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14 fixed wing fighter aircraft- flight performance - ii

Fighter Aircraft Performance, Part II of two, describes the parameters that affect aircraft performance.
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14 fixed wing fighter aircraft- flight performance - ii

  1. 1. Fixed Wing Fighter Aircraft Flight Performance Part II SOLO HERMELIN Updated: 04.12.12 28.02.15 1 http://www.solohermelin.com
  2. 2. Table of Content SOLO 2 Aerodynamics Introduction to Fixed Wing Aircraft Performance Earth Atmosphere Mach Number Shock & Expansion Waves Reynolds Number and Boundary Layer Knudsen Number Flight Instruments Aerodynamic Forces Aerodynamic Drag Lift and Drag Forces Wing Parameters Specific Stabilizer/Tail Configurations F i x e d W i n g P a r t I Fixed Wing Fighter Aircraft Flight Performance
  3. 3. Table of Content (continue – 1) SOLO 3 Specific Energy Aircraft Propulsion Systems Aircraft Propellers Aircraft Turbo Engines Afterburner Thrust Reversal Operation Aircraft Propulsion Summary Vertical Take off and Landing - VTOL Engine Control System Aircraft Flight Control Aircraft Equations of Motion Aerodynamic Forces (Vectorial) Three Degrees of Freedom Model in Earth Atmosphere F i x e d W i n g P a r t I Fixed Wing Fighter Aircraft Flight Performance
  4. 4. Table of Content (continue – 2) SOLO Fixed Wing Fighter Aircraft Flight Performance 4 Parameters defining Aircraft Performance Takeoff (no VSTOL capabilities) Landing (no VSTOL capabilities) Climbing Aircraft Performance Gliding Flight Level Flight Steady Climb (V, γ = constant) Optimum Climbing Trajectories using Energy State Approximation (ESA) Minimum Fuel-to- Climb Trajectories using Energy State Approximation (ESA) Maximum Range during Glide using Energy State Approximation (ESA) Aircraft Turn Performance Maneuvering Envelope, V – n Diagram
  5. 5. Table of Content (continue – 3) SOLO Fixed Wing Fighter Aircraft Flight Performance 5 Air-to-Air Combat Energy–Maneuverability Theory Supermaneuverability Constraint Analysis References Aircraft Combat Performance Comparison
  6. 6. SOLO This Presentation is about Fixed Wing Aircraft Flight Performance. The Fixed Wing Aircraft are •Commercial/Transport Aircraft (Passenger and/or Cargo) •Fighter Aircraft Fixed Wing Fighter Aircraft Flight Performance Continue from Part I
  7. 7. 7 Fixed Wing Fighter Aircraft Flight PerformanceSOLO The Aircraft Flight Performance is defined by the following parameters • Take-off distance • Landing distance • Maximum Endurance and Speed for Maximum Endurance • Maximum Range and Speed for Maximum Range • Ceiling(s) • Climb Performance • Turn Performance • Combat Radius • Maximum Payload Parameters defining Aircraft Performance
  8. 8. 8 Performance of an Aircraft with Parabolic PolarSOLO Assumptions: •Point mass model. •Flat earth with g = constant. •Three-dimensional aircraft trajectory. •Air density that varies with altitude ρ=ρ(h) •Drag that varies with altitude, Mach number and control effort D = D(h,M,n) and is given by a Parabolic Polar. •Thrust magnitude is controllable by the throttle. •No sideslip angle. •No wind. α T V L D Bx Wx Bz Wz Wy By Aircraft Coordinate System To understand how different parameters affect Aircraft Performance we start with a Simplified Model, where Analytical Solutions can be obtained. Results for real aircraft will then be presented. Return to Table of Content
  9. 9. 9 SOLO Aircraft Flight Performance Takeoff The Takeoff distance sTO is divided as the sum of the following distances: sg – Ground Run sr – Rotation Distance st – Transition Distance sc – Climb Distance to reach Screen Height ctrgTO sssss +++= Ground Run V = 0 sg sTO sr str V TO Rotation Transition sCL θ CL htr hobs R Takeoff htransition < hobstacle θ CL Ground Run V = 0 sg sTO sr sobs V TO Rotation Transition hobs R Takeoff htransition > hobstacle We distinguish between two cases of Takeoff •The Aircraft must passes over an obstacle at altitude hobs.. •The obstacle is cleared during the transition phase. Assume no Vertical Takeoff Capability.
  10. 10. 10 Takeoff (continue – 1) During the Ground Run there are additional effects than in free flight, that must be considered: -Friction between the tires and the ground during rolling. -Additional drag due to the landing gear fully extended. -Additional Lift Coefficient due to extended flaps. -Ground Effect due to proximity of the wings to the ground, that reduces the Induced Drag and the Lift. Ground Run SOLO Aircraft Flight Performance Ground run sg Transition distance st Climb distance sc Stall safety Take-off possible with one engine Continue take-off if engine fails after this point Stop take-off if engine fails before this point Acceleration at full power γ c Total take-off if distance VCRVMCG VTVS L W TD R μR The Aircraft can leave the ground when the velocity reaches the Stall Velocity where Lift equals Weight max, 2 0 2 1 Lstallstall CSVLW ρ== max,0 12 L stall CS W V ρ = The Liftoff Velocity is 1.1 to 1.2 Vstall.
  11. 11. 11 ReactionGroundLWR gW RDT td Vd V V td xd −= −− == = / µ ( ) ( )LWDT gW Vd td LWDT gWV Vd sd xs −−− = −−− = = µ µ / / Takeoff (continue – 2) Average Coefficient of Friction Values μ Ground Run SOLO Aircraft Flight Performance Ground run sg Transition distance st Climb distance sc Stall safety Take-off possible with one engine Continue take-off if engine fails after this point Stop take-off if engine fails before this point Acceleration at full power γ c Total take-off if distance VCRVMCG VTVS hc L W R μR D T V
  12. 12. 12 Takeoff (continue – 3) SOLO Aircraft Flight Performance Ground run sg Transition distance st Climb distance sc Stall safety Take-off possible with one engine Continue take-off if engine fails after this point Stop take-off if engine fails before this point Acceleration at full power γc Total take-off if distance VCRVMCG VTVS hc L W R μR D T V T (Jet) Lift,Drag,Thrust,Resistance–lb L,D,T,R T (Prop) D +μ R Ground Speed – ft/s Texcess(Prop)=T(Prop) -(D+μ R) Texcess(Jet)=T(Jet) -(D+μ R) Vground ( ) ReactionGroundLWR RDT g WV −= +−= µ  Ground Run (continue -1)
  13. 13. 13 2 0 VCVBTT ++= cVbVaVd td cVbVa V Vd sd xs ++ = ++ = = 2 2 1 Takeoff (continue – 4) Ground Run (continue – 2) To obtain an Analytic Solution assume that during the Ground Run the Thrust can be approximated by Using       = = L D CSVL CSVD 2 2 2 1 2 1 ρ ρ ( )       −= = +−−= µ µ ρ W T gc W gB b W gC CC W Sg a LD 0 : 2 : 2 : where SOLO Aircraft Flight Performance Ground run sg Transition distance st Climb distance sc Stall safety Take-off possible with one engine Continue take-off if engine fails after this point Stop take-off if engine fails before this point Acceleration at full power γ c Total take-off if distance VCRVMCG VTVS hc L W R μR D T V
  14. 14. 14 cVbVaVd td cVbVa V Vd sd xs ++ = ++ = = 2 2 1 Takeoff (continue – 5) Ground Run (continue – 3) Integrating those equations between two velocities V1 and V2 gives       − − ⋅ + + − + ++ ++ = 2 1 1 2 2 1 2 1 2 2 2 1 1 1 1 ln 42 ln 2 1 a a a a caba b cVbVa cVbVa a sg       − − ⋅ + + − = 1 2 2 1 2 1 1 1 1 ln 4 1 a a a a cab tg where cab bVa a cab bVa a 4 2 : 4 2 : 2 2 2 2 1 1 − + = − + = SOLO Aircraft Flight Performance Ground run sg Transition distance st Climb distance sc Stall safety Take-off possible with one engine Continue take-off if engine fails after this point Stop take-off if engine fails before this point Acceleration at full power γ c Total take-off if distance VCRVMCG VTVS hc L W R μR D T V 2 0 VCVBTT ++=
  15. 15. 15 Takeoff (continue – 6) Ground Run (continue – 4) then ( ) ( )             − − −− = + = TL LDLD g CWT CCCCg SW c cVa a s µ µµρ / 1 1 ln / ln 2 1 0 2 2 0,00 01 ==⇐== CBTTV Assume where 22 :& /2 : VV V SW C T T LT == ρ A further simplification, using , givesZ Z Z 1 1 1 ln << ≈ −       − = µρ W T Cg SW s TL g 0 / SOLO Aircraft Flight Performance gL sCg SW W T T ρ /0 > Ground run sg Transition distance st Climb distance sc Stall safety Take-off possible with one engine Continue take-off if engine fails after this point Stop take-off if engine fails before this point Acceleration at full power γc Total take-off if distance VCRVMCG VTVS hc L W R μR D T V
  16. 16. 16 Takeoff (continue – 7) Rotation Distance At the ground roll and just prior to going into transition phase, most aircraft are Rotated to achieve an Angle of Attack to obtain the desired Takeoff Lift Coefficient CL. Since the rotation consumes a finite amount of time (1 – 4 seconds), the distance traveled during rotation sr, must be accounted for by using where Δt is usually taken as 3 seconds. SOLO Aircraft Flight Performance tVs tr ∆= Ground Run V = 0 sg sTO sr str V TO Rotation Transition sCL θ CL htr hobs R L W R μR D T V
  17. 17. 17 Takeoff (continue – 8) Transition Distance In the Transition Phase the Aircraft is in the Air (μ = 0) and turn to the Climb Angle. The Equation of Motion are: SOLO Aircraft Flight Performance Ta Ta t Ta t VV DT VV g W t DT VV g W s >        − − = − − = 2 2 22 DT gW Vd td DT gWV Vd sd xs − = − = = / / Assuming T – D = const., we can Integrate the Equations of Motion (assuming Va > VT) Ground Run V = 0 sg sTO sr str V TO Rotation Transition sCL θ CL htr hobs R
  18. 18. 18 Takeoff (continue – 9) Climb Distance The Climb Distance is evaluated from the following (see Figure): SOLO Aircraft Flight Performance c c c c c hh s c γγ γ 1 tan << ≈= For small angles of Climb L = W. We can write Ground Run V = 0 sg sTO sr str V TO Rotation Transition sCL θ CL htr hobs R cL cLD c c c C CkC W T L D W T , 2 ,0 + −=−=γ We have cLcLD c c CkCCWT h s ,,0 // −− ≈
