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