Aircraft PerformanceRobert Stengel, Aircraft Flight Dynamics

MAE 331, 2008

• Cruising flight

• Flight and maneuveringenvelopes

• Climbing and diving

• Range and endurance

• Turning flight

Copyright 2008 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE331.html

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

Longitudinal Variables

Longitudinal Point-Mass

Equations of Motion

!

˙ V =CT cos" #CD( )

1

2$V

2S #mg sin%

m&

CT #CD( )1

2$V

2S #mg sin%

m

˙ % =CT sin" + CL( )

1

2$V

2S #mgcos%

mV&

CL

1

2$V

2S #mgcos%

mV

˙ h = #˙ z = #vz = V sin%

˙ r = ˙ x = vx = V cos%

!

V = velocity

" = flight path angle

h = height (altitude)

r = range

• Assume thrust is aligned with the velocityvector (small-angle approximation for !)

• Mass = constant

Steady, Level Flight

!

0 =CT "CD( )

1

2#V

2S

m

0 =CL

1

2#V

2S "mg

mV

˙ h = 0

˙ r = V

• Flight path angle = 0

• Altitude = constant

• Airspeed = constant

• Dynamic pressure = constant

• Thrust = Drag

• Lift = Weight

Simple Models of Subsonic

Aerodynamic Coefficients

!

CL

= CLo

+ CL""

!

CD

= CDo

+ "CL

2

• Lift coefficient

• Drag coefficient

• Subsonic flight, belowcritical Mach number

!

CLo

,CL",C

Do

,# $ constant

Subsonic

Supersonic

Transonic

Incompressible

Power and Thrust

• Propeller

• Turbojet

!

Power = P = T "V = CT

1

2#V

3S $ independent of airspeed

!

Thrust = T = CT

1

2"V

2S # independent of airspeed

• Throttle Effect

!

T = Tmax"T = CTmax

"Tq S, 0 # "T #1

Typical Effects of Altitude and

Velocity on Power and Thrust

• Propeller

• Turbojet

Thrust of a Propeller-

Driven Aircraft

!

T ="P"I

Pengine

V="net

Pengine

V

where

"P = propeller efficiency

"I = ideal propulsive efficiency

"netmax# 0.85 $ 0.9

• Efficiencies decrease with airspeed

• Engine power decreases with altitude

• With constant rpm, variable-pitch prop

• Advance Ratio

!

J =V

nD

where

V = airspeed,m /s

n = rotation rate, revolutions /s

D = propeller diameter,m

from McCormick

Propeller Efficiency, ",

and Advance Ratio, JEffect of propeller-blade pitch angle

Thrust of a

Turbojet

Engine

!

T = ˙ m V"o

"o #1

$

% &

'

( )

"t

"t #1

$

% &

'

( ) * c #1( ) +

"t

"o* c

+

, -

.

/ 0

1/ 2

#1

1

2 3

4 3

5

6 3

7 3

where

˙ m = ˙ m air + ˙ m fuel

"o =pstag

pambient

$

% &

'

( )

(8 #1)/8

; 8 = ratio of specific heats 91.4

"t =turbine inlet temperature

freestream ambient temperature

$

% &

'

( )

* c =compressor outlet temperature

compressor inlet temperature

$

% &

'

( )

• Little change in thrust with airspeed below Mcrit

• Decrease with increasing altitude

from Kerrebrock

Performance Parameters

• Lift-to-Drag Ratio

• Load Factor

!

LD

=CL

CD

!

n = LW

= Lmg,"g"s

• Thrust-to-Weight Ratio

!

TW

= Tmg,"g"s

• Wing Loading

!

WS, N m

2or lb ft

2

Trimmed CL and !

• Trimmed liftcoefficient, CL

– Proportional toweight

– Decrease with V2

– At constantairspeed, increaseswith altitude

• Trimmed angle ofattack, !– Constant if dynamic

pressure and weightare constant

!

CLtrim=

W S

q =2W

"V2S

=2W e

#h

"0V2S

!

" trim =2W #V

2S $CLo

CL"

=W q S $CLo

CL"

Thrust Required for

Steady, Level Flight

• Trimmed thrust

!

Ttrim

= Dcruise

= CDo

1

2"V 2

S#

$ %

&

' ( +2)W 2

"V 2S

• Minimum required thrust conditions

!

