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12 performance of an aircraft with parabolic polar

The document discusses the performance of an aircraft modeled with parabolic polar equations, focusing on the forces acting on the aircraft in a flat earth scenario. It includes equations for aerodynamic lift, drag, thrust, and energy per unit mass, while addressing constraints related to altitude, dynamic pressure, and control limits. The analysis emphasizes calculating maximum efficiencies and load factors during different flight trajectories.

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08/12/15 1 Performance of an Aircraft with Parabolic Polar SOLO HERMELIN Updated: 14.03.2004
08/12/15 2 Performance of an Aircraft with Parabolic PolarSOLO Table of Content Flat Earth Three Degrees of Freedom Aircraft Equations Performance of an Aircraft with Parabolic Polar Aircraft Drag Energy per unit mass E Load Factor n Aircraft Trajectories Summary Constraints Horizontal Plan Trajectory ( )0,0 == γγ  Horizontal Turn Rate as Function of ps, n Horizontal Turn Rate as Function of nV, References
08/12/15 3 SOLO 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) •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 Flat Earth Three Degrees of Freedom Aircraft Equations
4 SOLO • Rotation Matrix from Earth to Wind Coordinates [ ] [ ] [ ]321 χγσ=W EC where σ – Roll Angle γ – Elevation Angle of the Trajectory χ – Azimuth Angle of the Trajectory Force Equation: amgmTFA  =++ where: • Aerodynamic Forces (Lift L and Drag D) ( )           − − = L D F W A 0  • Thrust T ( )           = α α sin 0 cos T T T W  • Gravitation acceleration ( ) ( )                     −           −           − == g cs sc cs sc cs scgCg EW E W 0 0 100 0 0 0 010 0 0 0 001 χχ χχ γγ γγ σσ σσ ( ) g cc cs s g W          − = γσ γσ γ  α T V L D Bx Wx Bz Wz Wy By Flat Earth Three Degrees of Freedom Aircraft Equations
5 SOLO α T V L D Bx Wx Bz Wz Wy By • Aircraft Acceleration ( ) ( ) ( ) ( )WW W W VVa  ×+= → ω where: ( )           = 0 0 V V W  and ( )           = → 0 0 V V W   ( )                                         −+                     − +                     − =           = χ χχ χχ γ γγ γγσ σσ σσω     0 0 100 0 0 0 0 0 010 0 0 0 0 0 001 cs sc cs sc cs sc r q p W W W W or ( )           +− + − =           = γσχσγ γσχσγ γχσ ω ccs csc s r q p W W W W     therefore ( ) ( ) ( ) ( ) ( ) ( )           +− +−=           − =×+= → γσχσγ γσχσγω cscV ccsV V qV rV V VVa W W WW W W     Flat Earth Three Degrees of Freedom Aircraft Equations
6 SOLO α T V L D Bx Wx Bz Wz Wy By • Aircraft Acceleration Flat Earth Three Degrees of Freedom Aircraft Equations From the Force equation we obtain: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )WWW A WW W W gTF m VVa  ++=×+= → 1 ω or ( ) ( ) ( ) σ σ σ σ γσαγσχσγ γσγσχσγ γα s c c s ccgmLTcscVqV csgccsVrV sgmDTV W W −− −      ++−=+−=− =+−= −−= /sin /)cos(    from which we obtain: ( ) ( )     += −+= −−= msLTcV cgmcLTV sgmDTV /sin /sin /)cos( σαγχ γσαγ γα    Define the Load Factor gm LT n + = αsin :
7 SOLO α T V L D Bx Wx Bz Wz Wy By • Velocity Equation Flat Earth Three Degrees of Freedom Aircraft Equations ( ) ( )           ==           = 0 0 V CVC h y x V E W WE W E                          −           −          − =           0 0 0 0 001 0 010 0 100 0 0 V cs sc cs sc cs sc h y x σσ σσ γγ γγ χχ χχ         = = = γ χγ χγ sVh scVy ccVx    or • Energy per unit mass E g V hE 2 : 2 += Let differentiate this equation: ( ) W VDT W DT g g V V g VV hEps − =            − − +=+== α γ α γ cos sin cos sin:   Return to Table of Content
