SlideShare ist ein Scribd-Unternehmen logo

14 fixed wing fighter aircraft- flight performance - ii

•Als PPT, PDF herunterladen•
•
Solo Hermelin
Solo Hermelin

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.

Wissenschaft
1 von 257
Fixed Wing Fighter Aircraft Flight Performance Part II SOLO HERMELIN Updated: 04.12.12 28.02.15 1 http://www.solohermelin.com
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
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
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
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
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 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 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 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 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 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 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 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 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 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 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 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 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 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 // −− ≈
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 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 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 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 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 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 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 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 H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35 Return to Table of Content
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 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 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
32M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring” Comparison of Takeoff Weight and Empty Weight of different Aircraft
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
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 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 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 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 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 Drag Characteristics Fixed Wing Fighter Aircraft Flight Performance SOLO
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00-80T-80 1-1-1965, pg. 35 Fixed Wing Fighter Aircraft Flight Performance
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 H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00-80T-80 1-1-1965, pg. 35 Fixed Wing Fighter Aircraft Flight Performance
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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
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 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 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 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 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
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
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 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 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.
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 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 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
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
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 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 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 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 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 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 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 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
98 Vertical Plan Trajectory (σ = 0) SOLO Prof. Earll Murman, “Introduction to Aircraft Performance and Static Stability”, September 18, 2003
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
100Maneuverability Diagram R V =:χ 1'2 −= n V g χ Horizontal Plan Trajectory
101 F-5E Turn Performance Horizontal Plan Trajectory
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 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 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 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 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 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 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 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 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 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 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 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 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 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
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 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
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
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 SOLO Aircraft Flight Performance
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
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 http://www.airliners.net/aviation-forums/military/print.main?id=153429 SOLO Aircraft Flight Performance Return to Table of Content
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 Typical Maneuvering Envelope V – n Diagram Performance of an Aircraft with Parabolic PolarSOLO
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 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 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
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
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 http://www.aviationforum.org/military-aviation/16335-fighter-maneuverability-comparison.html F4 _Phantom versus MIG 21 MiG-21 SOLO Aircraft Flight Performance
133 Aircraft Flight Performance
134
135
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
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 http://www.alr-aerospace.ch/Performance_Mission_Analysis.php F-16 SOLO Aircraft Flight Performance
139Comparative Ps Diagram for Aircraft A and Aircraft B. Two Multi-Role Jet Fighters SOLO Aircraft Flight Performance
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 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 F-86F Sabre and MiG-15 performance comparison North American F-86 Sabre MiG-15 SOLO Aircraft Flight Performance
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 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 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 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 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 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 http://forums.eagle.ru/showthread.php?t=30263 SOLO Aircraft Flight Performance
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 Lockheed F-104 Starfighter SOLO Aircraft Flight Performance
152 Typical Ps Plot for Lockheed F-104 Starfighter Lockheed F-104 Starfighter SOLO Aircraft Flight Performance
153 SOLO Aircraft Flight Performance F-104 Flight Envelope Lockheed F-104 Starfighter
154F-104A flight envelope Lockheed F-104 Starfighter SOLO Aircraft Flight Performance Return to Table of Content
155 http://defence.pk/threads/design-characteristics-of-canard-non-canard-fighters.178592/ SOLO Aircraft Flight Performance Aircraft Combat Performance Comparison
156 http://defence.pk/threads/design-characteristics-of-canard-non-canard-fighters.178592/ SOLO Aircraft Flight Performance
157 http://img138.imageshack.us/img138/4146/image4u.jpg SOLO Aircraft Flight Performance
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 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 SOLO Aircraft Flight Performance
161 Sukhoi Su-30MKI SOLO Aircraft Flight Performance http://vayu-sena.tripod.com/interview-simonov1.html
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 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 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 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 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 ( )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
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 ( )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 ,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 ( )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 ( )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
173http://elpdefensenews.blogspot.co.il/2013_04_01_archive.html Constraint Analysis SOLO Aircraft Flight Performance
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 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 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
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 Comparison of Fighter Aircraft Propulsion Systems SOLO
179 Comparison of Fighter Aircraft Propulsion Systems SOLO
180 SOLO Aircraft Flight Performance Composite Thrust Loading versus Wing Loading – for different Aircraft Constraint Analysis
181 Constraint Diagram for F-16 SOLO Aircraft Flight Performance Constraint Analysis Return to Table of Content
182 Weapon System Agility Weapon System Agility Return to Table of Content
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii
14 fixed wing fighter aircraft- flight performance - ii

Empfohlen

7 air-to-air combat
7 air-to-air combat7 air-to-air combat
7 air-to-air combatSolo Hermelin
 
13 fixed wing fighter aircraft- flight performance - i
13 fixed wing fighter aircraft- flight performance - i13 fixed wing fighter aircraft- flight performance - i
13 fixed wing fighter aircraft- flight performance - iSolo Hermelin
 
10 fighter aircraft avionics - part iii
10 fighter aircraft avionics - part iii10 fighter aircraft avionics - part iii
10 fighter aircraft avionics - part iiiSolo Hermelin
 
Structural detailing of fuselage of aeroplane /aircraft.
Structural detailing of fuselage of aeroplane /aircraft.Structural detailing of fuselage of aeroplane /aircraft.
Structural detailing of fuselage of aeroplane /aircraft.PriyankaKg4
 
High Speed Aerodynamics
High Speed AerodynamicsHigh Speed Aerodynamics
High Speed Aerodynamicslccmechanics
 
Aircraft landing gear system
Aircraft landing gear systemAircraft landing gear system
Aircraft landing gear systemKrishikesh Singh
 
Introduction to Aircraft Structural Design
Introduction to Aircraft Structural DesignIntroduction to Aircraft Structural Design
Introduction to Aircraft Structural DesignSuthan Rajendran
 
Pressure Altitude | Flight Mechanics | GATE Aerospace
Pressure Altitude | Flight Mechanics | GATE AerospacePressure Altitude | Flight Mechanics | GATE Aerospace
Pressure Altitude | Flight Mechanics | GATE AerospaceAge of Aerospace
 

Empfohlen

7 air-to-air combat
7 air-to-air combat7 air-to-air combat
7 air-to-air combatSolo Hermelin
 
13 fixed wing fighter aircraft- flight performance - i
13 fixed wing fighter aircraft- flight performance - i13 fixed wing fighter aircraft- flight performance - i
13 fixed wing fighter aircraft- flight performance - iSolo Hermelin
 
10 fighter aircraft avionics - part iii
10 fighter aircraft avionics - part iii10 fighter aircraft avionics - part iii
10 fighter aircraft avionics - part iiiSolo Hermelin
 
Structural detailing of fuselage of aeroplane /aircraft.
Structural detailing of fuselage of aeroplane /aircraft.Structural detailing of fuselage of aeroplane /aircraft.
Structural detailing of fuselage of aeroplane /aircraft.PriyankaKg4
 
High Speed Aerodynamics
High Speed AerodynamicsHigh Speed Aerodynamics
High Speed Aerodynamicslccmechanics
 
Aircraft landing gear system
Aircraft landing gear systemAircraft landing gear system
Aircraft landing gear systemKrishikesh Singh
 
Introduction to Aircraft Structural Design
Introduction to Aircraft Structural DesignIntroduction to Aircraft Structural Design
Introduction to Aircraft Structural DesignSuthan Rajendran
 
Pressure Altitude | Flight Mechanics | GATE Aerospace
Pressure Altitude | Flight Mechanics | GATE AerospacePressure Altitude | Flight Mechanics | GATE Aerospace
Pressure Altitude | Flight Mechanics | GATE AerospaceAge of Aerospace
 
Lockheed Martin F35 Lightning II(Aerodynamics Presentation)
Lockheed Martin F35 Lightning II(Aerodynamics Presentation)Lockheed Martin F35 Lightning II(Aerodynamics Presentation)
Lockheed Martin F35 Lightning II(Aerodynamics Presentation)SYEDMOEEDHUSSAIN1
 
Flight control systems
Flight control systemsFlight control systems
Flight control systemsTalha Karim
 
