introduction to aerospace engineering - tu delft ocw · 2016-11-07 · introduction to aerospace...

1 Challenge the future

Introduction to Aerospace Engineering

Lecture slides

15-12-2012

Challenge the future

Delft University of Technology

Introduction Aerospace Engineering

Flight Mechanics

Dr. ir. Mark Voskuijl

2 Flight Mechanics

3.&4 Horizontal flight performance

3 Flight Mechanics

Flight mechanics

Hours 1 & 2 : Introduction, general equations of motion

Hours 3 & 4 : Horizontal flight performance

Hours 5 & 6 : Climbing and descending flight

Hours 7 & 8 : Flight envelope

Hours 9 & 10: Example questions and solutions

4 Flight Mechanics

Contents

1. Summary previous lecture

2. Introduction

3. Equations of motion horizontal steady symmetric flight

4. Performance diagram

5. Aircraft performance in horizontal flight

1. Minimum airspeed

2. Maximum airspeed

3. Range

4. Endurance

5. Speed instability / stability

6. Summary

7. Example exam question

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

General equations of motion in symmetric flight (2)

7 Flight Mechanics

Propulsive force

Thrust is assumed to be independent of airspeed

Typical analytical assumption – Jet

8 Flight Mechanics

Propulsive force

Pa is assumed to be constant and independent of airspeed

Typical analytical assumption – Propeller

9 Flight Mechanics

Propulsive force

Jet: Thrust is (assumed) independent of airspeed

Propeller: Power available is (assumed) independent of airspeed

Analytical assumptions - summary

airspeed airspeed

thrust power available

jet prop

10 Flight Mechanics

Summary previous lecture

• You should be able to derive the

equations of motion for 2

dimensional flight

• The thrust of a jet aircraft can be

assumed independent of airspeed

• The power available of a propeller

aircraft can be assumed

independent of airspeed

• The complete aircraft

aerodynamics can be represented

by 1 equation; the lift drag polar

// : cos sin

: cos sin

V T

V T

W dVF T D W

g dt

W dF V L W T

g dt

0

2

LD D

CC C

Ae

11 Flight Mechanics

Contents


2. Introduction




1. Minimum airspeed

2. Maximum airspeed

3. Range

4. Endurance


6. Summary


12 Flight Mechanics

Contents


2. Introduction




1. Minimum airspeed

2. Maximum airspeed

3. Range

4. Endurance


6. Summary


13 Flight Mechanics

Introduction

• How fast can an aircraft fly?

• How slow can a given aircraft fly?

• At what speed should be flown to be able to fly as far as possible?

(Interesting for airliners)

• At what speed should be flown to stay in the air as long as

possible? (Search and rescue, military purposes)

Horizontal flight performance

14 Flight Mechanics

Introduction

Most important are the lecture sheets!!!

Background material:

Introduction to Flight (Anderson)

Par. 6.1 – 6.2

Par. 6.3 – 6.6

Par. 6.12 – 6.14

What to learn?

15 Flight Mechanics

Contents


2. Introduction




1. Minimum airspeed

2. Maximum airspeed

3. Range

4. Endurance


6. Summary


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

Flight Condition: The aircraft is performing a horizontal (straight) and steady 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

20 Flight Mechanics

21 Flight Mechanics

http://www.youtube.com/watch?v=TCUHQ_-l6Qg

B52 landing in crosswind




22 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

How do these forces (aerodynamic, propulsive, gravitational) vary with airspeed?

23 Flight Mechanics

Contents


2. Introduction




1. Minimum airspeed

2. Maximum airspeed

3. Range

4. Endurance


6. Summary


24 Flight Mechanics

Performance diagram

• For a given altitude, configuration and engine control setting, the

drag or power required and thrust or power available are plotted

against airspeed

• Also called Pernaud diagram

• Very useful to determine aircraft performance parameters

25 Flight Mechanics

Performance diagram

• Typical shape:

26 Flight Mechanics

Performance diagram

a

r

P T V

P D V

27 Flight Mechanics

airspeed airspeed

Force Power

Jet (thrust / power) Propeller (power) Aerodynamics (power / drag)

28 Flight Mechanics

Contents


2. Introduction




1. Minimum airspeed

2. Maximum airspeed

3. Range

4. Endurance


6. Summary


29 Flight Mechanics

Horizontal flight performance

• Minimum airspeed (How slow)

• Maximum airspeed (How fast)

• Range (How far)

• Endurance (How long)

