16.885J/ESD.35J - Oct 1, 2002

Introduction to Aircraft Performance and Static Stability

16.885J/ESD.35JAircraft Systems Engineering

Prof. Earll MurmanSeptember 18, 2003

16.885J/ESD.35J - Oct 1, 2002

Today’s Topics• Specific fuel consumption and Breguet range

equation• Transonic aerodynamic considerations• Aircraft Performance

– Aircraft turning– Energy analysis– Operating envelope– Deep dive of other performance topics for jet transport

aircraft in Lectures 6 and 7

• Aircraft longitudinal static stability

16.885J/ESD.35J - Oct 1, 2002

Thrust Specific Fuel Consumption (TSFC)

• Definition:

• Measure of jet engine effectiveness at converting fuel to useable thrust

• Includes installation effects such as – bleed air for cabin, electric generator, etc..– Inlet effects can be included (organizational dependent)

• Typical numbers are in range of 0.3 to 0.9. Can be up to 1.5• Terminology varies with time units used, and it is not all

consistent.– TSFC uses hours– “c” is often used for TSFC– Another term used is

TSFC lb of fuel burned(lb of thrust delivered)(hour)

ctlb of fuel burned

(lb of thrust delivered)(sec)

16.885J/ESD.35J - Oct 1, 2002

Breguet Range Equation• Change in aircraft weight = fuel burned

• Solve for dt and multiply by V to get ds

• Set L/D, ct, V constant and integrate

dW ctTdt ct TSPC/3600 T thrust

ds V dt V dWctT

V WctT

dWW

V LctD

dWW

R 3600TSFCV

LDln

WTOWempty

16.885J/ESD.35J - Oct 1, 2002

Insights from Breguet Range Equation

R 3600TSFCV

LDln

WTOWempty

3600TSFC represents propulsion effects. Lower TSFC is better

V LD represents aerodynamic effect. L/D is aerodynamic efficiency

V LD a M L

D. a is constant above 36,000 ft. M LD important

lnWTOWempty

represents aircraft weight/structures effect on range

16.885J/ESD.35J - Oct 1, 2002

Optimized L/D - Transport A/C“Sweet spot” is in transonic range.

Lossesdue to shockwaves

Ref: Shevell

Max

(L/D

)

Mach No.1 2 3

10

20

Concorde

16.885J/ESD.35J - Oct 1, 2002

Transonic Effects on Airfoil Cd, ClCd

Mcr Mdrag divergence

M 81.0

M < Mcr8

V 8

V 8

Region I. II. III.

I.

II.

III. M > Mdrag divergence

8

Mcr < M < Mdrag divergence

8

M<1

M<1

M>1

M<1M>1

Separated flow

16.885J/ESD.35J - Oct 1, 2002

Strategies for Mitigating Transonic Effects

• Wing sweep– Developed by Germans. Discovered after WWII by Boeing– Incorporated in B-52

• Area Ruling, aka “coke bottling”– Developed by Dick Whitcomb at NASA Langley in 1954

• Kucheman in Germany and Hayes at North American contributors

– Incorporated in F-102• Supercritical airfoils

– Developed by Dick Whitcomb at NASA Langley in 1965• Percey at RAE had some early contributions

– Incorporated in modern military and commercial aircraft

16.885J/ESD.35J - Oct 1, 2002

Basic Sweep Concept• Consider Mach Number normal to leading edge

• For subsonic freestreams, Mn < M - Lower effective “freestream”Mach number delays onset of transonic drag rise.

• For supersonic freestreams– Mn < 1, > - Subsonic leading edge– Mn > 1, < - Supersonic leading edge

• Extensive analysis available, but this is gist of the concept

sin =1/ M

= Mach angle, the direction disturbancestravel in supersonic flow

MMn=M cos

16.885J/ESD.35J - Oct 1, 2002

Wing Sweep Considerations M > 1

• Subsonic leading edge– Can have rounded subsonic type wing section

• Thicker section• Upper surface suction• More lift and less drag

• Supersonic leading edge– Need supersonic type wing section

• Thin section• Sharp leading edge

16.885J/ESD.35J - Oct 1, 2002

Competing Needs

• Subsonic Mach number– High Aspect Ratio for low induced drag

• Supersonic Mach number– Want high sweep for subsonic leading edge

• Possible solutions– Variable sweep wing - B-1– Double delta - US SST– Blended - Concorde– Optimize for supersonic - B-58

16.885J/ESD.35J - Oct 1, 2002

Supercritical AirfoilSupercriticalairfoil shape keeps upper surface velocity from getting too large.

