16.885J/ESD.35J - Oct 1, 2002 Introduction to Aircraft Performance and Static Stability 16.885J/ESD.35J Aircraft Systems Engineering Prof. Earll Murman September 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) c t lb 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 f to get ds ? Set L/D, c t , V f constant and integrate dW c t Tdt c t TSPC/3600 T thrust ds V f dt  V f dW c t T  V f W c t T dW W  V f L c t D dW W R 3600 TSFC V f L D ln W TO W empty 16.885J/ESD.35J - Oct 1, 2002 Insights from Breguet Range Equation R 3600 TSFC V f L D ln W TO W empty 3600 TSFC represents propulsion effects. Lower TSFC is better V f L D represents aerodynamic effect. L/D is aerodynamic efficiency V f L D a f M f L D . a f is constant above 36,000 ft. M f L D important ln W TO W empty 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. Losses due to shock waves Ref: Shevell M ax (L/D) Mach No. 1 2 3 10 20 Concorde 16.885J/ESD.35J - Oct 1, 2002 Transonic Effects on Airfoil C d , C l C d M cr M drag divergence M 8 1.0 M < M cr8 V 8 V 8 Region I. II. III. I. II. III. M > M drag divergence 8 M cr < M < M drag divergence 8 M<1 M<1 M>1 M<1 M>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, M n < M f - Lower effective “freestream” Mach number delays onset of transonic drag rise. ? For supersonic freestreams –M n < 1, / > P - Subsonic leading edge –M n > 1, / < P - Supersonic leading edge ? Extensive analysis available, but this is gist of the concept sin P=1/ M f P = Mach angle, the direction disturbances travel in supersonic flow M f M n =M f cos/ P / 16.885J/ESD.35J - Oct 1, 2002 Wing Sweep Considerations M f > 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 Airfoil Supercritical airfoil shape keeps upper surface velocity from getting too large. Uses aft camber to generate lift. Gives nose down pitching moment. C p x/c C p, 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-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. 16.885J/ESD.35J - Oct 1, 2002 Horizontal Turn W = L cosI, I = bank angle Level turn, no loss of altitude F r = (L 2 -W 2 ) 1/2 =W(n 2 -1) 1/2 Where n { L/W = 1/ cosI is the load factor measured in “g’s”. But F r = (W/g)(V 2 ? /R) So radius of turn is R = V 2 ? /g(n 2 -1) 1/2 And turn rate Z = V ? /R is Z= g(n 2 -1) 1/2 / V ? Want high load factor, low velocity F li g h t p ath θ R φ φ R L W F r z z 16.885J/ESD.35J - Oct 1, 2002 Pull Up Pull Over Vertical F r = (L-W) =W(n-1) = (W/g)(V 2 ? /R) R = V 2 ? /g(n-1) Z = g(n-1)/ V ? F r = (L +W) =W(n+1) = (W/g)(V 2 ? /R) R = V 2 ? /g(n+1) Z = g(n+1)/ V ? F r = L =Wn = (W/g)(V 2 ? /R) R = V 2 ? /gn Z = gn/ V ? R L W θ R L W θ R L W θ Let Y T Z Pull Over K Z = (n+1)/(n 2 -1) 1/2 Vertical Maneuver K Z = n/(n 2 -1) 1/2 Pull Up K Z = (n-1)/(n 2 -1) 1/2 For large n, K Z ?1 and for all maneuvers Z#gn/ V ? Similarly for turn radius, for large n, R # V 2 ? /gn. For large Z and small R, want large n and small V ? Pu llo ve r fr om inv erte d at titud e Vertical maneuver Pull -up f rom l evel a ttitude 0 0.5 1.0 1.5 2.0 2.5 123456789 T u r n rate ratio , K θ Load factor, in 'g's Vertical Plan Turn Rates Vertical plane turn rate K θ = Horizontal plane turn rate 16.885J/ESD.35J - Oct 1, 2002 Z#gn/ V ? = gn/a ? M ? so Z ~ 1/ M ? at const h (altitude) & n Using R # V 2 ? /gn, Z#V ? /R = a ? M ? /R. So Z ~ M ? at const h & R For high Mach numbers, the turn radius gets large 16.885J/ESD.35J - Oct 1, 2002 R min and Z max Using V ? = (2L/U ? SC L ) 1/2 = (2nW/U ? SC L ) 1/2 R # V 2 ? /gn becomes R = 2(W/S)/ gU ? C L W/S = wing loading, an important performance parameter And using n = L/W = U ? V 2 ? SC L /2W Z# gn/ V ? = g U ? V ? C L /2(W/S) For each airplane, W/S set by range, payload, V max . Then, for a given airplane R min = 2(W/S)/ gU ? C L,max Z max = g U ? V ? C L,max /2(W/S) Higher C L,max gives superior turning performance. But does n CL,max = U ? V 2 ? C L,max /2(W/S) exceed structural limits? 