Flight and Orbital Mechanics Lecture slides Challenge the future 1
  2. 2. Flight and Orbital Mechanics Lecture hours 5, 6, 7 – Turning performance Mark Voskuijl Semester 1 - 2012 Delft University of Technology Challenge the future
  3. 3. Question Lockheed F-104 Starfighter Top 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? AE2104 Flight and Orbital Mechanics 2|
  4. 4. Turn at Mach 2.2, bank angle 30 deg Circle with radius 75 km… AE2104 Flight and Orbital Mechanics 3|
  5. 5. 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 AE2104 Flight and Orbital Mechanics 4|
  6. 6. 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 AE2104 Flight and Orbital Mechanics 5|
  7. 7. 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 AE2104 Flight and Orbital Mechanics 6|
  8. 8. 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 AE2104 Flight and Orbital Mechanics 7|
  9. 9. How to fly a turn? AE2104 Flight and Orbital Mechanics 8|
  10. 10. How to fly a turn Steady horizontal flight L L W T D W AE2104 Flight and Orbital Mechanics 9|
  11. 11. How to fly a turn Roll the aircraft L cos   W Lcos Only roll: Vertical component of the lift is smaller than the weight  W AE2104 Flight and Orbital Mechanics 10 |
  12. 12. How to fly a turn Roll the aircraft Lcos L W L cos   W D D T (increase angle of attack) Lcos  AE2104 Flight and Orbital Mechanics 11 |
  13. 13. How to fly a turn Roll the aircraft and increase the pitch AE2104 Flight and Orbital Mechanics 12 |
  14. 14. 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 AE2104 Flight and Orbital Mechanics 13 |
  15. 15. 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 AE2104 Flight and Orbital Mechanics 14 |
  16. 16. Equations of motion Horizontal sustained turn Free Body Diagram Lcos Kinetic Diagram F  ma 0 L mV2 / R Lsin W 0 T D WV 2  L sin  gR 0  L cos   W V T D T D Eq. of motion =d / dt Lsin R 0 mV2 / R WV 2  L sin  gR L cos   W AE2104 Flight and Orbital Mechanics 15 |
  17. 17. 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 AE2104 Flight and Orbital Mechanics 16 |
  18. 18. Load factor and performance diagrams Load factor Load factor: L W L  nW n Load factor during steady horizontal turn: W  L cos  nturn  L 1  L cos  cos  nturn ,30  1.15 nturn ,45  1.41 Apparent weight n 2 (C is a virtual / fictitious force) turn ,60 Video 1 F22 high g turn http://www.youtube.com/watch?v=n34RwIUlnAo&feature=related Video 2 (pilot’s perspective) http://www.youtube.com/watch?v=5LMVVey5bCM&feature=related AE2104 Flight and Orbital Mechanics 17 |
  19. 19. Load factor and performance diagrams Performance diagram Drag What will happen to this diagram in a turn? V AE2104 Flight and Orbital Mechanics 18 |
  20. 20. Load factor and performance diagrams Performance diagram Airspeed L  nW CL 1 V 2 S  nW 2 nW 2 1 V S  CL V  function  n, CL  Note: T D Pa  Pr Aerodynamic drag DD D L nW D L L CD nW CL D  function  n, CL  Power required Pr  DV C nW 2 1 Pr  D nW CL S  CL 2 n3W 3 2 CD Pr  3 S  CL Pr  function  n, CL  Conclusion: V, D, Pr are functions of lift coefficient and load factor (In symmetric flight, they only depend on the lift coefficient) AE2104 Flight and Orbital Mechanics 19 |
  21. 21. Load factor and performance diagrams Performance diagram Effect of load factor (n) on airspeed (V) (at constant angle of attack) V nW 2 1  n S  CL Effect of load factor (n) on aerodynamic drag (D) (at constant angle of attack) D CD nW  n CL Effect of load factor (n) on Power required (Pr) (at constant angle of attack) Pr  DV  n n AE2104 Flight and Orbital Mechanics 20 |