Airspeeds | Q & A | Question Analysis | Flight Mechanics | GATE Aerospace
Airspeeds | Q & A | Question Analysis | Flight Mechanics | GATE AerospaceAirspeeds | Q & A | Question Analysis | Flight Mechanics | GATE Aerospace
Airspeeds | Q & A | Question Analysis | Flight Mechanics | GATE AerospaceAge of Aerospace
 
Sideslip | Flight Mechanics | GATE Aerospace
Sideslip | Flight Mechanics | GATE AerospaceSideslip | Flight Mechanics | GATE Aerospace
Sideslip | Flight Mechanics | GATE AerospaceAge of Aerospace
 
032 aeroplane performance
032 aeroplane performance032 aeroplane performance
032 aeroplane performancechococrispis37
 
8 fighter aircraft avionics-part i
8 fighter aircraft avionics-part i8 fighter aircraft avionics-part i
8 fighter aircraft avionics-part iSolo Hermelin
 
Aircraft Systems and Instruments
Aircraft Systems and InstrumentsAircraft Systems and Instruments
Aircraft Systems and InstrumentsVarun Karthikeyan
 
Hydraulic system of landing gear in Aircraft
Hydraulic system of landing gear in Aircraft Hydraulic system of landing gear in Aircraft
Hydraulic system of landing gear in Aircraft Jordan Tractor and Equipment Co.
 
6 computing gunsight, hud and hms
6 computing gunsight, hud and hms6 computing gunsight, hud and hms
6 computing gunsight, hud and hmsSolo Hermelin
 
Principles of flight
Principles of flightPrinciples of flight
Principles of flightwillmac1wm
 
直升机飞行力学 Helicopter dynamics chapter 1
直升机飞行力学 Helicopter dynamics chapter 1直升机飞行力学 Helicopter dynamics chapter 1
直升机飞行力学 Helicopter dynamics chapter 1Falevai
 
Angle of attack | Flight Mechanics | GATE Aerospace
Angle of attack | Flight Mechanics | GATE AerospaceAngle of attack | Flight Mechanics | GATE Aerospace
Angle of attack | Flight Mechanics | GATE AerospaceAge of Aerospace
 
Advanced structures - wing section, beams, bending, shear flow and shear center
Advanced structures - wing section, beams, bending, shear flow and shear centerAdvanced structures - wing section, beams, bending, shear flow and shear center
Advanced structures - wing section, beams, bending, shear flow and shear centerRohan M Ganapathy
 
Helicopter vibration reduction techniques
Helicopter vibration reduction techniquesHelicopter vibration reduction techniques
Helicopter vibration reduction techniquesAnyket Mykr
 
helicopter-aerodynamics
helicopter-aerodynamics helicopter-aerodynamics
helicopter-aerodynamicsravi malla thakuri
 
Atmosphere: Properties and Standard Atmosphere | Flight Mechanics | GATE Aero...
Atmosphere: Properties and Standard Atmosphere | Flight Mechanics | GATE Aero...Atmosphere: Properties and Standard Atmosphere | Flight Mechanics | GATE Aero...
Atmosphere: Properties and Standard Atmosphere | Flight Mechanics | GATE Aero...Age of Aerospace
 
Fighter aircraft design adp 1
Fighter aircraft design adp 1Fighter aircraft design adp 1
Fighter aircraft design adp 1Dudekula Jamal
 
Final Report -Aircraft Design
Final Report -Aircraft DesignFinal Report -Aircraft Design
Final Report -Aircraft DesignThomas Spisak
 
Missile Technology
Missile TechnologyMissile Technology
Missile TechnologyAshwin Baindur
 
Aircraft basics
Aircraft basicsAircraft basics
Aircraft basicsRohit Ranjan
 
Aerodynamics part iii
Aerodynamics part iiiAerodynamics part iii
Aerodynamics part iiiSolo Hermelin
 
Stabilization of linear time invariant systems, Factorization Approach
Stabilization of linear time invariant systems, Factorization ApproachStabilization of linear time invariant systems, Factorization Approach
Stabilization of linear time invariant systems, Factorization ApproachSolo Hermelin
 

Weitere ähnliche Inhalte

Was ist angesagt?

Lockheed Martin F35 Lightning II(Aerodynamics Presentation)
Lockheed Martin F35 Lightning II(Aerodynamics Presentation)Lockheed Martin F35 Lightning II(Aerodynamics Presentation)
Lockheed Martin F35 Lightning II(Aerodynamics Presentation)SYEDMOEEDHUSSAIN1
 
Flight control systems
Flight control systemsFlight control systems
Flight control systemsTalha Karim
 
Airspeeds | Q & A | Question Analysis | Flight Mechanics | GATE Aerospace
Airspeeds | Q & A | Question Analysis | Flight Mechanics | GATE AerospaceAirspeeds | Q & A | Question Analysis | Flight Mechanics | GATE Aerospace
Airspeeds | Q & A | Question Analysis | Flight Mechanics | GATE AerospaceAge of Aerospace
 
Sideslip | Flight Mechanics | GATE Aerospace
Sideslip | Flight Mechanics | GATE AerospaceSideslip | Flight Mechanics | GATE Aerospace
Sideslip | Flight Mechanics | GATE AerospaceAge of Aerospace
 
032 aeroplane performance
032 aeroplane performance032 aeroplane performance
032 aeroplane performancechococrispis37
 
8 fighter aircraft avionics-part i
8 fighter aircraft avionics-part i8 fighter aircraft avionics-part i
8 fighter aircraft avionics-part iSolo Hermelin
 
Aircraft Systems and Instruments
Aircraft Systems and InstrumentsAircraft Systems and Instruments
Aircraft Systems and InstrumentsVarun Karthikeyan
 
6 computing gunsight, hud and hms
6 computing gunsight, hud and hms6 computing gunsight, hud and hms
6 computing gunsight, hud and hmsSolo Hermelin
 
Principles of flight
Principles of flightPrinciples of flight
Principles of flightwillmac1wm
 
直升机飞行力学 Helicopter dynamics chapter 1
直升机飞行力学 Helicopter dynamics chapter 1直升机飞行力学 Helicopter dynamics chapter 1
直升机飞行力学 Helicopter dynamics chapter 1Falevai
 
Angle of attack | Flight Mechanics | GATE Aerospace
Angle of attack | Flight Mechanics | GATE AerospaceAngle of attack | Flight Mechanics | GATE Aerospace
Angle of attack | Flight Mechanics | GATE AerospaceAge of Aerospace
 
Advanced structures - wing section, beams, bending, shear flow and shear center
Advanced structures - wing section, beams, bending, shear flow and shear centerAdvanced structures - wing section, beams, bending, shear flow and shear center
Advanced structures - wing section, beams, bending, shear flow and shear centerRohan M Ganapathy
 
Helicopter vibration reduction techniques
Helicopter vibration reduction techniquesHelicopter vibration reduction techniques
Helicopter vibration reduction techniquesAnyket Mykr
 
Atmosphere: Properties and Standard Atmosphere | Flight Mechanics | GATE Aero...
Atmosphere: Properties and Standard Atmosphere | Flight Mechanics | GATE Aero...Atmosphere: Properties and Standard Atmosphere | Flight Mechanics | GATE Aero...
Atmosphere: Properties and Standard Atmosphere | Flight Mechanics | GATE Aero...Age of Aerospace
 
Fighter aircraft design adp 1
Fighter aircraft design adp 1Fighter aircraft design adp 1
Fighter aircraft design adp 1Dudekula Jamal
 
Final Report -Aircraft Design
Final Report -Aircraft DesignFinal Report -Aircraft Design
Final Report -Aircraft DesignThomas Spisak
 

Was ist angesagt? (20)

Lockheed Martin F35 Lightning II(Aerodynamics Presentation)
Lockheed Martin F35 Lightning II(Aerodynamics Presentation)Lockheed Martin F35 Lightning II(Aerodynamics Presentation)
Lockheed Martin F35 Lightning II(Aerodynamics Presentation)
 