  19. 19. 19 Takeoff (continue – 10) SOLO Aircraft Flight Performance 19 ctrgTO sssss +++= sec41−=∆∆= ttVs tr Ta Ta t Ta t VV DT VV g W t DT VV g W s >        − − = − − = 2 2 22       − − ⋅ + + − + ++ ++ = 2 1 1 2 2 1 2 1 2 2 2 1 1 1 1 ln 42 ln 2 1 a a a a caba b cVbVa cVbVa a sg cab bVa a cab bVa a 4 2 : 4 2 : 2 2 2 2 1 1 − + = − + =       − − ⋅ + + − = 1 2 2 1 2 1 1 1 1 ln 4 1 a a a a cab tg Ground Run V = 0 sg sTO sr str V TO Rotation Transition sCL θ CL htr hobs R Takeoff Summary Rotation Phase Climb Phase Transition Phase Ground Run cLcLD c c CkCCWT h s ,,0 // −− ≈
  20. 20. 20Minimum required takeoff runway lengths. Summary of takeoff requirements In order to establish the allowable takeoff weight for a transport category airplane, at any airfield, the following must be considered: •Airfield pressure altitude •Temperature •Headwind component •Runway length •Runway gradient or slope •Obstacles in the flight path Return to Table of Content
  21. 21. 21 Landing Landing is similar to Takeoff, but in reverse. We assume again that the Aircraft doesn’t have VTOL capabilities. The Landing Phase can be divide in the following Phases: 1. The Final approach when the Aircraft Glides toward the runway at a steady speed and rate of descent. 2. The Flare, or Transition phase. The Pilot attempts to rotate the Aircraft nose up and reduce the Rate of Sink to zero and the forward speed to a minimum, that is larger than Vstall. When entering this phase the velocity is less than 1.3Vstall and 1.15 Vstall at touchdown. 3. The Floating Phase, which is necessary if at the end of Flare phase, when the rate of descent is zero, an additional speed reduction is necessary. The Float occurs when the Aircraft is subjected to ground effect which requires speed reduction for touchdown. 4. The Ground Run after the Touchdown the Aircraft must reduce the speed to reach a sufficient low one to be able to turn off the runway. For this it can use Thrust Reverse (if available), spoilers or drag parachutes (like F-15 or MIG-21) and brakes are applied. SOLO Aircraft Flight Performance Ground Run sgr Transition Airborne Phase Total Landing Distance Float sf Flare stGlide sg γ hg hf Touchdown
  22. 22. 22 Landing (continue – 1) Descending Phase SOLO Aircraft Flight Performance The Aircraft is aligned with the landing runaway at an altitude hg and a gliding angle γ. The Aircraft Glides toward the runway at a steady speed and rate of descent, until it reaches The altitude ht at which it goes to Transition Phase, turning with a Radius of Turn R. The Descending Range on the ground is : γγ γ γ γ RhRhhh s ggtg g − ≈ − = − = <<1 tan cos tan Ground Run sgr Transition Airborne Phase Total Landing Distance Float sf Flare stGlide sg γ hg hf Touchdown
  23. 23. 23 Landing (continue – 2) Transition Phase SOLO Aircraft Flight Performance If γ is the descent angle and R is the turn radius then the Aircraft must start the Transition Phase at an altitude ht, above the ground, given by: ( )γcos1−=Rht The Transition Range on the ground is γγ RRst ≈= sin To calculate the turn radius we must use the flight velocity which varies between 1.3 Vstall at the beginning to 1.1 Vstall at Touchdown. Let use an average velocity 3.11.1 −∈= tstalltt mVmV If the Transition Turn Acceleration is nt = 1.15 – 1.25 g than the Turn Radius is ( ) gn V R t t 1 2 − = The Transition Turn time is ( ) gn V RV t t t t t 1/ − == γγ Ground Run sgr Transition Airborne Phase Total Landing Distance Float sf Flare stGlide sg γ hg hf Touchdown
  24. 24. 24 Landing (continue – 3) Float Phase SOLO Aircraft Flight Performance In this phase the Pilot brings the nose wheel to the ground at the touchdown velocity Vt: tVs tf ∆= where Δt is between 2 to 3 seconds. Ground Run sgr Transition Airborne Phase Total Landing Distance Float sf Flare stGlide sg γ hg hf Touchdown
  25. 25. 25 Landing (continue – 4) Ground Run Phase SOLO Aircraft Flight Performance The equations of motion are the same as those developed for Takeoff, but with different parameters, adapted for Landing. Those equations are: cVbVaVd td cVbVa V Vd sd xs ++ = ++ = = 2 2 1 ( )       −= = +−−= µ µ ρ W T gc W gB b W gC CC W Sg a grLgrD 0 ,, : 2 : 2 : where       − − ⋅ + + − + ++ ++ = 2 1 1 2 2 1 2 1 2 2 2 1 1 1 1 ln 42 ln 2 1 a a a a caba b cVbVa cVbVa a sg       − − ⋅ + + − = 1 2 2 1 2 1 1 1 1 ln 4 1 a a a a cab tg where cab bVa a cab bVa a 4 2 : 4 2 : 2 2 2 2 1 1 − + = − + = 2 0 VCVBTT ++= Assume a constant Thrust T = T0: B = 0, C = 0. V1 = Vtouchdown, V2 = final velocity cVa cVa a sg + + −= 2 2 2 1 ln 2 1       − − ⋅ + + − = 1 2 2 1 1 1 1 1 ln 4 1 a a a a ca tg ( )       −==−−= µµ ρ W T gcbCC W Sg a LD 0 :,0, 2 : Ground Run sgr Transition Airborne Phase Total Landing Distance Float sf Flare stGlide sg γ hg hf Touchdown
  26. 26. 26 Landing (continue – 5) Ground Run Phase (continue – 1) SOLO Aircraft Flight Performance where ( )       −= −−= µ µ ρ W T gc CC W Sg a grLgrD 0 ,, : 2 : cab Va a ca Va a touchdown 4 2 : 4 2 : 2 1 1 − = − = Assume a constant Thrust T = T0: B = 0, C = 0. V1 = Vtouchdown, V2 = final velocity cVa cVa a sg + + −= 2 2 2 1 ln 2 1       − − ⋅ + + − = 1 2 2 1 1 1 1 1 ln 4 1 a a a a ca tg For the Landing Ground Run Phase the following must included: • if Thrust Reversal exists we must change T0 to – T0_reversal . •The Drag Coefficient CD0,gr must consider: - the landing gear fully extended. - spoilers or drag parachutes (if exist) •μ – the friction coefficient must be increased to describe the brakes effect. Ground Run sgr Transition Airborne Phase Total Landing Distance Float sf Flare stGlide sg γ hg hf Touchdown
  27. 27. 27 Landing (continue – 6) Summary SOLO Aircraft Flight Performance where ( )       −= −−= µ µ ρ W T gc CC W Sg a grLgrD 0 ,, : 2 : cab Va a ca Va a touchdown 4 2 : 4 2 : 2 1 1 − = − =cVa cVa a sg + + −= 2 2 2 1 ln 2 1       − − ⋅ + + − = 1 2 2 1 1 1 1 1 ln 4 1 a a a a ca tg Ground Run Phase tVs tf ∆= Float Phase ( ) gn V Rs t t t 1 2 − == γ γ ( ) gn V RV t t t t t 1/ − == γγ Transition Phase Descent Phase Ground Run sgr Transition Airborne Phase Total Landing Distance Float sf Flare stGlide sg γ hg hf Touchdown ( ) γγ γ γ 1/ tan cos tan 2 −− = − = − = ttggfg g nVhRhhh s
  28. 28. 28 H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35 Return to Table of Content
  29. 29. 29 Fixed Wing Fighter Aircraft Flight Performance SOLO Level Flight The forces acting on an airplane in Level Flight are shown in Figure 0= = h Vx   Lift and Drag Forces: ( ) TCkCSVCSVD WCSVL LDD L =+== == 2 0 22 2 2 1 2 1 2 1 ρρ ρ 2 2 VS W CL ρ =       +=      += SV Wk CSV SV Wk CSVD DD 2 2 0 2 242 2 0 2 2 2 14 2 1 ρ ρ ρ ρ Lift DragThrust Weight Equations of motion: 0 0 =− =− DT WL Quasi-Static
  30. 30. 30 Fixed Wing Fighter Aircraft Flight Performance SOLO Level Flight  DragInducedDragParasite D SV Wk CSVD 2 2 0 2 2 2 1 ρ ρ += Because of opposite trends in Parasite Drag and Induced Drag, with changes in velocity, the Total Drag assumes a minimum at a certain velocity. If we ignore the change in velocity of CD0 and k with velocity we obtain 0 4 3 2 0 =−= SV Wk CSV Vd Dd D ρ ρ The velocity of minimum Total Drag is * 4 0 2 V C k S W V D == ρ We see that the velocity of minimum Total Drag is equal to the Reference Velocity. 0 2 2 1 DCSVρ SV Wk 2 2 2 ρ* V Lift DragThrust Weight
  31. 31. 31 Fixed Wing Fighter Aircraft Flight Performance SOLO Level Flight For the velocity, V*, of minimum Total Drag we have 02* 2 2 Di CkW SV Wk D == ρ  DragInducedDragParasite D SV Wk CSVD 2 2 0 2 2 2 1 ρ ρ += 000min 2 DDD CkWCkWCkWD =+= and 0 2 2 1 DCSVρ SV Wk 2 2 2 ρ * V Lift DragThrust Weight
  32. 32. 32M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring” Comparison of Takeoff Weight and Empty Weight of different Aircraft
  33. 33. 33 Fixed Wing Fighter Aircraft Flight Performance SOLO Level Flight The Power Required, PR, for Level Flight is SV Wk CSVVDP DR ρ ρ 2 0 3 2 2 1 +=⋅= The Power Required for Level Flight assumes a minimum at a certain velocity Vmp. If we ignore the change in velocity of CD0 and k with velocity we obtain 0 2 2 3 2 2 0 2 =−= SV Wk CSV Vd Pd D R ρ ρ or * 4 0 3 1 3 2 V C k S W V D mp == ρ *0 2, 3 32 L D mp mpL C k C VS W C === ρ ( ) * 000 0 2 ,0 , 866.0 1 4 3 /3 /3 e CkkCkC kC CkC C e DDD D mpLD mpL mp == + = + = * 2 min, 866.03 8 e VW SV Wk P mp mp R == ρ 0 3 2 1 DCSVρ SV Wk ρ 2 2 * 3 1 V min,RP Lift DragThrust Weight
  34. 34. Fixed Wing Fighter Aircraft Flight Performance SOLO Available Aircraft Power and Thrust • Throttle Effect 10 ≤≤= ηη ATT • Propeller airspeedwithvariationsmallVTP propellerA ≈⋅=, V Pa, propeller Power Propeller Aircraft Available Power at Altitude (h) At a given Altitude h • Turbojet airspeedwithvariationsmallT jetA ≈, V Ta, jet Thrust Jet Aircraft Available Power at Altitude h At a given Altitude h Lift DragThrust Weight Lift DragThrust Weight Level Flight
  35. 35. 35 Fixed Wing Fighter Aircraft Flight Performance SOLO Vmin Vmax Pa, propeller PRPmin BA ηaPa, propeller Propeller Aircraft Vmin Vmax Ta, jet TR Dmin η Ta, jet A B Jet Aircraft Level Flight To have a Level Flight the requirement must be satisfied by the available propulsion performance. •For a Propeller Aircraft, the available power Pa,propeller , at a given altitude h, is almost insensitive with changes in velocity. The Velocity in Level Flight is steady when the graph of Required Power PR intersects the graph of Pa,propeller at points A and B. We get two velocities Vmin (h) at A and Vmax (h) at B. By controlling the Propeller Power ηa Pa,propeller (0< ηa <1) we can reach any velocity between Vmin (h) and Vmax (h). •For a Jet Aircraft, the available Thrust Ta,jet , at a given altitude h, is almost insensitive with changes in velocity. The Velocity in Level Flight is steady when the graph of Required Thrust TR intersects the graph of Ta,jet at points A and B. We get two velocities Vmin (h) at A and Vmax (h) at B. By controlling the Jet Thrust η Ta,jet (0< η<1) we can reach any velocity between Vmin (h) and Vmax (h).
