Aircraft Flight Dynamics Robert F. Stengel Lecture7 Gliding, Climbing, and Turning Performance

Gliding, Climbing, and Turning Flight Performance Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2012!. Copyright 2012 by Robert Stengel.. The Maneuvering Envelope• Maneuvering en

Gliding, Climbing, and Turning

Flight Performance


Robert Stengel, Aircraft Flight Dynamics,

MAE 331, 2012 !

Copyright 2012 by Robert Stengel All rights reserved For educational use only.!

http://www.princeton.edu/~stengel/MAE331.html ! http://www.princeton.edu/~stengel/FlightDynamics.html !

•  Flight envelope"

•  Minimum glide angle/rate"

•  Maximum climb angle/rate"

•  V-n diagram"

•  Energy climb"

•  Corner velocity turn"

•  Herbst maneuver "

The Flight Envelope

Flight Envelope Determined by

Available Thrust"

•   Flight ceiling defined by

available climb rate "

–  Absolute: 0 ft/min"

–  Service: 100 ft/min"

–  Performance: 200 ft/min" •  Excess thrust provides the ability to accelerate or climb"

•  Flight Envelope: Encompasses all altitudes

and airspeeds at which an aircraft can fly "

–  in steady, level flight "

–  at fixed weight "

Additional Factors Define the

Flight Envelope"

•   Maximum Mach number"

•   Maximum allowable aerodynamic heating"

•   Maximum thrust"

•   Maximum dynamic pressure"

•   Performance ceiling"

•   Wing stall"

•   Flow-separation buffet"

–   Angle of attack"

–   Local shock waves "

Piper Dakota Stall Buffet"

http://www.youtube.com/watch?v=mCCjGAtbZ4g !

Boeing 787 Flight

Best Cruise

Gliding Flight

h = V sinγ

r = V cosγ

•  Thrust = 0"

•  Flight path angle < 0 in gliding flight"

•  Altitude is decreasing"

•  Airspeed ~ constant"

•  Air density ~ constant "

tan γ = − D

CD

h

r =

dh

dr ; γ = − tan

−1 D L

#

$

% &

'

( = −cot−1 L

D

#

$

% & ' (

•  Gliding flight path angle "

•  Corresponding airspeed "

2

2

Maximum Steady Gliding Range"

•  Glide range is maximum when γ is least negative, i.e.,

most positive"

•  This occurs at (L/D)max

Maximum Steady Gliding Range"

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

•  This occurs at (L/D)max

tanγ = h

r = negative constant =

h − ho

r − ro

tanγ =

−Δh

− tan γ = maximum when

L

γmax= − tan−1 D

L

#

$

% &

'

(

min

= −cot−1 L

D

#

$

% &

'

(

max

Sink Rate "

•   Lift and drag define γ and V in gliding equilibrium "

sinγ = − D

W

L = CL

1

2 ρV2

S = W cosγ

CLρS

h =V sin γ

= − 2W cos γ

CLρ S

D W

$

%

& ' (

) = − 2W cos γ

CLρ S

L W

$

%

& ' (

L

$

%

& ' ( )

= − 2W cos γ

CLρ S cos γ

1

L D

$

%

& ' ( )

•  Minimum sink rate provides maximum endurance"

•  Minimize sink rate by setting ∂(dh/dt)/dC L = 0 ( cos γ ~1 )"

Conditions for Minimum Steady Sink Rate"

h = − 2W cosγ

CL

$

%

& ' ( )

= − 2W cos

3

γ

ρS

CD

CL3/2

$

%

& ' (

) ≈ − 2

ρ

W S

$

%

& ' (

) CD

CL3/2

$

%

& ' ( )

L/D and VME for Minimum Sink Rate"

VME= 2W

ρS CDME2+ CLME2 ≈

2 W S ( )

ρ

ε

3CDo ≈ 0.76VL Dmax

L

D

( )ME = 1

4

3

εCD o =

3 2

L D

( )max ≈ 0.86 L D ( )max

L/D for Minimum Sink Rate"

•  For L/D < L/Dmax , there are two solutions"

•  Which one produces minimum sink rate? "

L D

( )ME ≈ 0.86 L D( )max

V ME ≈ 0.76V L D

max

Gliding Flight of the

P-51 Mustang"

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

L / D

( )max = 1

2 εC D o

= 16.31

γMR= −cot −1 L

D

$

%

'

( max

= −cot −1 (16.31) = −3.51°

C D

( )L/Dmax= 2C D o= 0.0326

C L

( )L/Dmax = C D o

ε = 0.531

V L/Dmax =76.49

ρ m / s

h L/Dmax= V sinγ = −4.68

ρ m / s

R h o =10 km= 16.31( )( )10 = 163.1 km

Maximum Range Glide"

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

C D ME = 4CD o= 4 0.0163( )= 0.0652

3 0.0163( ) 0.0576 = 0.921

L D

( )ME= 14.13

h ME= − 2 ρ

W S

$

%

' (

ME

C L3/2ME

$

%

&& ' ( )) = −4.11ρ m / s

γME= −4.05°

V ME=58.12

Maximum Endurance Glide"

Climbing Flight

•  Rate of climb, dh/dt = Specific Excess Power "

Climbing Flight"



V = 0 =(T − D −W sinγ)

m

W

mV

P thrust − P drag

W

Specific Excess Power (SEP)=Excess Power

Unit Weight ≡

P thrust − P drag

W

•  Flight path angle •  Required lift"

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

Steady Rate of Climb"

h =V sinγ =V T

W

"

#

&

' − CD o+ εCL

2

W S

* +

, ,

-.

