Aircraft Performance Lecture6-7

7/30/2019 Aircraft Performance Lecture6-7

1/17

Aircraft PerformanceRobert Stengel, Aircraft Flight Dynamics

MAE 331, 2008

Cruising flight Flight and maneuvering

envelopes

Climbing and diving

Range and endurance

Turning flight

Copyright 2008 by Robert Stengel. All rights reserved. For education al 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=

CTcos"#CD( )

1

2$V2S#mg sin%

m&

CT #CD( )

1

2$V2S#mg sin%

m

%= CTsin"+ CL( )1

2$V2S#mgcos%

mV&

CL

1

2$V2S#mgcos%

mV

h = #z = #vz =Vsin%

r = x = vx =Vcos%

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 =

CL1

2#V

2S"mg

mV

h = 0

r =V

Flight path angle = 0

Altitude = constant

Airspeed = constant

Dynamic pressure = constant

Thrust = Drag

Lift = Weight


2/17

Simple Models of Subsonic

Aerodynamic Coefficients

CL=C

Lo+C

L""

CD=C

Do

+ "CL

2

Lift coefficient

Drag coefficient

Subsonic flight, belowcritical Mach number

CL

o

,CL"

,CD

o

,# $ constant

Subsonic

Supersonic

Transonic

Incompressible

Power and Thrust

Propeller

Turbojet

Power = P =T"V= CT1

2

#V3S$ independent of airspeed

Thrust=T=CT1

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


3/17

Advance Ratio

J= VnD

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= mV"

o

"o #1

$

%&

'

()

"t

"t #1

$

%&

'

()*c #1( )+

"t

"o*

c

+

,-

.

/0

1/ 2

#1123

43

563

73

wherem = mair + mfuel

"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 m2 orlb 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 Sq

= 2W"V2S

= 2W e#h

"0V2S

"trim =2W #V

2S$CLo

CL"

=

W q S$CLo

CL"


4/17

Thrust Required for


Trimmed thrust

Ttrim = Dcruise =CDo

1

2"V

2

S

#

$%

&

'(+

2)W2

"V2S

Minimum required thrust conditions

"Ttrim

"V=C

Do#VS( )$

4%W2

#V3S= 0

"2Ttrim

"V2

=CDo

#S( )+12%W

2

#V4S

> 0

Necessary Condition= Zero Slope

Sufficient Conditionfor a Minimum

Airspeed for

Minimum Thrust in


Fourth-order equation for velocity Choose the positive root

"Ttrim

"V=C

Do #VS

( )$4%W

2

#V3S= 0

or

V4=

4%W2

CD

o

#2S2

VMT

=

2

"

W

S

#

$%

&

'(

)

CD

o

Satisfy necessary condition

P-51 Mustang

Minimum-Thrust

Example

VMT

=

2

"

W

S

#

$%

&

'(

)

CD

o

=

2

"1555.7( )

0.947

0.0163=

76.49

"m /s

Wing Span = 37 ft(9.83 m)

Wing Area = 235 ft2(21.83 m

2)

Loaded Weight= 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

#

CD

o

CLMT

=

2W

"VMT

2S

=

CD

o

#

Airspeed for minimum thrust

Corresponding lift coefficient


5/17

Power Required for


Trimmed power

Ptrim

= TtrimV = D

cruiseV = C

Do

1

2"V

2S#

$% &

'(+ 2)W

2

"V2S

*

+, -

./V

Minimum required power conditions

"Ptrim

"V=C

Do

3

2#V

2S( )$

2%W2

#V2S

= 0

Airspeed for Minimum Power

in Steady, Level Flight

Fourth-order equation for velocity Choose the positive root

VMP

=

2

"

W

S

#

$%

&

'(

)

3CD

o

Satisfy necessary condition

"Ptrim

"V=C

Do

3

2#V

2S( )$

2%W2

#V2S

= 0

Corresponding lift anddrag coefficients

CLMP

=

3CD

o

"

CD

MP

= 4CD

o

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"V2S#

$% &

'(+

2)W

2

"V2S

CDo

1

2"V

4S

#

$%

&

'(*TavailV

2+

2)W2

"S= 0

V4 *

Tavail

V2

CDo

"S+

4)W2

CDo

"S( )2= 0

Pavail

= Tavail

V

V4"Pavail

V

CDo

#S+

4$W2

CDo

#S( )2= 0

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

Solutions for Vcan be put inquadratic form and solved easily

x "V2; V = x

ax2+ bx + c = 0

x =

#

b

2

b

2

$

%&

'

()

