flight and orbital mechanics - tu delft ocw · 2016-02-17 · ae2104 flight and orbital mechanics 2...

1Challenge the future

Flight and Orbital Mechanics

Lecture slides

Semester 1 - 2012

Challenge the future

DelftUniversity ofTechnology

Flight and Orbital MechanicsLecture hours 5, 6, 7 – Turning performance

Mark Voskuijl

2AE2104 Flight and Orbital Mechanics |

Question

Lockheed F-104 StarfighterTop 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?


Turn at Mach 2.2, bank angle 30 deg

Circle with radius 75 km…


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


Content

• Aim




diagrams





• Summary



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


Content

• Aim




diagrams





• Summary



How to fly a turn?


How to fly a turn

L

W

L W

T D

Steady horizontal flight


How to fly a turn

W

Lcos

Only roll:Vertical component of the lift is smaller than the weight

cosL W Roll the aircraft


How to fly a turn

cos

L W

L W

Lcos

Lcos

D

D T

Roll the aircraft(increase angle of attack)


How to fly a turnRoll the aircraft and increase the pitch


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


Content

• Aim




diagrams





• Summary



Equations of motion

0 T D

2

sin WV

LgR

T

Lcos

D

L

Lsin

W

V

Lsin

=d / dt

R

Free Body Diagram

0

mV2 / R

mV2 / R

0

Kinetic Diagram

F ma

Eq. of motion

2

sin

cos

T D

WVL

gR

L W

Horizontal sustained turn

0 cosL W


Content

• Aim




diagrams





• Summary



Load factor and performance diagrams

1

cos costurn

Ln

L

cosW L

Ln

W

L nW

Load factor:

Apparent weight(C is a virtual / fictitious force)

Load factor during steady horizontal turn:

,30

,45

,60

1.15

1.41

2

turn

turn

turn

n

n

n

Load factor

Video 1 F22 high g turnhttp://www.youtube.com/watch?v=n34RwIUlnAo&feature=related

Video 2 (pilot’s perspective)http://www.youtube.com/watch?v=5LMVVey5bCM&feature=related

http://www.youtube.com/watch?v=n34RwIUlnAo&feature=related

http://www.youtube.com/watch?v=n34RwIUlnAo&feature=related

http://www.youtube.com/watch?v=5LMVVey5bCM&feature=related



Drag

V

Performance diagram

What will happen to this diagram in a turn?



212LC V S nW

2 1

L

nWV

S C

, LV function n C

L nW L nWD D D

L L rP DV

Airspeed Aerodynamic drag Power required

Note:

a r

T D

P P

Conclusion:V, D, Pr are functions of lift coefficient and load factor(In symmetric flight, they only depend on the lift coefficient)

D

L

CD nW

C

, LD function n C

2 1Dr

L L

C nWP nW

C S C

23 3

3

2 Dr

L

Cn WP

S C

,r LP function n C

Performance diagram



2 1

L

nWV n

S C

D

L

CD nW n

C

Effect of load factor (n) on airspeed (V)(at constant angle of attack)

rP DV n n

Effect of load factor (n) on aerodynamic drag (D)(at constant angle of attack)

Effect of load factor (n) on Power required (Pr)(at constant angle of attack)

Performance diagram



Drag

V

Each point relates to 1 angle of attack (CL)

(note: this graph is purely qualitative)

Performance diagram



Tmax

Vnmax

T, D

V

1. Vmin increases when n increases

2. Vmin first aerodynamically limited then thrust limited

(safety: regulation for stick force/g)

3. Vmax decreases when n increases

4. At nmax Vmin = Vmax

Performance diagram



Tmax

n limited by engine

T, D

V

n limited by CLmax

2 speed ranges

Performance diagram


Content

• Aim




diagrams





• Summary



Turning performance





Assumption: sustained turns


Steepest turn

Tmax

T, D

V

Tmax

T, D

V

Aim: For each airspeed nmax

V1V2V3V4 V5 V6 V7 V8 V9


Steepest turn

V

n

1

Achievable load factor as function of airspeed


Steepest turn

2max

2

2 1

LCL nW n V

W

S V

D L

L D

C CTT D nW n

C W C

• Range A:

• Range B:

maxL LC C

max Vary LT T C

Tmax

n limited by engine

T, D

Vn limited by CLmax

Calculation of nmax(V)

L

2 1 at every C ,

L

nWV n V

S C


Steepest turn

max

L

D

C

C

Tmax

Vnmax

D at nmax

V

D, T

• Vnmax at Dmin

• D

L

CD nW

C

Solution for jet aircraft

Note: You must be able to derive the final step

0L DC C Ae


Steepest turn

0

max

maxmax

max

max 2 1

opt

opt

D

L

L

D

L D

n

L

CT D nW

C

T Cn

W C

C C Ae

n WV

S C

Solution for jet aircraft

(A numerical example will follow later)


Turning performance







Do not confuse turn radius with time to turn!