Flight control systems
Flight control systemsFlight control systems
Flight control systems
 
Airspeeds | Q & A | Question Analysis | Flight Mechanics | GATE Aerospace
Airspeeds | Q & A | Question Analysis | Flight Mechanics | GATE AerospaceAirspeeds | Q & A | Question Analysis | Flight Mechanics | GATE Aerospace
Airspeeds | Q & A | Question Analysis | Flight Mechanics | GATE Aerospace
 
Sideslip | Flight Mechanics | GATE Aerospace
Sideslip | Flight Mechanics | GATE AerospaceSideslip | Flight Mechanics | GATE Aerospace
Sideslip | Flight Mechanics | GATE Aerospace
 
032 aeroplane performance
032 aeroplane performance032 aeroplane performance
032 aeroplane performance
 
8 fighter aircraft avionics-part i
8 fighter aircraft avionics-part i8 fighter aircraft avionics-part i
8 fighter aircraft avionics-part i
 
Aircraft Systems and Instruments
Aircraft Systems and InstrumentsAircraft Systems and Instruments
Aircraft Systems and Instruments
 
Hydraulic system of landing gear in Aircraft
Hydraulic system of landing gear in Aircraft Hydraulic system of landing gear in Aircraft
Hydraulic system of landing gear in Aircraft
 
6 computing gunsight, hud and hms
6 computing gunsight, hud and hms6 computing gunsight, hud and hms
6 computing gunsight, hud and hms
 
Principles of flight
Principles of flightPrinciples of flight
Principles of flight
 
直升机飞行力学 Helicopter dynamics chapter 1
直升机飞行力学 Helicopter dynamics chapter 1直升机飞行力学 Helicopter dynamics chapter 1
直升机飞行力学 Helicopter dynamics chapter 1
 
Angle of attack | Flight Mechanics | GATE Aerospace
Angle of attack | Flight Mechanics | GATE AerospaceAngle of attack | Flight Mechanics | GATE Aerospace
Angle of attack | Flight Mechanics | GATE Aerospace
 
Advanced structures - wing section, beams, bending, shear flow and shear center
Advanced structures - wing section, beams, bending, shear flow and shear centerAdvanced structures - wing section, beams, bending, shear flow and shear center
Advanced structures - wing section, beams, bending, shear flow and shear center
 
Helicopter vibration reduction techniques
Helicopter vibration reduction techniquesHelicopter vibration reduction techniques
Helicopter vibration reduction techniques
 
helicopter-aerodynamics
helicopter-aerodynamics helicopter-aerodynamics
helicopter-aerodynamics
 
Atmosphere: Properties and Standard Atmosphere | Flight Mechanics | GATE Aero...
Atmosphere: Properties and Standard Atmosphere | Flight Mechanics | GATE Aero...Atmosphere: Properties and Standard Atmosphere | Flight Mechanics | GATE Aero...
Atmosphere: Properties and Standard Atmosphere | Flight Mechanics | GATE Aero...
 
Fighter aircraft design adp 1
Fighter aircraft design adp 1Fighter aircraft design adp 1
Fighter aircraft design adp 1
 
Final Report -Aircraft Design
Final Report -Aircraft DesignFinal Report -Aircraft Design
Final Report -Aircraft Design
 
Missile Technology
Missile TechnologyMissile Technology
Missile Technology
 
Aircraft basics
Aircraft basicsAircraft basics
Aircraft basics
 

Andere mochten auch

Aerodynamics part iii
Aerodynamics part iiiAerodynamics part iii
Aerodynamics part iiiSolo Hermelin
 
Stabilization of linear time invariant systems, Factorization Approach
Stabilization of linear time invariant systems, Factorization ApproachStabilization of linear time invariant systems, Factorization Approach
Stabilization of linear time invariant systems, Factorization ApproachSolo Hermelin
 
2 aircraft flight instruments
2 aircraft flight instruments2 aircraft flight instruments
2 aircraft flight instrumentsSolo Hermelin
 
5 introduction to quantum mechanics
5 introduction to quantum mechanics5 introduction to quantum mechanics
5 introduction to quantum mechanicsSolo Hermelin
 
11 fighter aircraft avionics - part iv
11 fighter aircraft avionics - part iv11 fighter aircraft avionics - part iv
11 fighter aircraft avionics - part ivSolo Hermelin
 
3 modern aircraft cutaway
3 modern aircraft cutaway3 modern aircraft cutaway
3 modern aircraft cutawaySolo Hermelin
 
9 fighter aircraft avionics-part ii
9 fighter aircraft avionics-part ii9 fighter aircraft avionics-part ii
9 fighter aircraft avionics-part iiSolo Hermelin
 
12 performance of an aircraft with parabolic polar
12 performance of an aircraft with parabolic polar12 performance of an aircraft with parabolic polar
12 performance of an aircraft with parabolic polarSolo Hermelin
 
1 radar basic -part i 1
1 radar basic -part i 11 radar basic -part i 1
1 radar basic -part i 1Solo Hermelin
 
1 radar basic - part ii
1 radar basic - part ii1 radar basic - part ii
1 radar basic - part iiSolo Hermelin
 
Aerodynamics part ii
Aerodynamics part iiAerodynamics part ii
Aerodynamics part iiSolo Hermelin
 
4 navigation systems
4 navigation systems4 navigation systems
4 navigation systemsSolo Hermelin
 
On Aircraft and Craft
On Aircraft and CraftOn Aircraft and Craft
On Aircraft and CraftDane Petersen
 
Aircraft Performance
Aircraft PerformanceAircraft Performance
Aircraft PerformanceAlline Louis
 
Aircraft Performance: Part I
Aircraft Performance: Part IAircraft Performance: Part I
Aircraft Performance: Part IMohammad Tawfik
 

Andere mochten auch (20)

Aerodynamics part iii
Aerodynamics part iiiAerodynamics part iii
Aerodynamics part iii
 
Stabilization of linear time invariant systems, Factorization Approach
Stabilization of linear time invariant systems, Factorization ApproachStabilization of linear time invariant systems, Factorization Approach
Stabilization of linear time invariant systems, Factorization Approach
 
Aerodynamics part i
Aerodynamics part iAerodynamics part i
Aerodynamics part i
 
2 aircraft flight instruments
2 aircraft flight instruments2 aircraft flight instruments
2 aircraft flight instruments
 
5 introduction to quantum mechanics
5 introduction to quantum mechanics5 introduction to quantum mechanics
5 introduction to quantum mechanics
 
11 fighter aircraft avionics - part iv
11 fighter aircraft avionics - part iv11 fighter aircraft avionics - part iv
11 fighter aircraft avionics - part iv
 
3 modern aircraft cutaway
3 modern aircraft cutaway3 modern aircraft cutaway
3 modern aircraft cutaway
 
9 fighter aircraft avionics-part ii
9 fighter aircraft avionics-part ii9 fighter aircraft avionics-part ii
9 fighter aircraft avionics-part ii
 
12 performance of an aircraft with parabolic polar
12 performance of an aircraft with parabolic polar12 performance of an aircraft with parabolic polar
12 performance of an aircraft with parabolic polar
 
1 radar basic -part i 1
1 radar basic -part i 11 radar basic -part i 1
1 radar basic -part i 1
 
1 radar basic - part ii
1 radar basic - part ii1 radar basic - part ii
1 radar basic - part ii
 
Aerodynamics part ii
Aerodynamics part iiAerodynamics part ii
Aerodynamics part ii
 
4 navigation systems
4 navigation systems4 navigation systems
4 navigation systems
 
3 earth atmosphere
3 earth atmosphere3 earth atmosphere
3 earth atmosphere
 
On Aircraft and Craft
On Aircraft and CraftOn Aircraft and Craft
On Aircraft and Craft
 
Prisms
Prisms Prisms
Prisms
 
Aircraft Performance
Aircraft PerformanceAircraft Performance
Aircraft Performance
 
Lens
LensLens
Lens
 
Aircraft Performance: Part I
Aircraft Performance: Part IAircraft Performance: Part I
Aircraft Performance: Part I
 
