Introduction to Aircraft Performance and Static Stability

16.885J/ESD.35J Aircraft Systems Engineering

Prof. Earll Murman September 18, 2003 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 Thrust Specific Fuel Consumption (TSFC) lb of fuel burned • Definition: TSFC (lb of thrust delivered)(hour) • 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 lb of fuel burned – Another term used is c t (lb of thrust delivered)(sec) Breguet Range Equation

• Change in aircraft weight = fuel burned

dW ctTdt ct TSPC/3600 T thrust

• Solve for dt and multiply by Vf to get ds VfdW VfW dW VfL dW ds Vfdt ctT ctT W ctD W

• Set L/D, ct, Vf constant and integrate W 3600 L TO R Vf ln TSFC DW empty Insights from Breguet Range Equation W 3600 L TO R Vf ln TSFC DW empty

3600 represents propulsion effects. Lower TSFC is better TSFC

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

V La M L . a is constant above 36,000 ft. M L important fD f fD f fD

W ln TO represents aircraft weight/structures effect on range Wempty Optimized L/D - Transport A/C “Sweet spot” is in transonic range.

20 Losses due to 10 shock Max (L/D) Concorde waves

1 2 3 Mach No.

Ref: Shevell Transonic EffectsonAirfoilC III. II. I. M

8< M

8> M cr drag

8< M drag C d V V 8 8 Region I. M<1 M>1 M>1 M cr I III. II. M<1 divergence M drag M<1 1.0 Separated flow Separated M d

8 , C l 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 Basic Sweep Concept

• Consider Mach Number normal to leading edge

sin P=1/ Mf P Mf Mn=Mfcos/ P = Mach angle, the direction / disturbances travel in supersonic flow

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

–Mn < 1, / > P - Subsonic leading edge –Mn > 1, / < P - Supersonic leading edge • Extensive analysis available, but this is gist of the concept Wing Sweep Considerations Mf > 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 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 Supercritical Airfoil

Supercritical airfoil shape C p keeps upper surface velocity C p, cr from getting too large. Uses aft camber x/c to generate lift. V 8 Gives nose down pitching moment. 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-Z 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-Z diagram - “Doghouse” chart – Maneuver limits and some example – Extensive notes from Lockheed available. Ask me. Horizontal Turn

ht pa t h W = L cosI, I = bank angle Flig Level turn, no loss of altitude R 2 2 1/2 2 1/2 F = (L -W) =W(n -1) θ r z Where n { L/W = 1/ cosI is the load factor measured in “g’s”.

2 But Fr = (W/g)(V /R) So radius of turn is φ L 2 2 1/2 R = V /g(n -1) z φ Fr And turn rate Z = V/R is 2 1/2 R Z= g(n -1) / V W Want high load factor, low velocity Pull Up Pull Over Vertical

W R R L

L θ L R

θ W

Fr = (L-W) =W(n-1) Fr = (L +W) =W(n+1) Fr = L =Wn 2 2 2 = (W/g)(V /R) = (W/g)(V /R) = (W/g)(V /R) 2 2 2 R = V /g(n-1) R = V /g(n+1) R = V /gn = = Z g(n-1)/ V Z g(n+1)/ V Z = gn/ V Vertical Plan Turn Rates

2.5

Let TÝ Z Vertical plane turn rate 2.0 Kθ = Horizontal plane turn rate Pull Over

2 1/2 KZ = (n+1)/(n -1) θ 1.5

Pullover fro m inverted attitude Vertical maneuver Vertical Maneuver 1.0 tude Turn rate ratio, K rate ratio, Turn Pull-up from level atti 2 1/2 KZ = n/(n -1)

0.5 Pull Up

2 1/2 0 KZ = (n-1)/(n -1) 1 2 3 4 5 6 7 8 9 Load factor, in 'g's

For large n, KZ » 1 and for all maneuvers Z#gn/ V 2 Similarly for turn radius, for large n, R # V /gn.

For large Z and small R, want large n and small V Z#gn/ V = gn/aM so Z ~ 1/ M at const h (altitude) & n 2 Using R # V /gn, Z#V/R = aM/R. So Z ~ M at const h & R For high Mach numbers, the turn radius gets large Rmin and Zmax 1/2 1/2 Using V = (2L/USCL) = (2nW/USCL) 2 R # V /gn becomes R = 2(W/S)/ gUCL W/S = wing loading, an important performance parameter

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

Z # gn/ V= g UV CL/2(W/S)

For each airplane, W/S set by range, payload, Vmax. Then, for a given airplane

Rmin = 2(W/S)/ gUCL,max

Zmax = g UV CL,max /2(W/S)

Higher CL,max gives superior turning performance. 2 But does n CL,max = UV CL,max/2(W/S) exceed structural limits? V-n diagram

CL < CLmax 2n W V* max UfCL, max S

Structural damage Load factor n Load Highest Positive limit load factor possible Z Stall area Lowest max q > q possible R 0 V V* 8

Negative limit load factor

Stall area Structural damage

CL < CLmax

Each airplane has a V-n diagram.

