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Introduction to AircraftPerformance and Static Stability
16.885J/ESD.35JAircraft Systems Engineering
Prof. Earll Murman
September 18, 2003
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Todays 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 transportaircraft in Lectures 6 and 7
Aircraft longitudinal static stability
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Thrust Specific Fuel Consumption (TSFC) Definition:
Measure of jet engine effectiveness at converting fuel touseable 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)
ct
lb of fuel burned
(lb of thrust delivered)(sec)
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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 ctTSPC/3600 T thrust
dsVdtV
dW
ctT
V
W
ctT dWW
V
L
ctD dWW
R 3600TSFC
VLD
ln WTOW
empty
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Insights from Breguet Range EquationR 3600
TSFCV
LD
lnW
TOW
empty
3600TSFC
represents propulsion effects. Lower TSFC is better
V
L
D
represents aerodynamic effect. L/D is aerodynamic efficiency
VLD
aMLD
. a is constant above 36,000 ft. MLD
important
lnW
TOW
emptyrepresents aircraft weight/structures effect on range
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Optimized L/D - Transport A/CSweet spot is in transonic range.
Lossesdue toshock
waves
Ref: Shevell
M
ax(L/D)
Mach No.1 2 3
10
20
Concorde
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Transonic Effects on Airfoil Cd, ClCd
Mcr
Mdragdivergence
M8
1.0
M < Mcr8
V8
V8
Region I. II. III.
I.
II.
III. M > Mdragdivergence
8
Mcr < M < Mdragdivergence
8
M
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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
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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 directiondisturbances
travel insupersonic flow
M Mn=Mcos
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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
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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
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Supercritical AirfoilSupercritical
airfoil shapekeeps uppersurface velocityfrom getting too
large.Uses aft camberto generate lift.
Gives nosedown pitchingmoment.
Cp
x/c
Cp, cr
V8
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Todays 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.
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Horizontal TurnW = L cos, = bank angleLevel turn, no loss of altitude
Fr= (L2
- W2
)1/2
=W(n2
-1)1/2
Where n L/W = 1/ cos is the loadfactor measured in gs.
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 pa
th
R
R
L
W
Fr
z
z
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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
R
L
W
R
L
W
V i l Pl T R
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Let
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
Pulloverfrom inverted attitude
Verticalmaneuver
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
ratio,K
Load factor, in 'g's
Vertical Plan Turn Rates
Vertical planeturn rateK=
Horizontal planeturn rate
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gn/ V
= gn/aM
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
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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 parameterAnd 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 airplane
Rmin
= 2(W/S)/ gC
L,maxmax
= g V
CL,max /2(W/S)
Higher CL,max gives superior turning performance.
But does n CL,max = V2CL,max/2(W/S) exceed structural limits?
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V-n diagram
V* 2nmax
CL,max
W
S
Highestpossible
Lowestpossible R
Each airplane has a V-n diagram.
Source: Anderson
Stall area
Stall area
Structural damage
Structural damage
q>
qmax
Loadfactorn
V8V*
CL < CLmax
CL < CLmax
Positive limit load factor
Negative limit load factor
0
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Summary on Turning Want large structural load factor n
Want large CL,MAX Want small V
Shortest turn radius, maximum turn rate isCorner Velocity
Question, does the aircraft have the powerto execute these maneuvers?
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Specific Energy and Excess PowerTotal aircraft energy = PE + KE
Etot = mgh + mV2/2
Specific energy = (PE + KE)/WHe = h + V2/2g energy height
Excess Power = (T-D)V
Specific excess power*
= (TV-DV)/W
= dHe/dtPs = dh/dt + V/g dV/dt
Ps may be used to change altitude, or accelerate, or both* Called specific power in Lockheed Martin notes.