2

#c , a =1

Solutions for Vcannot be putin quadratic form; solution ismore difficult, e.g., Ferrari!smethod

aV4+ 0( )V3 + 0( )V2 + dV+ e = 0

Best bet: rootsin MATLAB


6/17

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) = Tmax (SL)"#nh

"(SL), n


7/17

P-51 Mustang

Maximum L/D

Example

VL / Dmax

=VMT

=

76.49

"m /s

Wing Span = 37 ft(9.83 m)

Wing Area = 235 ft(21.83 m2)

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

CDo = 0.0163

"= 0.0576

W /S=1555.7 N/m2

CL( )L /Dmax =CD

o

"

=CLMT = 0.531

CD( )L /D

max

= 2CD

o

= 0.0326

L /D( )max

=

1

2 "CD

o

=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

2

S"

mgmV

h = 0

r =V

Thrust = Drag

Lift = Weight

Specific fuel consumption, SFC = cPor cT

Propeller aircraft

Jet aircraft

wf = "cPP proportional to power[ ]

wf = "cTT proportional to thrust[ ]

where

wf = 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

Wf

"dw

w

For constant true airspeed, V

R = "L

D

#

$%

&

'(V

cT

#

$%

&

'(ln w( )

Wi

Wf

=

L

D

#

$%

&

'(V

cT

#

$%

&

'(ln

Wi

Wf

#

$%%

&

'((=

CL

CD

#

$%

&

'(V

cT

#

$%

&

'(ln

Wi

Wf

#

$%%

&

'((

Breguet RangeEquation

Maximum Range of a Jet

Aircraft Flying at

Constant Airspeed

" VCL CD( )"CL

= 0 leading to CLMR =C

Do

3#

For given initial and final weight, range is maximized when

Breguet range equation for constant V

R =CL

CD

"

#$ %

&'V

cT

"

#$ %

&'ln

Wi

Wf

"

#$$

%

&''

Because weight decreases and Vis assumed constant,altitude must increase to hold CL constant at its best value(cruise-climb)

W= CLMR q S" q =W

S

#

$%

&

'(

3)

CDo

" *(t) =2

V2

W(t)

S

+

,-.

/03)

CDo


8/17

Maximum Range of a

Jet Aircraft Flying at

Constant Altitude

Range is maximized when

Range = "CL

CD

#

$%

&

'(

1

cT

#

$%

&

'(

2

CL)SWi

Wf

*dw

w1 2

=

CL

CD

#

$%%

&

'((

2

cT

#

$%

&

'(

2

)SWi

1 2 "Wf1 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

cP

TV= "

V

cP

DV= "

L

D

#

$%

&

'(

1

cP

W

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

Range traveled

Range = r = dr0

R

" = #L

D

$

%&

'

()

1

cP

$

%&

'

()

Wi

Wf

"dw

w

For constant true airspeed, V

R = "L

D

#

$%

&

'(

1

cP

#

$%

&

'(ln w( )

Wi

Wf

=

CL

CD

#

$%

&

'(

1

cP

#

$%

&

'(ln

Wi

Wf

#

$%%

&

'((

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

Wf

"

#$$

%

&''

= 16.31( )1

0.0017

"

#$

%

&'ln

3,465+ 600

3,465

"

#$

%

&'

=1,530 km (825 nm( )

Dynamic Pressure and Mach Number

"= airdensity, functionof height

= "sea levele#$h

a = speed of sound, linear functionof height

Dynamic pressure = q =1

2"V2

Mach number =V

a


9/17

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 PanelsBo ein g 74 7 St eam Ga uge Pan el Boe ing 777 G la ss Cock pit


10/17

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: 20 0 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 dynamic

pressure

Performance ceiling

Wing stall

Flow-separation buffet

Angle of attack

Local shock waves


11/17

Equilibrium Gliding Flight

D =CD

1

2"V

2S= #Wsin$

CL

1

2 "V

2S=Wcos

$

h =Vsin$

r =Vcos$

Gliding Flight

D =CD

1

2"V

2S=#Wsin$

CL

1

2"V

2S=Wcos$

h =Vsin$

r =Vcos$

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#1D

L

$

%&

'

()= #cot

#1 L

D

$

%&

'

()

Gliding flight path angle

Corresponding airspeed

Vglide =2W

"S CD2

+ CL2

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 Vin gliding equilibrium

D =CD

1

2

"V2S= #W sin$

sin$= #D

W

L =CL

1

2"V

2S=W cos#

V =2W cos#

CL"S

h =Vsin"

=#2Wcos"

CL$S

D

W

%

&'

(

)*=#

2Wcos"

CL$S

L

W

%

&'

(

)*D

L

%

&'

(

)*

=#2Wcos"

CL

$Scos"

CD

CL

%

&

'(

)

*

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


12/17

Minimum sink rate

provides maximumendurance

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

See Mathematicaperformance calculationsin Blackboard CourseMaterials

Conditions for Minimum

Steady Sink Rate

h =

"