"Ttrim

"V= C

Do

#VS( ) $4%W 2

#V 3S

= 0

" 2Ttrim

"V 2= C

Do

#S( ) +12%W 2

#V 4S

> 0

Necessary Condition= Zero Slope

Sufficient Conditionfor a Minimum

Airspeed for

Minimum Thrust in

Steady, Level Flight

• Fourth-order equation for velocity– Choose the positive root

!

"Ttrim

"V= C

Do

#VS( ) $4%W 2

#V 3S

= 0

or

V4 =

4%W 2

CDo

#2S2

!

VMT

=2

"

W

S

#

$ %

&

' (

)

CDo

• Satisfy necessary condition

P-51 Mustang

Minimum-Thrust

Example

!

VMT

=2

"

W

S

#

$ %

&

' (

)

CDo

=2

"1555.7( )

0.947

0.0163=76.49

"m /s

!

Wing Span = 37 ft (9.83m)

Wing Area = 235 ft2(21.83m

2)

LoadedWeight = 9,200 lb (3,465 kg)

CDo= 0.0163

" = 0.0576

W /S = 39.3 lb / ft2(1555.7 N /m

2)

Altitude, m

Air Density,

kg/m^3 VMT, m/s0 1.23 69.112,500 0.96 78.205,000 0.74 89.1510,000 0.41 118.87

Lift Coefficient in

Minimum-Thrust

Cruising Flight

!

VMT

=2W

"S

#

CDo

!

CLMT

=2W

"VMT

2

S=

CDo

#

• Airspeed for minimum thrust

• Corresponding lift coefficient

Power Required for

Steady, Level Flight

• Trimmed power

!

Ptrim

= TtrimV = D

cruiseV = C

Do

1

2"V 2

S#

$ %

&

' ( +2)W 2

"V 2S

*

+ ,

-

. / V

• Minimum required power conditions

!

"Ptrim

"V= C

Do

3

2#V 2

S( ) $2%W 2

#V 2S

= 0

Airspeed for Minimum Power

in Steady, Level Flight

• Fourth-order equation for velocity– Choose the positive root

!

VMP

=2

"

W

S

#

$ %

&

' (

)

3CDo

• Satisfy necessary condition

!

"Ptrim

"V= C

Do

3

2#V 2

S( ) $2%W 2

#V 2S

= 0

• Corresponding lift anddrag coefficients

!

CLMP

=3C

Do

"

CDMP

= 4CDo

Achievable Airspeeds

in Cruising Flight

• Two equilibrium airspeeds for a given thrust orpower setting– Low speed, high CL, high !

– High speed, low CL, low !

Achievable Airspeeds

in Cruising Flight

!

Tavail

= CDo

1

2"V 2

S#

$ %

&

' ( +2)W 2

"V 2S

CDo

1

2"V 4

S#

$ %

&

' ( *TavailV

2 +2)W 2

"S= 0

V4 *

TavailV2

CDo

"S+

4)W 2

CDo

"S( )2

= 0

!

Pavail

= TavailV

V4 "

PavailV

CDo

#S+

4$W 2

CDo

#S( )2

= 0

• Jet aircraft (Thrust = constant) • Propeller-driven aircraft(Power = constant)

• Solutions for V can be put inquadratic form and solved easily

!

x "V 2; V = ± x

ax2

+ bx + c = 0

x = #b

2±

b

2

$

% & '

( )

2

# c , a =1

• Solutions for V cannot be putin quadratic form; solution ismore difficult, e.g., Ferrari!smethod

!

aV4 + 0( )V 3 + 0( )V 2 + dV + e = 0

• Best bet: roots in MATLAB

• With increasing altitude, available thrust decreases,and range of achievable airspeeds decreases

• Stall limitation at low speed

• Mach number effect on lift and drag increases thrustrequired at high speed

Thrust Required and Thrust

Available for a Typical Bizjet

Stall

Limit

Typical Simplified JetThrust Model

!

Tmax(h) = T

max(SL)

"#nh

"(SL), n <1

= Tmax(SL)

"#$h

"(SL)

%

& '

(

) *

x

+ Tmax(SL), x

where

, ="#$h

"(SL), n or x is an empirical constant

Maximum Lift-to-Drag Ratio(with Simplified Aero Model)

!

CL( )

L /Dmax

=CDo

"= C

LMT

!

LD

=CL

CD

=CL

CDo

+ "CL

2

!