8 SOLO Flat Earth Three Degrees of Freedom Aircraft Equations Summary γ χγ χγ sin sincos coscos Vh Vy Vx = = =    ( ) ( ) σ γ σ γ α χ γσγσ α γ α γ α sin cos sin cos sin coscoscoscos sin cos sin cos n V g W LT V g n V g W LT V g W VDT Eor W DT gV = + = −=      − + = − =−      − =    where mgW = Aircraft Thrust( ) 10, ≤≤= ηη VhTT MAX ( ) ( )MSCVhL L , 2 1 2 αρ= ( ) ( )LD CMSCVhD , 2 1 2 ρ= ( ) ( ) ( ) 2 0, LDLD CMKMCCMC += ( ) 0/ 0 hh eh − = ρρ ( ) ( ) soundofspeedhaNumberMachMhaVM === &/ Aircraft Weight Aircraft Lift Aircraft Drag Parabolic Drag Polar Return to Table of Content
9 SOLO Flat Earth Three Degrees of Freedom Aircraft Equations Constraints: State Constraints • Minimum Altitude Limit minhh ≥ • Maximum dynamic pressure limit ( ) ( )hVVorqVhq MAXMAX ≤≤= 2 2 1 ρ • Maximum Mach Number limit ( ) MAXM ha V ≤ Aerodynamic or heat limitation Control Constraints • Maximum Lift Coefficient or Maximum Angle of Attack ( ) ( ) ( )VhorMCMC STALLMAXLL ,, _ ααα ≤≤ • Minimum Load Factor ( ) MAXn W VhL n ≤= , • Maximum Thrust ( )VhTT MAX ,≤
10 SOLO Flat Earth Three Degrees of Freedom Aircraft Equations MAXα MAXα αα LCDC MAXLC 0DC ( ) ( )αα 2 0 LDD kCCC += Drag and Lift Coefficients as functions of Angle of Attack
11 SOLO Flat Earth Three Degrees of Freedom Aircraft Equations ( ) ( ) ( ) Limit Vhor MCMC STALL MAXLL , , _ αα α = = ( ) ( ) Limit hVVor qVhq MAX MAX = == 2 2 1 ρ minhh = MAXMM = Mach Altitude Flight Envelope of the Aircraft Return to Table of Content
12 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 assumed 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
13 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 0 * 2 1 :* * : 2 * 2 1 :* Vq V V u CS kW CS W V D L ρ ρρ 2 0 : LD L D L CkC C CSq CSq D L e + === Climbing Aircraft Performance
14 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 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           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
15 Performance of an Aircraft with Parabolic PolarSOLO We obtained 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 u 0 - - - - 0 + + + + + D ↓ min ↑ n u D ∂ ∂       += 2 2 2 *2 u n u e W D
16 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 Return to Table of Content Climbing Aircraft Performance Maximum Load Factor           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
17 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 * −+− = nz nzzu nzzu nuzup constns >     −+= −−= →=+−→== 22 2 22 1224 020 ( ) ( ) 2 224 2 2243 23 * *244 * * u nuzu e V u nuzuuuzu e V u p constn s ++− = −+−−+− = ∂ ∂ = 3 3 0 22 nzz u u p MAX constn s ++ =→= ∂ ∂ = 2 21 2 uu uu MAX << + nz > Climbing Aircraft Performance
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 ==
19 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 nuzu e V ps 224 2 * * −+− = 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 =→=+−= ∂ ∂ = Return to Table of Content Climbing Aircraft Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
20 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 u V pe uzun s * *2 2 24 −+−= ( ) 3 0412 2 2 22 z uzu u n constps =→=+−= ∂ ∂ = ( ) * *2 44 3 2 V pe uzu u n s constps −+−= ∂ ∂ = Climbing Aircraft Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
21 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 Return to Table of Content Climbing Aircraft Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
22 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 Trajectories We found Aircraft Turn Performance
23 Performance of an Aircraft with Parabolic PolarSOLO ( ) ( ) ( ) γ σ φ 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  
24 Vertical Plan Trajectory (σ = 0) SOLO Return to Table of Content
25 Performance of an Aircraft with Parabolic PolarSOLO 2. Horizontal Plan Trajectory ( )0,0 == γγ  We can see that for n > 1 1 1 1' 1 11' 2 2 2 2 22 − ≈ − = −≈−= ng V ng V R n V g n V g χ 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
