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1 Challenge the future Introduction to Aerospace Engineering Lecture slides

Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

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Page 1: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

1 Challenge the future

Introduction to Aerospace Engineering

Lecture slides

Page 2: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

15-12-2012

Challenge the future

Delft University of Technology

Introduction Aerospace Engineering

Flight and Mechanics

Dr. ir. Mark Voskuijl

Page 3: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

2 Flight Mechanics

5.&6 Climbing and descending flight

Page 4: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

3 Flight Mechanics

Contents

1. Summary horizontal flight

2. Introduction climbing and descending flight

3. Equations of motion straight and symmetric flight

4. Gliding flight

5. Climbing flight

6. Example calculations

Page 5: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

4 Flight Mechanics

Contents

1. Summary horizontal flight

2. Introduction climbing and descending flight

3. Equations of motion straight and symmetric flight

4. Gliding flight

5. Climbing flight

6. Example calculations

Page 6: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

5 Flight Mechanics

Equations of motion General equations of motion in symmetric flight (1)

Free Body Diagram Kinetic Diagram

L

D

W

T V

T

Zb

Ze

Xe

V

Zb

Ze

Xe

dmV

dt

dVm

dt

Xb Xb

Page 7: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

6 Flight Mechanics

Equations of motion

: cos sin

: cos sin

T

T

F ma

dVV T D W m

dt

dV L W T mV

dt

Horizontal, steady, symmetric flight

= 1 = 0 = 0

= 1 = 0 = 0

:

:

V T D

V L W

Note: Clearly this is the most simple form of the equations of motion

Page 8: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

7 Flight Mechanics

Summary - Jet

Force

Drag

Airspeed

Max thrust

Max. Endurance (CL / CD)max

Max. Range (CL/CD

2)max

Max. Speed (depends on Tmax)

Min. speed CL,max

Page 9: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

8 Flight Mechanics

Summary – Propeller Aircraft

Power

Power required

Airspeed

Power available

Max. Endurance (CL

3 / CD2 )max

Max. Range (CL/CD)max

Max. Speed (depends on Pamax)

Min. speed CL,max

Page 10: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

9 Flight Mechanics

Summary – Horizontal flight

,maxLC min

,max

2 1

L

WV

S C

max

,max

(Jet)

(Prop)a r

T D

P P

Minimum airspeed

Maximum airspeed

Maximum range 2

max

L

D

C

C

L W

T D

...LC max

2 1

L

WV

S C

max

V

F

max

L

D

C

C

0

13L DC C Ae

0L DC C Ae

2 1

L

WV

S C

prop

jet

Page 11: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

10 Flight Mechanics

Summary – Horizontal flight

Maximum endurance

max

L

D

C

C

L W

T D

min

F

3

2

max

L

D

C

C

0L DC C Ae

03L DC C Ae

2 1

L

WV

S C

prop

jet

Page 12: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

11 Flight Mechanics

Contents

1. Summary horizontal flight

2. Introduction climbing and descending flight

3. Equations of motion straight and symmetric flight

4. Gliding flight

5. Climbing flight

6. Example calculations

Page 13: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

12 Flight Mechanics

Contents

1. Summary horizontal flight

2. Introduction climbing and descending flight

3. Equations of motion straight and symmetric flight

4. Gliding flight

5. Climbing flight

6. Example calculations

Page 14: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

13 Flight Mechanics

Introduction

• How fast can an aircraft climb?

• How steep can an aircraft climb?

• How far can an aircraft glide following an engine failure?

Page 15: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

14 Flight Mechanics

Rate of climb versus flight path angle

V

Vsin = RC

V1

RC1

V2

RC2

Example case 1 Example case 2

1 2

1 > 2 however, RC1 < RC2 ! So do not confuse the flight path angle with the rate of climb

Page 16: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

15 Flight Mechanics

Contents

1. Summary horizontal flight

2. Introduction climbing and descending flight

3. Equations of motion straight and symmetric flight

4. Gliding flight

5. Climbing flight

6. Example calculations

Page 17: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

16 Flight Mechanics

Equations of motion General equations of motion in symmetric flight (1)

Free Body Diagram Kinetic Diagram

L

D

W

T V

T

Zb

Ze

Xe

V

Zb

Ze

Xe

dmV

dt

dVm

dt

Xb Xb

Page 18: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

17 Flight Mechanics

Equations of motion

: cos sin

: cos sin

T

T

F ma

dVV T D W m

dt

dV L W T mV

dt

Simplification: The aircraft is performing a straight and symmetric flight. Assumption: The thrust is assumed to be in the direction of flight (T = 0).