2 estimators
2 estimators2 estimators
2 estimators
 

Ähnlich wie 14 fixed wing fighter aircraft- flight performance - ii

Take Off And Landing Performance
Take Off And Landing PerformanceTake Off And Landing Performance
Take Off And Landing Performanceahmad bassiouny
 
Performance.pdf
Performance.pdfPerformance.pdf
Performance.pdfClintTiu1
 
Experiments in Flight
Experiments in FlightExperiments in Flight
Experiments in FlightPritam Saha
 
Unmanned Airplane Autopilot Tuning
Unmanned Airplane Autopilot TuningUnmanned Airplane Autopilot Tuning
Unmanned Airplane Autopilot TuningIJERA Editor
 
Subsonic Airplane Design
Subsonic Airplane DesignSubsonic Airplane Design
Subsonic Airplane Designahmad bassiouny
 
Electric_Propeller_Aircraft_Sizing.pptx
Electric_Propeller_Aircraft_Sizing.pptxElectric_Propeller_Aircraft_Sizing.pptx
Electric_Propeller_Aircraft_Sizing.pptxssuser425dc3
 
Take Off and Landing Performance.pptx
Take Off and Landing Performance.pptxTake Off and Landing Performance.pptx
Take Off and Landing Performance.pptxMohaimenulIslam25
 
Flightpath+glossary+of+aviation+terms (1) (1)
Flightpath+glossary+of+aviation+terms (1) (1)Flightpath+glossary+of+aviation+terms (1) (1)
Flightpath+glossary+of+aviation+terms (1) (1)Francisco100
 
3 EDL.pdf
3 EDL.pdf3 EDL.pdf
3 EDL.pdfa a
 
AIRCRAFT WEIGHT AND BALANCE BASIC FOR LOAD CONTROL
AIRCRAFT WEIGHT AND BALANCE BASIC FOR LOAD CONTROLAIRCRAFT WEIGHT AND BALANCE BASIC FOR LOAD CONTROL
AIRCRAFT WEIGHT AND BALANCE BASIC FOR LOAD CONTROLjasmine jacob
 
Fighter jet Swept back wing design and Analysis by using of Xflr5
Fighter jet Swept back wing design and Analysis by using of Xflr5Fighter jet Swept back wing design and Analysis by using of Xflr5
Fighter jet Swept back wing design and Analysis by using of Xflr5Mani5436
 
Recap on Airbus A330 sim practice
Recap on Airbus A330 sim practice Recap on Airbus A330 sim practice
Recap on Airbus A330 sim practice careaviaitonhk
 
PPT-AIRCRAFT DESIGN PROJECT-II.pptx
PPT-AIRCRAFT DESIGN PROJECT-II.pptx PPT-AIRCRAFT DESIGN PROJECT-II.pptx
PPT-AIRCRAFT DESIGN PROJECT-II.pptxManojRasaily1
 
AIRCRAFT PITCH EECE 682 Computer Control Of Dynamic.docx
AIRCRAFT PITCH EECE 682 Computer Control Of Dynamic.docxAIRCRAFT PITCH EECE 682 Computer Control Of Dynamic.docx
AIRCRAFT PITCH EECE 682 Computer Control Of Dynamic.docxgalerussel59292
 
Drag polar | Flight Mechanics | GATE Aerospace
Drag polar | Flight Mechanics | GATE AerospaceDrag polar | Flight Mechanics | GATE Aerospace
Drag polar | Flight Mechanics | GATE AerospaceAge of Aerospace
 
Aircraft performance 2
Aircraft performance 2Aircraft performance 2
Aircraft performance 2Nazmul Alam
 
Aerodynamicsaeronauticsandflightmechanics 131107210632-phpapp01
Aerodynamicsaeronauticsandflightmechanics 131107210632-phpapp01Aerodynamicsaeronauticsandflightmechanics 131107210632-phpapp01
Aerodynamicsaeronauticsandflightmechanics 131107210632-phpapp01Mariutxy
 
Paper_Flutter
Paper_FlutterPaper_Flutter
Paper_FlutterRam Mohan
 

Ähnlich wie 14 fixed wing fighter aircraft- flight performance - ii (20)

Take Off And Landing Performance
Take Off And Landing PerformanceTake Off And Landing Performance
Take Off And Landing Performance
 
Performance.pdf
Performance.pdfPerformance.pdf
Performance.pdf
 
Experiments in Flight
Experiments in FlightExperiments in Flight
Experiments in Flight
 
Unmanned Airplane Autopilot Tuning
Unmanned Airplane Autopilot TuningUnmanned Airplane Autopilot Tuning
Unmanned Airplane Autopilot Tuning
 
Subsonic Airplane Design
Subsonic Airplane DesignSubsonic Airplane Design
Subsonic Airplane Design
 
Electric_Propeller_Aircraft_Sizing.pptx
Electric_Propeller_Aircraft_Sizing.pptxElectric_Propeller_Aircraft_Sizing.pptx
Electric_Propeller_Aircraft_Sizing.pptx
 
Take Off and Landing Performance.pptx
Take Off and Landing Performance.pptxTake Off and Landing Performance.pptx
Take Off and Landing Performance.pptx
 
Flightpath+glossary+of+aviation+terms (1) (1)
Flightpath+glossary+of+aviation+terms (1) (1)Flightpath+glossary+of+aviation+terms (1) (1)
Flightpath+glossary+of+aviation+terms (1) (1)
 
3 EDL.pdf
3 EDL.pdf3 EDL.pdf
3 EDL.pdf
 
AIRCRAFT WEIGHT AND BALANCE BASIC FOR LOAD CONTROL
AIRCRAFT WEIGHT AND BALANCE BASIC FOR LOAD CONTROLAIRCRAFT WEIGHT AND BALANCE BASIC FOR LOAD CONTROL
AIRCRAFT WEIGHT AND BALANCE BASIC FOR LOAD CONTROL
 
Fighter jet Swept back wing design and Analysis by using of Xflr5
Fighter jet Swept back wing design and Analysis by using of Xflr5Fighter jet Swept back wing design and Analysis by using of Xflr5
Fighter jet Swept back wing design and Analysis by using of Xflr5
 
Recap on Airbus A330 sim practice
Recap on Airbus A330 sim practice Recap on Airbus A330 sim practice
Recap on Airbus A330 sim practice
 
PPT-AIRCRAFT DESIGN PROJECT-II.pptx
PPT-AIRCRAFT DESIGN PROJECT-II.pptx PPT-AIRCRAFT DESIGN PROJECT-II.pptx
PPT-AIRCRAFT DESIGN PROJECT-II.pptx
 
AIRCRAFT PITCH EECE 682 Computer Control Of Dynamic.docx
AIRCRAFT PITCH EECE 682 Computer Control Of Dynamic.docxAIRCRAFT PITCH EECE 682 Computer Control Of Dynamic.docx
AIRCRAFT PITCH EECE 682 Computer Control Of Dynamic.docx
 
Drag polar | Flight Mechanics | GATE Aerospace
Drag polar | Flight Mechanics | GATE AerospaceDrag polar | Flight Mechanics | GATE Aerospace
Drag polar | Flight Mechanics | GATE Aerospace
 
ADP II Lab.pdf
ADP II Lab.pdfADP II Lab.pdf
ADP II Lab.pdf
 
Aircraft performance 2
Aircraft performance 2Aircraft performance 2
Aircraft performance 2
 
Mel341 3
Mel341 3Mel341 3
Mel341 3
 
Aerodynamicsaeronauticsandflightmechanics 131107210632-phpapp01
Aerodynamicsaeronauticsandflightmechanics 131107210632-phpapp01Aerodynamicsaeronauticsandflightmechanics 131107210632-phpapp01
Aerodynamicsaeronauticsandflightmechanics 131107210632-phpapp01
 
Paper_Flutter
Paper_FlutterPaper_Flutter
Paper_Flutter
 

Mehr von Solo Hermelin

Slide Mode Control (S.M.C.)
Slide Mode Control (S.M.C.)Slide Mode Control (S.M.C.)
Slide Mode Control (S.M.C.)Solo Hermelin
 