  36. 36. 36 Fixed Wing Fighter Aircraft Flight Performance SOLO Vmin Vmax Ta, jet TR Dmin η Ta, jet A B Jet Aircraft Level Flight We have Analytical Solution for Jet Aircraft ( ) SV Wk CSVCkCSVDT DLD 2 2 0 22 0 2 2 2 1 2 1 ρ ρρ +=+== Define 0 * 0 0 * * * 4 0 2 : 2*,*,: 2 :*, * : D DD D L D L D CkW T W eT z CC k C C C C e C k S W V V V u == === == ρ    2 2 /1 2 0 0 2 2 0 2 2 2 u D u Dz D V C k S W C k S W V T CkW ρ ρ += 012 24 =+− uzu Lift DragThrust Weight
  37. 37. 37 Fixed Wing Fighter Aircraft Flight Performance SOLO Vmin Vmax Ta, jet TR Dmin η Ta, jet A B Jet Aircraft Level Flight Analytical Solution for Jet Aircraft 012 24 =+− uzu Solving we obtain 1 1 2 max 2 min −+= −−= zzu zzu 4 0 maxmaxmax 4 0 minminmin 2 * 2 * D D C k S W uVuV C k S W uVuV ρ ρ == == Lift DragThrust Weight 0 * 0 0 * * * 4 0 2 : 2*,*,: 2 :*, * : D DD D L D L D CkW T W eT z CC k C C C C e C k S W V V V u == === == ρ
  38. 38. 38 Fixed Wing Fighter Aircraft Flight Performance SOLO Level Flight Analytical Solution for Jet Aircraft 1 1 2 max 2 min −+= −−= zzu zzu 12 min −−= zzu 12 max −+= zzu At the absolute Ceiling (when is only one possible velocity) we have umax = umin, therefore z = 1. max, 2 L stall CS W V ρ = Lift DragThrust Weight 0 * 0 0 * * * 4 0 2 : 2*,*,: 2 :*, * : D DD D L D L D CkW T W eT z CC k C C C C e C k S W V V V u == === == ρ
  39. 39. 39 Drag Characteristics Fixed Wing Fighter Aircraft Flight Performance SOLO
  40. 40. 40 Fixed Wing Fighter Aircraft Flight Performance SOLO Level Flight Aircraft Range in Level Flight Lift DragThrust Weight Range in Level Flight of Jet Aircraft Equations of motion: 0 0 =− =− DT WL 0= = h Vx   We add the equation of fuel consumption TcW −= c – specific fuel consumption We assume that fuel consumption is constant for a given altitude. V td Wd Wd xd td xd == Dc V Tc V W V Wd xd DT −=−== = 
  41. 41. 41 Fixed Wing Fighter Aircraft Flight Performance SOLO Level Flight Aircraft Range in Level Flight Lift DragThrust Weight Range in Level Flight of Jet Aircraft Dc V Wd xd −= The quantity dx/dW is called the “Instantaneous Range” and is equal to the Horizontal Range traveled per unit load of fuel or the “Specific Range”. Multiply and divide by L = W Wc V C C Wc V D L Wd xd D L       −=            −= Integrating we obtain ∫       −=−= f i W W D L if W Wd V cC C xxR 1 :
  42. 42. 42 Fixed Wing Fighter Aircraft Flight Performance SOLO Level Flight Aircraft Range in Level Flight Lift DragThrust Weight Range in Level Flight of Jet Aircraft To perform the integration we must specify the variation of CL, CD and V. Let consider two cases: ∫       −=−= f i W W D L if W Wd V cC C xxR 1 : a. Range at Constant Altitude of Jet Aircraft We have LCVSLW 2 2 1 ρ== LCS W V ρ 2 = The velocity changes (decreases) since the weight W decreases due to fuel consumption. [ ]if D L W W D L WW C C cW Wd ScC C R f i −         =         −= ∫ 221 ρ
  43. 43. 43 Fixed Wing Fighter Aircraft Flight Performance SOLO Level Flight Aircraft Range in Level Flight Lift DragThrust Weight a. Range at Constant Altitude of Jet Aircraft [ ]if D L W W D L WW C C cW Wd ScC C R f i −         =         −= ∫ 221 ρ The maximum range is obtained when [ ]if D L WW C C c R −         = max max 2 max 2 0max         + =         LD L D L CkC C C C ( ) 030 2 2 1 2 022 0 2 0 2 0 =−⇒= + − + =         + LD LD LL L LD LD L L CkC CkC CkC C CkC CkC C Cd d The maximum range is obtained when *0 3 1 3 1 L D L C k C C == The Velocity at maximum range is ( ) ( ) ( ) ( )tV CS tW CS tW tV LL *4 * 4 * 3 2 3 3/ 2 === ρρ
  44. 44. 44 Fixed Wing Fighter Aircraft Flight Performance SOLO Level Flight Aircraft Range in Level Flight Lift DragThrust Weight b. Range at Constant Velocity of Jet Aircraft               =      −= ∫ f i D L W W D L W W C C c V W Wd V cC C R f i ln 1 The Velocity V is constant and equal to V* corresponding to initial weight Wi. 4 0 * * 22 D i L i C k S W CS W V ρρ == The maximum range is obtained when         =               = f i f i D L W W e c V W W C C V c R lnln 1 * * max max To keep Velocity V constant when weight W decreases, the air density ρ must also decrease, hence the Aircraft will gain (qvasistatic) altitude ( ) Pc td Wd td hd e td hd p hh −==−= − 0/ 0ρ ρ
  45. 45. 45 Fixed Wing Fighter Aircraft Flight Performance SOLO Level Flight Aircraft Range in Level Flight Range in Level Flight of Propeller Aircraft Lift DragThrust Weight The equation of fuel consumption PcW P−= cp – specific fuel consumption (consumed per unit power developed by the engine per unit time We assume that fuel consumption is constant for a given altitude. V td Wd Wd xd td xd == Pc V W V Wd xd p −==  - Required PowerVDPR ⋅= PP pA ⋅=η - Available Power ηp – propulsive efficiency AR PP = p VD P η ⋅ =
  46. 46. 46 Fixed Wing Fighter Aircraft Flight Performance SOLO Level Flight Aircraft Range in Level Flight Range in Level Flight of Propeller Aircraft Lift DragThrust Weight WcC C WcD L DcPc V Wd xd p p D L p p WL p p p ηηη       −=−=−=−= = Integration gives ∫       −=−= f i W W p p D L ff W Wd cC C xxR η : We assume • Angle of Attack is kept constant throughout cruise, therefore e = CL/CD is constant •ηp is independent on flight velocity f i p p W W e c R ln η = Bréguet Range Equation The maximum range of Propeller Aircraft in Level Flight is f i Dp p f i p p W W CkcW W e c R ln 2 1 ln 0 * max ηη ==
  47. 47. 47 Louis Charles Bréguet (1880 – 1955) The Bréguet Range Equation The Bréguet range equation determines the maximum flight distance. The key assumptions are that SFC, L/D, and flight speed, V are constant, and therefore take-off, climb, and descend portions of flights are not well modeled (McCormick, 1979; Houghton, 1982). ( )         ⋅ = final initial W W SFCg DLV Range ln / Winitial = Wfuel + Wpayload + Wstructure + Wreserve Wfinal = Wpayload + Wstructure + Wreserve where ( )         ++ + ⋅ = reservestructurepayload fuel WWW W SFCg DLV Range 1ln / where SFC, L/D, and Wstructure are technology parameters while Wfuel, Wpayload, and Wreserve are operability parameters. Fixed Wing Fighter Aircraft Flight Performance SOLO
  48. 48. 48 Fixed Wing Fighter Aircraft Flight Performance SOLO Level Flight Aircraft Range in Level Flight Range in Level Flight of Propeller Aircraft Lift DragThrust Weight Let assume that the flight to maximum range is performed in one of two ways 1. Propeller Aircraft Flight at Constant Altitude In Constant Altitude Flight the velocity changes with the decrease of weight such that ( ) ( ) 4 0 * 2 DC k S tW VtV ρ == 2. Propeller Aircraft Flight with Constant Velocity In Constant Velocity Flight the velocity is the V* velocity based on the initial weight of the Aircraft . 2 4 0 * const C k S W VV D i === ρ
  49. 49. 49 Fixed Wing Fighter Aircraft Flight Performance SOLO Level Flight Aircraft Endurance in Level Flight Lift DragThrust Weight The Endurance of an Airplane remains in the air and is usually expressed in hours. Endurance of Jet Aircraft in Level Flight We have TcW −= c – specific fuel consumption W Wd c e W Wd D L cDc Wd Tc Wd td WLDT −=−=−=−= == 1 Integrating we obtain ∫−= f i W W W Wd c e t Assuming that the Angle of Attack is held constant throughout the flight, e =CL/CD is constant f i W W c e t ln= f i Df i W W CkcW W c e t ln 2 1 ln 0 * max == The Maximum Endurance for Jet Aircraft occurs for e = e*, CL = CL*, V = V*, D = Dmin.
  50. 50. 50 Fixed Wing Fighter Aircraft Flight Performance SOLO Level Flight Aircraft Endurance in Level Flight The Endurance of an Airplane remains in the air and is usually expressed in hours. Endurance of Propeller Aircraft in Level Flight We have ppp VDcPcW η/⋅−=−= W Wd V e cW Wd VD L cVD Wd c td p p p p WL p p 11 ηηη −=−= ⋅ −= = Assuming that the Angle of Attack is held constant throughout the flight, e =CL/CD is constant Lift DragThrust Weight cp – specific fuel consumption (consumed per unit power developed by the engine per unit time. ηp – propulsive efficiency Integrating we obtain ∫−= f i W W p p W Wd V e c t 1η The Endurance of Propeller Aircraft depends on Velocity, therefore we will assume two cases 1.Flight at Constant Altitude 2.Flight with Constant Velocity
  51. 51. 51 Fixed Wing Fighter Aircraft Flight Performance SOLO Level Flight Endurance of Propeller Aircraft in Level Flight Lift DragThrust Weight The velocity will change to compensate for the decrease in weight ∫ =−= f i W W D L p p C C e W Wd V e c t 1η 1. Propeller Aircraft Flight at Constant Altitude We have LCVSLW 2 2 1 ρ== LCS W V ρ 2 =         −        = ifD L p p WW S C C c t 11 2 2 2/3 ρη For Maximum Endurance Propeller Aircraft has to fly at that Angle of Attack such that (CL 3/2 /CD) is maximum, which occurs when CL=√3 CL * and V = 0.76 V* .         −        = ifDp p WW S Ckc t 11 2 27 4 12 0 3max ρη
  52. 52. 52 Fixed Wing Fighter Aircraft Flight Performance SOLO Level Flight Endurance of Propeller Aircraft in Level Flight Lift DragThrust Weight ∫ =−= f i W W D L p p C C e W Wd V e c t 1η 2. Propeller Aircraft Flight with Constant Velocity f i p p W W V e c t ln 1η = For Maximum Endurance Propeller Aircraft has to fly at a velocity such that e=(CL/CD) is maximum, which occurs when CL=CL * and V = V* , which is based on initial weight Wi 4 0 * * 22 D i L i C k S W CS W V ρρ == 0 * 2 1 DCk e = f i D i p p f i D i Dp p f i p p W W CkS W cW W C k S W CkcW W V e c t ln 1 2 ln 2 2 1 ln 1 4 3 0 4 00 * * max ρ η ρ ηη ===
  53. 53. 53 D=TR V V* tmax Slope min(PR/V) Bréguet Velocities for Maximum Range and Maximum Endurance of Propeller Aircraft Fixed Wing Fighter Aircraft Flight Performance SOLO Graphical Finding of Maximum Range and Endurance of Jet Aircraft in Level Flight       =      ⇔ V D D V R VV minmaxmax Maximum Range From Figure we can see that min (D/V) is obtained by taking the tangent to D graph that passes through origin. The point of tangency will give D and V for (D)min. Maximum Endurance  ∫∫ < = < −=−= 00 111 Wd Dc Wd Tc t DT ( )D D t VV min 1 maxmax =      ⇔ From Figure we can see that min (PR) is obtained by taking the PR and V for (PR)min. Lift DragThrust Weight ∫∫ < −== 0 Wd Dc V xdR
  54. 54. 54 PR V V* Rmax 0.866 V* tmax Slope min(PR/V) Velocities for Maximum Range and Maximum Endurance of Propeller Aircraft Lift DragThrust Weight Fixed Wing Fighter Aircraft Flight Performance SOLO Graphical Finding of Maximum Range and Endurance of Propeller Aircraft in Level Flight   ∫∫∫∫ >>> ⋅ −=−=−== 000 Wd VD V c Wd P V c Wd Pc V xdR p p Rp p p ηη D V P P V R V R V R V minminmaxmax =      =      ⇔ Maximum Range From Figure we can see that min (PR/V) is obtained by taking the tangent to PR graph that passes through origin. The point of tangency will give PR and V for (PR/V)min. Maximum Endurance  ∫∫ << −=−= 00 11 Wd Pc Wd Pc t Rp p p η ( ) ( )VDP P t V R V R V ⋅==      ⇔ minmin 1 maxmax From Figure we can see that min (PR) is obtained by taking the PR and V for (PR)min.