Uses aft camber to generate lift.

Gives nose down pitching moment.

Cp

x/c

Cp, cr

V 8

16.885J/ESD.35J - Oct 1, 2002

Today’s Performance Topics• Turning analysis

– Critical for high performance military a/c. Applicable to all.– Horizontal, pull-up, pull-down, pull-over, vertical– Universal M- turn rate chart , V-n diagram

• Energy analysis– Critical for high performance military a/c. Applicable to all.– Specific energy, specific excess power– M-h diagram, min time to climb

• Operating envelope• Back up charts for fighter aircraft

– M- diagram - “Doghouse” chart– Maneuver limits and some example– Extensive notes from Lockheed available. Ask me.

16.885J/ESD.35J - Oct 1, 2002

Horizontal TurnW = L cos , = bank angle

Level turn, no loss of altitude

Fr = (L2 - W2)1/2 =W(n2-1)1/2

Where n L/W = 1/ cos is the load factor measured in “g’s”.

But Fr = (W/g)(V2 /R)

So radius of turn is

R = V2 /g(n2-1)1/2

And turn rate = V /R is

= g(n2-1)1/2 / V

Want high load factor, low velocity

Flight path

θ

R

φ

φ

R

L

W

Fr

z

z

16.885J/ESD.35J - Oct 1, 2002

Pull Up Pull Over Vertical

Fr = (L-W) =W(n-1)

= (W/g)(V2 /R)

R = V2 /g(n-1)

= g(n-1)/ V

Fr = (L +W) =W(n+1)

= (W/g)(V2 /R)

R = V2 /g(n+1)

= g(n+1)/ V

Fr = L =Wn

= (W/g)(V2 /R)

R = V2 /gn

= gn/ V

R

L

W

θ

R

L

W

θR

L

W

θ

Let Ý

Pull Over

K = (n+1)/(n2-1)1/2

Vertical Maneuver

K = n/(n2-1)1/2

Pull Up

K = (n-1)/(n2-1)1/2

For large n, K 1 and for all maneuvers gn/ V

Similarly for turn radius, for large n, R V2 /gn.

For large and small R, want large n and small V

Pullover from inverted attitude

Vertical maneuver

Pull-up from level attitude

0

0.5

1.0

1.5

2.0

2.5

1 2 3 4 5 6 7 8 9

Turn

rate

rati

o, K

θ

Load factor, in 'g's

Vertical Plan Turn Rates

Vertical planeturn rateKθ = Horizontal planeturn rate

16.885J/ESD.35J - Oct 1, 2002

gn/ V = gn/a M so ~ 1/ M at const h (altitude) & n

Using R V2 /gn, V /R = a M /R. So ~ M at const h & R

For high Mach numbers, the turn radius gets large

16.885J/ESD.35J - Oct 1, 2002

Rmin and maxUsing V = (2L/ SCL)1/2 = (2nW/ SCL)1/2

R V2 /gn becomes R = 2(W/S)/ g CL

W/S = wing loading, an important performance parameter

And using n = L/W = V2 SCL/2W

gn/ V = g V CL/2(W/S)

For each airplane, W/S set by range, payload, Vmax.

Then, for a given airplaneRmin = 2(W/S)/ g CL,max

max = g V CL,max /2(W/S)

Higher CL,max gives superior turning performance.

But does n CL,max = V2 CL,max/2(W/S) exceed structural limits?

16.885J/ESD.35J - Oct 1, 2002

V-n diagram

V* 2nmax

CL, max

WS

Highestpossible

Lowestpossible R

Each airplane has a V-n diagram.Source: Anderson

Stall area

Stall area

Structural damage

Structural damage

q >

qm

ax

Load

fact

or n

V 8V*

CL < CLmax

CL < CLmax

Positive limit load factor

Negative limit load factor

0

16.885J/ESD.35J - Oct 1, 2002

Summary on Turning• Want large structural load factor n• Want large CL,MAX

• Want small V• Shortest turn radius, maximum turn rate is

“Corner Velocity”

• Question, does the aircraft have the power to execute these maneuvers?