Sliding Mode Observers
Sliding Mode ObserversSliding Mode Observers
Sliding Mode ObserversSolo Hermelin
 
Reduced order observers
Reduced order observersReduced order observers
Reduced order observersSolo Hermelin
 
Inner outer and spectral factorizations
Inner outer and spectral factorizationsInner outer and spectral factorizations
Inner outer and spectral factorizationsSolo Hermelin
 
Keplerian trajectories
Keplerian trajectoriesKeplerian trajectories
Keplerian trajectoriesSolo Hermelin
 
Anti ballistic missiles ii
Anti ballistic missiles iiAnti ballistic missiles ii
Anti ballistic missiles iiSolo Hermelin
 
Anti ballistic missiles i
Anti ballistic missiles iAnti ballistic missiles i
Anti ballistic missiles iSolo Hermelin
 
2Anti-aircraft Warhead
2Anti-aircraft Warhead2Anti-aircraft Warhead
2Anti-aircraft WarheadSolo Hermelin
 
1 susceptibility vulnerability
1 susceptibility vulnerability1 susceptibility vulnerability
1 susceptibility vulnerabilitySolo Hermelin
 
Calculus of variation problems
Calculus of variation problemsCalculus of variation problems
Calculus of variation problemsSolo Hermelin
 
Calculus of variations
Calculus of variationsCalculus of variations
Calculus of variationsSolo Hermelin
 

Mehr von Solo Hermelin (13)

Slide Mode Control (S.M.C.)
Slide Mode Control (S.M.C.)Slide Mode Control (S.M.C.)
Slide Mode Control (S.M.C.)
 
Sliding Mode Observers
Sliding Mode ObserversSliding Mode Observers
Sliding Mode Observers
 
Reduced order observers
Reduced order observersReduced order observers
Reduced order observers
 
Inner outer and spectral factorizations
Inner outer and spectral factorizationsInner outer and spectral factorizations
Inner outer and spectral factorizations
 
Keplerian trajectories
Keplerian trajectoriesKeplerian trajectories
Keplerian trajectories
 
Anti ballistic missiles ii
Anti ballistic missiles iiAnti ballistic missiles ii
Anti ballistic missiles ii
 
Anti ballistic missiles i
Anti ballistic missiles iAnti ballistic missiles i
Anti ballistic missiles i
 
Analytic dynamics
Analytic dynamicsAnalytic dynamics
Analytic dynamics
 
2Anti-aircraft Warhead
2Anti-aircraft Warhead2Anti-aircraft Warhead
2Anti-aircraft Warhead
 
1 susceptibility vulnerability
1 susceptibility vulnerability1 susceptibility vulnerability
1 susceptibility vulnerability
 
15 sky cars
15 sky cars15 sky cars
15 sky cars
 
Calculus of variation problems
Calculus of variation problemsCalculus of variation problems
Calculus of variation problems
 
Calculus of variations
Calculus of variationsCalculus of variations
Calculus of variations
 

Kürzlich hochgeladen

well logging & petrophysical analysis.pptx
well logging & petrophysical analysis.pptxwell logging & petrophysical analysis.pptx
well logging & petrophysical analysis.pptxzaydmeerab121
 
User Guide: Capricorn FLX™ Weather Station
User Guide: Capricorn FLX™ Weather StationUser Guide: Capricorn FLX™ Weather Station
User Guide: Capricorn FLX™ Weather StationColumbia Weather Systems
 
Pests of Blackgram, greengram, cowpea_Dr.UPR.pdf
Pests of Blackgram, greengram, cowpea_Dr.UPR.pdfPests of Blackgram, greengram, cowpea_Dr.UPR.pdf
Pests of Blackgram, greengram, cowpea_Dr.UPR.pdfPirithiRaju
 
User Guide: Orion™ Weather Station (Columbia Weather Systems)
User Guide: Orion™ Weather Station (Columbia Weather Systems)User Guide: Orion™ Weather Station (Columbia Weather Systems)
User Guide: Orion™ Weather Station (Columbia Weather Systems)Columbia Weather Systems
 
Microphone- characteristics,carbon microphone, dynamic microphone.pptx
Microphone- characteristics,carbon microphone, dynamic microphone.pptxMicrophone- characteristics,carbon microphone, dynamic microphone.pptx
Microphone- characteristics,carbon microphone, dynamic microphone.pptxpriyankatabhane
 
Manassas R - Parkside Middle School 🌎🏫
Manassas R - Parkside Middle School 🌎🏫Manassas R - Parkside Middle School 🌎🏫
Manassas R - Parkside Middle School 🌎🏫qfactory1
 
User Guide: Pulsar™ Weather Station (Columbia Weather Systems)
User Guide: Pulsar™ Weather Station (Columbia Weather Systems)User Guide: Pulsar™ Weather Station (Columbia Weather Systems)
User Guide: Pulsar™ Weather Station (Columbia Weather Systems)Columbia Weather Systems
 
GENERAL PHYSICS 2 REFRACTION OF LIGHT SENIOR HIGH SCHOOL GENPHYS2.pptx
GENERAL PHYSICS 2 REFRACTION OF LIGHT SENIOR HIGH SCHOOL GENPHYS2.pptxGENERAL PHYSICS 2 REFRACTION OF LIGHT SENIOR HIGH SCHOOL GENPHYS2.pptx
GENERAL PHYSICS 2 REFRACTION OF LIGHT SENIOR HIGH SCHOOL GENPHYS2.pptxRitchAndruAgustin
 
OECD bibliometric indicators: Selected highlights, April 2024
OECD bibliometric indicators: Selected highlights, April 2024OECD bibliometric indicators: Selected highlights, April 2024
OECD bibliometric indicators: Selected highlights, April 2024innovationoecd
 
User Guide: Magellan MX™ Weather Station
User Guide: Magellan MX™ Weather StationUser Guide: Magellan MX™ Weather Station
User Guide: Magellan MX™ Weather StationColumbia Weather Systems
 
Pests of castor_Binomics_Identification_Dr.UPR.pdf
Pests of castor_Binomics_Identification_Dr.UPR.pdfPests of castor_Binomics_Identification_Dr.UPR.pdf
Pests of castor_Binomics_Identification_Dr.UPR.pdfPirithiRaju
 
Pests of soyabean_Binomics_IdentificationDr.UPR.pdf
Pests of soyabean_Binomics_IdentificationDr.UPR.pdfPests of soyabean_Binomics_IdentificationDr.UPR.pdf
Pests of soyabean_Binomics_IdentificationDr.UPR.pdfPirithiRaju
 
Dubai Calls Girl Lisa O525547819 Lexi Call Girls In Dubai
Dubai Calls Girl Lisa O525547819 Lexi Call Girls In DubaiDubai Calls Girl Lisa O525547819 Lexi Call Girls In Dubai
Dubai Calls Girl Lisa O525547819 Lexi Call Girls In Dubaikojalkojal131
 
Thermodynamics ,types of system,formulae ,gibbs free energy .pptx
Thermodynamics ,types of system,formulae ,gibbs free energy .pptxThermodynamics ,types of system,formulae ,gibbs free energy .pptx
Thermodynamics ,types of system,formulae ,gibbs free energy .pptxuniversity
 
Environmental Biotechnology Topic:- Microbial Biosensor
Environmental Biotechnology Topic:- Microbial BiosensorEnvironmental Biotechnology Topic:- Microbial Biosensor
Environmental Biotechnology Topic:- Microbial Biosensorsonawaneprad
 
Microteaching on terms used in filtration .Pharmaceutical Engineering
Microteaching on terms used in filtration .Pharmaceutical EngineeringMicroteaching on terms used in filtration .Pharmaceutical Engineering
Microteaching on terms used in filtration .Pharmaceutical EngineeringPrajakta Shinde
 
GenAI talk for Young at Wageningen University & Research (WUR) March 2024
GenAI talk for Young at Wageningen University & Research (WUR) March 2024GenAI talk for Young at Wageningen University & Research (WUR) March 2024
GenAI talk for Young at Wageningen University & Research (WUR) March 2024Jene van der Heide
 