  55. 55. 55 H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00-80T-80 1-1-1965, pg. 35 Fixed Wing Fighter Aircraft Flight Performance
  56. 56. 56 H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00-80T-80 1-1-1965, pg. 35 Fixed Wing Fighter Aircraft Flight Performance
  57. 57. 57 H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00-80T-80 1-1-1965, pg. 35 Fixed Wing Fighter Aircraft Flight Performance
  58. 58. 58 Flight Ceiling by the available Climb Rate - Absolute 0 ft/min - Service 100 ft/min - Performance 200 ft/min True Airspeed Altitude Absolute Ceiling Service Ceiling Performance Ceiling Excess Thrust provides the ability to accelerate or climb True Airspeed Thrust Available Thrust Required Thrust True Airspeed Thrust Available Thrust Required Thrust A AB B C D E E Thrust True Airspeed Available Thrust Required Thrust C D Jet Aircraft Flight Envelope Determined by Available Thrust Flight Envelope: Encompasses all Altitudes and Airspeeds at which Aircraft can Fly Stengel, MAE331, Lecture 7, Gliding, Climbing and Turning Performance Fixed Wing Fighter Aircraft Flight Performance SOLO Lift DragThrust Weight Changes in Jet Aircraft Thrust with Altitude
  59. 59. 59 Propeller Aircraft Ceiling Determined by Available Power To find graphically the maximum Flight Altitude (Ceiling) for a Propeller Aircraft we use the PR (Power Required) versus V (Velocity) graph. The maximum Flight Altitude corresponds to maximum Range Rmax. We have shown that to find Rmax we draw the Tangent Line to PR Graph, passing trough the origin. Fixed Wing Fighter Aircraft Flight Performance SOLO Lift DragThrust Weight Changes in Propeller Aircraft Power and Thrust with Altitude VC Pa, propeller PR hcruise A h2 h1 h0 h0 < h1 <h2 < hcruise The intersection point A with PR Graph defines the Ceiling Velocity VC, and the Pa (Available Power – function of Altitude) with this point defines the Ceiling Altitude. Return to Table of Content
  60. 60. 60 Fixed Wing Fighter Aircraft Flight Performance SOLO Gliding Flight A Glider is an unpowered airplane. 0 sin cos = = = W Vh Vx    γ γ 1<<γ 0=+ = γWD WL .constW Vh Vx = = = γ  Lift and Drag Forces: ( ) γρρ ρ WCkCSVCSVD WCSVL LDD L −=+== == 2 0 22 2 2 1 2 1 2 1 LCS W V ρ 2 = eC C L D W D L D LW 1 −=−=−=−= = γ Equations of motion: 0sin 0cos =+ =− γ γ WD WLQuasi-Steady Flight
  61. 61. 61 Fixed Wing Fighter Aircraft Flight Performance SOLO Gliding Flight We found LCS W V ρ 2 = eC C L D W D L D LW 1 −=−=−=−= = γ Flattest Glide” (γ = γmin) The Flattest Glide (γ = γmin) is given by: 0 * max min min 22 1 DL CkCk eW D −=−=−=−=γ e LC*LC *2 1 LCk CL/CD as a function of CL k C C D L 0 * = The flight velocity for the Flattest Glide is given by: 4 0 *.. 2 * 2 DL GF C k S W V CS W V ρρ === The flight velocity for the Flattest Glide is equal to the reference velocity V* or u = 1. The Flattest Glide is conducted at constant dynamic pressure. . 2 1 0 * 2 .. const C k W C W VSq DL GFG ==== ρ
  62. 62. 62 H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35 Gliding Flight CLEAR CONFIGURATION LANDING CONFIGURATION LIFT to DRAG RATIO L/D (L/D)max LIFT COEFFICIENT, CL CLEAR CONFIGURATION LANDING CONFIGURATION RATEOF SINK VELOCITY (L/D)max TANGENT TO RATE OF SINK GRAPH AT THE ORIGIN Gliding Performance Fixed Wing Fighter Aircraft Flight Performance SOLO
  63. 63. 63 Fixed Wing Fighter Aircraft Flight Performance SOLO Gliding Flight We have: Distance Covered with respect to Ground The maximum Ground Range is covered for the Flattest Glide at the reference velocity V* or u = 1. γV td hd V td xd = = D L e V V hd xd −=−=== γγ 1 Assuming a constant Angle of Attack during Glide, e is constant and the Ground Range R, to descend from altitude hi to altitude hf is given by: ( ) hehhehdexxR fi h h if f i ∆=−=−=−= ∫: and 0 maxmax 2 DCk h heR ∆ =∆= e LC*LC *2 1 LCk CL/CD as a function of CL
  64. 64. 64 Fixed Wing Fighter Aircraft Flight Performance SOLO Gliding Flight Rate of sink is defined as: Rate of Sink         == ⋅ =−=−= = = 2/3 2 22 L D L D L L D W D CS W V s C C S W C C CS W W VD V td hd h L ρρ γ ρ  The term DV = PR represents the Power Required to sustain the Gliding Flight. Therefore the Rate of Sink is minimum when the Power Required is minimum, or (CD/CL 3/2 ) is minimum ( ) ( ) 0 2 3 2 342 3 2 2/5 0 2 2/5 2 0 2 3 2 0 2/12/3 2/3 2 0 2/3 = − = +− = +− =        + =        L DL L LDL L LDLLL L LD LL D L C CCk C CkCCk C CkCCCCk C CkC Cd d C C Cd d Denote by CL,m the value of Lift Coefficient CL for which (CD/CL 3/2 ) is minimum *0 , 3 3 0* L k C C D mL C k C C D L = == 27 4 3 3 0 3 2/3 0 0 0 min 2/3 D D D D L D Ck k C k C kC C C =       + =       
  65. 65. 65 Fixed Wing Fighter Aircraft Flight Performance SOLO Gliding Flight Rate of Sink *0 , 3 3 0* L k C C D mL C k C C D L = == 0 3 max 2/3 27 4 1 DD L CkC C =        We found: The velocity Vm for glide with minimum sink rate is given by:  * 4 0 76.0~ 4 4 0, 76.0 2 3 1 3 22 * V C k S W C k S W CS W V V D DmL m ≈        = ==    ρ ρρ S CkW C C S W h D L D s ρρ 27 22 0 3 min 2/3min, =        = The minimum sink rate is given by:
  66. 66. 66 Fixed Wing Fighter Aircraft Flight Performance SOLO Gliding Flight Endurance The Endurance is the total time the glider remains in the air. Minimum Sink Rate tmax Flatest Glide Rmax         −== 2/3 2 L D C C S W V td hd ρ γ         −== D L C C W S V hd td 2/3 2 ρ γ ( )fi D L h h D L hh C C W S hd C C W S t f i −        =        −= ∫ 2/32/3 22 ρρ Assuming that the Angle of Attack is held constant during the glide and ignoring the variation in density as function of altitude, we have For Maximum Endurance the Glider has to fly at that Angle of Attack such that (CL 3/2 /CD) is maximum, which occurs when CL=√3 CL * and V = 0.76 V* .       − = 4 27 2 4 0 3max fi D hh CkW S t ρ Return to Table of Content
  67. 67. 67 Performance of an Aircraft with Parabolic PolarSOLO W LT n + = αsin :' W L n =: 2 0 : LD L D L CkC C CSq CSq D L e + === We assume a Parabolic Drag Polar: 2 0 LDD CkCC += Let find the maximum of e as a function of CL ( ) ( ) 0 2 22 0 2 0 22 0 22 0 = + − = + −+ = ∂ ∂ LD LD LD LLD L CkC CkC CkC CkCkC C e e LC*LC *2 1 LCk CL/CD as a function of CL The maximum of e is obtained for k C C D L 0 * = ( ) 0 0 0 2 0 2** D D DLDD C k C kCCkCC =+=+= Start with Load Factor Total Load Number Lift to Drag Ratio Climbing Aircraft Performance
  68. 68. 68 Performance of an Aircraft with Parabolic PolarSOLO e LC*LC *2 1 LCk CL/CD as a function of CL The maximum of e is obtained for k C C D L 0 * = ( ) 0 0 0 2 0 2** D D DLDD C k C kCCkCC =+=+= *2 1 *2 1 2 1 2* * * 22 00 0 LLDD D D L CkCkCkC k C C C e ===== We have WnCSVCSqL LL === 2 2 1 ρ Let define for n = 1             = = == 2 4 0 * 2 1 :* * : 2 * 2 1 :* Vq V V u C k S W CS W V D L ρ ρρ 2 0 : LD L D L CkC C CSq CSq D L e + === Climbing Aircraft Performance
  69. 69. 69 Performance of an Aircraft with Parabolic PolarSOLO Using those definitions we obtain L L L L C C nqq WCSq WnCSqL * * ** =→    = == 2 2 2 1 2 1 * 2 1 * uV V n q q == ρ ρ 2 * * * u C nC q q nC L LL == ( )       +=      +=       +=+= = 2 2 2 04 02 0 2 * 4 2 2 0 22 0 ** * * 0 2 u n uCSq u C nCuSq u C nkCuSqCkCSqD D D D CCk L DLD DL           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ *2 1 * *** 0 0 e W C C CSqCSq L D LD ==       += 2 2 2 *2 u n u e W D Therefore Return to Table of Content Climbing Aircraft Performance
  70. 70. 70 Performance of an Aircraft with Parabolic PolarSOLO We obtained       += 2 2 2 *2 u n u e W D u 0 - - - - 0 + + + + + D ↓ min ↑ n u D ∂ ∂ Let find the minimum of D as function of u. nu u nu e W u n u e W u D =→ = − =      −= ∂ ∂ 2 3 24 3 2 0 * 22 *2 * 2min e Wn DD nu == = Aircraft Drag Climbing Aircraft Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
  71. 71. 71 Performance of an Aircraft with Parabolic PolarSOLO Aircraft Drag ( ) MAXn W VhL n ≤= ,         +== 2 2 2 *2 u n u e W D MAX nn MAX Maximum Lift Coefficient or Maximum Angle of Attack ( ) ( ) ( )VhorMCMC STALLMAXLL ,, _ ααα ≤≤ We have u C C u n u C nC q q nC L MAXL CC L LL MAXLL * * * * _ 2 _ =→== = 2 2 _ 2 2 _2 * 1 *2 **2_ u C C e W u C C u e W D L MAXL L MAXL CC MAXLL               +=               +== Maximum dynamic pressure limit ( ) ( ) MAX MAX MAXMAX u V V uhVVorqVhq =<→≤≤= : *2 1 2 ρ *e W D MAXLC _ 2 2 _ 1 2 1 u C C L MAXL               +         += 2 2 2 2 1 * u n ue W D MAX LIMIT nn MAX= 2min * ue W D =       += 2 2 2 2 1 * u n ue W D MAXuu =MAX MAXL L CORNER n C C u _ * = n LIMIT u MAXnu = as a function of u*e W D Maximum Load Factor Climbing Aircraft Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
  72. 72. 72 Performance of an Aircraft with Parabolic PolarSOLO Energy per unit mass E Let define Energy per unit mass E: g V hE 2 : 2 += Let differentiate this equation: ( ) ( ) W VDT W VDT W DT g g V V g VV hEps − ≈ − =            − − +=+== α γ α γ cos sin cos sin:   *& *2 2 2 2 VuV u n u e W D =      += Define *: e W T z       = We obtain ( )             +−=             +−      = − = 2 2 2 2 2 2 2 1 * * * 2 1 * * u n uzu e V W Vu u n ue W T e W W VDT ps or ( ) u nuzu e V ps 224 2 *2 * −+− = 020 224 =+−→== nuzup constns ( ) ( ) 2 224 2 2243 23 * *244 * * u nuzu e V u nuzuuuzu e V u p constn s ++− = −+−−+− = ∂ ∂ = 0= ∂ ∂ =constn s u p 2 21 2 uu uu MAX << + nz > Climbing Aircraft Performance nz nzzu nzzu >     −+= −−= 22 2 22 1 3 3 22 nzz uMAX ++ =           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
  73. 73. 73 Performance of an Aircraft with Parabolic PolarSOLO Energy per unit mass E sp 2u1u MAXu 2 21 uu + u MAXn n 1=n ( ) u nuzu e V ps 224 2 * * −+− = ps as a function of u u V pe uzunnuzuu V pe ss * *2 22 * *2 242224 −+−=→−+−= From which u V pe uzun s * *2 2 24 −+−= ( ) * *2 44 3 2 V pe uzu u n s constps −+−= ∂ ∂ = ( ) 3 0412 2 2 22 z uzu u n constps =→=+−= ∂ ∂ = ( ) u nuzu e V ps 224 2 *2 * −+− = Climbing Aircraft Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
  74. 74. 74 Performance of an Aircraft with Parabolic PolarSOLO Load Factor n u 3 z z z2z 3 z u 2 n 0=sp 0>sp 0<sp 0<sp 0=sp 0>sp ( ) u n ∂ ∂ 2 ( ) 2 22 u n ∂ ∂ 3 z u ( ) ( ) 2 2 2 22 ,, n u n u n ∂ ∂ ∂ ∂ as a function of u ( ) 3 0412 2 2 22 z uzu u n constps =→=+−= ∂ ∂ = ( ) * *2 44 3 2 V pe uzu u n s constps −+−= ∂ ∂ = Integrating once u V pe uzun s * *2 2 24 −+−= Integrating twice Climbing Aircraft Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
  75. 75. 75 Performance of an Aircraft with Parabolic PolarSOLO Load Factor n For ps = 0 we have zuuzun 202 24 ≤≤+−= Let find the maximum of n as function of u. 0 22 44 24 3 = +− +− = ∂ ∂ uzu uzu u n Therefore the maximum value for n is achieved for zu = ( ) zn MAXps ==0 u 0 √z √2z ∂ n/∂u | + + + 0 - - - - | - - n ↑ Max ↓ z2z u n 0=sp 0>sp 0<sp MAXn z MAX MAXL L n C C _ * n as a function of u Climbing Aircraft Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
  76. 76. Performance of an Aircraft with Parabolic PolarSOLO Energy per unit mass E g V hE 2 : 2 += Climbing Aircraft Performance Energy Height versus Mach NumberEnergy Height versus True Airspeed ( )hV V M sound =:( ) 00 : T T V T T MhVTAS sound == Return to Table of Content