/ /

€

L = C L q S = W cosγ

C L= W

S

#

$

&

'

cos γ

q

V = 2 W S

#

$

&

'

cos γ

C Lρ

h =V T

W

!

"

%

& − CD oq

W S

ε ( W S ) cos2

γ

q

* +

- /

W

!

"

%

& − CD oρ ( ) h V3

2 ε ( W S ) cos2γ

ρ ( ) h V

•  Climb rate

respect to airspeed"

Condition for Maximum Steady Rate of Climb"

h =V T

W

!

"

%

& − CD oρV3

2 W S ( ) −

2ε W S ( ) cos2

γ

ρV

T

W

"

#

&

'+V ∂ T / ∂ V

W

"

#

&

'

(

)

, -− 3CD oρ V

2

2 W S ( ) +

2 ε ( W S ) cos2

γ

ρ V2

Maximum Steady " Rate of Climb: "

Propeller-Driven Aircraft"

∂ P thrust

T W

"

#

&

W

"

#

&

' (

)

+ ,

•  At constant power"

∂ h

∂V = 0 = −

3CDoρV2

2 W S ( ) +

2ε W S ( )

ρV2

•  With cos 2γ ~ 1, optimality condition reduces to"

•  Airspeed for maximum rate of climb at maximum power, P max"

V4

3

!

"

%

& ε ( W S )2

CD

oρ2 ; V = 2

W S

ρ

ε

3CD o

= VME

Maximum Steady Rate

of Climb: "

Jet-Driven Aircraft"

•  Condition for a maximum at constant thrust and cos 2γ ~ 1 "

∂ h

0 = ax2

+ bx + c and V = + x

= − 3CD oρ

2 W S ( ) V

4

W

#

$

% &

'

(V2+ 2ε W S ( )

ρ

= − 3CD oρ

2 W S ( ) V

2

W

#

$

% &

'

( V ( )2 + 2ε W S ( )

ρ

Optimal Climbing Flight

What is the Fastest Way to Climb from

One Flight Condition to Another?" •   Specific Energy "

•  = (Potential + Kinetic Energy) per Unit Weight"

•  = Energy Height "

Energy Height"

height if thrust and drag were zero "

Total Energy Unit Weight ≡ Specific Energy =

mgh + mV2

2

V2

2g

Specific Excess Power"

dEh

d

2

2g

!

"

%

& = dh

V g

!

"

%

& dV

dt

= V sinγ + V

g

"

#

&

' T − D − mgsinγ

m

"

#

&

' = V ( T − D )

W = V

CT− CD

2 ρ(h)V

W

= Specific Excess Power (SEP) = Excess Power

Unit Weight ≡

Pthrust− Pdrag

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

Subsonic Energy Climb"

and airspeed "

Supersonic Energy Climb"

and airspeed "

The Maneuvering Envelope

•  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:

•   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"

C-130 exceeds maneuvering envelope"

http://www.youtube.com/watch?v=4bDNCac2N1o&feature=related !

Maneuvering Envelopes (V-n Diagrams)

for Three Fighters of the Korean War Era"

Republic F-84"

North American F-86"

Lockheed F-94"

Turning Flight

•  Vertical force equilibrium "

Level Turning Flight"

L cos µ = W

n = L W = L mg = sec µ,"g"s

T req = C D o+εC L

2

( )12ρV2S = D o+ 2ε

W

cos µ

#

$%

&

'(

2

µ : Bank Angle

•  Level flight = constant altitude"

•  Sideslip angle = 0 "

•  Bank angle"

Maximum Bank Angle in Level Flight"

cosµ = W

C L qS=

1

n = W

2ε

T req − D o

µ = cos−1 W

C L qS

$

%

& ' ( ) = cos−1 1

n

$

%

' (

T req − D o

* +

, ,

-

/ /

•  Bank angle is limited by "

€

µ : Bank Angle

CLmax or Tmax or nmax

Turning Rate and Radius in Level Flight"

W tan µ

g tan µ

L2

mV

2

− 1

T req − D o

( )ρV2S 2ε − W2

mV

CL

max or Tmax or nmax

R turn=Vξ= V

2

g n2− 1

Maximum Turn Rates"

“Wind-up turns”"

•   Corner velocity"

Corner Velocity Turn"

•   Turning radius "

Rturn= V

γ

g nmax2

− cos2γ

C L

mas ρS

•   For steady climbing or diving flight"

sinγ =Tmax− D

W

Corner Velocity Turn"

•   Time to complete a full circle "

t2π= V cosγ

g nmax2

•   Altitude gain/loss "

Δh2π = t2πV sinγ

•   Turning rate "

ξ = g nmax

2

− cos2γ

V cosγ

Not a turning rate comparison "

http://www.youtube.com/watch?v=z5aUGum2EiM!

Herbst Maneuver"

•  Minimum-time reversal of direction"

•  Kinetic-/potential-energy exchange"

•  Yaw maneuver at low airspeed"

Aircraft Equations of Motion

Reading