FREE NURSING BUNDLE FOR NURSES.PDF by na
FREE NURSING BUNDLE FOR NURSES.PDF by naFREE NURSING BUNDLE FOR NURSES.PDF by na
FREE NURSING BUNDLE FOR NURSES.PDF by naJASISJULIANOELYNV
 

Kürzlich hochgeladen (20)

well logging & petrophysical analysis.pptx
well logging & petrophysical analysis.pptxwell logging & petrophysical analysis.pptx
well logging & petrophysical analysis.pptx
 
User Guide: Capricorn FLX™ Weather Station
User Guide: Capricorn FLX™ Weather StationUser Guide: Capricorn FLX™ Weather Station
User Guide: Capricorn FLX™ Weather Station
 
Pests of Blackgram, greengram, cowpea_Dr.UPR.pdf
Pests of Blackgram, greengram, cowpea_Dr.UPR.pdfPests of Blackgram, greengram, cowpea_Dr.UPR.pdf
Pests of Blackgram, greengram, cowpea_Dr.UPR.pdf
 
User Guide: Orion™ Weather Station (Columbia Weather Systems)
User Guide: Orion™ Weather Station (Columbia Weather Systems)User Guide: Orion™ Weather Station (Columbia Weather Systems)
User Guide: Orion™ Weather Station (Columbia Weather Systems)
 
Microphone- characteristics,carbon microphone, dynamic microphone.pptx
Microphone- characteristics,carbon microphone, dynamic microphone.pptxMicrophone- characteristics,carbon microphone, dynamic microphone.pptx
Microphone- characteristics,carbon microphone, dynamic microphone.pptx
 
Manassas R - Parkside Middle School 🌎🏫
Manassas R - Parkside Middle School 🌎🏫Manassas R - Parkside Middle School 🌎🏫
Manassas R - Parkside Middle School 🌎🏫
 
User Guide: Pulsar™ Weather Station (Columbia Weather Systems)
User Guide: Pulsar™ Weather Station (Columbia Weather Systems)User Guide: Pulsar™ Weather Station (Columbia Weather Systems)
User Guide: Pulsar™ Weather Station (Columbia Weather Systems)
 
Let’s Say Someone Did Drop the Bomb. Then What?
Let’s Say Someone Did Drop the Bomb. Then What?Let’s Say Someone Did Drop the Bomb. Then What?
Let’s Say Someone Did Drop the Bomb. Then What?
 
GENERAL PHYSICS 2 REFRACTION OF LIGHT SENIOR HIGH SCHOOL GENPHYS2.pptx
GENERAL PHYSICS 2 REFRACTION OF LIGHT SENIOR HIGH SCHOOL GENPHYS2.pptxGENERAL PHYSICS 2 REFRACTION OF LIGHT SENIOR HIGH SCHOOL GENPHYS2.pptx
GENERAL PHYSICS 2 REFRACTION OF LIGHT SENIOR HIGH SCHOOL GENPHYS2.pptx
 
OECD bibliometric indicators: Selected highlights, April 2024
OECD bibliometric indicators: Selected highlights, April 2024OECD bibliometric indicators: Selected highlights, April 2024
OECD bibliometric indicators: Selected highlights, April 2024
 
User Guide: Magellan MX™ Weather Station
User Guide: Magellan MX™ Weather StationUser Guide: Magellan MX™ Weather Station
User Guide: Magellan MX™ Weather Station
 
Pests of castor_Binomics_Identification_Dr.UPR.pdf
Pests of castor_Binomics_Identification_Dr.UPR.pdfPests of castor_Binomics_Identification_Dr.UPR.pdf
Pests of castor_Binomics_Identification_Dr.UPR.pdf
 
Pests of soyabean_Binomics_IdentificationDr.UPR.pdf
Pests of soyabean_Binomics_IdentificationDr.UPR.pdfPests of soyabean_Binomics_IdentificationDr.UPR.pdf
Pests of soyabean_Binomics_IdentificationDr.UPR.pdf
 
Dubai Calls Girl Lisa O525547819 Lexi Call Girls In Dubai
Dubai Calls Girl Lisa O525547819 Lexi Call Girls In DubaiDubai Calls Girl Lisa O525547819 Lexi Call Girls In Dubai
Dubai Calls Girl Lisa O525547819 Lexi Call Girls In Dubai
 
Thermodynamics ,types of system,formulae ,gibbs free energy .pptx
Thermodynamics ,types of system,formulae ,gibbs free energy .pptxThermodynamics ,types of system,formulae ,gibbs free energy .pptx
Thermodynamics ,types of system,formulae ,gibbs free energy .pptx
 
Environmental Biotechnology Topic:- Microbial Biosensor
Environmental Biotechnology Topic:- Microbial BiosensorEnvironmental Biotechnology Topic:- Microbial Biosensor
Environmental Biotechnology Topic:- Microbial Biosensor
 
Microteaching on terms used in filtration .Pharmaceutical Engineering
Microteaching on terms used in filtration .Pharmaceutical EngineeringMicroteaching on terms used in filtration .Pharmaceutical Engineering
Microteaching on terms used in filtration .Pharmaceutical Engineering
 
AZOTOBACTER AS BIOFERILIZER.PPTX
AZOTOBACTER AS BIOFERILIZER.PPTXAZOTOBACTER AS BIOFERILIZER.PPTX
AZOTOBACTER AS BIOFERILIZER.PPTX
 
GenAI talk for Young at Wageningen University & Research (WUR) March 2024
GenAI talk for Young at Wageningen University & Research (WUR) March 2024GenAI talk for Young at Wageningen University & Research (WUR) March 2024
GenAI talk for Young at Wageningen University & Research (WUR) March 2024
 
FREE NURSING BUNDLE FOR NURSES.PDF by na
FREE NURSING BUNDLE FOR NURSES.PDF by naFREE NURSING BUNDLE FOR NURSES.PDF by na
FREE NURSING BUNDLE FOR NURSES.PDF by na
 

14 fixed wing fighter aircraft- flight performance - ii

  • 1. Fixed Wing Fighter Aircraft Flight Performance Part II SOLO HERMELIN Updated: 04.12.12 28.02.15 1 http://www.solohermelin.com
  • 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. 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. 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. 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. 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 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 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 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 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 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 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 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 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 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 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 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 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 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. 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 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 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 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 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 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 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 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 H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35 Return to Table of Content
  • 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 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 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. 32M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring” Comparison of Takeoff Weight and Empty Weight of different Aircraft
  • 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. 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 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 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 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 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 Drag Characteristics Fixed Wing Fighter Aircraft Flight Performance SOLO
  • 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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. 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 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 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 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 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. 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. 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 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 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. 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 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 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. 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. 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 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 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 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 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 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 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 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
  • 98. 98 Vertical Plan Trajectory (σ = 0) SOLO Prof. Earll Murman, “Introduction to Aircraft Performance and Static Stability”, September 18, 2003
  • 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
  • 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 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 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 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 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 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 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 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 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 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 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 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 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 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. 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 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. 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. 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
  • 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
  • 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 Typical Maneuvering Envelope V – n Diagram Performance of an Aircraft with Parabolic PolarSOLO
  • 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 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 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. 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
  • 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
  • 134. 134
  • 135. 135
  • 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. 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
  • 139. 139Comparative Ps Diagram for Aircraft A and Aircraft B. Two Multi-Role Jet Fighters SOLO Aircraft Flight Performance
  • 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 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 F-86F Sabre and MiG-15 performance comparison North American F-86 Sabre MiG-15 SOLO Aircraft Flight Performance
  • 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 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 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 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 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 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
  • 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 Lockheed F-104 Starfighter SOLO Aircraft Flight Performance
  • 152. 152 Typical Ps Plot for Lockheed F-104 Starfighter Lockheed F-104 Starfighter SOLO Aircraft Flight Performance
  • 153. 153 SOLO Aircraft Flight Performance F-104 Flight Envelope Lockheed F-104 Starfighter
  • 154. 154F-104A flight envelope Lockheed F-104 Starfighter SOLO Aircraft Flight Performance Return to Table of Content
  • 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
  • 161. 161 Sukhoi Su-30MKI SOLO Aircraft Flight Performance http://vayu-sena.tripod.com/interview-simonov1.html
  • 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 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 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 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 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 ( )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. 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 ( )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 ,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 ( )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 ( )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
  • 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 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 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. 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 Comparison of Fighter Aircraft Propulsion Systems SOLO
  • 179. 179 Comparison of Fighter Aircraft Propulsion Systems SOLO
  • 180. 180 SOLO Aircraft Flight Performance Composite Thrust Loading versus Wing Loading – for different Aircraft Constraint Analysis
  • 181. 181 Constraint Diagram for F-16 SOLO Aircraft Flight Performance Constraint Analysis Return to Table of Content
  • 182. 182 Weapon System Agility Weapon System Agility Return to Table of Content