• Speed stability / instability

Assumptions: Horizontal steady symmetric flight

( L = W, T = D, Pa = Pr ), except for the speed stability section

30 Flight Mechanics

Contents


2. Introduction




1. Minimum airspeed

2. Maximum airspeed

3. Range

4. Endurance


6. Summary


31 Flight Mechanics

Minimum airspeed

212

2 1

L

L

L W

C V S W

WV

S C

max

min

2 1

L

WV

S C

32 Flight Mechanics

Minimum airspeed

Flaps

33 Flight Mechanics

Minimum airspeed Performance diagram

Vmin

34 Flight Mechanics

Minimum airspeed

• Question: An aircraft has a wing loading (W/S) 2400 N/m2 and

CLmax = 1.4. Find the airspeed at which stall occurs (minimum airspeed)

at (i) sea level ( = 1.225 kg/m3) and (ii) at 5000m ( = 0.737 kg/m3)

• Answer:

21

L 2

min

L,max

L W

C V S W

W 2 1V

S C

min

min

2 1(i)V 2400 51.8 m/s

1.225 1.4

2 1(ii)V 2400 66.8 m/s

0.737 1.4

Example calculation

35 Flight Mechanics

Contents


2. Introduction




1. Minimum airspeed

2. Maximum airspeed

3. Range

4. Endurance


6. Summary


36 Flight Mechanics

Maximum airspeed

• Pilot applies maximum throttle

(max. thrust or Power available)

maxT D

L W

Vmax

Force

Maximum thrust

Drag

Dmax

L

CLT D W

L C

Jet aircraft

0

2

D D LC C k C

0D maxL

L

C Tk C

C W

Solution

0

2 maxL L D

TkC C C 0

W

2

L

ax bx c 0

C ...

L

W 2 1L W V

S CV

37 Flight Mechanics

Maximum airspeed

The following data is known of the Cessna Citation II (subsonic jet)

Calculate (1) the maximum airspeed of this aircraft when flying at H = 0 m and (2) the corresponding Mach number

Example

Aircraft Weight : W = 60 kN, Wing area : S = 30 m2, (Parabolic) Lift-Drag polar : CD = CDo + kCL2; CDo = 0.022, k = 0.047, CLmax = 1.35, Maximum Thrust at 0 m ISA :T0 = 12 kN. The aircraft is flying at an altitude of H = 0 m in the International Standard Atmosphere (0 = 1.225 kg/m3) Thrust is assumed to be independent of the airspeed

38 Flight Mechanics

Maximum airspeed

2 21 1L D2 2

Steady symmetric horizontal flight:

L W, T D

With

L C V S, D C V S

maxmax

Maximum Mach number?

VM

a

Solution

0

L LD

D

2

D D L

Thrust must be at maximum to achieve Vmax

Combining the above yields:

C CW 600005 C

C T 12000 5

Lift drag polar:

C C k C

0D2 LL

2 LL

CCC 0

5k k

C 0.022C 0

5 0.047 0.047

LC 0.11

max

L

W 2 1 60000 2 1V 172.1 m/s

S C 30 1.225 0.11

a RT 1.4 287.05 288.15 340.3 m/s

max

172.1M 0.5

340.3

39 Flight Mechanics

Contents


2. Introduction




1. Minimum airspeed

2. Maximum airspeed

3. Range

4. Endurance


6. Summary


40 Flight Mechanics

Range

• How can we fly as far as possible for a given amount of fuel?

• How can we fly a given distance with the minimum amount of fuel?

/

V m sVm N

F F N s

Specific range is defined as the ratio V/F, where F equals the fuel flow

Dimensional analysis:

Or, put in words, specific range is the distance that can be flown for 1 Newton of fuel. Clearly, this variable must be maximised

41 Flight Mechanics

Range

a

P br P

j

PF c P F c

(steady horizontal flight)a rP P

1

rP P

j j

j

P

P DVF c c

V

F c D

Maximum range propeller aircraft

j P

max min

max max

and c can be assumed to be constant as a function of airspeed

(over the range of cruising speeds) for propeller aircraft

V

FL

D

CR D

C

42 Flight Mechanics

Range Maximum range propeller aircraft

43 Flight Mechanics

Problem!

Result:

Prop: maximum range at maximum CL / CD

Dear pilot, could you please fly at CL / CD max?

