Gliding, Climbing, and Turning Flight Performance!

Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2016

Copyright 2016 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE331.html

http://www.princeton.edu/~stengel/FlightDynamics.html

•! Conditions for gliding flight•! Parameters for maximizing climb angle and rate•! Review the V-n diagram•! Energy height and specific excess power•! Alternative expressions for steady turning flight•! The Herbst maneuver

Learning Objectives

Reading:!Flight Dynamics !

Aerodynamic Coefficients, 130-141!

1

Review Questions!!! How does air density decrease with altitude?!!! What are the different definitions of airspeed?!!! What is a “lift-drag polar”?!!! Power and thrust: How do they vary with

altitude?!!! What factors define the “flight envelope”?!!! What were some features of the first

commercial transport aircraft?!!! What are the important parameters of the

“Breguet Range Equation”?!!! What is a “step climb”, and why is it important?!

2

Gliding Flight!

3

D = CD12!V 2S = "W sin#

CL12!V 2S =W cos#

!h =V sin#!r =V cos#

Equilibrium Gliding Flight

4

Gliding Flight•! Thrust = 0•! Flight path angle < 0 in gliding flight•! Altitude is decreasing•! Airspeed ~ constant•! Air density ~ constant

tan! = "D

L= "

CD

CL

=!h!r=dhdr; ! = " tan"1 D

L#

$%

&

'(= "cot"1

LD#

$%

&

'(

Gliding flight path angle

Corresponding airspeed

Vglide =2W

!S CD2 + CL

25

Maximum Steady Gliding Range

•! Glide range is maximum when !! is least negative, i.e., most positive

•! This occurs at (L/D)max 6

Maximum Steady Gliding Range

•! Glide range is maximum when !! is least negative, i.e., most positive

•! This occurs at (L/D)max

tan! =!h!r= negative constant =

h " ho( )r " ro( )

#r = #htan!

= "#h" tan!

= maximum when LD

= maximum

!max = " tan"1 D

L#

$%

&

'(min

= "cot"1 LD#

$%

&

'(max

7

Sink Rate, m/s •! Lift and drag define !! and V in gliding equilibrium

D = CD12!V 2S = "W sin#

sin# = " DW

L = CL12!V 2S =W cos"

V = 2W cos"CL!S

!h =V sin!

= "2W cos!CL#S

DW$

%&

'

()= "

2W cos!CL#S

LW$

%&

'

()DL

$

%&

'

()

= "2W cos!CL#S

cos! 1L D$

%&

'

()

•! Sink rate = altitude rate, dh/dt (negative)

8

•! Minimum sink rate provides maximum endurance•! Minimize sink rate by setting !(dh/dt)/!CL = 0 (cos !! ~1)

Conditions for Minimum Steady Sink Rate

!h = ! 2W cos"CL#S

cos" CD

CL

$

%&

'

()

= !2W cos3"

#SCD

CL3/2

$

%&

'

() * !

2#WS

$

%&

'

()CD

CL3/2

$

%&

'

()

CLME=

3CDo

!and CDME

= 4CDo9

L/D and VME for Minimum Sink Rate

VME =2W

!S CDME

2 + CLME2"

2 W S( )!

#3CDo

" 0.76VL Dmax

LD( )

ME=14

3!CDo

=32

LD( )

max" 0.86 L

D( )max

10

L/D for Minimum Sink Rate•! For L/D < L/Dmax, there are

two solutions•! Which one produces smaller

sink rate?

LD( )

ME! 0.86 L

D( )max

VME ! 0.76VL Dmax

11

Checklist!"!Steady flight path angle?!"!Corresponding airspeed?!"!Sink rate?!"!Maximum-range glide?!"!Maximum-endurance glide?!

12

HL-10M2-F1

M2-F2

M2-F3

X-24A

X-24B

!!iiss""rriiccaall FFaacc""iiddss##iiff$$nngg--BBooddyy RReeeenn%%yy VVeehhiicclleess

13

Climbing Flight!

14

Rate of climb, dh/dt = Specific Excess Power

Climbing Flight

!V = 0 =T ! D !W sin"( )

m

sin" =T ! D( )W

; " = sin!1 T ! D( )W

!! = 0 =L "W cos!( )

mVL =W cos!

!h =V sin! =VT " D( )W

=Pthrust " Pdrag( )

W

Specific Excess Power (SEP) = Excess PowerUnit Weight

#Pthrust " Pdrag( )

W

•! Flight path angle •! Required lift

15

Note significance of thrust-to-weight ratio and wing loading

Steady Rate of Climb

!h =V sin! =V TW"

#$

%

&'(

CDo+)CL

2( )qW S( )

*

+,,

-

.//

L = CLqS =W cos!