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Excess PowerPower Required
PR= DV
= q
S(CD,0 + C2L/ARe)V
= qSCD,0V + qSVC2L/ARe
= SCD,0V3/2 + 2n2W2/SVARe
Parasite powerrequired
Induced powerrequired
Power Available
PA = TV and Thrust is approximatelyconstant with velocity, but varieslinearly with density.
Excess powerdepends uponvelocity, altitudeand load factor
P
ower
Excesspower
V8
PR
PA
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Altitude Effects on Excess PowerPR= DV = (nW/L) DV
= nWV
CD/CL
From L=
SV2
CL/2 = nW, getV
= (2nW/ SCL)1/2
Substitute in PRto get
PR= (2n3W3C2D/ SC3L)1/2
So can scale between sea level 0 andaltitude 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)
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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
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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 Major performance
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.
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S&C Definitions L - rolling moment Lateral motion/stability
M - pitching moment
Longitudinal
motion/control
N - rolling moment
Directional motion/control
CM
qSc
Moment coefficient:
L'
M
Rudder deflection
Elevator deflection
N
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Aircraft Moments
Aerodynamic center (ac): forces and moments can be completelyspecified by the lift and drag acting through the ac plus a moment aboutthe 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
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Absolute Angle of Attack
Stability and control analysis simplified by usingthe absolute angle of attack which is 0 at CL = 0.
a = + L=0
Lift s
lope =
CL,max
L=0
dC L
d
Lift s
lope =
CL,max
dC L
d
Lift coefficient vs geometric angle of attack, Lift coefficient vs absolute angle of attack,a
CL
a
CL
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Criteria for Longitudinal Static Stability
CM,0
must be positive
C
M,cga must be negativeImplied that e is within flight range of angle of attackfor the airplane, i.e. aircraft can be trimmed
CM,0
a
CM,cg
e
Slope= dCM,cgda
(Trimmed)
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Moment Around cg
Mcg
Mac
wb
Lwb
(hc hac
c) ltL
t
Divide by qSc and note that CL ,t L
t
qSt
CM,cg
CM,ac
wb C
L wb(h h
ac) ltSt
cSC
L ,t, or
C
M,cg C
M,acwb C
L wb
(h hac
) VH
C
L ,t, where V
H
ltS
t
cS
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CM,cg
CM,ac
wb C
L wb(h h
ac) V
HC
L ,t
CLwb
dCL wb
d a, wb awba, wb
Cl, t att at(wb it )
where is the downwash at the
tail due to the lift on the wing 0
a, wb
CL, t
at
a, wb1
a
t(i
t
0)
At this point, the
convention is dropthe wb on awb
CM,cg
CM,ac
wb a
a(h h
ac) V
H
at
a1
VH
at
it
0
Lift
slope
=
Cl, max
L=0
dC l
d
Cl
stall
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Eqs for Longitudinal Static Stability
CM,cg
CM,ac
wb a
a(h h
ac) V
H
at
a1
VH
at
it
0
CM,0 CM,cg
l 0 CM,ac
wb VHat it0
CM,cg
a
a (h hac
) VH
at
a1
CM,acwb < 0, VH> 0, t> 0 it> 0 forCM,0 > 0
Tail must be angled down to generate negative liftto trim airplane
Major effect of cg location (h) and tail parameterVH=
(lS)t/(cs) in determining longitudinal static stability
CM,0
a
CM,cg
e
Slope= dCM,cgda
(Trimmed)
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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
a
a (h hac
) VH
at
a1
hn hac VHat
a 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
a
a h hn a hn h a static margin
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Longitudinal Static Stability
Aerodynamic centerlocation moves aft forsupersonic 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
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Todays 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, PrenticeHall, 2nd Edition, 1989
Bertin, John J. and Smith, Michael L., Aerodynamicsfor Engineers, Prentice Hall, 3rd edition, 1998
Daniel Raymer, Aircraft Design: A ConceptualApproach, AIAA Education Series, 3rd edition, 1999,Particularly Chapter 17
Note: There are extensive cost and weight estimationrelationships in Raymer for military aircraft.
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