We must give an airspeed to the pilot

44 Flight Mechanics

Solution Result:

Prop: maximum range at maximum CL / CD

( )D LC f C y f x

2 1

L

L W

WV

S C

2

( ) ' '0

( )

L

L D

Cd d g x hg gh

dC C dx h x h

2

1

0

DD L

L

D

dCC C

dC

C

D D

L L

dC C

dC C

0

2

2L

DL

L

CC

C Ae

Ae C

0L DC C Ae

0

2

LD D

CC C

Ae

45 Flight Mechanics

Range

• Charles Lindbergh (1927)

• First solo, non stop flight across the

Atlantic ocean

Example: Spirit of St. Louis

46 Flight Mechanics

Performance diagram Spirit of St Louis (ref: NASA)

Note: The Pr curve shifts down when weight decreases. This means that Charles had to decrease airspeed during the flight

47 Flight Mechanics

Contents


2. Introduction

3. Equations of motion horizontal symmetric flight



1. Minimum airspeed

2. Maximum airspeed

3. Range

1. Propeller

2. Jet

4. Endurance


6. Summary


48 Flight Mechanics

Range

So, at what speed should the pilot fly to maximise specific range?

TF c T

Typically, cT can be considered to be constant

T D

Jet aircraft

T TF c T c D

T

VV F

c D

max max min

V F V D D V

49 Flight Mechanics

Range

max max

max max

1

1D

L

D DL

V L V

L W

CDW

V C V

212

2 1L

L

WC V S W V

S C

2

1

2 1 2

D

L L

L D

CD WW

V C W CW

S C S C

Jet aircraft

Performance diagram (jet)

2

min max

L

D

CD

V C

50 Flight Mechanics

Problem!

Result:

Jet: maximum range at maximum CL / CD2

Dear pilot, could you please fly at CL / CD2 max?

We must give an airspeed to the pilot

51 Flight Mechanics

Solution Result:

Jet: maximum range at maximum CL / CD2

( )D LC f C y f x

2 1

L

L W

WV

S C

2 2

( ) ' '0

( )

L

L D

Cd d g x hg gh

dC C dx h x h

2

4

1 2

0

DD L D

L

D

dCC C C

dC

C

12

D D

L L

dC C

dC C

0

2

12

2L

DL

L

CC

C Ae

Ae C

0

13L DC C Ae

0

2

LD D

CC C

Ae

52 Flight Mechanics

How environmentally (un)friendly are

aircraft?

• Example: Boeing 747:

• W = 3000 kN

• S = 510 m2

• b = 60 m

• hcr = 11000 m

• Mcr = 0.85

• Estimated:

• CD0 = 0.02; e = 0.85; tot = 0.40

53 Flight Mechanics


aircraft really

• W/S = 5882 N/m2; A = 7.06

• a = 295 m/s; V = 250 m/s (=900 km/hr)

2

2 10.517L

WC

S V

0

2

0.0342LD D

CC C

Ae

15.1L

D

C

C

198.4 kND

L

CD W

C

49591 kJ/sa rP P DV

54 Flight Mechanics


aircraft really?

0.40 estimated (in the order of efficiency power plant)atot

P

Q

28.3 N/sa

tot

PgF

H

Fuel consumption: 12980 l/hr (in accordance with data manufacturer)

Every hour 900 km is travelled and 12980 l of fuel is used:

69 m/l

With 400 passengers: 27.7 pax km/l

3

43000 kJ/kg

0.8 kg/mfuel

H

55 Flight Mechanics

Comparison to a car

• Car driving 1:15 and 4 passengers: 60 pax km/l

• Conclusion: consumption of car is half the consumption of a 747

despite the speed ratio 9:1

• In practice ( seat occupation < 100% ) consumption per

passenger km is similar for car and aviation traffic

56 Flight Mechanics

Contents


2. Introduction




1. Minimum airspeed

2. Maximum airspeed

3. Range

4. Endurance


6. Summary


57 Flight Mechanics

Endurance

• Endurance: time duration spent

in flight

• Do not confuse endurance with

range!

• Solution: maximum endurance for

minimum fuel flow

• Fmin

58 Flight Mechanics

Endurance

min min

max

Horizontal steady flight

Assumption: constant

T

T

T

T

L

D

Fc

T

F c T

T D

F c D

c

CF D

C

Jet aircraft

Performance diagram (jet)

59 Flight Mechanics

Endurance

a

P P

br j

PFc F c

P

(steady horizontal flight)a rP P

r

P P

j j

P DVF c c

Propeller aircraft

23

3

2 1

2 1 2

P

j L

P D P D

j L L j L

D WF c

S C

c C c CW WF W

C S C S C

j P

3

max min ,min2 min

max

and c can be assumed to be constant as a function of airspeed

(over the range of cruising speeds) for propeller aircraft

F Lr

D

CE DV P

C

60 Flight Mechanics

Endurance Propeller aircraft

Pr should be minimal!