" CL

CD

# $ %

& ' (

"CL

=1

CDo

+ )CL

2*

2)CL

2

CDo

+ )CL

2( )2

= 0

• Satisfy necessary condition for a maximum

• Lift-to-drag ratio

• Lift coefficient for maximum L/D and minimumthrust are the same

Airspeed, Drag Coefficient, and

Lift-to-Drag Ratio for L/Dmax

!

VL /D

max

=VMT

=2W

"S

#

CDo

!

CD( )

L /Dmax

= CDo

+ CDo

= 2CDo

!

L /D( )max

=CDo

"

2CDo

=1

2 "CDo

• Maximum L/D depends only on induced dragfactor and zero-! drag coefficient

Lift-Drag Polar for

a Typical Bizjet• L/D equals slope of line drawn from the origin

– Single maximum for a given polar

– Two solutions for lower L/D (high and low airspeed)

– Available L/D decreases with Mach number

• Intercept for L/Dmax depends only on # and zero-lift drag

Note different scalesfor lift and drag

P-51 Mustang

Maximum L/D

Example

!

VL /Dmax

=VMT

=76.49

"m /s

!

Wing Span = 37 ft (9.83m)

Wing Area = 235 ft (21.83m2)

LoadedWeight = 9,200 lb (3,465 kg)

CDo= 0.0163

" = 0.0576

W /S =1555.7 N /m2

!

CL( )

L /Dmax

=CDo

"= C

LMT

= 0.531

!

CD( )

L /Dmax

= 2CDo

= 0.0326

!

L /D( )max

=1

2 "CDo

=16.31

Altitude, m

Air Density,

kg/m^3 VMT, m/s0 1.23 69.112,500 0.96 78.205,000 0.74 89.1510,000 0.41 118.87

Cruising Range and

Specific Fuel Consumption

!

0 =CT "CD( )

1

2#V

2S

m

0 =CL

1

2#V

2S "mg

mV

˙ h = 0

˙ r = V

• Thrust = Drag

• Lift = Weight

• Specific fuel consumption, SFC = cP or cT

• Propeller aircraft

• Jet aircraft

!

˙ w f = "cP P proportional to power[ ]

˙ w f = "cTT proportional to thrust[ ]

where

w f = fuel weight

!

cP :kg s

kWor

lb s

HP

cT :kg s

kNor

lb s

lbf

Breguet Range Equation for

Jet Aircraft

!

dr

dw=dr dt

dw dt= "

V

cTT

= "V

cTD

= "L

D

#

$ %

&

' ( V

cTW

• Rate of change of range with respect to weight of fuel burned

• Range traveled

!

Range = r = dr0

R

" = #L

D

$

% &

'

( ) V

cT

$

% &

'

( )

Wi

W f

"dw

w

• For constant true airspeed, V

!

R = "L

D

#

$ %

&

' ( V

cT

#

$ %

&

' ( ln w( )

Wi

W f

=L

D

#

$ %

&

' ( V

cT

#

$ %

&

' ( ln

Wi

W f

#

$ % %

&

' ( ( =

CL

CD

#

$ %

&

' ( V

cT

#

$ %

&

' ( ln

Wi

W f

#

$ % %

&

' ( (

Breguet RangeEquation

Maximum Range of a Jet

Aircraft Flying at

Constant Airspeed

!

" VCL CD( )"CL

= 0 leading to CLMR=

CDo

3#

• For given initial and final weight, range is maximized when

• Breguet range equation for constant V

!

R =CL

CD

"

# $

%

& ' V

cT

"

# $

%

& ' ln

Wi

W f

"

# $ $

%

& ' '

• Because weight decreases and V is assumed constant,altitude must increase to hold CL constant at its best value(“cruise-climb”)

!

W = CL MRq S " q =

W

S

#

$ %

&

' (

3)

CDo

" *(t) =2

V2

W (t)

S

+

, - .

/ 0 3)

CDo

Maximum Range of a

Jet Aircraft Flying at

Constant Altitude

• Range is maximized when

!

Range = "CL

CD

#

$ %

&

' ( 1

cT

#

$ %

&

' (

2

CL)SWi

W f

*dw

w1 2

=CL

CD

#

$ % %

&

' ( ( 2

cT

#

$ %

&

' (

2

)SWi

1 2

"Wf

1 2( )!

V =2W

CL"S

• At constant altitude

!