  22. 22. Load factor and performance diagrams Performance diagram Drag Each point relates to 1 angle of attack (CL) (note: this graph is purely qualitative) V AE2104 Flight and Orbital Mechanics 21 |
  23. 23. Load factor and performance diagrams AE2104 Flight and Orbital Mechanics 22 |
  24. 24. Load factor and performance diagrams Performance diagram 1. T, D 2. 3. Tmax 4. Vnmax Vmin increases when n increases Vmin first aerodynamically limited then thrust limited (safety: regulation for stick force/g) Vmax decreases when n increases At nmax Vmin = Vmax V AE2104 Flight and Orbital Mechanics 23 |
  25. 25. Load factor and performance diagrams Performance diagram T, D 2 speed ranges Tmax V n limited by CLmax n limited by engine AE2104 Flight and Orbital Mechanics 24 |
  26. 26. 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 AE2104 Flight and Orbital Mechanics 25 |
  27. 27. 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 AE2104 Flight and Orbital Mechanics 26 |
  28. 28. 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 AE2104 Flight and Orbital Mechanics 27 |
  29. 29. Steepest turn T, D Aim: For each airspeed nmax Tmax V1V34 V5 V6 V7 V8 V9 VV 2 V AE2104 Flight and Orbital Mechanics 28 |
  30. 30. Steepest turn Achievable load factor as function of airspeed n 1 V AE2104 Flight and Orbital Mechanics 29 |
  31. 31. Steepest turn T, D Calculation of nmax(V) Tmax • Range A: CL  CL max L  nW  n  • Range B: CL max V 2 W 2 1 S  V2 T  Tmax Vary CL T D V n limited n limited by engine by CLmax V CD T CL nW  n  CL W CD nW 2 1 S  CL at every CL   n,V  AE2104 Flight and Orbital Mechanics 30 |
  32. 32. Steepest turn Solution for jet aircraft • Vnmax at Dmin D, T D at nmax CD nW • D CL  CL     CD max Tmax  CL  CD0  Ae Vnmax V Note: You must be able to derive the final step AE2104 Flight and Orbital Mechanics 31 |
  33. 33. Steepest turn Solution for jet aircraft CD T D nW CL nmax Tmax  W (A numerical example will follow later)  CL     CD  max CLopt  CD0  Ae Vnmax nmaxW 2 1  S  CLopt AE2104 Flight and Orbital Mechanics 32 |
  34. 34. 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 AE2104 Flight and Orbital Mechanics 33 |
  35. 35. Do not confuse turn radius with time to turn! AE2104 Flight and Orbital Mechanics 34 |
  36. 36. Minimum turn radius (tightest turn) First we must have an equation for the turn radius Using the equilibrium equations: W V2 L sin   g R W  L cos  L 1 n  W cos  1 cos x  sin x  1  sin   1  2 n 1 W V2 V2 nW 1  2  R n g R g n2  1 2 2 AE2104 Flight and Orbital Mechanics 35 |
  37. 37. Minimum turn radius R V 2 R g n2  1 V 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) AE2104 Flight and Orbital Mechanics 36 |
  38. 38. nmax V R  R V2 g n2 1 Tightest turn V AE2104 Flight and Orbital Mechanics 37 |
  39. 39. Turn at Mach 2.2, bank angle 30 deg Circle with radius 75 km… AE2104 Flight and Orbital Mechanics 38 |
  40. 40. MMO VMO AE2104 Flight and Orbital Mechanics 39 |
  41. 41. 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 AE2104 Flight and Orbital Mechanics 40 |
  42. 42. Fastest turn Minimize time to achieve fastest turn Time to complete a full circle: 2 R T2  V n 1 V R (Circumference of turn: 2R) T V V AE2104 Flight and Orbital Mechanics 41 |
  43. 43. 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 AE2104 Flight and Orbital Mechanics 42 |
  44. 44. 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 AE2104 Flight and Orbital Mechanics 43 |
  45. 45. Rate one turn Example: Approach: V = 80 m/s, T = 60 [s] V V 80  R   1502 [m] R   / 60 V R 2 V  R n    1  1.09 g n2  1  gR  V 2 2  1   arccos    23 n AE2104 Flight and Orbital Mechanics 44 |