26 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
27 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 a 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
28 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
29 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
30 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
31 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 s χ Aircraft Turn Performance
32 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
33 Performance of an Aircraft with Parabolic PolarSOLO Horizontal Turn Rate as Function of ps, n u 0<sp 0=sp 0>sp χ ( )12 * −z V g 1=u1u 2u 2 24 1 * *2 2 * u up V e uzu V g s −−+− =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 nMAX LIMIT C MAXL_ MAX MAX L MAXL n n C C V g 1 ** 2 _ − 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
34 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
35 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
36 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 The minimum of 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
37 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
38 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
39 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
08/12/15 40 Performance of an Aircraft with Parabolic PolarSOLO References Angelo Miele, “Flight Mechanics, Volume I, Theory of Flight Paths”, Addison-Wesley, 1962 Return to Table of Content S. Hermelin, “3 DOF Model of Aircraft in Earth Atmosphere”
41 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA

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12 performance of an aircraft with parabolic polar

  • 1.
    08/12/15 1 Performance ofan Aircraft with Parabolic Polar SOLO HERMELIN Updated: 14.03.2004
  • 2.
    08/12/15 2 Performance ofan Aircraft with Parabolic PolarSOLO Table of Content Flat Earth Three Degrees of Freedom Aircraft Equations Performance of an Aircraft with Parabolic Polar Aircraft Drag Energy per unit mass E Load Factor n Aircraft Trajectories Summary Constraints Horizontal Plan Trajectory ( )0,0 == γγ  Horizontal Turn Rate as Function of ps, n Horizontal Turn Rate as Function of nV, References
  • 3.
    08/12/15 3 SOLO Assumptions: •Point massmodel. •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) •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 Flat Earth Three Degrees of Freedom Aircraft Equations
  • 4.
    4 SOLO • Rotation Matrixfrom Earth to Wind Coordinates [ ] [ ] [ ]321 χγσ=W EC where σ – Roll Angle γ – Elevation Angle of the Trajectory χ – Azimuth Angle of the Trajectory Force Equation: amgmTFA  =++ where: • Aerodynamic Forces (Lift L and Drag D) ( )           − − = L D F W A 0  • Thrust T ( )           = α α sin 0 cos T T T W  • Gravitation acceleration ( ) ( )                     −           −           − == g cs sc cs sc cs scgCg EW E W 0 0 100 0 0 0 010 0 0 0 001 χχ χχ γγ γγ σσ σσ ( ) g cc cs s g W          − = γσ γσ γ  α T V L D Bx Wx Bz Wz Wy By Flat Earth Three Degrees of Freedom Aircraft Equations
  • 5.
    5 SOLO α T V L D Bx Wx Bz Wz Wy By • Aircraft Acceleration () ( ) ( ) ( )WW W W VVa  ×+= → ω where: ( )           = 0 0 V V W  and ( )           = → 0 0 V V W   ( )                                         −+                     − +                     − =           = χ χχ χχ γ γγ γγσ σσ σσω     0 0 100 0 0 0 0 0 010 0 0 0 0 0 001 cs sc cs sc cs sc r q p W W W W or ( )           +− + − =           = γσχσγ γσχσγ γχσ ω ccs csc s r q p W W W W     therefore ( ) ( ) ( ) ( ) ( ) ( )           +− +−=           − =×+= → γσχσγ γσχσγω cscV ccsV V qV rV V VVa W W WW W W     Flat Earth Three Degrees of Freedom Aircraft Equations
  • 6.