CL

CD

"

# $ $

%

& ' ' = maximum and (= minimum

Breguet Range Equation for

Propeller-Driven Aircraft

!

dr

dw=dr dt

dw dt= "

V

cPTV

= "V

cPDV

= "L

D

#

$ %

&

' ( 1

cPW

• Rate of change of range with respect to weight of fuel burned

• Range traveled

!

Range = r = dr0

R

" = #L

D

$

% &

'

( ) 1

cP

$

% &

'

( )

Wi

W f

"dw

w

• For constant true airspeed, V

!

R = "L

D

#

$ %

&

' ( 1

cP

#

$ %

&

' ( ln w( )

Wi

W f

=CL

CD

#

$ %

&

' ( 1

cP

#

$ %

&

' ( ln

Wi

W f

#

$ % %

&

' ( (

Breguet RangeEquation

• Range is maximized when

!

CL

CD

"

# $

%

& ' = maximum

P-51 Mustang

Maximum Range

(Internal Tanks only)

!

LoadedWeight = 9,200 lb (3,465 kg)

FuelWeight =1,320 lb (600 kg)

L /D( )max

=16.31

cP = 0.0017kg /s

kW

!

R =CL

CD

"

# $

%

& ' max

1

cP

"

# $

%

& ' ln

Wi

W f

"

# $ $

%

& ' '

= 16.31( )1

0.0017

"

# $

%

& ' ln

3,465 + 600

3,465

"

# $

%

& '

=1,530 km (825 nm( )

Dynamic Pressure and Mach Number

!

" = air density, functionof height

= "sea levele#$h

a = speed of sound, linear functionof height

Dynamic pressure = q =1

2"V

2

Mach number =V

a

Air Data System

Air Speed IndicatorAltimeterVertical Speed Indicator

from Kayton & Fried

Air Data Instruments(“Steam Gauges”)

Calibrated Airspeed Indicator Altimeter Vertical Speed Indicator

Variometer/Altimeter

True Airspeed Indicator

Machmeter

Private Aircraft

Cockpit Panels

Cirrus SR-22 PanelPiper J-3 Cub Panel

Jet Transport

Cockpit PanelsBoeing 747 “Steam Gauge” Panel Boeing 777 “Glass Cockpit”

Definitions of Airspeed

• True Airspeed (TAS)

• Indicated Airspeed (IAS)

• Calibrated Airspeed (CAS)

• Equivalent Airspeed (EAS)

• Airspeed is speed of aircraft measured withrespect to the air mass– Airspeed = Velocity (Inertial speed) if wind speed = 0

!

IAS = 2 pstag " pstatic( ) #SL

!

CAS = IAS corrected for instrument and position errors

!

EAS = CAS corrected for compressibility effects

!

TAS = EAS "SL

"(z)

• Mach number

!

M =TAS

a

Dynamic Pressure and Mach Number

Flight Envelope Determined by

Available Thrust

• Flight ceiling defined byavailable climb rate– Absolute: 0 ft/min

– Service: 100 ft/min

– Performance: 200 ft/min

• Excess thrust providesthe ability to accelerateor climb

• Flight Envelope: Encompasses all altitudes and airspeeds atwhich an aircraft can fly in steady, level flight

Additional Factors Define the

Flight Envelope• Maximum Mach number

• Maximum allowableaerodynamic heating

• Maximum thrust

• Maximum dynamicpressure

• Performance ceiling

• Wing stall

• Flow-separation buffet– Angle of attack

– Local shock waves

Equilibrium Gliding Flight

!

D = CD

1

2"V

2S = #W sin$

CL

1

2"V

2S = W cos$

˙ h = V sin$

˙ r = V cos$

Gliding Flight

!

D = CD

1

2"V

2S = #W sin$

CL

1

2"V

2S = W cos$

˙ h = V sin$

˙ r = V cos$

• Thrust = 0

• Flight path angle < 0 in gliding flight

• Altitude is decreasing

• Airspeed ~ constant

• Air density ~ constant

!

tan" = #D

L= #

CD

CL

=˙ h

˙ r =

dh

dr

" = #tan#1 D

L

$

% &

'

( ) = #cot

#1 L

D

$

% &

'

( )

• Gliding flight path angle

• Corresponding airspeed

!

Vglide =2W

"S CD

2

+ CL

2

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( )

$r =$h

tan"=

#$h

#tan"= maximum when

L

D= maximum

!