General equations of motion in symmetric flight (2)

Page 19: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

18 Flight Mechanics

Equations of motion

• Straight flight: flight in which the centre of gravity of the

aircraft travels along a straight line (d/dt = 0)

• Steady flight: Flight in which the forces and moments acting on

the aircraft do not vary in time, neither in magnitude, nor in

direction (dV/dt = 0)

• Horizontal flight: The aircraft remains at a constant altitude

(= 0)

• Symmetric flight: flight in which both the angle of sideslip is

zero and the plane of symmetry of the aircraft is perpendicular to

the earth ( = 0 and the aircraft is not turning)

definitions

Page 20: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

19 Flight Mechanics

Equations of motion

: cos sin

: cos sin

T

T

F ma

dVV T D W m

dt

dV L W T mV

dt

Straight, symmetric flight

= 1

1 = 0 = 0

: sin

:

W dV

V T D Wg dt

V L W

Page 21: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

20 Flight Mechanics

Power equation Straight, symmetric flight

: sin

:

W dV

V T D Wg dt

V L W

Multiply first equation with airspeed

21sin potential kinetic energy increase

2a rP P dV

VW g dt

L W

So, excess power (Pa – Pr) can be used to climb or to increase airspeed

Page 22: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

21 Flight Mechanics

Contents

1. Summary horizontal flight – (Vmin, Vmax, Range)

2. Endurance

3. Speed instability

4. Introduction climbing and descending flight

5. Equations of motion straight and symmetric flight

6. Gliding flight

7. Climbing flight

8. Example calculations

Page 23: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

22 Flight Mechanics

Gliding flight

0 aT P

Question: How far can a Boeing 747 glide from 10km altitude?

Answer: Approximately 180 km!

Page 24: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

23 Flight Mechanics

Page 25: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

24 Flight Mechanics

Gliding flight

2

0

2

a

r

T P

dH W dVP W

dt g dt

2

aircraft decelerates2

r

W dVP

g dt

What if the pilot tries to fly a horizontal path?

Clearly this is not an option… glide at constant airspeed:

sinr

dHP W WV

dt

Power equation

Page 26: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

25 Flight Mechanics

Best gliding performance

min ,min

sinr

r

P WV

RC P

Conclusion: 1. To glide as far as possible, one must glide at the

condition for minimum drag (e.g. useful for engine failure)

2. To glide as long (time wise) as possible, one must glide at the condition for minimum power required (e.g. useful for gliders)

Note: This is not treated in ‘introduction to flight’

1. RCmin (long time)

2. min (long distance)

Option 1

min min

sinDV

WV

D

Option 2

Page 27: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

26 Flight Mechanics

Glide as far as possible

arcsin arcsinD DW L

arcsin D

L

CC

0

max

LL D

D

CC C Ae

C

Question: Does the aircraft weight influence the minimum glide angle a. Yes b. No c. ?

2 1

L

WV

S C

Page 28: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

27 Flight Mechanics

Glide as far as possible

arcsin arcsinD DW L

arcsin D

L

CC

0

max

LL D

D

CC C Ae

C

Question: Does the aircraft weight influence the minimum glide angle a. Yes b. No c. ?

2 1

L

WV

S C

Page 29: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

28 Flight Mechanics

Glide as far as possible

Strangely enough, the glide angle is independent of the aircraft weight!

V1, W1

V2,W2

W1 > W2 1 = 2 and V1 < V2

Page 30: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

29 Flight Mechanics

Story: We have lost both engines

Page 31: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

30 Flight Mechanics

Story: We have lost both engines

Question: Could the aircraft have made it to Dobbins Air Force Base

if the pilots had decided to glide there?

W = 90,000 lb (= 400500 N)

S = 1001 sq ft (=93 m2)

b = 93.3 ft (=28.4 m)

CD0 = 0.02

e = 0.85

Distance to Dobbins AFB: 20 nautical miles (=36 km)

Altitude: 7000 ft (=2134 m)

Page 32: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

31 Flight Mechanics

Story: We have lost both engines

Answer: Yes!