16.885J/ESD.35J - Oct 1, 2002 V-n diagram V* 2n max U f C L, max W S Highest possible Z Lowest possible R Each airplane has a V-n diagram. Source: Anderson Stall area Stall area Structural damage Structural damage q > q max L oad fac t or n V 8 V* C L < C Lmax C L < C Lmax 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 C L,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 Power Total aircraft energy = PE + KE E tot = mgh + mV 2 /2 Specific energy = (PE + KE)/W H e = h + V 2 /2g “energy height” Excess Power = (T-D)V Specific excess power* = (TV-DV)/W = dH e /dt P s = dh/dt + V/g dV/dt P s 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 Power Power Required P R = DV ? = q ? S(C D,0 + C 2 L /SARe)V ? = q ? SC D,0 V ? + q ? SV ? C 2 L /SARe = U ? SC D,0 V 3 ? /2 + 2n 2 W 2 /U ? SV ? SARe Parasite power required Induced power required Power Available P A = TV ? and Thrust is approximately constant with velocity, but varies linearly with density. Excess power depends upon velocity, altitude and load factor Po w e r Excess power V 8 P R P A 16.885J/ESD.35J - Oct 1, 2002 Altitude Effects on Excess Power P R = DV ? = (nW/L) DV ? = nWV ? C D /C L From L= U ? SV 2 ? C L /2 = nW, get V ? = (2nW/ U ? SC L ) 1/2 Substitute in P R to get P R = (2n 3 W 3 C 2 D / U ? SC 3 L ) 1/2 So can scale between sea level “0” and altitude “alt” assuming C D ,C L const. V alt = V 0 (U 0 /U alt ) 1/2 , P R,alt = P R,0 (U 0 /U alt ) 1/2 Thrust scales with density, so P A,alt = P A,0 (U alt /U 0 ) 16.885J/ESD.35J - Oct 1, 2002 Summary of Power Characteristics ?H e = specific energy represents “state” of aircraft. Units are in feet. – Curves are universal ?P s = (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 P s diagrams in the cockpit”, Anderson 16.885J/ESD.35J - Oct 1, 2002 A/C Performance Summary Factor Commercial Transport Military Transport Fighter General Aviation LiebeckTake-off h obs = 35’ h obs = 50’ h obs = 50’ h obs = 50’ LiebeckLanding V app = 1.3 V stall V app = 1.2 V stall V app = 1.2 V stall V app = 1.3 V stall Climb Liebeck Level 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 Emergency handling Supersonic Dash N/A N/A Important N/A Service Ceiling 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 C M M q f 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 –C M,ac is the aircraft pitching moment at L = 0 around any point ? Contributions to pitching moment about cg, C M,cg come from – Lift and C M,ac – Thrust and drag - will neglect due to small vertical separation from cg – Lift on tail ? Airplane is “trimmed” when C M,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 C L = 0. ? D a = D + D L=0 Li f t slope = C L, max α L=0 dC L d α Li ft s l o pe = C L, max dC L d α Lift coefficient vs geometric angle of attack, α Lift coefficient vs absolute angle of attack, α a α C L α a C L 16.885J/ESD.35J - Oct 1, 2002 Criteria for Longitudinal Static Stability C M,0 must be positive wC M,cg wD a must be negative Implied that D e is within flight range of angle of attack for the airplane, i.e. aircraft can be trimmed C M,0 α a C M,cg α e Slope = dC M,cg d α a (Trimmed) 16.885J/ESD.35J - Oct 1, 2002 Moment Around cg M cg M ac wb L wb (hc h ac c) l t L t Divide by q f Sc and note that C L,t L t q f S t C M,cg C M,ac wb C L wb (h h ac ) l t S t cS C L,t , or C M,cg C M,ac wb C L wb (h h ac )V H C L,t , where V H l t S t cS 16.885J/ESD.35J - Oct 1, 2002 C M,cg C M,ac wb C L wb (h h ac )V H C L,t C L wb dC L wb dD D a, wb a wb D a, wb C l, t a t D t a t (D wb  i t H) where H is the downwash at the tail due to the lift on the wing H H 0  wH wD § ? ¨ · 1 ? D a, wb C L, t a t D a, wb 1 wH wD § ? ¨ · 1 ?  a t (i t H 0 ) At this point, the convention is drop the wb on a wb C M,cg C M,ac wb aD a (h h ac )V H a t a 1 wH wD § ? ¨ · 1 ? a ? ? ? o ? ? ? V H a t i t H 0 L i f t slo p e = C l, max α L=0 dC l d α α C l α stall 16.885J/ESD.35J - Oct 1, 2002 Eqs for Longitudinal Static Stability C M,cg C M,ac wb aD a (h h ac )V H a t a 1 wH wD § ? ¨ · 1 ? a ? ? ? o ? ? ? V H a t i t H 0 C M,0 { C M,cg § ? ¨ · 1 ? l 0 C M ,ac wb V H a t i t H 0 wC M,cg wD a a (h h ac )V H a t a 1 wH wD § ? ¨ · 1 ? a ? ? ? o ? ? ? ? C M,acwb < 0, V H > 0, D t > 0 ? i t > 0 for C M,0 > 0 – Tail must be angled down to generate negative lift to trim airplane ? Major effect of cg location (h) and tail parameter V H = (lS) t /(cs) in determining longitudinal static stability C M,0 α a C M,cg α e Slope = dC M,cg d α a (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 wC M,cg wD a a (h h ac )V H a t a 1 wH wD § ? ¨ · 1 ? a ? ? ? o ? ? ? h n h ac V H a t a 1 wH wD § ? ¨ · 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 wC M,cg wD a ah h n ah n  h a u 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.