"max

= #tan#1D

L

$

% &

'

( ) min

= #cot#1L

D

$

% &

'

( ) max

Sink Rate

• Lift and drag define $ and V in gliding equilibrium

!

D = CD

1

2"V 2

S = #W sin$

sin$ = #D

W

!

L = CL

1

2"V 2

S =W cos#

V =2W cos#

CL"S

!

˙ h = V sin"

= #2W cos"

CL$S

D

W

%

& '

(

) * = #

2W cos"

CL$S

L

W

%

& '

(

) *

D

L

%

& '

(

) *

= #2W cos"

CL$S

cos"C

D

CL

%

& '

(

) *

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

• Minimum sink rateprovides maximumendurance

• Minimize sink rate bysetting ! (dh/dt)/dCL = 0(cos $ ~1)

• See Mathematicaperformance calculationsin Blackboard CourseMaterials

Conditions for Minimum

Steady Sink Rate

!

˙ h = "2W cos#

CL$S

cos#C

D

CL

%

& '

(

) *

= "2W cos

3 #

$S

CD

CL

3 / 2

%

& '

(

) *

!

CLME

=3C

Do

"and C

DME

= 4CDo

!

LD( )

ME

=1

4

3

"CDo

=3

2

LD( )

max

# 0.86 LD( )

max

Minimum

Sink Rate

!

˙ h ME

= "2cos

3 #

$

W

S

%

& '

(

) *

CDME

CL ME

3 / 2

%

& ' '

(

) * * + "

2

$

W

S

%

& '

(

) *

CDME

CL ME

3 / 2

%

& ' '

(

) * *

!

CLME

=3C

Do

"and C

DME

= 4CDo

!

VME

=2W

"S CDME

2+ C

LME

2#

2 W S( )"

$

3CDo

# 0.76VL D

max

• For small $, maximum-endurance airspeed andsink rate

L/D for Minimum

Sink Rate

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

• Which one produces minimum sink rate?

!

LD( )

ME

" 0.86 LD( )

max

VME

" 0.76VL D

max

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 L

D

%

& '

(

) *

max

= $cot$1

(16.31) = $3.51°

CD( )L / Dmax

= 2CDo= 0.0326

CL( )L / Dmax

=CDo

"= 0.531

VL / Dmax=

76.49

+m /s

˙ h L / 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 = 21.83 m2

CDME= 4CDo

= 4 0.0163( ) = 0.0652

CL ME=

3CDo

"=

3 0.0163( )0.0576

= 0.921

L D( )ME

=14.13

˙ h ME = #2

$

W

S

%

& '

(

) *

CDME

CL ME

3 / 2

%

& ' '

(

) * * = #

4.11

$m /s

+ME = #4.05°

VME =58.12

$m /s

Maximum Endurance Glide

• 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"( )

mV

L =W cos"

!

˙ h = V sin" = VT #D( )

W=

Pthrust # Pdrag( )W

Specific Excess Power (SEP) =Excess Power

Unit Weight$

Pthrust # Pdrag( )W

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

Steady Rate of Climb

!

˙ h = V sin" = VT

W

#

$ %

&

' ( )

CDo+ *CL

2( )q

W S( )

+

,

- -

.

/

0 0

!

L = CLq S = W cos"

CL =W

S

#

$ %

&

' ( cos"

q

V = 2W

S

#

$ %

&

' ( cos"

CL)

!

˙ h = VT

W

"

# $

%

& ' (

CDoq

W S( )() W S( )cos

2 *

q

+

, -

.

/ 0

= VT

W

"

# $

%

& ' (

CDo1V

3

2 W S( )(

2) W S( )cos2 *

1V

• Necessary condition for a maximum with respectto airspeed

Condition for Maximum

Steady Rate of Climb

!

˙ h = VT

W

"

# $

%

& ' (

CDo

)V3

2 W S( )(

2* W S( )cos2 +

)V

!

" ˙ h

"V= 0 =

T

W

#

$ %

&

' ( + V

"T /"V

W

#

$ %

&

' (

)

* +

,

- . /

3CDo

0V2

2 W S( )+

21 W S( )cos2 2

0V2

Maximum Steady

Rate of Climb:

Propeller-Driven Aircraft

!

"Pthrust

"V=

T

W

#

$ %

&

' ( +V

"T /"V

W

#

$ %

&

' (

)

* +

,

- . = 0

• At constantpower

!