CL =WS

"#$

%&'cos!q

V = 2 WS

"#$

%&'cos!CL(

!h =V TW

!"#

$%& '

CDoq

W S( ) '( W S( )cos2 )

q*

+,

-

./

=VT h( )W

!"#

$%&'CDo

0 h( )V 3

2 W S( ) '2( W S( )cos2 )

0 h( )V

Climb rate

16

Necessary condition for a maximum with respect to airspeed

Condition for Maximum Steady Rate of Climb

!h =V TW!

"#

$

%&'

CDo(V 3

2 W S( )'2) W S( )cos2 *

(V

! !h!V

= 0 = TW"

#$

%

&'+V

!T /!VW

"

#$

%

&'

(

)*

+

,-.3CDo

/V 2

2 W S( )+20 W S( )cos2 1

/V 2

17

Maximum Steady Rate of Climb:

Propeller-Driven Aircraft

!Pthrust!V

= 0 = TW"

#$

%

&'+V

!T /!VW

"

#$

%

&'

(

)*

+

,-•! At constant power

! !h!V

= 0 = "3CDo

#V 2

2 W S( )+2$ W S( )#V 2

•! With cos2!! ~ 1, optimality condition reduces to

•! Airspeed for maximum rate of climb at maximum power, Pmax

V 4 =43!

"#$

%&' W S( )2

CDo(2

; V = 2W S( )(

'3CDo

=VME

18

Maximum Steady Rate of Climb:

Jet-Driven AircraftCondition for a maximum at constant thrust and cos2!! ~ 1

Airspeed for maximum rate of climb at maximum thrust, Tmax

! !h!V

= 0

0 = ax2 + bx + c and V = + x

!3CDo

"2 W S( )V

4 + TW

#$%

&'(V

2 +2) W S( )

"= 0

19

!3CDo

"2 W S( ) V

2( )2 + TW

#$%

&'( V

2( ) + 2) W S( )"

= 0

Quadratic in V2

Checklist!"!Specific excess power?!"!Maximum steady rate of climb?!"!Velocity for maximum climb rate?!

20

Optimal Climbing Flight!

21

What is the Fastest Way to Climb from One Flight Condition to

Another?

22

•! Specific Energy •! = (Potential + Kinetic

Energy) per Unit Weight•! = Energy Height

Energy Height

Can trade altitude for airspeed with no change in energy height if thrust and drag are zero

Total EnergyUnit Weight

! Specific Energy

= mgh +mV2 2

mg= h + V

2

2g

23

! Energy Height, Eh , ft or m

Specific Excess Power

dEh

dt=ddt

h+V2

2g!

"#

$

%&=

dhdt+Vg!

"#

$

%&dVdt

Rate of change of Specific Energy

=V sin! + Vg

"#$

%&'T ( D (mgsin!

m"#$

%&' =V

T ( D( )W

= Specific Excess Power (SEP)

= Excess PowerUnit Weight

!Pthrust " Pdrag( )

W

24

=VCT !CD( ) 1

2"(h)V 2S

W

Contours of Constant Specific Excess Power

•! Specific Excess Power is a function of altitude and airspeed•! SEP is maximized at each altitude, h, when

d SEP(h)[ ]dV

= 0

25

maxwrt V

SEP(h)[ ]

Subsonic Minimum-Time Energy ClimbObjective: Minimize time to climb to desired

altitude and airspeed

26

•! Minimum-Time Strategy:•! Zoom climb/dive to intercept SEPmax(h) contour•! Climb at SEPmax(h)•! Zoom climb/dive to intercept target SEPmax(h) contour

Bryson, Desai, Hoffman, 1969

Subsonic Minimum-Fuel Energy ClimbObjective: Minimize fuel to climb to desired

altitude and airspeed

27

•! Minimum-Fuel Strategy:•! Zoom climb/dive to intercept [SEP (h)/(dm/dt)] max contour•! Climb at [SEP (h)/(dm/dt)] max•! Zoom climb/dive to intercept target[SEP (h)/(dm/dt)] max contour

Bryson, Desai, Hoffman, 1969

Supersonic Minimum-Time Energy Climb

Objective: Minimize time to climb to desired altitude and airspeed

28

•! Minimum-Time Strategy:•! Intercept subsonic SEPmax(h) contour•! Climb at SEPmax(h) to intercept matching zoom climb/dive contour•! Zoom climb/dive to intercept supersonic SEPmax(h) contour•! Climb at SEPmax(h) to intercept target SEPmax(h) contour•! Zoom climb/dive to intercept target SEPmax(h) contour

Bryson, Desai, Hoffman, 1969

Checklist!"!Energy height?!"!Contours?!"!Subsonic minimum-time climb?!"!Supersonic minimum-time climb?!"!Minimum-fuel climb?!

