introduction to aircraft performance and static stability · pdf file16.885j/esd.35j - oct 1,...
TRANSCRIPT
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.