Maximum distance when CL/CD is max. So:

Now the glide angle and distance can be calculated:

0

0

2 2

2 2

28.4( 8.7)

93

0.02 8.7 0.85 0.68

0.680.02 0.04

8.7 0.85

L D

LD D

bA

S

C C Ae

CC C

Ae

0.04arcsin arcsin 3.4 [deg]

0.68

distance = / tan 2134 / tan 3.4 36 km

D

L

C

C

h

Page 33: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

32 Flight Mechanics

Glide as long as possible

0

3

,min 2

max

3Lr L D

D

CP C C Ae

C

sinrP WV

Minimum rate of descent

rP DV

D

L

CD W

C

2 1

L

WV

S C

2

3

2 Dr

L

CWP W

S C

Page 34: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

33 Flight Mechanics

Summary gliding flight

Minimum glide angle

V

D

Minimum rate of descent

V

Pr

Minimum glide angle

Page 35: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

34 Flight Mechanics

Contents

1. Summary horizontal flight – (Vmin, Vmax, Range)

2. Endurance

3. Speed instability

4. Introduction climbing and descending flight

5. Equations of motion straight and symmetric flight

6. Gliding flight

7. Climbing flight

8. Example calculations

Page 36: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

35 Flight Mechanics

Climbing performance

P

V

Pr

Pa (at a certain setting of )

Vmin Vmax, (hor,steady)

Excess power

21sin potential kinetic energy increase

2a rP P dV

VW g dt

L W

Page 37: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

36 Flight Mechanics

Climbing performance

• Maximum rate of climb (propeller)

• Steepest climb (jet)

Steady flight

Page 38: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

37 Flight Mechanics

Climbing performance

• Maximum rate of climb (propeller)

• Steepest climb (jet)

Steady flight

Page 39: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

38 Flight Mechanics

Maximum rate of climb

Maximum rate of climb maximum

excess power (Pa – Pr)max

sina rP PV RC

W

L W

Steady flight

Page 40: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

39 Flight Mechanics

Solutions Maximum rate of climb propeller aircraft

P

V

Maximum excess power Pr

Pa

RCmax at Pr,min This corresponds to the following condition:

0

3

2

max

3

L

D

L D

C

C

C C Ae

Assumption: Pa is independent of airspeed

Page 41: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

40 Flight Mechanics

Climbing performance

• Maximum rate of climb (propeller)

• Steepest climb (jet)

Steady flight

Page 42: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

41 Flight Mechanics

Steepest climb Jet aircraft

T D

W

L W

sin

T

D

Maximum excess thrust

Page 43: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

42 Flight Mechanics

Solutions Steepest climb for jet aircraft

max at Dmin This corresponds to the following condition:

L

D

L D

C

C

C C Ae

0

max

Assumption: T is independent of airspeed

T

D

Maximum excess thrust

Page 44: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

43 Flight Mechanics

Contents

1. Summary horizontal flight

2. Introduction climbing and descending flight

3. Equations of motion straight and symmetric flight

4. Gliding flight

5. Climbing flight

6. Example calculations

Page 45: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

44 Flight Mechanics

Example 1 Climbing performance of the Beach King Air

Two engine propeller aircraft CD = CD0 + kCL

2 CD0 = 0.02 k = 0.04 W = 60 [kN] S = 28.2 [m2]

Maximum power available (741 kW) can be assumed independent of airspeed. The aircraft is performing a steady symmetrical climb Question: What is the maximum rate of climb of this aircraft at sea-level ( = 1.225 [kg/m3] and what is the corresponding airspeed

Page 46: Introduction to Aerospace Engineering · PP ar dV V W g dt ar 741000 206000 8.9 [m/s] 60000 PP RC W J ar steady flight sin PP V RC W U ½ ° ¾ D ° ¿ max r,min 2 3 Maximum power

45 Flight Mechanics

Example 1 Solution

2

Power equation:

1sin

2a rP P dV

VW g dt

741000 2060008.9 [m/s]

60000a rP P

RCW

steady flight

sina rP PV RC

W

max r,min

2

3

Maximum power available is independent of airspeed (741 kW)

RC at P

2r

DrD

LL

P DVCW

P WCD W S C

C

0

3

,min 2

max

0.023 3 1.22

0.04

DLr L

D

CCP C

kC

2 1 60000 2 153 [m/s] (=190 km/h)

28.2 1.225 1.22L

WL W V

S C

0

2 20.02 0.04 0.41 0.08D D LC C kC

3 31 12 2

0.08 1.225 53 28.2 206 [kW]r DP DV C V S