16.885J/ESD.35J - Oct 1, 2002

Specific Energy and Excess PowerTotal aircraft energy = PE + KE

Etot = mgh + mV2/2

Specific energy = (PE + KE)/W

He = h + V2/2g “energy height”

Excess Power = (T-D)V

Specific excess power*

= (TV-DV)/W

= dHe/dt

Ps = dh/dt + V/g dV/dt

Ps may be used to change altitude, or accelerate, or both* Called specific power in Lockheed Martin notes.

16.885J/ESD.35J - Oct 1, 2002

Excess PowerPower Required

PR = DV = q S(CD,0 + C2L/ ARe)V

= q SCD,0V + q SV C2L/ ARe

= SCD,0V3 /2 + 2n2W2/ SV ARe

Parasite power required

Induced power required

Power Available

PA = TV and Thrust is approximately constant with velocity, but varies linearly with density.

Excess power depends upon velocity, altitude and load factor

Pow

er

Excesspower

V 8

PR

PA

16.885J/ESD.35J - Oct 1, 2002

Altitude Effects on Excess PowerPR = DV = (nW/L) DV

= nWV CD/CL

From L= SV2 CL/2 = nW, get

V = (2nW/ SCL)1/2

Substitute in PR to get

PR = (2n3W3C2D/ SC3

L)1/2

So can scale between sea level “0” and altitude “alt” assuming CD,CL const.

Valt = V0( 0/ alt)1/2, PR,alt = PR,0( 0/ alt)1/2

Thrust scales with density, so

PA,alt = PA,0( alt/ 0)

16.885J/ESD.35J - Oct 1, 2002

Summary of Power Characteristics• He = specific energy represents “state” of aircraft. Units are in

feet.– Curves are universal

• Ps = (T/W-D/W)V= specific excess power– Represents ability of aircraft to change energy state.– Curves depend upon aircraft (thrust and drag)– Maybe used to climb and/or accelerate– Function of altitude– Function of load factor

• “Military pilots fly with Ps diagrams in the cockpit”, Anderson

16.885J/ESD.35J - Oct 1, 2002

A/C Performance SummaryFactor Commercial

TransportMilitary

TransportFighter General Aviation

LiebeckTake-offhobs = 35’ hobs = 50’ hobs = 50’ hobs = 50’

LiebeckLandingVapp = 1.3 Vstall Vapp = 1.2 Vstall Vapp = 1.2 Vstall Vapp = 1.3 Vstall

Climb LiebeckLevel Flight Liebeck

Range Breguet Range Radius of action*.Uses refueling

Breguet for prop

Endurance,Loiter

E (hrs) = R (miles)/V(mph), where R = Breguet Range

Turning,Maneuver

Emergency handling Majorperformance

factor

Emergencyhandling

SupersonicDash

N/A N/A Important N/A

ServiceCeiling

100 fpm climb

Lectures 6 and 7 for commercial and military transport* Radius of action comprised of outbound leg, on target leg, and return.

16.885J/ESD.35J - Oct 1, 2002

Stability and Control• Performance topics

deal with forces and translational motion needed to fulfill the aircraft mission

• Stability and control topics deal with moments and rotational motion needed for the aircraft to remain controllable.

16.885J/ESD.35J - Oct 1, 2002

S&C Definitions• L’ - rolling moment• Lateral motion/stability

• M - pitching moment• Longitudinal

motion/control

• N - rolling moment• Directional motion/control

CMMq Sc

Moment coefficient:

L'

M

Rudder deflection

Elevator deflection

N

16.885J/ESD.35J - Oct 1, 2002

Aircraft Moments

• Aerodynamic center (ac): forces and moments can be completely specified by the lift and drag acting through the ac plus a moment about the ac

– CM,ac is the aircraft pitching moment at L = 0 around any point• Contributions to pitching moment about cg, CM,cg come from

– Lift and CM,ac– Thrust and drag - will neglect due to small vertical separation from cg– Lift on tail

• Airplane is “trimmed” when CM,cg = 0

16.885J/ESD.35J - Oct 1, 2002

Absolute Angle of Attack

• Stability and control analysis simplified by using the absolute angle of attack which is 0 at CL = 0.