  77. 77. 77 Performance of an Aircraft with Parabolic PolarSOLO Steady Climb (V, γ = constant) Climbing Aircraft Performance 0sin 0cos ==−− ==− td Vd g W WDT td d V g W WL γ γ γ Equation of Motion for Steady Climb: γ γ sin cos Vh Vx = =   Define the Rate of Climb: ( ) s Ra C p W PP W DTV Vh = − = −⋅ == γsin where Pa = V T - available power PR = V D - required power ps - excess power per unit weight Weight ThrustExcess W DT = − =γsin C C WL const γ γγ cos . = == Lift Drag Thrust Weight
  78. 78. 78 Performance of an Aircraft with Parabolic PolarSOLO Climbing Aircraft Performance LC CSVW 2 2 1 cos ργ = ( ) s C D LD C p SV W kCSVVT WW CkCSVVT h =             −−= +− = ρ γ ρ ρ 2 1 cos 2 112 1 22 0 3 2 0 3  Let find the velocity V for which the Rate of Climb is maximum, for the Propeller Aircraft: 0 cos2 2 31 2 22 0 2 =      +−== SV Wk CSV Wtd pd td hd C D sC,Prop ρ γ ρ  Steady Climb (V, γ = constant) For a Propeller Aircraft we assume that Pa=T V= constant. or ** 4 4 0 4 76.0 3 12 3 1 VV C k S W V D Climb.Prop === ρ s C DaPropC p SV W kCSVP W h =      −−= ρ γ ρ 22 0 3 , cos 2 2 11 We can see that the velocity at which the Rate of Climb of Propeller Aircraft is maximum is the same as the velocity at which the Required Power in Level Flight is maximum. Lift Drag Thrust Weight
  79. 79. 79 Performance of an Aircraft with Parabolic PolarSOLO Climbing Aircraft Performance LC CSVW 2 2 1 cos ργ = ( )             −−= +− = SV W kCSVVT WW CkCSVVT h C D LD C ρ γ ρ ρ 2 1 cos 2 112 1 22 0 3 2 0 3  Let find the velocity V for which the Rate of Climb is maximum, for the Jet Aircraft: 0 cos2 2 31 2 22 0 2 =      +−= SV Wk CSVT Wtd hd C D C ρ γ ρ  Steady Climb (V, γ = constant) For a Jet Aircraft we assume that T = constant. Define 0 * 0 0 * * * 4 0 2 :2*,*,: 2 :*, * : D DD D L D L D CkW T W eT zCC k C C C C e C k S W V V V u ======= ρ 0cos 2 2 3 2 2 /1 2 0 0 2 2 0 2 2 =+− C u D u Dz D V C k S W C k S W V T CkW γ ρ ρ    0cos23 224 =−− Cuzu γ Czzu γ22 cos3++=
  80. 80. 80 Performance of an Aircraft with Parabolic PolarSOLO Climbing Aircraft Performance Steady Climb (V, γ = constant) ps versus the nondimensional velocity u ps versus the velocity V 0sin ==−− td Vd g W WDT γ 1 2 2 2 *2 =       += n u n u e W D Define 0 * 0 0 * * * 4 0 2 :2*,* ,: 2 :*, * : D DD D L D L D CkW T W eT zCC k C C C C e C k S W V V V u ==== === ρ             +−== − = 2 2 * 1 2 2 1 sin u uz eV p W DT s γ To find the maximum γ we must have 0 2 2 2 1sin 3* =      −−= u u eud d γ 4 0 2 *max DC k S W VV ρ γ == ( ) ( )1 * *2 *2 * 1 1 224 , max −= −+− = = = z e V u nuzu e V p u n s γ * , max 1 sin max max e z V ps − == γ γ γ 1max =γu
  81. 81. SOLO 81 Aircraft Flight Performance Construction of the Specific Excess Power contours ps in the Altitude-Mach Number map for a Subsonic Aircraft below the Drag-divergence Mach Number. These contour are constructed for a fixed load factor W/S and Thrust factor T/S, if the load or thrust factor change, the ps contours will shift. ( ) ( ) W VDT W VDT W DT g g V V g VV hEps − ≈ − =            − − +=+== α γ α γ cos sin cos sin:   In Figure (a) is a graph of Specific Excess Power contours ps versus Mach Number. Each curve is for a specific altitude h. In Figure (b) each curve is for a given Specific Excess Power ps in Altitude versus Mach Number coordinates. The points a, b, c, d, e, f for ps = 0 in Figure (a) are plotted on the curve for ps = 0 in Figure (b). Similarly all points ps = 200 ft/sec in Figure (a) on the line AB are projected on the curve ps = 200 ft/sec in Figure (b). Specific Excess Power contours ps for a Subsonic Aircraft Specific Excess Power contours ps
  82. 82. SOLO 82 Aircraft Flight Performance Specific Excess Power contours ps for a Supersonic Aircraft In the graphs of Specific Excess Power ps versus Mach Number Figure (a) for a Supersonic Aircraft we see a “dent” in h contour in the Transonic Region. This is due to the increase in Drag in this region.2 In Figure (b) the graphs of Altitude versus Mach Number we see a “closed” ps = 400 ft/sec contour due to the increase in Drag in this Transonic Region. Specific Excess Power contours ps ( ) ( ) W VDT W VDT W DT g g V V g VV hEps − ≈ − =            − − +=+== α γ α γ cos sin cos sin:   Return to Table of Content
  83. 83. 83 Performance of an Aircraft with Parabolic PolarSOLO Climbing Aircraft Performance Optimum Climbing Trajectories using Energy State Approximation (ESA) We defined the Energy per unit mass E (Specific Energy): g V hE 2 : 2 += Differentiate this equation: ( ) ( ) W VDT W VDT W DT g g V V g VV h td Ed ps − ≈ − =            − − +=+== α γ α γ cos sin cos sin:   Minimum Time-to-Climb The time to reach a given Energy Height Ef is computed as follows E Ed td  = ∫= fE E f E Ed t 0  The minimum time to reach the given Energy Height Ef is obtained by using at each level. ( )∫= fE E f E Ed t 0 max max,  ( )maxE
  84. 84. 84 Performance of an Aircraft with Parabolic PolarSOLO Climbing Aircraft Performance Optimum Climbing Trajectories using Energy State Approximation (ESA) Minimum Time Climb Profiles for Subsonic Speed ( ) ( ) W VDT W VDT W DT g g V V g VV hEps − ≈ − =            − − +=+== α γ α γ cos sin cos sin:   Stengel, MAE331, Lecture 7, Gliding, Climbing and Turning Performance The minimum time to reach the given Energy Height Ef is obtained by using at each level. ( )maxE Energy can be converted from potential to kinetic or vice versa along lines of constant energy in zero time with zero fuel expended. This is physically not possible so the method gives only an approximation of real paths.
  85. 85. SOLO 85 Aircraft Flight Performance Stengel, MAE331, Lecture 7, Gliding, Climbing and Turning Performance Minimum Time Climb Profiles for Supersonic Speed ( ) ( ) W VDT W VDT W DT g g V V g VV hEps − ≈ − =            − − +=+== α γ α γ cos sin cos sin:   The minimum time to reach the given Energy Height Ef is obtained by using at each level. ( )maxE The optimum flight profile for the fastest time to altitude or time to speed involves climbing to maximal altitude at subsonic speed, then diving in order to get through the transonic speed range as quickly as possible, and than climbing at supersonic speeds again using .( )maxE
  86. 86. 86 SOLO Climbing Aircraft Performance Optimum Climbing Trajectories using Energy State Approximation (ESA) Shaw, “Fighter Combats – Tactics and Maneuvering” Minimum Time Climb Profiles Aircraft Flight Performance The minimum time to reach the given Energy Height Ef is obtained by using at each level . ( )maxE
  87. 87. 87 SOLO Climbing Aircraft Performance Optimum Climbing Trajectories using Energy State Approximation (ESA) A.E. Bryson, Course “Performance Analysis of Flight Vehicles”, AA200, Stanford University, Winter 1977-1978 Aircraft Flight Performance A.E. Bryson, Jr., “Energy-State Approximation in Performance Optimization of Supersonic Aircraft”, Journal of Aircraft, Vol.6, No. 5, Nov-Dec 1969, pp. 481-488 Approximate (ESA) Solutions. Implicit to ESA Approximation is the possibility of instantaneous jump between kinetic to potential energy (from A to B ). This non physical situation is called a “zoom climb” or “zoom dive”. A B The minimum time to reach the given Energy Height Ef is obtained by using at each level. ( )maxE
  88. 88. SOLO Climbing Aircraft Performance Optimum Climbing Trajectories using Energy State Approximation (ESA) “Exact” calculated using Optimization Methods Computations Aircraft Flight Performance Comparison between “Exact” and Approximate (ESA) Solutions. Implicit to ESA Approximation is the possibility of instantaneous jump between kinetic to potential energy (from A to B , and from C to D). This non physical situation is called a “zoom climb” or “zoom dive”. We can see the “exact” solution in those cases. A B C D The minimum time to reach the given Energy Height Ef is obtained by using at each level. ( )maxE 88 A.E. Bryson, Course “Performance Analysis of Flight Vehicles”, AA200, Stanford University, Winter 1977-1978 A.E. Bryson, Jr., “Energy-State Approximation in Performance Optimization of Supersonic Aircraft”, Journal of Aircraft, Vol.6, No. 5, Nov-Dec 1969, pp. 481-488
  89. 89. 89http://msflights.net/forum/showthread.php?1184-Supersonic-Level-Flight-Envelopes-in-FSX F-15 Streak Eagle Time to Climb Records, which follow the ideal path to reach set altitudes in a minimal amount of time. The Streak Eagle could break the sound barrier in a vertical climb, so the ideal flightpath to 30000m involved a large Immelmann. https://www.youtube.com/watch?v=HLka4GoUbLo https://www.youtube.com/watch?v=S7YAN9--3MA F-15 Streak Eagle Record Flights part 2F-15 Streak Eagle Record Flights part 1 SOLO Climbing Aircraft Performance Optimum Climbing Trajectories using Energy State Approximation (ESA) Aircraft Flight Performance
  90. 90. 90 How to climb as fast as possible Takeoff and pull up: You want to build energy (kinetic or potential) as quickly as you can. Peak acceleration is at mach 0.9, which is the speed that energy is gained the fastest. You should first accelerate to near that speed. Avoid bleeding off energy in a high-g pull up. Start a smooth pull up before at mach 0.7-0.8 and accelerate to mach 0.9 during the pull. http://msflights.net/forum/showthread.php?1184-Supersonic-Level-Flight-Envelopes-in-FSX SOLO Climbing Aircraft Performance Optimum Climbing Trajectories using Energy State Approximation (ESA) F-15 Streak Eagle Time to Climb Records, which follow the ideal path to reach set altitudes in a minimal amount of time. The Streak Eagle could break the sound barrier in a vertical climb, so the ideal flightpath to 30000m involved a large Immelmann. Aircraft Flight Performance Climb again: to 36000ft for maximum speed, or higher as to not exceed design limits or to save fuel for a longer run Climb: Adjust your climb angle to maintain mach 0.9. In a modern fighter, the climb angle may be 45-60 degrees. If you need a heading change, during the pull and climb is a good time to make it. Level off: between 25000 and 36000ft by rolling inverted. Maximum speed is reached at 36000, but remember the engines produce more thrust at higher KIAS, so slightly denser air may not hurt acceleration through the sound barrier. Break the mach barrier: Accelerate to mach 1.25 with minimal wing loading (don't turn, try to set 0AoA) Return to Table of Content
  91. 91. 91 SOLO Climbing Aircraft Performance Optimum Climbing Trajectories using Energy State Approximation (ESA) Aircraft Flight Performance Minimum Fuel-to- Climb Trajectories using Energy State Approximation (ESA) The Rate of Fuel consumed by the Aircraft is given by:    =−= AircraftJetTc AircraftPropellerPc td Wd td fd T p We can write ( )DTV EdW E Ed td − ==  The fuel consumed in a flight time , tf for a Jet Aircraft is: ( )∫∫∫ − === fff t T t T t f Ed TDV Wc E Ed Tctd td fd f 000 /1 The minimum fuel consumed in a flight time tf is obtained when using Maximum Thrust and the Mach Number that minimize the integrand: ( )∫ − = ft T M f Ed TDV Wc f 0 max min, /1 minarg for each level of E.