    6 SOLO α T V L D Bx Wx Bz Wz Wy By • Aircraft Acceleration FlatEarth Three Degrees of Freedom Aircraft Equations From the Force equation we obtain: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )WWW A WW W W gTF m VVa  ++=×+= → 1 ω or ( ) ( ) ( ) σ σ σ σ γσαγσχσγ γσγσχσγ γα s c c s ccgmLTcscVqV csgccsVrV sgmDTV W W −− −      ++−=+−=− =+−= −−= /sin /)cos(    from which we obtain: ( ) ( )     += −+= −−= msLTcV cgmcLTV sgmDTV /sin /sin /)cos( σαγχ γσαγ γα    Define the Load Factor gm LT n + = αsin :
  • 7.
    7 SOLO α T V L D Bx Wx Bz Wz Wy By • Velocity Equation FlatEarth Three Degrees of Freedom Aircraft Equations ( ) ( )           ==           = 0 0 V CVC h y x V E W WE W E                          −           −          − =           0 0 0 0 001 0 010 0 100 0 0 V cs sc cs sc cs sc h y x σσ σσ γγ γγ χχ χχ         = = = γ χγ χγ sVh scVy ccVx    or • Energy per unit mass E g V hE 2 : 2 += Let differentiate this equation: ( ) W VDT W DT g g V V g VV hEps − =            − − +=+== α γ α γ cos sin cos sin:   Return to Table of Content
  • 8.
    8 SOLO Flat EarthThree Degrees of Freedom Aircraft Equations Summary γ χγ χγ sin sincos coscos Vh Vy Vx = = =    ( ) ( ) σ γ σ γ α χ γσγσ α γ α γ α sin cos sin cos sin coscoscoscos sin cos sin cos n V g W LT V g n V g W LT V g W VDT Eor W DT gV = + = −=      − + = − =−      − =    where mgW = Aircraft Thrust( ) 10, ≤≤= ηη VhTT MAX ( ) ( )MSCVhL L , 2 1 2 αρ= ( ) ( )LD CMSCVhD , 2 1 2 ρ= ( ) ( ) ( ) 2 0, LDLD CMKMCCMC += ( ) 0/ 0 hh eh − = ρρ ( ) ( ) soundofspeedhaNumberMachMhaVM === &/ Aircraft Weight Aircraft Lift Aircraft Drag Parabolic Drag Polar Return to Table of Content
  • 9.
    9 SOLO Flat EarthThree Degrees of Freedom Aircraft Equations Constraints: State Constraints • Minimum Altitude Limit minhh ≥ • Maximum dynamic pressure limit ( ) ( )hVVorqVhq MAXMAX ≤≤= 2 2 1 ρ • Maximum Mach Number limit ( ) MAXM ha V ≤ Aerodynamic or heat limitation Control Constraints • Maximum Lift Coefficient or Maximum Angle of Attack ( ) ( ) ( )VhorMCMC STALLMAXLL ,, _ ααα ≤≤ • Minimum Load Factor ( ) MAXn W VhL n ≤= , • Maximum Thrust ( )VhTT MAX ,≤
  • 10.
    10 SOLO Flat EarthThree Degrees of Freedom Aircraft Equations MAXα MAXα αα LCDC MAXLC 0DC ( ) ( )αα 2 0 LDD kCCC += Drag and Lift Coefficients as functions of Angle of Attack
  • 11.
    11 SOLO Flat EarthThree Degrees of Freedom Aircraft Equations ( ) ( ) ( ) Limit Vhor MCMC STALL MAXLL , , _ αα α = = ( ) ( ) Limit hVVor qVhq MAX MAX = == 2 2 1 ρ minhh = MAXMM = Mach Altitude Flight Envelope of the Aircraft Return to Table of Content
  • 12.
    12 Performance of anAircraft with Parabolic PolarSOLO W LT n + = αsin :' W L n =: 2 0 : LD L D L CkC C CSq CSq D L e + === We assumed 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
  • 13.
    13 Performance of anAircraft 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 0 * 2 1 :* * : 2 * 2 1 :* Vq V V u CS kW CS W V D L ρ ρρ 2 0 : LD L D L CkC C CSq CSq D L e + === Climbing Aircraft Performance
  • 14.
    14 Performance of anAircraft 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 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           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
  • 15.