• a = + L=0

Liftslope =

CL, max

αL=0

dC Ldα

Liftslope =

CL, max

dC Ldα

Lift coefficient vs geometric angle of attack, α Lift coefficient vs absolute angle of attack, αa

α

CL

αa

CL

16.885J/ESD.35J - Oct 1, 2002

Criteria for Longitudinal Static Stability

CM,0 must be positiveCM,cga

must be negative

Implied that e is within flight range of angle of attack for the airplane, i.e. aircraft can be trimmed

CM,0

αa

CM,cg

αe

Slope = dCM,cgdαa

(Trimmed)

16.885J/ESD.35J - Oct 1, 2002

Moment Around cg

Mcg MacwbLwb(hc hacc) ltLt

Divide by q Sc and note that CL ,tLtq St

CM ,cg CM ,acwbCLwb

(h hac )ltStcSCL ,t , or

CM ,cg CM ,acwbCLwb

(h hac ) VHCL ,t , where VHltStcS

16.885J/ESD.35J - Oct 1, 2002

CM ,cg CM ,acwbCLwb

(h hac ) VHCL ,tCLwb

dCLwbd a,wb awb a,wb

Cl, t at t at( wb it )where is the downwash at the tail due to the lift on the wing

0 a,wb

CL, t at a,wb 1 at(it 0 )

At this point, the convention is drop the wb on awb

CM ,cg CM ,acwba a (h hac ) VH

ata

1 VH at it 0

Liftslope =

Cl, max

αL=0

dC ldα

α

Cl

αstall

16.885J/ESD.35J - Oct 1, 2002

Eqs for Longitudinal Static StabilityCM ,cg CM ,acwb

a a (h hac ) VHata

1 VH at it 0

CM ,0 CM ,cg l 0CM ,acwb

VH at it 0

CM ,cg

aa (h hac ) VH

ata

1

• CM,acwb < 0, VH> 0, t> 0 it> 0 for CM,0> 0– Tail must be angled down to generate negative lift

to trim airplane• Major effect of cg location (h) and tail parameter VH=(lS)t/(cs) in determining longitudinal static stability

CM,0

αa

CM,cg

αe

Slope = dCM,cgdαa

(Trimmed)

16.885J/ESD.35J - Oct 1, 2002

Neutral Point and Static Margin

• The slope of the moment curve will vary with h, the location of cg.

• If the slope is zero, the aircraft has neutral longitudinal static stability.

• Let this location be denote by

• or

CM ,cg

aa (h hac ) VH

ata

1

hn hac VHata

1

• For a given airplane, the neutral point is at a fixed location.• For longitudinal static stability, the position of the center of

gravity must always be forward of the neutral point.• The larger the static margin, the more stable the airplane

CM ,cg

aa h hn a hn h a static margin

16.885J/ESD.35J - Oct 1, 2002

Longitudinal Static Stability

Aerodynamic center location moves aft for

supersonic flight

cg shifts with fuel burn, stores separation,

configuration changes

• “Balancing” is a significant design requirement

• Amount of static stability affects handling qualities

• Fly-by-wire controls required for statically unstable aircraft

16.885J/ESD.35J - Oct 1, 2002

Today’s References• Lockheed Martin Notes on “Fighter Performance”

• John Anderson Jr. , Introduction to Flight, McGraw-Hill, 3rd ed, 1989, Particularly Chapter 6 and 7

• Shevell, Richard S., “Fundamentals of Flight”, Prentice Hall, 2nd Edition, 1989

• Bertin, John J. and Smith, Michael L., Aerodynamics for Engineers, Prentice Hall, 3rd edition, 1998

• Daniel Raymer, Aircraft Design: A Conceptual Approach, AIAA Education Series, 3rd edition, 1999, Particularly Chapter 17– Note: There are extensive cost and weight estimation

relationships in Raymer for military aircraft.