  92. 92. 92 SOLO Climbing Aircraft Performance Optimum Climbing Trajectories using Energy State Approximation (ESA) Aircraft Flight Performance Minimum Fuel-to- Climb Trajectories using Energy State Approximation (ESA) Assuming W nearly constant, during the climb period, contours of constant ( ) max max Tc DTV T − can be computed, as we see in the Figure A.E. Bryson, Course “Performance Analysis of Flight Vehicles”, AA200, Stanford University, Winter 1977-1978 A.E. Bryson, Jr., “Energy-State Approximation in Performance Optimization of Supersonic Aircraft”, Journal of Aircraft, Vol.6, No. 5, Nov-Dec 1969, pp. 481-488 The Minimum Fuel-to- Climb Trajectory is obtained by choosing at each state. ( ) max max Tc DTV T − The Minimum Time-to- Climb Path is also displayed. Implicit to ESA Approximation is the possibility of instantaneous jump between kinetic to potential energy (from A to B) where the Total Energy is constant. A B Return to Table of Content
  93. 93. 93 SOLO Climbing Aircraft Performance Optimum Climbing Trajectories using Energy State Approximation (ESA) Aircraft Flight Performance Maximum Range during Glide using Energy State Approximation (ESA) Equations of motion A.E. Bryson, Jr., “Energy-State Approximation in Performance Optimization of Supersonic Aircraft”, Journal of Aircraft, Vol.6, No. 5, Nov-Dec 1969, pp. 481-488 γ γ γ sin 0cos WDTV g W WL td d V g W −−= ≈−=  ( ) W VDT Eps − == : g V W DT  − − =γsin γ γ sin cos Vh Vx = =         − − = g V W DT Vh   γγ cos 1 cos       − − === g V W DT V h x h xd hd    During Glide we have: T = 0, W = constant, dE≤0, |γ| <<1, therefore       +−= g V W D xd hd  ( ) γcos 1 VW DT xd Ed − = 2 2 1 : VhE += ( ) ( ) ( )EL VED W VED td Ed −≈−= V td xd = ( ) ( ) ( )ED EL ED W Ed xd −≈−= ( ) ( ) ( )∫∫∫ −≈−== Ed ED EL Ed ED W xdR
  94. 94. 94 SOLO Climbing Aircraft Performance Optimum Climbing Trajectories using Energy State Approximation (ESA) Aircraft Flight Performance Maximum Range during Glide using Energy State Approximation (ESA) We found A.E. Bryson, Jr., “Energy-State Approximation in Performance Optimization of Supersonic Aircraft”, Journal of Aircraft, Vol.6, No. 5, Nov-Dec 1969, pp. 481-488 ( ) ( ) ( )∫∫ −≈−= Ed ED EL Ed ED W R Using the first integral we see that to maximize R we must choose the path that minimizes the drag D (E). The approximate optimal trajectory can be divided in: 1.If the initial conditions are not on the maximum range glide path the Aircraft shall either “zoom dive” or “zoom climb” at constant E0, A to B path in Figure . 2.The Aircraft will dive on the min D (E) until it reaches the altitude h = 0 at a velocity V and Specific Energy E1=V2 /2, B to C in the Figure. 3.Since h=0 no optimization is possible and to stay airborne one must keep the drag such that L = W, by increasing the Angle of Attack and decreasing velocity until it reaches Vstall and Es=Vstall 2 /2, C to D in Figure Since h=0, d E=V dV. ( ) ( ) ( )[ ]∫ ∫∫ = = −−−= 1 0 1 0 0 0min 10 max E E E E h pathon E E s Vd VD VW Ed ED W Ed ED W R    Return to Table of Content
  95. 95. 95 Performance of an Aircraft with Parabolic PolarSOLO       − + = = γσ α γσ coscos sin cossin V g Vm LT q V g r W W n W L W LT n =≈ + = αsin :' Therefore ( )      −= = γσ γσ coscos' cossin n V g q V g r W W γσγσγσω 2222222 coscoscoscos'2'cossin +−+=+= nn V g qr WW or γγσω 22 coscoscos'2' +−= nn V g γγσω 22 2 coscoscos'2' 1 +− == nng VV R Aircraft Turn Performance
  96. 96. 96 Performance of an Aircraft with Parabolic PolarSOLO ( ) ( ) ( ) γ σ σ γ α χ γσγσ α γ cos sin sin cos sin coscos'coscos sin V gLT n V g V g Vm LT = + = −=− + =   2. Horizontal Plan Trajectory ( )0,0 == γγ  ( ) 1' 1 1' ' 1 1'sin' cos 1 '01cos' 2 2 2 2 − = −=      −== =→=−= ng V R n V g n n V g n V g nn V g σχ σ σγ   Aircraft Turn Performance 1. Vertical Plan Trajectory (σ = 0) ( ) γ γγ χ cos' 1 cos' 0 2 − = −= = ng V R n V g  
  97. 97. 97
  98. 98. 98 Vertical Plan Trajectory (σ = 0) SOLO Prof. Earll Murman, “Introduction to Aircraft Performance and Static Stability”, September 18, 2003
  99. 99. 99 R V =:χ1'2 −= n V g χ Contours of Constant n and Contours of Constant Turn Radius in Turn-Rate in Horizontal Plan versus Mach coordinates Horizontal Plan TrajectorySOLO
  100. 100. 100Maneuverability Diagram R V =:χ 1'2 −= n V g χ Horizontal Plan Trajectory
  101. 101. 101 F-5E Turn Performance Horizontal Plan Trajectory
  102. 102. 102 Performance of an Aircraft with Parabolic PolarSOLO 2. Horizontal Plan Trajectory ( )0,0 == γγ  We can see that for n > 1 We found that 2 2 * * u C C n u C nC L LL L =→= n 1n 2n MAXn u u LC MAXLC _ 1 _ n C C MAXL L MAX MAXL L corner n C C u _ * = *2 L MAX L C u n C = MAX MAXL L corner n C C u _ * = MAX L L n C C 1 * MAXLC _ 2LC 1LC 2 * 1 u C C n L L = MAXn n, CL as a function of u Aircraft Turn Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ 1 1 1' 1 11' 2 2 2 2 22 − ≈ − = −≈−= ng V ng V R n V g n V g χ Horizontal Turn Rate Horizontal Turn Radius
  103. 103. 103 Performance of an Aircraft with Parabolic PolarSOLO MAX MAX L MAXL n n C C V g 1 ** 2 _ − MAX MAXL L corner n C C V g u _ * * = MAXL L C C V g u _ 1 * * = MAXn 2n 1n MAXLC _ 2LC 1LC u χ MAXu Horizontal Turn Rate as function of u, with n and CL as parametersχ We defined 2 * & * : u C C n V V u L L == We found 2 2 2 22 1 ** 1 * 1 u u C C V g n Vu g n V g L L −      =−=−=χ This is defined for 1: ** 1 __ <=≥≥= u C C un C C u MAXL L MAX MAXL L corner 2. Horizontal Plan Trajectory ( )0,0 == γγ  Aircraft Turn Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
  104. 104. 104 Performance of an Aircraft with Parabolic PolarSOLO From 2 2 2 22 1 ** 1 * 1 u u C C V g n Vu g n V g L L −      =−=−=χ 4 2 2 2 22 1 * 1* 1 * : uC Cg V n u g VV R L L −      = − == χ Therefore cornerMAX MAXL L MAXL L L MAXL C un C C u C C u uC Cg V R MAXL =≤≤= −      = __ 1 4 2 _ 2 ** 1 * 1* _ cornerMAX MAXL L MAX n un C C u n u g V R MAX =≥ − = _ 2 22 * 1 * MAX L L L L L L C n C C u C C u uC Cg V R L ** 1 * 1* 1 4 2 2 ≤≤= −      = n C C u n u g V R MAXL L n _ 2 22 * 1 * ≥ − = 2. Horizontal Plan Trajectory ( )0,0 == γγ  Aircraft Turn Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
  105. 105. 105 Performance of an Aircraft with Parabolic PolarSOLO R (Radius of Turn) a function of u, with n and CL as parameters 1 ** 2 _ 2 −MAX MAX MAXL L n n C C g V MAX MAXL L corner n C C V g u _ * * = MAXL L C C V g u _ 1 * * = MAXn 2n 1nMAXLC _ 2LC 1LC u R 4 2 2 2 22 1 * 1* 1 * : uC Cg V n u g VV R L L −      = − == χ 2. Horizontal Plan Trajectory ( )0,0 == γγ  Return to Table of Content Aircraft Turn Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
  106. 106. 106 Performance of an Aircraft with Parabolic PolarSOLO Horizontal Turn Rate as Function of ps, n ( ) u nuzu e V ps 224 2 *2 * −+− = up V e uzun s * *2 2 242 −+−= 2 24 2 2 1 * *2 2 * 1 * u up V e uzu V g u n V g s −−+− = − =χ 2 24 4 2423 1 * *2 2 2 1 * *2 22 * *2 44 * u up V e uzu u up V e uzuuup V e uzu V g u s ss −−+−       −−+−−      −+− = ∂ ∂ χ Therefore       −−+− ++− = ∂ ∂ 1 * *2 2 1 * * * 244 4 up V e uzuu up V e u V g u s s χ For ps = 0 2 22 12 24 0 11 12 * uzzuzzu u uzu V g sp =−+<<−−= −+− == χ ( ) 2 22 1 244 4 0 11 12 1 * uzzuzzu uzuu u V g u sp =−+<<−−= −+− +− = ∂ ∂ = χ Aircraft Turn Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
  107. 107. 107 Performance of an Aircraft with Parabolic PolarSOLO Horizontal Turn Rate as Function of ps, n For ps = 0 2 22 12 24 0 11 12 * uzzuzzu u uzu V g sp =−+<<−−= −+− == χ ( ) 2 22 1 244 4 0 11 12 1 * uzzuzzu uzuu u V g u sp =−+<<−−= −+− +− = ∂ ∂ = χ Let find the maximum of as a function of uχ ( )12 1 * 244 4 0 −+− +− = ∂ ∂ = uzuu u V g u sp χ ( ) ( )12 * 1 00 −=== == z V g u ss ppMAX χχ  u 0 u1 1 (u1+u2)/2 u2 ∞ + + 0 - - - - - - -∞ ↑ Max ↓ u∂ ∂ χ χ From 2 24 1 * *2 2 * u up V e uzu V g s −−+− =χ       −−+− ++− = ∂ ∂ 1 * *2 2 1 * * * 244 4 up V e uzuu up V e u V g u s sχ Aircraft Turn Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
  108. 108. 108 Performance of an Aircraft with Parabolic PolarSOLO Horizontal Turn Rate as Function of ps, n u u 0<sp 0<sp 0=sp 0=sp 0>sp 0>sp χ u∂ ∂ χ ( )12 * −z V g 1=u1u 2u as a function of u with ps as parameter u∂ ∂ χ χ  ,       −−+− ++− = ∂ ∂ 1 * *2 2 1 * * * 244 4 up V e uzuu up V e u V g u s sχ 2 24 1 * *2 2 * u up V e uzu V g s −−+− =χ Because ,we have0 * * >u V e 000 >=< >> sss ppp χχχ  0 1 0 1 0 1 0 > = = = < = ∂ ∂ <= ∂ ∂ < ∂ ∂ sss p u p u p u uuu χχχ  Aircraft Turn Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
  109. 109. 109 Performance of an Aircraft with Parabolic PolarSOLO Horizontal Turn Rate as Function of ps, n a function of u, with ps as parameter χ 2 24 1 * *2 2 * u up V e uzu V g s −−+− =χ Aircraft Turn Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ Sustained Turn Instantaneous Turn
  110. 110. 110 Performance of an Aircraft with Parabolic PolarSOLO Horizontal Turn Rate as Function of ps, n 2 24 1 * *2 2 * u up V e uzu V g s −−+− =χ ( ) ( )ss s puupu up V e uzu u g VV R 21 24 42 1 * *2 2 * << −−+− == χ 3 242 23 2 24 4 2 24 34243 2 1 * *2 22 2 * *3 22 * 1 * *2 2 2 1 * *2 2 * *2 441 * *2 24 *       −−+−       −− = −−+−       −−+−       −+−−      −−+− = ∂ ∂ up V e uzuu up V e uzu g V up V e uzu u up V e uzu p V e uzuuup V e uzuu g V u R s s s s ss Aircraft Turn Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
  111. 111. 111 Performance of an Aircraft with Parabolic PolarSOLO Horizontal Turn Rate as Function of ps, n 3 24 2 2 1 * *2 2 2 * *3 2 *       −−+−       −− = ∂ ∂ up V e uzu up V e uzu g V u R s s or We have            > +      + = < +      − = →= ∂ ∂ 0 4 16 * * 9 * *3 0 4 16 * * 9 * *3 0 2 2 2 1 z zp V e up V e u z zp V e up V e u u R ss R ss R u 0 u1 uR2 u2 ∞ - - - 0 + + ∞ ↓ min ↑ u R ∂ ∂ R 2 22 124 42 0 11 12 * uzzuzzu uzu u g V R sp =−+<<−−= −+− == ( ) ( ) 2 22 1 324 22 0 11 12 1*2 uzzuzzu uzu uzu g V u R sp =−+<<−−= −+− − = ∂ ∂ = Aircraft Turn Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
  112. 112. 112 Performance of an Aircraft with Parabolic PolarSOLO Horizontal Turn Rate as Function of ps, n u R 0>sp 0=sp 0<sp MAXL L C C _ * 1 ** 2 _ −MAX MAX MAXL L n n C C g V 1 1* 2 −zg V 4 2 _ 1* 1* uC C g V MAXL L −         1 * 2 22 −MAXn u g V MAX MAXL L n C C _ * LIMIT C MAXL_ LIMIT nMAX z 1 12 −− zz 12 −+ zz 1 * *2 2 * 24 42 −−+− = up V e uzu u g V R s Minimum Radius of Turn R is obtained for zu /1= 1 1* 2 2 0 − == zg V R sp R (Radius of Turn) a function of u, with ps as parameter ( ) ( )ss s puupu up V e uzu u g VV R 21 24 42 1 * *2 2 * << −−+− == χ Return to Table of Content Because ,we have0 * * >u V e 000 >=< << sss ppp RRR 000 minminmin >=< << sss pRpRpR uuu Aircraft Turn Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
  113. 113. 113 Performance of an Aircraft with Parabolic PolarSOLO Horizontal Turn Rate as Function of nV, ( ) W VDT g VV hEps − ≈+==  : For an horizontal turn 0=h V g Vu g VV ps   * == We found 2 24 1 * *2 2 * u up V e uzu V g s −−+− =χ from which 2 24 1*2 * u ue g V zu V g −      −+− =  χ defined for 2 22 1 :1**1**: ue g V ze g V zue g V ze g V zu =−      −+      −≤≤−      −−      −=  Aircraft Turn Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
  114. 114. 114 Performance of an Aircraft with Parabolic PolarSOLO Horizontal Turn Rate as Function of nV, Let compute 2 24 4 2423 1*2 2 1*22*44 * u ue g V zu u ue g V zuuuue g V zu V g u −      −+−       −      −+−−            −+− = ∂ ∂   χ       −      −+− +− = ∂ ∂ 1*2 1 * 244 4 ue g V zuu u V g u  χ or u 0 u1 1 (u1+u2)/2 u2 ∞ + + 0 - - - - - - -∞ ↑ Max ↓ u∂ ∂ χ χ       −−= 1*2 * e g V z V g MAX  χ Aircraft Turn Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
  115. 115. 115 Performance of an Aircraft with Parabolic PolarSOLO Horizontal Turn Rate as Function of nV, u 0<V 0=V 0>V χ ( )12 * −z V g 1=u1u 2u 2 24 1*2 * u ue g V zu V g −      −+− =  χ 1 * 2 −MAXn uV g 2 2 2 _ 1 ** u u C C V g L MAXL −      MAXL L C C _ * MAX MAXL L n C C _ * LIMIT nMAXLIMIT C MAXL _ MAX MAX L MAXL n n C C V g 1 ** 2 _ − as function of u and as parameter χ V Return to Table of Content Aircraft Turn Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