Source: Anderson 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? Specific Energy and Excess Power Total aircraft energy = PE + KE

2 Etot = mgh + mV /2 Specific energy = (PE + KE)/W

2 He = h + V /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. Excess Power

Power Required

2 PR = DV = qS(CD,0 + C L/SARe)V Power PA Excess power 2 PR = qSCD,0V + qSVC L/SARe 3 2 2 = USCD,0V /2 + 2n W /USVSARe

Parasite power Induced power required required V 8

Power Available Excess power PA = TV and Thrust is approximately depends upon constant with velocity, but varies velocity, altitude linearly with density. and load factor Altitude Effects on Excess Power

PR = DV = (nW/L) DV

= nWV CD/CL 2 From L= USV CL/2 = nW, get 1/2 V = (2nW/ USCL)

Substitute in PR to get 3 3 2 3 1/2 PR = (2n W C D/ USC L) So can scale between sea level “0” and

altitude “alt” assuming CD,CL const. 1/2 1/2 Valt = V0(U0/Ualt) , PR,alt = PR,0(U0/Ualt) Thrust scales with density, so

PA,alt = PA,0(Ualt/U0) 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 A/C Performance Summary

Factor Commercial Military Fighter General Aviation Transport Transport Take-off Liebeck

hobs = 35’ hobs = 50’ hobs = 50’ hobs = 50’ Landing Liebeck

Vapp = 1.3 Vstall Vapp = 1.2 Vstall Vapp = 1.2 Vstall Vapp = 1.3 Vstall Climb Liebeck Level Flight Liebeck Range Breguet Range Radius of action*. Breguet for prop Uses refueling Endurance, E (hrs) = R (miles)/V(mph), where R = Breguet Range Loiter Turning, Emergency handling Major Emergency Maneuver performance handling factor Supersonic N/A N/A Important N/A Dash Service 100 fpm climb Ceiling

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

L' • L’ - rolling moment • Lateral motion/stability

M • M - pitching moment • Longitudinal Elevator deflection motion/control

N • N - rolling moment • Directional motion/control

M Moment coefficient:C M q Sc f Rudder deflection 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 Absolute Angle of Attack

CL CL CL, max CL, max

L L dC dC dα dα pe = lo ft slope = ft s Li Li

α L=0 α α a

α α Lift coefficient vs geometric angle of attack, Lift coefficient vs absolute angle of attack, a • Stability and control analysis simplified by using

the absolute angle of attack which is 0 at CL = 0.

• Da = D + DL=0 Criteria for Longitudinal Static Stability

CM,cg

C M,0 Slope = dC M,cg dα a α α a e ( Trimmed)

C must be positive M,0 wC M,cg must be negative wDa Implied that De is within flight range of angle of attack for the airplane, i.e. aircraft can be trimmed Moment Around cg

M M L (hc h c) l L cg ac wb ac t t wb L Divide by q Sc and note that C t f L,t q S f t l S C C C (h h ) t t C , or M ,cg M ,ac L ac L,t wb wb cS l S C C C (h h ) V C , where V t t M ,cg M ,ac L ac H L,t H wb wb cS C C C (h h ) V C M ,cg M ,ac L ac H L,t wb wb dC C L wb D a D Lwb a, wb wb a, wb Cl dD Cl, max

Cl, t atDt at (Dwb it H ) l dC dα = where H is the downwash at the pe slo ft Li tail due to the lift on the wing

At this point, the convention is drop

the wb on awb

ª a § wH ·º C C aD «(h h ) V t 1 »V a i H M ,cg M ,ac a ac H ¨ ¸ H t t 0 wb ¬« a © wD¹¼» Eqs for Longitudinal Static Stability

§ · C { C C V a i H M ,0 ¨ M ,cg ¸ M ,ac H t t 0 © ¹l 0 wb C M,0 Slope = dC M,cg α wC ª a º d M ,cg t § wH · a a«(h h ) V ¨1 ¸» wD ac H a © wD¹ α a ¬« ¼» α a e ( Trimmed)

• CM,acwb < 0, VH > 0, Dt > 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 Neutral Point and Static Margin wC ª a º M ,cg t § wH · a«(h h ) V ¨1 ¸» wD ac H a © wD¹ a ¬« ¼» • 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. a § wH · • Let this location be denote by h h V t ¨1 ¸ n ac H a © wD¹ wC M ,cg ah h ah h a u static margin •or wD n n a • 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 Longitudinal Static Stability

Aerodynamic center cg shifts with fuel burn, location moves aft for stores separation, supersonic flight configuration changes

• “Balancing” is a significant design requirement • Amount of static stability affects handling qualities • Fly-by-wire controls required for statically unstable aircraft 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.