2Wcos#

CL$Scos

#

CD

CL

%

&'

(

)*

= "2Wcos

3#

$S

CD

CL

3/ 2

%

&'

(

)*

CL

ME=

3CD

o

"

and C D

ME= 4C

Do

LD( )ME =

1

4

3

"CD

o

=

3

2

LD( )max # 0.86

LD( )max

Minimum

Sink Rate

hME

= "2cos

3#

$

W

S

%

&'

(

)*

CD

ME

CLME

3/ 2

%

&''

(

)**+ "

2

$

W

S

%

&'

(

)*

CDME

CLME

3/ 2

%

&''

(

)**

CLME

=

3CDo

"

and C DME

= 4CDo

VME

=

2W

"S CD

ME

2+ C

LME

2#

2 W S( )"

$

3CD

o

# 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


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

hL /Dmax =Vsin#= $4.68

+m /s

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

Maximum Range Glide


S= 21.83 m2

CDME = 4CDo = 4 0.0163( ) = 0.0652

CLME =3CDo"

=3 0.0163( )

0.0576= 0.921

L D( )ME

=14.13

hME = #2

$

W

S

%

&'

(

)*

CDMECLME

3/ 2

%

&''

(

)**= #

4.11

$m /s

+ME = #4.05

VME =58.12

$m /s

Maximum Endurance Glide


13/17

Rate of climb, dh/dt= Specific Excess Power

Climbing

Flight

V= 0 =T"D"Wsin#( )

m

sin#=T"D( )W

; #= sin"1 T"D( )

W

"= 0 = L #W cos"( )mV

L =Wcos"

h =Vsin"=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 =Vsin"=VT

W

#

$

%&

'

()CDo

+ *CL2( )q

W S( )

+

,

-

-

.

/

0

0

L =CLq S=Wcos"

CL =

W

S

#

$%

&

'(cos"

q

V= 2 WS

#

$% &

'(cos

"C

L)

h =VT

W

"

#$

%

&'(

CDoq

W S( )()W S( )cos2 *

q

+

,-

.

/0

=VT

W

"

#$

%

&'(

CDo

1V3

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( )+

21W S( )cos2 20V

2

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 = 2W S( ))

(

3CDo

= VME


14/17

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 energy

height if thrust and drag were zero

Total EnergyUnit Weight

" Specific Energy = mgh + mV2

2mg

= h + V2

2g

" Energy Height, Eh, ft or m

Specific Excess Power

dEh

dt=

d

dth +

V2

2g

"

#$

%

&'=

dh

dt+

V

g

"

#$

%

&'

dV

dt

=Vsin(+V

g

"

#$

%

&'

T) D) mgsin(

m

"

#$

%

&'=V

T) D( )W

=VCT ) CD( )

1

2*(h)V2S

W

= Specific Excess Power (SEP) =Excess Power

Unit Weight+

Pthrust ) Pdrag( )W


15/17

Contours of Constant

Specific Excess Power

Specific Excess Power is a function of altitude and airspeed

SEPis 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 envelope

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


16/17

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

Lcos=W

Load factor

n = LW

=Lmg

= sec,"g"s

Thrust required to maintain level flight

Treq =

CDo

+ "CL2( )1

2#V2S= Do +

2"

#V2S

W

cos

$

%&

'

()

2

= Do +2"

#V2SnW( )

2

:Bank Angle

Level flight = constant altitude

Sideslip angle = 0

Bank angle

Maximum Bank Angle

in Level Flight

cos=W

CL

q S=

1

n=W

2"

Treq

#Do( )$V2S

= cos#1 W

CL

q S

%

&'

(

)*= cos

#1 1

n

%

&'

(

)*= cos

#1 W2"

Treq

#Do( )$V2S

+

,

--

.

/

00

Bank angle is limited by

:Bank Angle

CLmax

or Tmax

or nmax

Turning rate

Turning Rate and Radius in Level Flight

"=CLq Ssin

mV

=

Wtan

mV

=

g tan

V

=

L2#W

2

mV

=

W n2#1

mV=

Treq #Do( )$V2S 2%#W2

mV

Turning rate is limited by

CLmax

or Tmax

or nmax

Turning radius

Rturn =

V

" =V

2

g n

2#1


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Corner velocity

Corner Velocity Turn

Turning radius

Rturn =V

2cos

2"

g nmax

2# cos

2"

Vcorner

=

2nmaxW

CLmas

"S

For steady climbing or diving flight

sin"=Tmax

# D

W

Turning rate

"= g nmax2

# cos2 $( )

Vcos$

Time to complete a full circle

t2"

=

Vcos#

g nmax

2

$ cos2#

Altitude gain/loss

"h2#

= t2#Vsin$

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

Documents

Aircraft Performance Lecture6-7