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# Ae2104 flight-mechanics-slides 3

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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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