  116. 116. 116http://forum.keypublishing.com/showthread.php?69698-Canards-and-the-4-Gen-aircraft/page11 Example of Horizontal Turn, versus Mach, Performance of an Aircraft SOLO Aircraft Flight Performance
  117. 117. 117 Mirage 2000 at 15000ft. http://forums.eagle.ru/showthread.php?t=98497 Max sustained rate (at around 6.5G on the 0 Ps line) occurring at around 0.9M/450KCAS looking at around 12.5 deg sec 9G Vc (Max instant. Rate) is around 0.65M/320KCAS looking at 23.5 deg sec SOLO Aircraft Flight Performance
  118. 118. 118http://n631s.blogspot.co.il/2011/03/book-review-boyd-fighter-pilot-who.html Example of Horizontal Turn, versus Mach, Performance of MiG-21 SOLO Aircraft Flight Performance
  119. 119. SOLO 119 Aircraft Flight Performance Comparison of Sustained ( ) Turn Performance of three Fightry Aircrafts F-16, F-4 and MiG-21 at Altitude h = 11 km = 36000 ft 0=V
  120. 120. 120 SOLO Aircraft Flight Performance
  121. 121. 121 The black lines are the F-4D, the dark orange lines are the heavy F-4E, and the blue lines are the lightweight F-4E (same weight as F-4D). Up to low transonic mach numbers and up to medium altitudes, the F-4E is about 7% better than the F-4D (15% better with the same weight). At higher mach numbers, the F-4 doesn't have to pull as much AoA to get the same lift, so the slats actually cause a drag penalty that allows the F- 4D to perform better. For reference, the F-14 is known to turn about 20% better than the unslatted F-4J. So, if the slats made the F-4S turn about 15% better, sustained turn rates would almost be pretty close between the F-14 and F- 4S. The F-4E, being heavier, would still be significantly under the F-14. However, with numbers this close, pilot quality is everything rather than precise performance figures. http://combatace.com/topic/71161-beating-a-dead-horse-us-fighter-turn-performance/ F-4 SOLO Aircraft Flight Performance
  122. 122. 122http://www.worldaffairsboard.com/military-aviation/62863-comparing-fighter-performance-same- generations-important-factor-war-2.html F-15 F-4 SOLO Aircraft Flight Performance
  123. 123. 123 http://www.airliners.net/aviation-forums/military/print.main?id=153429 SOLO Aircraft Flight Performance Return to Table of Content
  124. 124. 124 Corner Speed Maximum Positive Capability (CL) max Maximum Negative Capability (CL) min LoadFactor-n Structural Limit Structural Limit Limit Airspeed Area of Structural Damage of Failure Vmin V n Operational Load Limit Operational Load Limit Structural Load Limit Structural Load Limit Typical Maneuvering Envelope V – n Diagram Maneuvering Envelope: Limits on Normal Load Factor and Allowable Equivalent Airspeed -Structural Factor -Maximum and Minimum allowable Lift Coefficient -Maximum and Minimum Airspeeds -Corner Velocity: Intersection of Maximum Lift Coefficient and Maximum Load Factor SOLO Aircraft Flight Performance
  125. 125. 125 Typical Maneuvering Envelope V – n Diagram Performance of an Aircraft with Parabolic PolarSOLO
  126. 126. 126R.W. Pratt, Ed., “Flight Control Systems, Practical issues in design and implementation”, AIAA Publication, 2000 SOLO Aircraft Flight Performance Return to Table of Content
  127. 127. 127 Air-to-Air Combat Destroy Enemy Aircraft to achieve Air Supremacy in order to prevent the enemy to perform their missions and enable to achieve tactical goals. SOLO See S. Hermelin, “Air Combat”, Presentation, http://www.solohermelin.com
  128. 128. 128 http://forum.warthunder.com/index.php?/topic/110779-taktik-ve-manevralar-hakk%C4%B1ndaki-e% Air-to-Air Combat Before the introduction of all-aspect Air-to-Air Missiles destroying an Enemy Aircraft was effective only from the tail zone of the Enemy Aircraft, so the pilots had to maneuver to reach this position, for the minimum time necessary to activate the guns or launch a Missile. Return to Table of Content
  129. 129. SOLO 129 Energy–Maneuverability Theory Aircraft Flight Performance Energy–maneuverability theory is a model of aircraft performance. It was promulgated by Col. John Boyd, and is useful in describing an aircraft's performance as the total of kinetic and potential energies or aircraft specific energy. It relates the thrust, weight, drag, wing area, and other flight characteristics of an aircraft into a quantitative model. This allows combat capabilities of various aircraft or prospective design trade-offs to be predicted and compared. Colonel John Richard Boyd (1927 –1997) Boyd, a skilled U.S. jet fighter pilot in the Korean War, began developing the theory in the early 1960s. He teamed with mathematician Thomas Christie at Eglin Air Force Base to use the base's high-speed computer to compare the performance envelopes of U.S. and Soviet aircraft from the Korean and Vietnam Wars. They completed a two-volume report on their studies in 1964. Energy Maneuverability came to be accepted within the U.S. Air Force and brought about improvements in the requirements for the F-15 Eagle and later the F-16 Fighting Falcon fighters
  130. 130. 130
  131. 131. 131 Turning Capability Comparison of F4E and MiG21 at Sea Level http://forum.keypublishing.com/showthread.php?96201-fighter-maneuverability- comparison F-4E MiG-21 Aircraft Flight Performance
  132. 132. 132 http://www.aviationforum.org/military-aviation/16335-fighter-maneuverability-comparison.html F4 _Phantom versus MIG 21 MiG-21 SOLO Aircraft Flight Performance
  133. 133. 133 Aircraft Flight Performance
  134. 134. 134
  135. 135. 135
  136. 136. SOLO 136 Aircraft Flight Performance In combat, a pilot is faced with a variety of limiting factors. Some limitations are constant, such as gravity, drag, and thrust-to-weight ratio. Other limitations vary with speed and altitude, such as turn radius, turn rate, and the specific energy of the aircraft. The fighter pilot uses Basic Fighter Maneuvers (BFM) to turn these limitations into tactical advantages. A faster, heavier aircraft may not be able to evade a more maneuverable aircraft in a turning battle, but can often choose to break off the fight and escape by diving or using its thrust to provide a speed advantage. A lighter, more maneuverable aircraft can not usually choose to escape, but must use its smaller turning radius at higher speeds to evade the attacker's guns, and to try to circle around behind the attacker.[13] BFM are a constant series of trade-offs between these limitations to conserve the specific energy state of the aircraft. Even if there is no great difference between the energy states of combating aircraft, there will be as soon as the attacker accelerates to catch up with the defender. Instead of applying thrust, a pilot may use gravity to provide a sudden increase in kinetic energy (speed), by diving, at a cost in the potential energy that was stored in the form of altitude. Similarly, by climbing the pilot can use gravity to provide a decrease in speed, conserving the aircraft's kinetic energy by changing it into altitude. This can help an attacker to prevent an overshoot, while keeping the energy available in case one does occur Energy Management
  137. 137. SOLO 137 Aircraft Flight Performance Energy Management Colonel J. R. Boyd: In an air-to-air battle offensive maneuvering advantage will belong to the pilot who can enter an engagement at a higher energy level and maintain more energy than his opponent while locked into a maneuver and counter-maneuver duel. Maneuvering advantage will also belong to the pilot who enters an air-to-air battle at a lower energy level, but can gain more energy than his opponent during the course of the battle, From a performance standpoint, such an advantage is clear because the pilot with the most energy has a better opportunity to engage or disengage at his own choosing. On the other hand, energy-loss maneuvers can be employed defensively to nullify an attack or to gain a temporary offensive maneuvering position. http://www.ausairpower.net/JRB/fast_transients.pdf “New Conception for Air-to-Air Combat”, J. Boyd, 4 Aug. 1976
  138. 138. 138 http://www.alr-aerospace.ch/Performance_Mission_Analysis.php F-16 SOLO Aircraft Flight Performance
  139. 139. 139Comparative Ps Diagram for Aircraft A and Aircraft B. Two Multi-Role Jet Fighters SOLO Aircraft Flight Performance
  140. 140. 140 http://www.simhq.com/_air/air_065a.html http://en.wikipedia.org/wiki/Lavochkin_La-5 Comparison of Turn Performance of two WWII Fighter Aircraft: Russian Lavockin La5 vs German Messershmitt Bf 109 http://en.wikipedia.org/wiki/Messerschmitt_Bf_109 SOLO Aircraft Flight Performance
  141. 141. 141 Comparison of Turn Performance of two WWII Fighter Aircraft: Russian Lavockin La5 vs German Messershmitt Bf 109 http://en.wikipedia.org/wiki/Lavochkin_La-5http://en.wikipedia.org/wiki/Messerschmitt_Bf_109 http://www.simhq.com/_air/air_065a.html SOLO Aircraft Flight Performance
  142. 142. 142 F-86F Sabre and MiG-15 performance comparison North American F-86 Sabre MiG-15 SOLO Aircraft Flight Performance
  143. 143. 143 Falcon F-16C versus Fulcrum MIG 29, left is w/o afterburner, right is with it, fuel reserves 50% http://forum.keypublishing.com/showthread.php?47529-MiG-29-kontra-F-16-(aerodynamics-) FulcrumMiG-29F-16 SOLO Aircraft Flight Performance
  144. 144. 144 Comparison of Turn Performance of two Modern Fighter Aircraft: Russian MiG-29 vs USA F-16 FulcrumMiG-29 F-16 http://www.simhq.com/_air/air_012a.html http://www.evac-fr.net/forums/lofiversion/index.php?t984.html
  145. 145. 145 Comparison of Turn Performance of two Modern Fighter Aircraft: Russian MiG-29 vs USA F-16 FulcrumMiG-29 F-16 http://www.simhq.com/_air/air_012a.html http://www.evac-fr.net/forums/lofiversion/index.php?t984.html
  146. 146. 146 http://www.simhq.com/_air/air_012a.html Comparison of Turn Performance of two Modern Fighter Aircraft: Russian MiG-29 vs USA F-16 Fulcrum MiG-29 F-16 http://www.evac-fr.net/forums/lofiversion/index.php?t984.html
  147. 147. 147 http://www.simhq.com/_air3/air_117e.html While the turn radius of both aircraft is very similar, the MiG-29 has gained a significant angular advantage. Comparison of Turn Performance of two Modern Fighter Aircraft: Russian MiG-29 vs USA F-16 MiG-29 F-16 SOLO Aircraft Flight Performance
  148. 148. 148 http://www.evac-fr.net/forums/lofiversion/index.php?t984.html Comparison of Turn Performance of two Modern Fighter Aircraft: Russian MiG-29 vs USA F-16 With afterburner, fuel reserves 50% Without afterburner, fuel reserves 50% MiG-29 F-16 SOLO Aircraft Flight Performance
  149. 149. 149 http://forums.eagle.ru/showthread.php?t=30263 SOLO Aircraft Flight Performance
  150. 150. 150 An assessment is made of the applicability of Energy Maneuverability techniques (EM) to flight path optimization. A series of minimum time and fuel maneuvers using the F-4C aircraft were established to progressively violate the assumptions inherent in the EM program and comparisons were made with the Air Force Flight Dynamics Laboratory's (AFFDL) Three-Degree-of-Freedom Trajectory Optimization Program and a point mass option of the Six-Degree-of-Freedom flight path program. It was found the EM results were always optimistic in the value of the payoff functions with the optimism increasing as the percentage of the maneuver involving constant energy transitions Increases. For the minimum time paths the resulting optimism was less than 27%f1o r the maneuvers where the constant energy percentage was less than 35.',", followed by a rather steeply rising curve approaching in the limit 100% error for paths which are comprised entirely of constant energy transitions. Two new extensions are developed in the report; the first is a varying throttle technique for use on minimum fuel paths and the second a turning analysis that can be applied in conjunction with a Rutowski path. Both extensions were applied to F-4C maneuvers in conjunction with 'Rutowski’s paths generated from the Air Force Armament Laboratory's Energy Maneuverability program. The study findings are that energy methods offer a tool especially useful in the early stages of preliminary design and functional performance studies where rapid results with reasonable accuracy are adequate. If the analyst uses good judgment in its applications to maneuvers the results provide a good qualitative insight for comparative purposes. The paths should not, however, be used as a source of maneuver design or flight schedule without verification especially on relatively dynamic maneuvers where the accuracy and optimality of the method decreases. David T. Johnson, “Evaluation of Energy Maneuverability Procedures in Aircraft Flight Path Optimization and Performance Estimation”, November 1972, AFFDL-TR-72-53 SOLO Aircraft Flight Performance
