introduction to aerospace engineering · pp ar dv v w g dt ar 741000 206000 8.9 [m/s] 60000 pp rc w...

1 Challenge the future

Introduction to Aerospace Engineering

Lecture slides

15-12-2012

Challenge the future

Delft University of Technology

Introduction Aerospace Engineering

Flight and Mechanics

Dr. ir. Mark Voskuijl

2 Flight Mechanics

5.&6 Climbing and descending flight

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

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

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

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

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

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

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

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

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

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

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?

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

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

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

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)

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

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

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

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

22 Flight Mechanics

Gliding flight

0 aT P

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

Answer: Approximately 180 km!

23 Flight Mechanics

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

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

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

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

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

29 Flight Mechanics

Story: We have lost both engines

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)

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

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

33 Flight Mechanics

Summary gliding flight

Minimum glide angle

V

D

Minimum rate of descent

V

Pr

Minimum glide angle

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

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

36 Flight Mechanics

Climbing performance

• Maximum rate of climb (propeller)

• Steepest climb (jet)

Steady flight

37 Flight Mechanics

Climbing performance

• Maximum rate of climb (propeller)

• Steepest climb (jet)

Steady flight

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

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

40 Flight Mechanics

Climbing performance

• Maximum rate of climb (propeller)

• Steepest climb (jet)

Steady flight

41 Flight Mechanics

Steepest climb Jet aircraft

T D

W

L W

sin

T

D

Maximum excess thrust

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

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

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

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