" ˙ h

"V= 0 = #

3CDo

$V2

2 W S( )+

2% W S( )$V

2

• Therefore, with cos2$ ~ 1

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

!

V4 =

4

3

"

# $ %

& ' ( W S( )

2

CDo

)2; V = 2

W S( ))

(

3CDo

=VME

Maximum Steady

Rate of Climb:

Jet-Driven Aircraft

• Condition for a maximum at constant thrust and cos2$ ~ 1

• Airspeed for maximum rate of climb at maximum thrust, Tmax,found by solving equation of the form!

" ˙ h

"V= 0

0 = #3C

Do

$

2 W S( )V

4 +T

W

%

& '

(

) * V

2 +2+ W S( )

$

= #3C

Do

$

2 W S( )V

2( )2

+T

W

%

& '

(

) * V

2( ) +2+ W S( )

$

!

0 = ax2

+ bx + c and V = + x

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

• Could trade altitude with airspeed with no change in energyheight if thrust and drag were zero

!

Total Energy

Unit Weight" Specific Energy =

mgh + mV22

mg= h +

V2

2g

" Energy Height, Eh, ft or m

Specific Excess Power

!

dEh

dt=d

dth +

V2

2g

"

# $

%

& ' =

dh

dt+V

g

"

# $

%

& ' dV

dt

=V sin( +V

g

"

# $

%

& ' T )D)mgsin(

m

"

# $

%

& ' =V

T )D( )W

=VCT )CD( )

1

2*(h)V 2

S

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

• Objective: Minimize time or fuel to climb to desired altitudeand airspeed

Supersonic Energy Climb

• Objective: Minimize time or fuel to climb to desired altitudeand airspeed

• “V-n diagram”

• Maneuvering envelopedescribes limits on normalload factor and allowableequivalent airspeed– Structural factors

– Maximum and minimumachievable lift coefficients

– Maximum and minimumairspeeds

– Protection againstoverstressing due to gusts

– Corner Velocity:Intersection of maximum liftcoefficient and maximumload 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 envelopehttp://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

• Vertical force equilibrium

Level Turning Flight

!

Lcosµ =W

• Load factor

!

n = LW

= Lmg

= secµ,"g"s

• Thrust required to maintain level flight

!

Treq = CDo+ "CL

2( )1

2#V 2

S = Do +2"

#V 2S

W

cosµ

$

% &

'

( )

2

= Do +2"

#V 2SnW( )

2

!

µ :Bank Angle

• Level flight = constant altitude

• Sideslip angle = 0

• Bank angle

Maximum Bank Angle

in Level Flight

!

cosµ =W

CLq S=1

n= W

2"

Treq #Do( )$V2S

µ = cos#1W

CLq S

%

& '

(

) * = cos

#1 1

n

%

& ' (

) * = cos

#1W

2"

Treq #Do( )$V2S

+

,

- -

.

/

0 0

• Bank angle is limited by

!

µ :Bank Angle

!

CLmax

or Tmax

or nmax

• Turning rate

Turning Rate and Radius in Level Flight

!

˙ " =CLq S sinµ

mV=

W tanµ

mV=

gtanµ

V=

L2 #W

2

mV

=W n

2 #1

mV=

Treq #Do( )$V2S 2% #W

2

mV

• Turning rate is limited by

!

CLmax

or Tmax

or nmax

• Turning radius

!

Rturn =V

˙ " =

V2

g n2#1

• Corner velocity

Corner Velocity Turn

• Turning radius

!

Rturn =V2cos

2 "

g nmax

2# cos2 "

!

Vcorner

=2n

maxW

CLmas

"S

• For steady climbing or diving flight

!

sin" =Tmax

#D

W

• Turning rate

!

˙ " =g n

max

2

# cos2 $( )

V cos$

• Time to complete a full circle

!

t2" =

V cos#

g nmax

2$ cos2 #

• Altitude gain/loss

!

"h2# = t

2#V sin$

F-16 in 9-g turnhttp://www.youtube.com/watch?v=zT-pJDo6nkw&feature=related

Pilot in 9-g centrifuge traininghttp://www.youtube.com/watch?v=3rQexWEwV6M&feature=related

Maximum Turn Rates

Herbst Maneuver

• Minimum-time reversal of direction

• Kinetic-/potential-energy exchange

• Yaw maneuver at low airspeed

• X-31 performing the maneuver

Next Time:Aircraft Equations of Motion