29

dEh

dmfuel

= dEh

dtdt

dmfuel

= 1!mfuel

dhdt

+ Vg

!"#

$%&dVdt

'

()

*

+,

SpaceShipOne"Ansari X Prize, December 17, 2003

Brian Binnie, *78Pilot, Astronaut

30

SpaceShipOne Altitude vs. Range!MAE 331 Assignment #4, 2010

31

SpaceShipOne State Histories

32

SpaceShipOne Dynamic Pressure and Mach Number Histories

33

The Maneuvering Envelope!

34

•! Maneuvering envelope: limits on normal load factor and allowable equivalent airspeed–! Structural factors–! Maximum and minimum

achievable lift coefficients–! Maximum and minimum

airspeeds–! Protection against

overstressing due to gusts–! Corner Velocity: Intersection

of maximum lift coefficient and maximum load factor

Typical Maneuvering Envelope: V-n Diagram

•! Typical positive load factor limits–! Transport: > 2.5–! Utility: > 4.4–! Aerobatic: > 6.3–! Fighter: > 9

•! Typical negative load factor limits–! Transport: < –1–! Others: < –1 to –3

35

Maneuvering Envelopes (V-n Diagrams) for Three Fighters of the Korean War Era

Republic F-84

North American F-86

Lockheed F-94

36

Turning Flight!

37

•! Vertical force equilibrium

Level Turning Flight

L cosµ =W•! Load factor

n = LW = L mg = secµ,"g"s

•! Thrust required to maintain level flight

Treq = CDo+ !CL

2( ) 12 "V2S = Do +

2!"V 2S

Wcosµ

#$%

&'(

2

= Do +2!

"V 2SnW( )2

µ :Bank Angle

•! Level flight = constant altitude•! Sideslip angle = 0

38

Bank angle

Maximum Bank Angle in Steady Level Flight

cosµ = WCLqS

= 1n

=W 2!Treq " Do( )#V 2S

Bank angle is limited by

µ :Bank Angle

CLmaxor Tmax or nmax

39

µ = cos!1 WCLqS

"#$

%&'

= cos!1 1n

"#$

%&'

= cos!1 W 2(Treq ! Do( ))V 2S

*

+,,

-

.//

Turning rate

Turning Rate and Radius in Level Flight

!! = CLqS sinµmV

= W tanµmV

= g tanµV

= L2 "W 2

mV

= W n2 "1mV

=Treq " Do( )#V 2S 2$ "W 2

mV

Turning rate is limited by

CLmaxor Tmax or nmax

Turning radius

Rturn =

V!!=

V 2

g n2 "140

Maximum Turn Rates

“Wind-up turns”

41

•! Corner velocity

Corner Velocity Turn

•! Turning radius Rturn =V 2 cos2 !

g nmax2 " cos2 !

Vcorner =2nmaxWCLmas

!S

•! For steady climbing or diving flight sin! = Tmax " D

W

42

Corner Velocity Turn

•! Time to complete a full circle

t2! =V cos"

g nmax2 # cos2 "

•! Altitude gain/loss !h2" = t2"V sin#

•! Turning rate

!! =g nmax

2 " cos2 #( )V cos#

43

Checklist!"!V-n diagram?!"!Maneuvering envelope?!"! Level turning flight?!"! Limiting factors?!"!Wind-up turn?!"! Corner velocity?!

44

Herbst Maneuver•! Minimum-time reversal of direction•! Kinetic-/potential-energy exchange•! Yaw maneuver at low airspeed•! X-31 performing the maneuver

45

Next Time:!Aircraft Equations of Motion!

Reading:!Flight Dynamics, !

Section 3.1, 3.2, pp. 155-161!

46

What use are the equations of motion?How is the angular orientation of the airplane described?

What is a cross-product-equivalent matrix?What is angular momentum?

How are the inertial properties of the airplane described?How is the rate of change of angular momentum calculated?

Learning Objectives

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47

Gliding Flight of the P-51 Mustang

Loaded Weight = 9,200 lb (3,465 kg)

L /D( )max =1

2 !CDo

= 16.31

" MR = #cot#1 LD

$%&

'() max

= #cot#1(16.3) = #3.5°

CD( )L/Dmax = 2CDo= 0.0326

CL( )L/Dmax =CDo

!= 0.531

VL/Dmax =76.49

*m/s

!hL/Dmax =V sin" = # 4.68*m/s

Rho=10km = 16.31( ) 10( ) = 163.1 km

Maximum Range Glide

Loaded Weight = 9,200 lb (3,465 kg)S = 21.83m2

CDME= 4CDo

= 4 0.0163( ) = 0.0652

CLME=

3CDo

!=

3 0.0163( )0.0576

= 0.921

L D( )ME = 14.13

!hME = " 2#

WS

$%&

'()

CDME

CLME3/2

$

%&'

()= " 4.11

#m/s

* ME = "4.05°

VME =58.12

#m/s

Maximum Endurance Glide

48