  46. 46. Rate one turn AE2104 Flight and Orbital Mechanics 45 |
  47. 47. Rate one turn AE2104 Flight and Orbital Mechanics 46 |
  48. 48. 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 AE2104 Flight and Orbital Mechanics 47 |
  49. 49. Example calculations Numerical example – turning performance at 7000 [m] altitude Lift drag polar is known for this aircraft 2 CL CD  CD0   Ae Aircraft weight, maximum thrust and dimensions are known as well AE2104 Flight and Orbital Mechanics 48 |
  50. 50. Example calculations Aircraft and atmospheric data at 7000 [m]   0.59 [kg/m3 ] (ISA) Tmax  8.67 [kN] CD0  0.021 Ae  7 [-] W  60000 [N] S  30 [m 2 ] Calculate: 1. Maximum load factor and corresponding airspeed 2. Radius of tightest turn (Rmin) and corresponding airspeed 3. Time for fastest turn (T2,min) and corresponding airspeed AE2104 Flight and Orbital Mechanics 49 |
  51. 51. Example calculations Maximum load factor nmax Tmax  CL     W  CD max 0.682 CD  0.021   0.042 [-] 7 Tmax  8670 [N] nmax  CL     CL  CD0  Ae  CD max Vn max  CL  0.021   7  0.68 [-] 2 CL CD  CD0   Ae 8670 0.68   2.34 60000 0.042 nmaxW 2 1 S  CL 2.34  60000 2 1  30 0.59 0.68  152.7 [m/s] AE2104 Flight and Orbital Mechanics 50 |
  52. 52. Example calculations Maximum load factor AE2104 Flight and Orbital Mechanics 51 |
  53. 53. Example calculations Maximum load factor Vnmax , “steepest turn” AE2104 Flight and Orbital Mechanics 52 |
  54. 54. Example calculations Minimum turn radius Vrmin – “tightest turn” AE2104 Flight and Orbital Mechanics 53 |
  55. 55. Example calculations Minimum time to turn (fastest turn) VTmin – fastest turn AE2104 Flight and Orbital Mechanics 54 |
  56. 56. Example calculations All results combined nmax  max tan   T2  2 V gT2 2 R V AE2104 Flight and Orbital Mechanics 55 |
  57. 57. 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 AE2104 Flight and Orbital Mechanics 56 |
  58. 58. Advanced manoeuvres Immelmann AE2104 Flight and Orbital Mechanics 57 |
  59. 59. http://www.youtube.com/watch?v=CGbOs0vgYOA&NR=1 (thrust vecotring turn at 2:50) Advanced Manoeuvres Thrust vectoring AE2104 Flight and Orbital Mechanics 58 |
  60. 60. 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 AE2104 Flight and Orbital Mechanics 59 |
  61. 61. Altitude effects Equations  CL    0.75    CD max nmax T  W Vnmax nW 2 1   S  CL Rnmax  V2 g n 1 2  if   AE2104 Flight and Orbital Mechanics 60 |
  62. 62. Altitude effects Example AE2104 Flight and Orbital Mechanics 61 |
  63. 63. Altitude effects Example AE2104 Flight and Orbital Mechanics 62 |
  64. 64. 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 AE2104 Flight and Orbital Mechanics 63 |
  65. 65. Summary • Be able to derive the equations of motion for a horizontal sustained turn (what is the definition of horizontal and sustained?) T D mV 2  L sin  R L cos   W • Load factor is defined as n = L/W • In a horizontal sustained turn, n = 1/cos AE2104 Flight and Orbital Mechanics 64 |
  66. 66. Summary • Maximum load factor for jet aircraft: • Turn radius: R  nmax Tmax  W  CL    CD max  V2 g 1  n2 • You must be able to derive the equations mentioned above • Time to turn 360 deg: T2  2 R V • Rate one turn is defined as 180 deg / min AE2104 Flight and Orbital Mechanics 65 |
  67. 67. 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 AE2104 Flight and Orbital Mechanics 66 |
  68. 68. 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. AE2104 Flight and Orbital Mechanics 67 |
  69. 69. Questions? AE2104 Flight and Orbital Mechanics 68 |
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