Hinweis der Redaktion

  1. J.D. Anderson, Jr, “Fundamentals of Aerodynamics”, McGraw-Hill, 3th Ed., 1984, 1991, 2001
  2. M. Saarlas, “Aircraft Performance”, John Wiley &amp; Sons, 2007 http://www.ctie.monash.edu.au/hargrave/breguet.html
  3. J.D. Mattingly, W.H. Heiser, D.T. Pratt, “Aircraft Engine Design”, 2nd Es., AIAA Education Series, 2002
  4. M. Saarlas, “Aircraft Performance”, John Wiley &amp; Sons, 2007
  5. M. Saarlas, “Aircraft Performance”, John Wiley &amp; Sons, 2007
  6. M. Saarlas, “Aircraft Performance”, John Wiley &amp; Sons, 2007
  7. M. Saarlas, “Aircraft Performance”, John Wiley &amp; Sons, 2007
  8. M. Saarlas, “Aircraft Performance”, John Wiley &amp; Sons, 2007
  9. M. Saarlas, “Aircraft Performance”, John Wiley &amp; Sons, 2007
  10. M. Saarlas, “Aircraft Performance”, John Wiley &amp; Sons, 2007
  11. M. Saarlas, “Aircraft Performance”, John Wiley &amp; Sons, 2007
  12. M. Saarlas, “Aircraft Performance”, John Wiley &amp; Sons, 2007
  13. M. Saarlas, “Aircraft Performance”, John Wiley &amp; Sons, 2007
  14. http://www.free-online-private-pilot-ground-school.com/aircraft_performance.html
  15. M. Saarlas, “Aircraft Performance”, John Wiley &amp; Sons, 2007
  16. M. Saarlas, “Aircraft Performance”, John Wiley &amp; Sons, 2007 J. D. Anderson, Jr., “Aircraft Performance and Design”, McGraw-Hill, 1999
  17. M. Saarlas, “Aircraft Performance”, John Wiley &amp; Sons, 2007 J. D. Anderson, Jr., “Aircraft Performance and Design”, McGraw-Hill, 1999
  18. M. Saarlas, “Aircraft Performance”, John Wiley &amp; Sons, 2007 J. D. Anderson, Jr., “Aircraft Performance and Design”, McGraw-Hill, 1999
  19. M. Saarlas, “Aircraft Performance”, John Wiley &amp; Sons, 2007 J. D. Anderson, Jr., “Aircraft Performance and Design”, McGraw-Hill, 1999
  20. M. Saarlas, “Aircraft Performance”, John Wiley &amp; Sons, 2007 J. D. Anderson, Jr., “Aircraft Performance and Design”, McGraw-Hill, 1999
  21. M. Saarlas, “Aircraft Performance”, John Wiley &amp; Sons, 2007 J. D. Anderson, Jr., “Aircraft Performance and Design”, McGraw-Hill, 1999
  22. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  23. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  24. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  25. http://personal.mecheng.adelaide.edu.au/maziar.arjomandi/Aeronautical%20Engineering%20Projects/2006/group17.pdf
  26. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  27. Stengel, MAE331, Lecture 6
  28. Stengel, MAE331, Lecture 6 A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  29. Stengel, MAE331, Lecture 6 A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  30. Stengel, MAE331, Lecture 6 A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  31. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  32. H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35 http://surroundedbyair.blogspot.co.il/2010_10_01_archive.html
  33. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  34. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  35. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  36. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  37. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  38. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  39. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  40. http://www.southampton.ac.uk/~jps7/D8%20website/future%20trends%20in%20aircraft%20costs.pdf
  41. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  42. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  43. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  44. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  45. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  46. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962, Ch.9 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998, Ch.2
  47. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  48. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  49. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  50. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  51. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  52. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  53. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  54. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  55. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  56. J.D. Anderson, Jr., “Aircraft Performance and Design”, McGraw Hill, 1999
  57. J.D. Anderson, Jr., “Aircraft Performance and Design”, McGraw Hill, 1999
  58. J.D. Anderson, Jr., “Aircraft Performance and Design”, McGraw Hill, 1999
  59. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  60. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  61. Prof. Earll Murman, “Introduction to Aircraft Performance and Static Stability”, September 18, 2003
  62. Prof. Earll Murman, “Introduction to Aircraft Performance and Static Stability”, September 18, 2003
  63. http://scilib.narod.ru/Avia/Bonanni/bonanni.htm
  64. Brandt, “Introduction to Aerodynamics – A Design Perspective”, Ch. 5 , Performance and Constraint Analysis
  65. Randal K. Liefer, John Valasek, David P. Eggold, “Fighter Aircraft Metrics, Research , and Test”, Phase I Report, KU-FRL-831-2
  66. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  67. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  68. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  69. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  70. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  71. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  72. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  73. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  74. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  75. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  76. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  77. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  78. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  79. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  80. “Climb and Turn Performance”, G. Leng, NUS
  81. Klaus Huenecke, “Modern Combat Aircraft Design”, Airlife, 1984, 1987
  82. “Climb and Turn Performance”, G. Leng, NUS
  83. Chapter 4 Aerodynamics of Flight http://www.gov/library/manuals/aviation/pilot_handbook/media/PHAK%20-%20Chapter%2004.pdf
  84. http://en.wikipedia.org/wiki/Energy%E2%80%93maneuverability_theory J.R. Boyd, “Energy Maneuvrability”, AD 372 287, March 1966
  85. “Climb and Turn Performance”, G. Leng, NUS
  86. “Certain to Win, John Boyd ‘s Strategy in the 21st Strategy”, Chet Richards,
  87. “Certain to Win, John Boyd ‘s Strategy in the 21st Strategy”, Chet Richards,
  88. http://en.wikipedia.org/wiki/Basic_fighter_maneuvers
  89. http://en.wikipedia.org/wiki/Basic_fighter_maneuvers
  90. Brandt, “Introduction to Aerodynamics – A Design Perspective”, Ch. 5 , Performance and Constraint Analysis
  91. http://www.simhq.com/_air/air_065d.html