61 Flight Mechanics

Problem!

Result:

Propeller: maximum endurance at

maximum CL3 / CD

2

Dear pilot, could you please fly at CL3 / CD

2 max? We must give an airspeed to the pilot

62 Flight Mechanics

Solution Result:

Prop: maximum endurance at maximum CL3 / CD

2

( )D LC f C y f x

2 1

L

L W

WV

S C

3

2 2

( ) ' '0

( )

L

L D

Cd d g x hg gh

dC C dx h x h

2 2 3

4

3 2

0

DD L L D

L

D

dCC C C C

dC

C

32

D D

L L

dC C

dC C

0

2

32

2

LD

L

L

CC

C Ae

Ae C

03 L DC C Ae

0

2

LD D

CC C

Ae

63 Flight Mechanics

Contents


2. Introduction




1. Minimum airspeed

2. Maximum airspeed

3. Range

4. Endurance


6. Summary


64 Flight Mechanics

Speed instability

• General equations of motion

• Equations of motion horizontal

symmetric flight (thrust parallel

to airspeed

: cos sin

: cos sin

T

T

F ma

dVV T D W m

dt

dV L W T mV

dt

:

:

F ma

dVV T D m

dt

V L W

65 Flight Mechanics

Speed instability

Force

Drag

Thrust

Airspeed Vdesired

Disturbance (gust of wind for example)

D>T

T - D = mdV/dt

dV/dt < 0

Conclusion: the aircraft is ‘speed unstable’ and will deviate from the original position

following a disturbance. This makes it difficult for the pilot

66 Flight Mechanics

Speed stability

Force

Drag

Thrust

Airspeed Vdesired

Disturbance (gust of wind for example)

D>T

T - D = mdV/dt

dV/dt < 0

Conclusion: the aircraft is speed stable and the aircraft will automatically

return to the original airspeed. Very nice for the pilot!

67 Flight Mechanics

Speed stability - instability

Force

Drag

Airspeed

Speed unstable Speed stable

68 Flight Mechanics

Contents


2. Introduction




1. Minimum airspeed

2. Maximum airspeed

3. Range

4. Endurance


6. Summary


69 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

70 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

71 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

72 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

73 Flight Mechanics

Contents

1. Previous lecture – Propeller - simplification

2. Introduction




1. Minimum airspeed

2. Maximum airspeed

3. Range

4. Endurance


6. Summary


74 Flight Mechanics

Example exam question (homework)

The Gulfstream IV, indicated in Figure 1 is a twin-turbofan executive transport aircraft. Data for this aircraft are given below: S = 88.3 [m2] b = 23.7 [m] A = 6.36 [-] e = 0.67 [-] CD0 = 0.015 [-] CD = CD0 + CL

2 / Ae W = 311000 [N] This aircraft has two Rolls Royce Tay turbofan engines. Fuel consumption can be represented with the following equation: F = cTT, where cT = 0.69 [N/hr/N]. The thrust specific fuel consumption is assumed to be independent of airspeed and altitude Question: What is the maximum specific range of this aircraft when flying at 8000m (ρ = 0.5252 kg/m3) ?

Problem

75 Flight Mechanics


2

2 1

1

2 1 2

D

L

L

D

L L

L D

CLD D W

L C

WV

S C

CD WW

V C W CW

S C S C

Solution (part 1) Specific range = V/F (m/kg) = V/cTT Steady horizontal flight: T = D; L = W V/F = V/cTD cT is a constant so maximum specific range when (D/V) is minimal So CL / CD

2 should be maximal. This is the case when CL = √(CD0Ae/3 ) (you should derive this on the exam, see sheet 48)

0

13

2 2

0.015 6.36 0.67 0.26

0.260.015 0.02

6.36 0.67

L

LD D

C

CC C

Ae

76 Flight Mechanics


2 1 311000 2 1227 [m/s]

88.3 0.5252 0.26L

WV

S C

0.02311000 23.9 [kN]

0.26

D

L

CT D W

C

Solution (part 2) From the lift coefficient, we can calculate the airspeed

The aerodynamic drag can be calculated with the drag coefficient. This must be equal to the thrust

22750 [m/N]

(0.69/3600) 23900T

V V

F c T

Now, all parameters are known to calculate the maximum specific range for this altitude and aircraft weight

introduction to aerospace engineering - tu delft ocw · 2016-11-07 · introduction to aerospace...

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