Minimum turn radius (tightest turn)

First we must have an equation for the turn radius

Using the equilibrium equations:

1

cos

Ln

W

2

sin

cos

W VL

g R

W L

2 2

2

1cos sin 1 sin 1x x

n

2 2

2 2

11

1

W V VnW R

n g R g n


Minimum turn radius

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)

2

2 1

VR

g n

V

R


2

2 1

VR

g n

nmax

R

V

V

Tightest turn


Turn at Mach 2.2, bank angle 30 deg

Circle with radius 75 km…


MMO

VMO


Turning performance







Fastest turn

V

n

1

V

R

V

T

2

2 RT

V

Minimize time to achieve fastest turnTime to complete a full circle:

(Circumference of turn: 2R)


Turning performance







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


Rate one turn

Example:

Approach: V = 80 m/s, T = 60 [s]

801502 [m]

/ 60

V VR

R

22 2

21 1.09

1

V VR n

gRg n

V

R

1arccos 23

n


Rate one turn


Content

• Aim




diagrams





• Summary



Example calculations

Numerical example – turning performance at 7000 [m] altitude

Lift drag polar is known for this aircraft

Aircraft weight, maximum thrust and dimensions are known as well

0

2

LD D

CC C

Ae



0

3

max

2

0.59 [kg/m ] (ISA)

8.67 [kN]

0.021

7 [-]

60000 [N]

30 [m ]

D

T

C

Ae

W

S

Aircraft and atmospheric data at 7000 [m]

Calculate:1. Maximum load factor and corresponding airspeed2. Radius of tightest turn (Rmin) and corresponding airspeed3. Time for fastest turn (T2,min) and corresponding airspeed



max 8670 [N]T

maxmax

max

L

D

T Cn

W C

0

max

LL D

D

CC C Ae

C

0.021 7 0.68 [-]LC

0

2

LD D

CC C

Ae

max

8670 0.682.34

60000 0.042 n

maxmax

2 1

2.34 60000 2 1

30 0.59 0.68

152.7 [m/s]

n

L

n WV

S C

Maximum load factor20.68

0.021 0.042 [-]7

DC


Example calculationsMaximum load factor



Vnmax , “steepest turn”

Maximum load factor


Example calculationsMinimum turn radius

Vrmin – “tightest turn”



VTmin – fastest turn

Minimum time to turn (fastest turn)



2

2tan

V

gT

max maxn

2

2 RT

V

All results combined


Content

• Aim




diagrams





• Summary



Advanced manoeuvresImmelmann


Advanced ManoeuvresThrust vectoring

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

http://www.youtube.com/watch?v=CGbOs0vgYOA&NR=1


Content

• Aim




diagrams





• Summary



Altitude effects

0.75

max

max

L

D

CTn

W C

max

2 1

n

L

nWV

S C

max

2

2 if

1n

VR

g n

Equations


Altitude effectsExample


Content

• Aim




diagrams





• Summary



Summary

• Be able to derive the equations of motion for a horizontal sustained

turn (what is the definition of horizontal and sustained?)

• Load factor is defined as n = L/W

• In a horizontal sustained turn, n = 1/cos

2

sin

cos

T D

mVL

R

L W


Summary

• Maximum load factor for jet aircraft:

• Turn radius:

• You must be able to derive the equations mentioned above

• Time to turn 360 deg:

• Rate one turn is defined as 180 deg / min

2

2 RT

V

maxmax

max

L

D

T Cn

W C

2

21

VR

g n


Content

• Aim




diagrams





• Summary



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.


Questions?

flight and orbital mechanics - tu delft ocw · 2016-02-17 · ae2104 flight and orbital mechanics 2...

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