  92. http://www.sci.fi/~fta/JohnBoyd.htm
  93. http://en.wikipedia.org/wiki/Mikoyan_MiG-29 http://en.wikipedia.org/wiki/General_Dynamics_F-16_Fighting_Falcon
  94. http://en.wikipedia.org/wiki/Mikoyan_MiG-29 http://en.wikipedia.org/wiki/General_Dynamics_F-16_Fighting_Falcon
  95. http://en.wikipedia.org/wiki/Mikoyan_MiG-29 http://en.wikipedia.org/wiki/General_Dynamics_F-16_Fighting_Falcon
  96. http://en.wikipedia.org/wiki/Mikoyan_MiG-29 http://en.wikipedia.org/wiki/General_Dynamics_F-16_Fighting_Falcon
  97. http://en.wikipedia.org/wiki/Mikoyan_MiG-29 http://en.wikipedia.org/wiki/General_Dynamics_F-16_Fighting_Falcon
  98. “Climb and Turn Performance”, G. Leng, NUS
  99. Randal K. Liefer, John Valasek, David P. Eggold, “Fighter Aircraft Metrics, Research , and Test”, Phase I Report, KU-FRL-831-2
  100. http://en.wikipedia.org/wiki/Lockheed_F-104_Starfighter#mediaviewer/File:F-104A_flight_envelope.jpg
  101. Asselin, M., “Introduction to Aircraft Performance”, AIAA Education Series, 1997 http://forum.worldofwarplanes.com/index.php?/topic/5146-sukhoi-su-37-terminator/
  102. J.D. Mattingly, W.H. Heiser, D.T. Pratt, “Aircraft Engine Design”, 2nd Ed., AIAA Education Series, 2002 Brandt, “Introduction to Aerodynamics – A Design Perspective”, Ch. 5 , Performance and Constraint Analysis J.D. Anderson, Jr., “Aircraft Performance and Design”, McGraw-Hill, 1999
  103. J.D. Mattingly, W.H. Heiser, D.T. Pratt, “Aircraft Engine Design”, 2nd Ed., AIAA Education Series, 2002 Brandt, “Introduction to Aerodynamics – A Design Perspective”, Ch. 5 , Performance and Constraint Analysis
  104. J.D. Mattingly, W.H. Heiser, D.T. Pratt, “Aircraft Engine Design”, 2nd Es., AIAA Education Series, 2002
  105. J.D. Mattingly, W.H. Heiser, D.T. Pratt, “Aircraft Engine Design”, 2nd Ed., AIAA Education Series, 2002 Brandt, “Introduction to Aerodynamics – A Design Perspective”, Ch. 5 , Performance and Constraint Analysis
  106. J.D. Mattingly, W.H. Heiser, D.T. Pratt, “Aircraft Engine Design”, 2nd Ed., AIAA Education Series, 2002 Brandt, “Introduction to Aerodynamics – A Design Perspective”, Ch. 5 , Performance and Constraint Analysis
  107. http://personal.mecheng.adelaide.edu.au/maziar.arjomandi/Aeronautical%20Engineering%20Projects/2006/group17.pdf
  108. J.D. Mattingly, W.H. Heiser, D.T. Pratt, “Aircraft Engine Design”, 2nd Ed., AIAA Education Series, 2002 Brandt, “Introduction to Aerodynamics – A Design Perspective”, Ch. 5 , Performance and Constraint Analysis
  109. J.D. Mattingly, W.H. Heiser, D.T. Pratt, “Aircraft Engine Design”, 2nd Ed., AIAA Education Series, 2002 Brandt, “Introduction to Aerodynamics – A Design Perspective”, Ch. 5 , Performance and Constraint Analysis
  110. J.D. Mattingly, W.H. Heiser, D.T. Pratt, “Aircraft Engine Design”, 2nd Ed., AIAA Education Series, 2002 Brandt, “Introduction to Aerodynamics – A Design Perspective”, Ch. 5 , Performance and Constraint Analysis
  111. J.D. Mattingly, W.H. Heiser, D.T. Pratt, “Aircraft Engine Design”, 2nd Ed., AIAA Education Series, 2002 Brandt, “Introduction to Aerodynamics – A Design Perspective”, Ch. 5 , Performance and Constraint Analysis
  112. J.D. Mattingly, W.H. Heiser, D.T. Pratt, “Aircraft Engine Design”, 2nd Ed., AIAA Education Series, 2002 Brandt, “Introduction to Aerodynamics – A Design Perspective”, Ch. 5 , Performance and Constraint Analysis J.D. Anderson, Jr., “Aircraft Performance and Design”, McGraw-Hill, 1999, pp. 392 - 395
  113. J.D. Mattingly, W.H. Heiser, D.T. Pratt, “Aircraft Engine Design”, 2nd Ed., AIAA Education Series, 2002 Brandt, “Introduction to Aerodynamics – A Design Perspective”, Ch. 5 , Performance and Constraint Analysis J.D. Anderson, Jr., “Aircraft Performance and Design”, McGraw-Hill, 1999, pp. 392 - 395
  114. M. Saarlas, “Aircraft Performance”, John Wiley &amp; Sons, 2007 J. D. Anderson, Jr., “Aircraft Performance and Design”, McGraw-Hill, 1999
  115. Brandt, “Introduction to Aerodynamics – A Design Perspective”, Ch. 5 , Performance and Constraint Analysis J.D. Mattingly, W.H. Heiser, D.T. Pratt, “Aircraft Engine Design”, 2nd Ed., AIAA Education Series, 2002
  116. Randal K. Liefer, John Valasek, David P. Eggold, “Fighter Aircraft Metrics, Research , and Test”, Phase I Report, KU-FRL-831-2
  117. Randal K. Liefer, John Valasek, David P. Eggold, “Fighter Aircraft Metrics, Research , and Test”, Phase I Report, KU-FRL-831-2
  118. J.D. Mattingly, W.H. Heiser, D.T. Pratt, “Aircraft Engine Design”, 2nd Es., AIAA Education Series, 2002
  119. Brandt, “Introduction to Aerodynamics – A Design Perspective”, Ch. 5 , Performance and Constraint Analysis J.D. Mattingly, W.H. Heiser, D.T. Pratt, “Aircraft Engine Design”, 2nd Es., AIAA Education Series, 2002
  120. Patrick Le Blaye, “Agility: Definitions, Basic Concepts, History”, ONERA
  121. http://en.wikipedia.org/wiki/OODA_loop
  122. Asselin, M., “Introduction to Aircraft Performance”, AIAA Education Series, 1997
  123. http://personal.mecheng.adelaide.edu.au/maziar.arjomandi/Aeronautical%20Engineering%20Projects/2006/group17.pdf
  124. Asselin, M., “Introduction to Aircraft Performance”, AIAA Education Series, 1997
  125. Klaus Huenecke, “Modern Combat Aircraft Design”, Airlife, 1984, 1987
  126. A. Miele, “Flight Mechanics”, Vol. 1,”Theory of Flight Paths”, Addison-Wesley , 1962 B.N. Pamadi, “Performance, Stability, Dynamics and Control of Airplanes”, AIAA Education Series, 1998
  127. http://en.wikipedia.org/wiki/Basic_fighter_maneuvers
  128. http://en.wikipedia.org/wiki/Basic_fighter_maneuvers
  129. http://en.wikipedia.org/wiki/Basic_fighter_maneuvers
  130. “Supplemental FLIGHT MANUAL F-16C/D Blocks 50 and 52+”, Lockheed Martin Corporation, T.O. GR1F-16CJ-1-1
  131. “Supplemental FLIGHT MANUAL F-16C/D Blocks 50 and 52+”, Lockheed Martin Corporation, T.O. GR1F-16CJ-1-1
  132. “Supplemental FLIGHT MANUAL F-16C/D Blocks 50 and 52+”, Lockheed Martin Corporation, T.O. GR1F-16CJ-1-1
  133. “Supplemental FLIGHT MANUAL F-16C/D Blocks 50 and 52+”, Lockheed Martin Corporation, T.O. GR1F-16CJ-1-1
  134. Klaus Huenecke, “Modern Combat Aircraft Design”, Airlife, 1984, 1987
  135. Klaus Huenecke, “Modern Combat Aircraft Design”, Airlife, 1984, 1987
  136. Klaus Huenecke, “Modern Combat Aircraft Design”, Airlife, 1984, 1987
  137. R.M. Engelbeck, “Investigation Into the Impact of Agility in Conceptual Fighter Design”, NASA Contractor Report 195079, Boeing Defense &amp; Space Group, Seattle Washington, Contract NAS1=18762, June 1995
  138. Brandt, “Introduction to Aerodynamics – A Design Perspective”, Ch. 5 , Performance and Constraint Analysis
  139. Brandt, “Introduction to Aerodynamics – A Design Perspective”, Ch. 5 , Performance and Constraint Analysis
  140. J.D. Mattingly, W.H. Heiser, D.T. Pratt, “Aircraft Engine Design”, 2nd Es., AIAA Education Series, 2002
  141. Asselin, M., “Introduction to Aircraft Performance”, AIAA Education Series, 1997
  142. Naval Air Training Command, “Air Combat Maneuvering”, CNATRA P-1289 (Rev. 08-09)