  151. 151. 151 Lockheed F-104 Starfighter SOLO Aircraft Flight Performance
  152. 152. 152 Typical Ps Plot for Lockheed F-104 Starfighter Lockheed F-104 Starfighter SOLO Aircraft Flight Performance
  153. 153. 153 SOLO Aircraft Flight Performance F-104 Flight Envelope Lockheed F-104 Starfighter
  154. 154. 154F-104A flight envelope Lockheed F-104 Starfighter SOLO Aircraft Flight Performance Return to Table of Content
  155. 155. 155 http://defence.pk/threads/design-characteristics-of-canard-non-canard-fighters.178592/ SOLO Aircraft Flight Performance Aircraft Combat Performance Comparison
  156. 156. 156 http://defence.pk/threads/design-characteristics-of-canard-non-canard-fighters.178592/ SOLO Aircraft Flight Performance
  157. 157. 157 http://img138.imageshack.us/img138/4146/image4u.jpg SOLO Aircraft Flight Performance
  158. 158. 158 https://s3-eu-west-1.amazonaws.com/rbi-blogs/wp-content/uploads/mt/flightglobalweb/blogs/the-dewline/assets_c/2011/05/chart%20 Aircraft Combat Performance Comparison SOLO Aircraft Flight Performance Return to Table of Content
  159. 159. 159 Supermaneuverability is defined as the ability of an aircraft to perform high alpha maneuvers that are impossible for most aircraft is evidence of the aircraft's supermaneuverability. Such maneuvers include Pugachev's Cobra and the Herbst maneuver (also known as the "J-turn"). Some aircraft are capable of performing Pugachev's Cobra without the aid of features that normally provide post-stall maneuvering such as thrust vectoring. Advanced fourth generation fighters such as the Su-27, MiG-29 along with their variants have been documented as capable of performing this maneuver using normal, non-thrust vectoring engines. The ability of these aircraft to perform this maneuver is based in inherent instability like that of the F-16; the MiG-29 and Su-27 families of jets are designed for desirable post-stall behavior. Thus, when performing a maneuver like Pugachev's Cobra the aircraft will stall as the nose pitches up and the airflow over the wing becomes separated, but naturally nose down even from a partially inverted position, allowing the pilot to recover complete control. http://en.wikipedia.org/wiki/Supermaneuverability Supermaneuverability SOLO Aircraft Flight Performance
  160. 160. 160 SOLO Aircraft Flight Performance
  161. 161. 161 Sukhoi Su-30MKI SOLO Aircraft Flight Performance http://vayu-sena.tripod.com/interview-simonov1.html
  162. 162. 162 SOLO Aircraft Flight Performance The Herbst maneuver or "J-Turn" named after Wolfgang Herbst is the only thrust vector post stall maneuver that can be used in actual combat but very few air frames can sustain the stress of this violent maneuver. Herbst Maneuver http://en.wikipedia.org/wiki/Herbst_maneuver Return to Table of Content
  163. 163. 163 Constraint Analysis SOLO Aircraft Flight Performance The Performance Requirements can be translated into functional relationship between the Thrust-to- Weight or Thrust Loading at Sea Level Takeoff (TSL/WTO) and the Wing Loading at Takeoff (WTO/S). The keys to the development are •Reasonable assumption hor Aircraft Lift-to-Drag Polar. •The low sensibility of Engine Thrust with Flight Altitude and Mach Number. The minimum of TSL/WTO as functions of WTO/S are required for: •Takeoff from a Runway of a specified length. •Flight at a given Altitude and Required Speed. •Climb at a Required Speed. •Turn at a given Altitude, Speed and a required Rate. •Acceleration capability at constant Altitude. •Landing without reverse Thrust on a Runway of a given length.
  164. 164. 164 Energy per unit mass E Let define Energy per unit mass E: g V hE 2 : 2 += Let differentiate this equation: ( ) ( ) W VDT W VDT W DT g g V V g VV hEps − ≈ − =            − − +=+== α γ α γ cos sin cos sin:   define 10 ≤<= ββ TOWW WTO – Take-off Weight ( ) ( ) ( ) SLThThhT αα === 0 TSL – Sea Level Thrust V p W D W T s += Load Factor W CSq W L n L ==: SOLO Aircraft Flight Performance TOL W Sq n W Sq n C β==       += V p W D W T s TO SL α β Constraint Analysis
  165. 165. 165 SOLO Aircraft Flight Performance General Mission Description of a Typical Fighter Aircraft 10: ≤<= ββ TOW W WTO – Take-off Weight W – Aircraft Weight during Flight Constraint Analysis
  166. 166. 166 Assume a General Lift-to-Drag Polar Relationship Total DragRD CSqCSqRD +=+ D, CD - Clean Aircraft Drag and Drag Coefficient R, CR – Additional Aircraft Drag and Additional Drag Coefficient caused by External Stores, Bracking Parachute, Flaps, External Hardware 02 2 102 2 1 D TOTO DLLD C S W q n K S W q n KCCKCKC +      +      =++= ββ TOL W Sq n W Sq n C β== ( )       ++= V p CC W Sq W T s RD TOTO SL βα β         +         ++      +      = V p CC S W q n K S W q n K W Sq W T s DRD TOTO TOTO SL 02 2 1 ββ βα β SOLO Aircraft Flight Performance Constraint Analysis
  167. 167. 167 ( )WLn td Vd td hd ==== ,1,0,0 Case 1: Constant Altitude/Speed Cruise (ps = 0) Given:                   + ++      = S W q CC K S W q K W T TO DRDTO TO SL β β α β 0 21 We obtain: We can see that TSL/WTO → ∞ for WTO/S → 0 and WTO/S→∞, therefore a minimum exist. By differentiating TSL/WTO with respect to WTO/S and setting the result equal to zero, we obtain: 1 0 /min K CCq S W DRD WT TO + =      β ( )[ ]210 min 2 KKCC W T DRD TO SL ++=      α β Lift DragThrust Weight SOLO Aircraft Flight Performance Constraint Analysis
  168. 168. 168M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring” Case 1: Constant Altitude/Speed Cruise (ps = 0) SOLO Aircraft Flight Performance Constraint Analysis
  169. 169. 169 ( )WLn td hd ≈≈= ,1,0 Case 2: Constant Speed Climb (ps = dh/dt) Given: We can see that TSL/WTO → ∞ for WTO/S → 0 and WTO/S→∞, therefore a minimum exist. By differentiating TSL/WTO with respect to WTO/S and setting the result equal to zero, we obtain: 1 0 /min K CCq S W DRD WT TO + =      β ( )       +++=      td hd V KKCC W T DRD TO SL 1 2 210 min α β We obtain:             +       + ++      = td hd V S W q CC K S W q K W T TO DRDTO TO SL 10 21 β β α β SOLO Aircraft Flight Performance
  170. 170. 170 ,1,0,0 ,, >== n td hd td Vd givenhVgivenhV Case 3: Constant Altitude/Speed Turn (ps = 0) Given: We can see that TSL/WTO → ∞ for WTO/S → 0 and WTO/S→∞, therefore a minimum exist. By differentiating TSL/WTO with respect to WTO/S and setting the result equal to zero, we obtain: 1 0 /min K CC n q S W DRD WT TO + =      β ( )[ ]210 min 2 KKCC n W T DRD TO SL ++=      α β We obtain:             +       + ++      = td hd V S W q CC nK S W q nK W T TO DRDTO TO SL 10 2 2 1 β β α β 2 0 2 2 0 11       +=      Ω += cRg V g V n SOLO Aircraft Flight Performance Constraint Analysis
  171. 171. 171 ( )WLn td hd givenh === ,1,0 Case 4: Horizontal Acceleration (ps = (V/g0) (dV/dt) ) Given: We obtain:             +       + ++      = td Vd g S W q CC K S W q K W T TO DRDTO TO SL 0 0 21 1 β β α β SOLO Aircraft Flight Performance Lift DragThrust Weight This can be rearranged to give:                   + ++      = S W q CC K S W q K W T td Vd g TO DRDTO TO SL β β β α 0 21 0 1 Constraint Analysis
  172. 172. 172 ( )WLn td hd givenh === ,1,0 Case 4: Horizontal Acceleration (ps = (V/g0) (dV/dt) ) (continue – 1) Given: SOLO Aircraft Flight Performance Lift DragThrust Weight We obtain:                   + ++      = S W q CC K S W q K W T td Vd g TO DRDTO TO SL β β β α 0 21 0 1 This equation can be integrated from initial velocity V0 to final velocity Vf, from initial t0 to final tf times. ( )∫=− fV V s f Vp VdV g tt 0 0 0 1 where                                 + ++      −= S W q CC K S W q K W T Vp TO DRDTO TO SL s β β β α 0 21 The solutions of TSL/WTO for different WTO/S are obtained iteratively. Constraint Analysis
  173. 173. 173http://elpdefensenews.blogspot.co.il/2013_04_01_archive.html Constraint Analysis SOLO Aircraft Flight Performance
  174. 174. 174 0= givenh td hd Case 5: Takeoff (sg given and TSL >> (D+R) ) Given: SOLO Aircraft Flight Performance Ground Run V = 0 sg sTO sr str V TO Rotation Transition sCL θ CL htr hobs R Start from:  ( ) TO T SL s W VRDT td Vd g V td hd p SL β α α         +− ≈+= ≈    0        == TO SL V W Tg td sd sd Vd td Vd β α 0 /1 VdV T W g sd SL TO       = 0α β max,2 2 0max, 2 0 2 1 2 1 L TO TO LstallstallTO CS k V CSVLW ρρβ === The take-off velocity VTO is VTO = kTO Vstall Where Vstall is the minimum velocity at at which Lift equals weight and kTO ≈ 1.1 to 1.2:       == S W C kV k V TO L TOstall TO TO max,0 22 2 2 22 ρ β Integration from: s = 0 to s = sg V = 0 to V = VTO 2 2 0 TO SL TO g V T W g s       = α β sg – Ground Run Constraint Analysis
  175. 175. 175 Case 5: Takeoff (sg given and TSL >> (D+R) ) (continue – 1) SOLO Aircraft Flight Performance Ground Run V = 0 sg sTO sr str V TO Rotation Transition sCL θ CL htr hobs R 2 2 0 TO SL TO g V T W g s       = α β       == S W C kV k V TO L TOstall TO TO max,0 22 2 2 22 ρ β       = S W Cgs k W T TO Lg TO TO SL max,00 22 ρ β α β We obtained: from which:             = S W C k T W g s TO L TO SL TO g max,0 2 0 ρ β α β We have a Linear Relation between TSL/WTO and WTO/S Constraint Analysis
  176. 176. 176 Case 6: Landing SOLO Aircraft Flight Performance where ( ) ( )       −= −−= µ µ β ρ W T gc CC SW g a grLgrD TO 0 ,, : /2 : cab Va a ca Va a touchdown 4 2 : 4 2 : 2 1 1 − = − =cVa cVa a sg + + −= 2 2 2 1 ln 2 1       − − ⋅ + + − = 1 2 2 1 1 1 1 1 ln 4 1 a a a a ca tg Ground Run Phase We found Ground Run sgr Transition Airborne Phase Total Landing Distance Float sf Flare stGlide sg γ hg hf Touchdown 2 0 VCVBTT ++= For a given value of sg , there is only one value of WTO/S that satisfies this equation. ( )gTO sfSW =/ This constraint is represented in the TSL/WTO versus WTO/S plane as a vertical line, at WTO/S corresponding to the required sg. Constraint Analysis
  177. 177. 177Constraint Diagram SOLO Aircraft Flight Performance                   + ++      = S W q CC K S W q K W T TO DRDTO TO SL β β α β 0 21             +       + ++      = td hd V S W q CC nK S W q nK W T TO DRDTO TO SL 10 2 2 1 β β α β       = S W Cgs k W T TO Lg TO TO SL max,00 22 ρ β α β ( )gTO sfSW =/ Constraint Analysis
  178. 178. 178 Comparison of Fighter Aircraft Propulsion Systems SOLO
  179. 179. 179 Comparison of Fighter Aircraft Propulsion Systems SOLO
  180. 180. 180 SOLO Aircraft Flight Performance Composite Thrust Loading versus Wing Loading – for different Aircraft Constraint Analysis
  181. 181. 181 Constraint Diagram for F-16 SOLO Aircraft Flight Performance Constraint Analysis Return to Table of Content
  182. 182. 182 Weapon System Agility Weapon System Agility Return to Table of Content

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Fighter Aircraft Performance, Part II of two, describes the parameters that affect aircraft performance. For comments please contact me at solo.hermelin@gmail.com. For more presentations on different subjects visit my website at http://www.solohermelin.com.

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