    15 Performance of anAircraft with Parabolic PolarSOLO We obtained 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 u 0 - - - - 0 + + + + + D ↓ min ↑ n u D ∂ ∂       += 2 2 2 *2 u n u e W D
  • 16.
    16 Performance of anAircraft 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 Return to Table of Content Climbing Aircraft Performance Maximum Load Factor           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
  • 17.
    17 Performance of anAircraft 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 * −+− = nz nzzu nzzu nuzup constns >     −+= −−= →=+−→== 22 2 22 1224 020 ( ) ( ) 2 224 2 2243 23 * *244 * * u nuzu e V u nuzuuuzu e V u p constn s ++− = −+−−+− = ∂ ∂ = 3 3 0 22 nzz u u p MAX constn s ++ =→= ∂ ∂ = 2 21 2 uu uu MAX << + nz > Climbing Aircraft Performance
  • 18.
    Performance of anAircraft 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 ==
  • 19.
    19 Performance of anAircraft 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 nuzu e V ps 224 2 * * −+− = 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 =→=+−= ∂ ∂ = Return to Table of Content Climbing Aircraft Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
  • 20.
    20 Performance of anAircraft 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 u V pe uzun s * *2 2 24 −+−= ( ) 3 0412 2 2 22 z uzu u n constps =→=+−= ∂ ∂ = ( ) * *2 44 3 2 V pe uzu u n s constps −+−= ∂ ∂ = Climbing Aircraft Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
  • 21.
    21 Performance of anAircraft 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 Return to Table of Content Climbing Aircraft Performance           = = = 2 0 * 2 1 :* * : 2 :* Vq V V u CS kW V D ρ ρ
  • 22.
    22 Performance of anAircraft 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 Trajectories We found Aircraft Turn Performance
  • 23.
    23 Performance of anAircraft with Parabolic PolarSOLO ( ) ( ) ( ) γ σ φ 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  
  • 24.
    24 Vertical Plan Trajectory(σ = 0) SOLO Return to Table of Content
  • 25.
    25 Performance of anAircraft with Parabolic PolarSOLO 2. Horizontal Plan Trajectory ( )0,0 == γγ  We can see that for n > 1 1 1 1' 1 11' 2 2 2 2 22 − ≈ − = −≈−= ng V ng V R n V g n V g χ 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
  • 26.
    26 R V =:χ1'2 −= n V g χ Contours ofConstant n and Contours of Constant Turn Radius in Turn-Rate in Horizontal Plan versus Mach coordinates Horizontal Plan TrajectorySOLO
  • 27.
    27 Performance of anAircraft 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 a 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
  • 28.
    28 Performance of anAircraft 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
  • 29.
    29 Performance of anAircraft 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
  • 30.
    30 Performance of anAircraft 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
  • 31.
    31 Performance of anAircraft 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 s χ Aircraft Turn Performance
  • 32.
    32 Performance of anAircraft 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
  • 33.
    33 Performance of anAircraft with Parabolic PolarSOLO Horizontal Turn Rate as Function of ps, n u 0<sp 0=sp 0>sp χ ( )12 * −z V g 1=u1u 2u 2 24 1 * *2 2 * u up V e uzu V g s −−+− =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 nMAX LIMIT C MAXL_ MAX MAX L MAXL n n C C V g 1 ** 2 _ − 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
  • 34.
    34 Performance of anAircraft 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
  • 35.
    35 Performance of anAircraft 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
  • 36.
    36 Performance of anAircraft 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 The minimum of 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
  • 37.
    37 Performance of anAircraft 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
  • 38.
    38 Performance of anAircraft 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
  • 39.
    39 Performance of anAircraft 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
  • 40.
    08/12/15 40 Performance ofan Aircraft with Parabolic PolarSOLO References Angelo Miele, “Flight Mechanics, Volume I, Theory of Flight Paths”, Addison-Wesley, 1962 Return to Table of Content S. Hermelin, “3 DOF Model of Aircraft in Earth Atmosphere”
  • 41.
    41 SOLO Technion Israeli Institute ofTechnology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA

Editor's Notes

  • #25 Prof. Earll Murman, “Introduction to Aircraft Performance and Static Stability”, September 18, 2003